mechan design
vickrei-clark-grove
redistribut payment
effici mechan
strategi-proof
individu ration mechan
mechan
linear vcg redistribut mechan
transform to linear program
analyt character
worst-case optim mechan
vickrei-clark-grove mechan
payment redistribut

Worst-Case Optimal Redistribution of VCG Payments Mingyu Guo Duke University Department of Computer Science Durham, NC, USA mingyu@cs.duke.edu Vincent Conitzer Duke University Department of Computer Science Durham, NC, USA conitzer@cs.duke.edu ABSTRACT For allocation problems with one or more items, the wellknown Vickrey-Clarke-Groves (VCG) mechanism is efficient, strategy-proof, individually rational, and does not incur a deficit.
However, the VCG mechanism is not (strongly) budget balanced: generally, the agents'' payments will sum to more than 0.
If there is an auctioneer who is selling the items, this may be desirable, because the surplus payment corresponds to revenue for the auctioneer.
However, if the items do not have an owner and the agents are merely interested in allocating the items efficiently among themselves, any surplus payment is undesirable, because it will have to flow out of the system of agents.
In 2006, Cavallo [3] proposed a mechanism that redistributes some of the VCG payment back to the agents, while maintaining efficiency, strategy-proofness, individual rationality, and the non-deficit property.
In this paper, we extend this result in a restricted setting.
We study allocation settings where there are multiple indistinguishable units of a single good, and agents have unit demand.
(For this specific setting, Cavallo``s mechanism coincides with a mechanism proposed by Bailey in 1997 [2].)
Here we propose a family of mechanisms that redistribute some of the VCG payment back to the agents.
All mechanisms in the family are efficient, strategyproof, individually rational, and never incur a deficit.
The family includes the Bailey-Cavallo mechanism as a special case.
We then provide an optimization model for finding the optimal mechanism-that is, the mechanism that maximizes redistribution in the worst case-inside the family, and show how to cast this model as a linear program.
We give both numerical and analytical solutions of this linear program, and the (unique) resulting mechanism shows significant improvement over the Bailey-Cavallo mechanism (in the worst case).
Finally, we prove that the obtained mechanism is optimal among all anonymous deterministic mechanisms that satisfy the above properties.
Categories and Subject Descriptors J.4 [Computer Applications]: Social and Behavioral Sciences-Economics; I.2.11 [Distributed Artificial Intelligence]: Multiagent Systems General Terms Algorithms, Economics, Theory 1.
INTRODUCTION Many important problems in computer science and electronic commerce can be modeled as resource allocation problems.
In such problems, we want to allocate the resources (or items) to the agents that value them the most.
Unfortunately, agents'' valuations are private knowledge, and self-interested agents will lie about their valuations if this is to their benefit.
One solution is to auction off the items, possibly in a combinatorial auction where agents can bid on bundles of items.
There exist ways of determining the payments that the agents make in such an auction that incentivizes the agents to report their true valuations-that is, the payments make the auction strategy-proof.
One very general way of doing so is to use the VCG mechanism [23, 4, 12].
(The VCG mechanism is also known as the Clarke mechanism or, in the specific context of auctions, the Generalized Vickrey Auction.)
Besides strategy-proofness, the VCG mechanism has several other nice properties in the context of resource allocation problems.
It is efficient: the chosen allocation always maximizes the sum of the agents'' valuations.
It is also (expost) individually rational: participating in the mechanism never makes an agent worse off than not participating.
Finally, it has a no-deficit property: the sum of the agents'' payments is always nonnegative.
In many settings, another property that would be desirable is (strong) budget balance, meaning that the payments sum to exactly 0.
Suppose the agents are trying to distribute some resources among themselves that do not have a previous owner.
For example, the agents may be trying to allocate the right to use a shared good on a given day.
Or, the agents may be trying to allocate a resource that they have collectively constructed, discovered, or otherwise obtained.
If the agents use an auction to allocate these resources, and the sum of the agents'' payments in the auction is positive, then this surplus payment must leave the system 30 of the agents (for example, the agents must give the money to an outside party, or burn it).
Na¨ıve redistribution of the surplus payment (e.g. each of the n agents receives 1/n of the surplus) will generally result in a mechanism that is not strategy-proof (e.g. in a Vickrey auction, the second-highest bidder would want to increase her bid to obtain a larger redistribution payment).
Unfortunately, the VCG mechanism is not budget balanced: typically, there is surplus payment.
Unfortunately, in general settings, it is in fact impossible to design mechanisms that satisfy budget balance in addition to the other desirable properties [16, 11, 21].
In light of this impossibility result, several authors have obtained budget balance by sacrificing some of the other desirable properties [2, 6, 22, 5].
Another approach that is perhaps preferable is to use a mechanism that is more budget balanced than the VCG mechanism, and maintains all the other desirable properties.
One way of trying to design such a mechanism is to redistribute some of the VCG payment back to the agents in a way that will not affect the agents'' incentives (so that strategy-proofness is maintained), and that will maintain the other properties.
In 2006, Cavallo [3] pursued exactly this idea, and designed a mechanism that redistributes a large amount of the total VCG payment while maintaining all of the other desirable properties of the VCG mechanism.
For example, in a single-item auction (where the VCG mechanism coincides with the second-price sealed-bid auction), the amount redistributed to bidder i by Cavallo``s mechanism is 1/n times the second-highest bid among bids other than i``s bid.
The total redistributed is at most the second-highest bid overall, and the redistribution to agent i does not affect i``s incentives because it does not depend on i``s own bid.
In this paper, we restrict our attention to a limited setting, and in this setting we extend Cavallo``s result.
We study allocation settings where there are multiple indistinguishable units of a single good, and all agents have unit demand, i.e. they want only a single unit.
For this specific setting, Cavallo``s mechanism coincides with a mechanism proposed by Bailey in 1997 [2].
Here we propose the family of linear VCG redistribution mechanisms.
All mechanisms in this family are efficient, strategy-proof, individually rational, and never incur a deficit.
The family includes the Bailey-Cavallo mechanism as a special case (with the caveat that we only study allocation settings with multiple indistinguishable units of a single good and unit demand, while Bailey``s and Cavallo``s mechanisms can be applied outside these settings as well).
We then provide an optimization model for finding the optimal mechanism inside the family, based on worst-case analysis.
Both numerical and analytical solutions of this model are provided, and the resulting mechanism shows significant improvement over the BaileyCavallo mechanism (in the worst case).
For example, for the problem of allocating a single unit, when the number of agents is 10, our mechanism always redistributes more than 98% of the total VCG payment back to the agents (whereas the Bailey-Cavallo mechanism redistributes only 80% in the worst case).
Finally, we prove that our mechanism is in fact optimal among all anonymous deterministic mechanisms (even nonlinear ones) that satisfy the desirable properties.
Around the same time, the same mechanism has been independently derived by Moulin [19].1 Moulin actually pursues a different objective (also based on worst-case analysis): whereas our objective is to maximize the percentage of VCG payments that are redistributed, Moulin tries to minimize the overall payments from agents as a percentage of efficiency.
It turns out that the resulting mechanisms are the same.
Towards the end of this paper, we consider dropping the individual rationality requirement, and show that this does not change the optimal mechanism for our objective.
For Moulin``s objective, dropping individual rationality does change the optimal mechanism (but only if there are multiple units).
2.
PROBLEM DESCRIPTION Let n denote the number of agents, and let m denote the number of units.
We only consider the case where m < n (otherwise the problem becomes trivial).
We also assume that m and n are always known.
(This assumption is not harmful: in environments where anyone can join the auction, running a redistribution mechanism is typically not a good idea anyway, because everyone would want to join to collect part of the redistribution.)
Let the set of agents be {a1, a2, ... , an}, where ai is the agent with ith highest report value ˆvi-that is, we have ˆv1 ≥ ˆv2 ≥ ... ≥ ˆvn ≥ 0.
Let vi denote the true value of ai.
Given that the mechanism is strategy-proof, we can assume vi = ˆvi.
Under the VCG mechanism, each agent among a1, ... , am wins a unit, and pays ˆvm+1 for this unit.
Thus, the total VCG payment equals mˆvm+1.
When m = 1, this is the second-price or Vickrey auction.
We modify the mechanism as follows.
After running the original VCG mechanism, the center returns to each agent ai some amount zi, agent ai``s redistribution payment.
We do not allow zi to depend on ˆvi; because of this, ai``s incentives are unaffected by this redistribution payment, and the mechanism remains strategy-proof.
3.
LINEAR VCG REDISTRIBUTION MECHANISMS We are now ready to introduce the family of linear VCG redistribution mechanisms.
Such a mechanism is defined by a vector of constants c0, c1, ... , cn−1.
The amount that the mechanism returns to agent ai is zi = c0 + c1ˆv1 + c2ˆv2 + ... + ci−1ˆvi−1 + ciˆvi+1 + ... + cn−1ˆvn.
That is, an agent receives c0, plus c1 times the highest bid other than the agent``s own bid, plus c2 times the second-highest other bid, etc..
The mechanism is strategy-proof, because for all i, zi is independent of ˆvi.
Also, the mechanism is anonymous.
It is helpful to see the entire list of redistribution payments: z1 = c0 + c1ˆv2 + c2ˆv3 + c3ˆv4 + ... + cn−2ˆvn−1 + cn−1ˆvn z2 = c0 + c1ˆv1 + c2ˆv3 + c3ˆv4 + ... + cn−2ˆvn−1 + cn−1ˆvn z3 = c0 + c1ˆv1 + c2ˆv2 + c3ˆv4 + ... + cn−2ˆvn−1 + cn−1ˆvn z4 = c0 + c1ˆv1 + c2ˆv2 + c3ˆv3 + ... + cn−2ˆvn−1 + cn−1ˆvn ... zi = c0 + c1ˆv1 + c2ˆv2 + ... + ci−1ˆvi−1 + ciˆvi+1 + ... + cn−1ˆvn ... zn−2 = c0 + c1ˆv1 + c2ˆv2 + c3ˆv3 + ... + cn−2ˆvn−1 + cn−1ˆvn zn−1 = c0 + c1ˆv1 + c2ˆv2 + c3ˆv3 + ... + cn−2ˆvn−2 + cn−1ˆvn zn = c0 + c1ˆv1 + c2ˆv2 + c3ˆv3 + ... + cn−2ˆvn−2 + cn−1ˆvn−1 1 We thank Rakesh Vohra for pointing us to Moulin``s working paper.
31 Not all choices of the constants c0, ... , cn−1 produce a mechanism that is individually rational, and not all choices of the constants produce a mechanism that never incurs a deficit.
Hence, to obtain these properties, we need to place some constraints on the constants.
To satisfy the individual rationality criterion, each agent``s utility should always be non-negative.
An agent that does not win a unit obtains a utility that is equal to the agent``s redistribution payment.
An agent that wins a unit obtains a utility that is equal to the agent``s valuation for the unit, minus the VCG payment ˆvm+1, plus the agent``s redistribution payment.
Consider agent an, the agent with the lowest bid.
Since this agent does not win an item (m < n), her utility is just her redistribution payment zn.
Hence, for the mechanism to be individually rational, the ci must be such that zn is always nonnegative.
If the ci have this property, then it actually follows that zi is nonnegative for every i, for the following reason.
Suppose there exists some i < n and some vector of bids ˆv1 ≥ ˆv2 ≥ ... ≥ ˆvn ≥ 0 such that zi < 0.
Then, consider the bid vector that results from replacing ˆvj by ˆvj+1 for all j ≥ i, and letting ˆvn = 0.
If we omit ˆvn from this vector, the same vector results that results from omitting ˆvi from the original vector.
Therefore, an``s redistribution payment under the new vector should be the same as ai``s redistribution payment under the old vector-but this payment is negative.
If all redistribution payments are always nonnegative, then the mechanism must be individually rational (because the VCG mechanism is individually rational, and the redistribution payment only increases an agent``s utility).
Therefore, the mechanism is individually rational if and only if for any bid vector, zn ≥ 0.
To satisfy the non-deficit criterion, the sum of the redistribution payments should be less than or equal to the total VCG payment.
So for any bid vector ˆv1 ≥ ˆv2 ≥ ... ≥ ˆvn ≥ 0, the constants ci should make z1 + z2 + ... + zn ≤ mˆvm+1.
We define the family of linear VCG redistribution mechanisms to be the set of all redistribution mechanisms corresponding to constants ci that satisfy the above constraints (so that the mechanisms will be individually rational and have the no-deficit property).
We now give two examples of mechanisms in this family.
Example 1 (Bailey-Cavallo mechanism): Consider the mechanism corresponding to cm+1 = m n and ci = 0 for all other i. Under this mechanism, each agent receives a redistribution payment of m n times the (m+1)th highest bid from another agent.
Hence, a1, ... , am+1 receive a redistribution payment of m n ˆvm+2, and the others receive m n ˆvm+1.
Thus, the total redistribution payment is (m+1)m n ˆvm+2 +(n−m− 1)m n ˆvm+1.
This redistribution mechanism is individually rational, because all the redistribution payments are nonnegative, and never incurs a deficit, because (m + 1) m n ˆvm+2 + (n−m−1)m n ˆvm+1 ≤ nm n ˆvm+1 = mˆvm+1.
(We note that for this mechanism to make sense, we need n ≥ m + 2.)
Example 2: Consider the mechanism corresponding to cm+1 = m n−m−1 , cm+2 = − m(m+1) (n−m−1)(n−m−2) , and ci = 0 for all other i.
In this mechanism, each agent receives a redistribution payment of m n−m−1 times the (m + 1)th highest reported value from other agents, minus m(m+1) (n−m−1)(n−m−2) times the (m+2)th highest reported value from other agents.
Thus, the total redistribution payment is mˆvm+1 − m(m+1)(m+2) (n−m−1)(n−m−2) ˆvm+3.
If n ≥ 2m+3 (which is equivalent to m n−m−1 ≥ m(m+1) (n−m−1)(n−m−2) ), then each agent always receives a nonnegative redistribution payment, thus the mechanism is individually rational.
Also, the mechanism never incurs a deficit, because the total VCG payment is mˆvm+1, which is greater than the amount mˆvm+1 − m(m+1)(m+2) (n−m−1)(n−m−2) ˆvm+3 that is redistributed.
Which of these two mechanisms is better?
Is there another mechanism that is even better?
This is what we study in the next section.
4.
OPTIMAL REDISTRIBUTION MECHANISMS Among all linear VCG redistribution mechanisms, we would like to be able to identify the one that redistributes the greatest percentage of the total VCG payment.2 This is not a well-defined notion: it may be that one mechanism redistributes more on some bid vectors, and another more on other bid vectors.
We emphasize that we do not assume that a prior distribution over bidders'' valuations is available, so we cannot compare them based on expected redistribution.
Below, we study three well-defined ways of comparing redistribution mechanisms: best-case performance, dominance, and worst-case performance.
Best-case performance.
One way of evaluating a mechanism is by considering the highest redistribution percentage that it achieves.
Consider the previous two examples.
For the first example, the total redistribution payment is (m + 1)m n ˆvm+2 + (n − m − 1)m n ˆvm+1.
When ˆvm+2 = ˆvm+1, this is equal to the total VCG payment mˆvm+1.
Thus, this mechanism redistributes 100% of the total VCG payment in the best case.
For the second example, the total redistribution payment is mˆvm+1 − m(m+1)(m+2) (n−m−1)(n−m−2) ˆvm+3.
When ˆvm+3 = 0, this is equal to the total VCG payment mˆvm+1.
Thus, this mechanism also redistributes 100% of the total VCG payment in the best case.
Moreover, there are actually infinitely many mechanisms that redistribute 100% of the total VCG payment in the best case-for example, any convex combination of the above two will redistribute 100% if both ˆvm+2 = ˆvm+1 and ˆvm+3 = 0.
Dominance.
Inside the family of linear VCG redistribution mechanisms, we say one mechanism dominates another mechanism if the first one redistributes at least as much as the other for any bid vector.
For the previous two examples, neither dominates the other, because they each redistribute 100% in different cases.
It turns out that there is no mechanism in the family that dominates all other mechanisms in the family.
For suppose such a mechanism exists.
Then, it should dominate both examples above.
Consider the remaining VCG payment (the VCG payment failed to be redistributed).
The remaining VCG payment of the dominant mechanism should be 0 whenever ˆvm+2 = ˆvm+1 or ˆvm+3 = 0.
Now, the remaining VCG payment is a linear function of the ˆvi (linear redistribution), and therefore also a polynomial function.
The above implies that this function can be written as (ˆvm+2 − ˆvm+1)(ˆvm+3)P(ˆv1, ˆv2, ... , ˆvn), where P is a 2 The percentage redistributed seems the natural criterion to use, among other things because it is scale-invariant: if we multiply all bids by the same positive constant (for example, if we change the units by re-expressing the bids in euros instead of dollars), we would not want the behavior of our mechanism to change.
32 polynomial function.
But since the function must be linear (has degree at most 1), it follows that P = 0.
Thus, a dominant mechanism would always redistribute all of the VCG payment, which is not possible.
(If it were possible, then our worst-case optimal redistribution mechanism would also always redistribute all of the VCG payment, and we will see later that it does not.)
Worst-case performance.
Finally, we can evaluate a mechanism by considering the lowest redistribution percentage that it guarantees.
For the first example, the total redistribution payment is (m+1)m n ˆvm+2 +(n−m−1)m n ˆvm+1, which is greater than or equal to (n−m−1) m n ˆvm+1.
So in the worst case, which is when ˆvm+2 = 0, the percentage redistributed is n−m−1 n .
For the second example, the total redistribution payment is mˆvm+1 − m(m+1)(m+2) (n−m−1)(n−m−2) ˆvm+3, which is greater than or equal to mˆvm+1(1− (m+1)(m+2) (n−m−1)(n−m−2) ).
So in the worst case, which is when ˆvm+3 = ˆvm+1, the percentage redistributed is 1 − (m+1)(m+2) (n−m−1)(n−m−2) .
Since we assume that the number of agents n and the number of units m are known, we can determine which example mechanism has better worst-case performance by comparing the two quantities.
When n = 6 and m = 1, for the first example (Bailey-Cavallo mechanism), the percentage redistributed in the worst case is 2 3 , and for the second example, this percentage is 1 2 , which implies that for this pair of n and m, the first mechanism has better worst-case performance.
On the other hand, when n = 12 and m = 1, for the first example, the percentage redistributed in the worst case is 5 6 , and for the second example, this percentage is 14 15 , which implies that this time the second mechanism has better worst-case performance.
Thus, it seems most natural to compare mechanisms by the percentage of total VCG payment that they redistribute in the worst case.
This percentage is undefined when the total VCG payment is 0.
To deal with this, technically, we define the worst-case redistribution percentage as the largest k so that the total amount redistributed is at least k times the total VCG payment, for all bid vectors.
(Hence, as long as the total amount redistributed is at least 0 when the total VCG payment is 0, these cases do not affect the worst-case percentage.)
This corresponds to the following optimization problem: Maximize k (the percentage redistributed in the worst case) Subject to: For every bid vector ˆv1 ≥ ˆv2 ≥ ... ≥ ˆvn ≥ 0 zn ≥ 0 (individual rationality) z1 + z2 + ... + zn ≤ mˆvm+1 (non-deficit) z1 + z2 + ... + zn ≥ kmˆvm+1 (worst-case constraint) We recall that zi = c0 + c1ˆv1 + c2ˆv2 + ... + ci−1ˆvi−1 + ciˆvi+1 + ... + cn−1ˆvn.
5.
TRANSFORMATION TO LINEAR PROGRAMMING The optimization problem given in the previous section can be rewritten as a linear program, based on the following observations.
Claim 1.
If c0, c1, ... , cn−1 satisfy both the individual rationality and the non-deficit constraints, then ci = 0 for i = 0, ... , m. Proof.
First, let us prove that c0 = 0.
Consider the bid vector in which ˆvi = 0 for all i. To obtain individual rationality, we must have c0 ≥ 0.
To satisfy the non-deficit constraint, we must have c0 ≤ 0.
Thus we know c0 = 0.
Now, if ci = 0 for all i, there is nothing to prove.
Otherwise, let j = min{i|ci = 0}.
Assume that j ≤ m.
We recall that we can write the individual rationality constraint as follows: zn = c0 +c1ˆv1 +c2ˆv2 +c3ˆv3 +...+cn−2ˆvn−2 +cn−1ˆvn−1 ≥ 0 for any bid vector.
Let us consider the bid vector in which ˆvi = 1 for i ≤ j and ˆvi = 0 for the rest.
In this case zn = cj, so we must have cj ≥ 0.
The non-deficit constraint can be written as follows: z1 + z2 + ... + zn ≤ mˆvm+1 for any bid vector.
Consider the same bid vector as above.
We have zi = 0 for i ≤ j, because for these bids, the jth highest other bid has value 0, so all the ci that are nonzero are multiplied by 0.
For i > j, we have zi = cj, because the jth highest other bid has value 1, and all lower bids have value 0.
So the non-deficit constraint tells us that cj(n − j) ≤ mˆvm+1.
Because j ≤ m, ˆvm+1 = 0, so the right hand side is 0.
We also have n − j > 0 because j ≤ m < n.
So cj ≤ 0.
Because we have already established that cj ≥ 0, it follows that cj = 0; but this is contrary to assumption.
So j > m. Incidentally, this claim also shows that if m = n − 1, then ci = 0 for all i. Thus, we are stuck with the VCG mechanism.
From here on, we only consider the case where m < n − 1.
Claim 2.
The individual rationality constraint can be written as follows: Pj i=m+1 ci ≥ 0 for j = m + 1, ... , n − 1.
Before proving this claim, we introduce the following lemma.
Lemma 1.
Given a positive integer k and a set of real constants s1, s2, ... , sk, (s1t1 + s2t2 + ... + sktk ≥ 0 for any t1 ≥ t2 ≥ ... ≥ tk ≥ 0) if and only if ( Pj i=1 si ≥ 0 for j = 1, 2, ... , k).
Proof.
Let di = ti −ti+1 for i = 1, 2, ... , k−1, and dk = tk.
Then (s1t1 +s2t2 +...+sktk ≥ 0 for any t1 ≥ t2 ≥ ... ≥ tk ≥ 0) is equivalent to (( P1 i=1 si)d1 + ( P2 i=1 si)d2 + ... + ( Pk i=1 si)dk ≥ 0 for any set of arbitrary non-negative dj).
When Pj i=1 si ≥ 0 for j = 1, 2, ... , k, the above inequality is obviously true.
If for some j, Pj i=1 si < 0, if we set dj > 0 and di = 0 for all i = j, then the above inequality becomes false.
So Pj i=1 si ≥ 0 for j = 1, 2, ... , k is both necessary and sufficient.
We are now ready to present the proof of Claim 2.
Proof.
The individual rationality constraint can be written as zn = c0 + c1ˆv1 + c2ˆv2 + c3ˆv3 + ... + cn−2ˆvn−2 + cn−1ˆvn−1 ≥ 0 for any bid vector ˆv1 ≥ ˆv2 ≥ ... ≥ ˆvn−1 ≥ ˆvn ≥ 0.
We have already shown that ci = 0 for i ≤ m. Thus, the above can be simplified to zn = cm+1ˆvm+1 + cm+2ˆvm+2+...+cn−2ˆvn−2+cn−1ˆvn−1 ≥ 0 for any bid vector.
By the above lemma, this is equivalent to Pj i=m+1 ci ≥ 0 for j = m + 1, ... , n − 1.
Claim 3.
The non-deficit constraint and the worst-case constraint can also be written as linear inequalities involving only the ci and k. Proof.
The non-deficit constraint requires that for any bid vector, z1 +z2 +...+zn ≤ mˆvm+1, where zi = c0 +c1ˆv1 + 33 c2ˆv2 +...+ci−1ˆvi−1 +ciˆvi+1 +...+cn−1ˆvn for i = 1, 2, ... , n. Because ci = 0 for i ≤ m, we can simplify this inequality to qm+1ˆvm+1 + qm+2ˆvm+2 + ... + qnˆvn ≥ 0 qm+1 = m − (n − m − 1)cm+1 qi = −(i−1)ci−1 −(n−i)ci, for i = m+2, ... , n−1 (when m + 2 > n − 1, this set of equalities is empty) qn = −(n − 1)cn−1 By the above lemma, this is equivalent to Pj i=m+1 qi ≥ 0 for j = m + 1, ... , n. So, we can simplify further as follows: qm+1 ≥ 0 ⇐⇒ (n − m − 1)cm+1 ≤ m qm+1 + ... + qm+i ≥ 0 ⇐⇒ n Pj=m+i−1 j=m+1 cj + (n − m − i)cm+i ≤ m for i = 2, ... , n − m − 1 qm+1 + ... + qn ≥ 0 ⇐⇒ n Pj=n−1 j=m+1 cj ≤ m So, the non-deficit constraint can be written as a set of linear inequalities involving only the ci.
The worst-case constraint can be also written as a set of linear inequalities, by the following reasoning.
The worstcase constraint requires that for any bid input z1 +z2 +...+ zn ≥ kmˆvm+1, where zi = c0 +c1ˆv1 +c2ˆv2 +...+ci−1ˆvi−1 + ciˆvi+1 + ... + cn−1ˆvn for i = 1, 2, ... , n. Because ci = 0 for i ≤ m, we can simplify this inequality to Qm+1ˆvm+1 + Qm+2ˆvm+2 + ... + Qnˆvn ≥ 0 Qm+1 = (n − m − 1)cm+1 − km Qi = (i − 1)ci−1 + (n − i)ci, for i = m + 2, ... , n − 1 Qn = (n − 1)cn−1 By the above lemma, this is equivalent to Pj i=m+1 Qi ≥ 0 for j = m + 1, ... , n. So, we can simplify further as follows: Qm+1 ≥ 0 ⇐⇒ (n − m − 1)cm+1 ≥ km Qm+1 + ... + Qm+i ≥ 0 ⇐⇒ n Pj=m+i−1 j=m+1 cj + (n − m − i)cm+i ≥ km for i = 2, ... , n − m − 1 Qm+1 + ... + Qn ≥ 0 ⇐⇒ n Pj=n−1 j=m+1 cj ≥ km So, the worst-case constraint can also be written as a set of linear inequalities involving only the ci and k. Combining all the claims, we see that the original optimization problem can be transformed into the following linear program.
Variables: cm+1, cm+2, ... , cn−1, k Maximize k (the percentage redistributed in the worst case) Subject to:Pj i=m+1 ci ≥ 0 for j = m + 1, ... , n − 1 km ≤ (n − m − 1)cm+1 ≤ m km ≤ n Pj=m+i−1 j=m+1 cj + (n − m − i)cm+i ≤ m for i = 2, ... , n − m − 1 km ≤ n Pj=n−1 j=m+1 cj ≤ m 6.
NUMERICAL RESULTS For selected values of n and m, we solved the linear program using Glpk (GNU Linear Programming Kit).
In the table below, we present the results for a single unit (m = 1).
We present 1−k (the percentage of the total VCG payment that is not redistributed by the worst-case optimal mechanism in the worst case) instead of k in the second column because writing k would require too many significant digits.
Correspondingly, the third column displays the percentage 5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of AgentsWorst−caseRedistributionPercentage 1 Unit WO 1 Unit BC 2 Units WO 2 Units BC 3 Units WO 3 Units BC 4 Units WO 4 Units BC Figure 1: A comparison of the worst-case optimal mechanism (WO) and the Bailey-Cavallo mechanism (BC).
of the total VCG payment that is not redistributed by the Bailey-Cavallo mechanism in the worst case (which is equal to 2 n ).
n 1 − k Bailey − Cavallo Mechanism 3 66.7% 66.7% 4 42.9% 50.0% 5 26.7% 40.0% 6 16.1% 33.3% 7 9.52% 28.6% 8 5.51% 25.0% 9 3.14% 22.2% 10 1.76% 20.0% 20 3.62e − 5 10.0% 30 5.40e − 8 6.67e − 2 40 7.09e − 11 5.00e − 2 The worst-case optimal mechanism significantly outperforms the Bailey-Cavallo mechanism in the worst case.
Perhaps more surprisingly, the worst-case optimal mechanism sometimes does better in the worst case than the BaileyCavallo mechanism does on average, as the following example shows.
Recall that the total redistribution payment of the BaileyCavallo mechanism is (m + 1)m n ˆvm+2 + (n − m − 1)m n ˆvm+1.
For the single-unit case, this simplifies to 2 n ˆv3 + n−2 n ˆv2.
Hence the percentage of the total VCG payment that is not redistributed is ˆv2− 2 n ˆv3− n−2 n ˆv2 ˆv2 = 2 n − 2 n ˆv3 ˆv2 , which has an expected value of E( 2 n − 2 n ˆv3 ˆv2 ) = 2 n − 2 n E ˆv3 ˆv2 .
Suppose the bid values are drawn from a uniform distribution over [0, 1].
The theory of order statistics tells us that the 34 joint probability density function of ˆv2 and ˆv3 is f(ˆv3, ˆv2) = n(n − 1)(n − 2)ˆvn−3 3 (1 − ˆv2) for ˆv2 ≥ ˆv3.
Now, E ˆv3 ˆv2 = R 1 0 R ˆv2 0 ˆv3 ˆv2 f(ˆv3, ˆv2)dˆv3dˆv2 = n−2 n−1 .
So, the expected value of the remaining percentage is 2 n − 2 n n−2 n−1 = 2 n(n−1) .
For n = 20, this is 5.26e − 3, whereas the remaining percentage for the worst-case optimal mechanism is 3.62e−5 in the worst case.
Let us present the optimal solution for the case n = 5 in detail.
By solving the above linear program, we find that the optimal values for the ci are c2 = 11 45 , c3 = −1 9 , and c4 = 1 15 .
That is, the redistribution payment received by each agent is: 11 45 times the second highest bid among the other agents, minus 1 9 times the third highest bid among the other agents, plus 1 15 times the fourth highest bid among the other agents.
The total amount redistributed is 11 15 ˆv2 + 4 15 ˆv3 − 4 15 ˆv4 + 4 15 ˆv5; in the worst case, 11 15 ˆv2 is redistributed.
Hence, the percentage of the total VCG payment that is not redistributed is never more than 4 15 = 26.7%.
Finally, we compare the worst-case optimal mechanism to the Bailey-Cavallo mechanism for m = 1, 2, 3, 4, n = m + 2, ... , 30.
These results are in Figure 1.
We see that for any m, when n = m + 2, the worst-case optimal mechanism has the same worst-case performance as the Bailey-Cavallo mechanism (actually, in this case, the worst-case optimal mechanism is identical to the BaileyCavallo mechanism).
When n > m + 2, the worst-case optimal mechanism outperforms the Bailey-Cavallo mechanism (in the worst case).
7.
ANALYTICAL CHARACTERIZATION OF THE WORST-CASE OPTIMAL MECHANISM We recall that our linear program has the following form: Variables: cm+1, cm+2, ... , cn−1, k Maximize k (the percentage redistributed in the worst case) Subject to:Pj i=m+1 ci ≥ 0 for j = m + 1, ... , n − 1 km ≤ (n − m − 1)cm+1 ≤ m km ≤ n Pj=m+i−1 j=m+1 cj + (n − m − i)cm+i ≤ m for i = 2, ... , n − m − 1 km ≤ n Pj=n−1 j=m+1 cj ≤ m A linear program has no solution if and only if either the objective is unbounded, or the constraints are contradictory (there is no feasible solution).
It is easy to see that k is bounded above by 1 (redistributing more than 100% violates the non-deficit constraint).
Also, a feasible solution always exists, for example, k = 0 and ci = 0 for all i.
So an optimal solution always exists.
Observe that the linear program model depends only on the number of agents n and the number of units m. Hence the optimal solution is a function of n and m.
It turns out that this optimal solution can be analytically characterized as follows.
Theorem 1.
For any m and n with n ≥ m+2, the worstcase optimal mechanism (among linear VCG redistribution mechanisms) is unique.
For this mechanism, the percentage redistributed in the worst case is k∗ = 1 − `n−1 m ´ Pn−1 j=m `n−1 j ´ The worst-case optimal mechanism is characterized by the following values for the ci: c∗ i = (−1)i+m−1 (n − m) `n−1 m−1 ´ i Pn−1 j=m `n−1 j ´ 1 `n−1 i ´ n−1X j=i n − 1 j !
for i = m + 1, ... , n − 1.
It should be noted that we have proved ci = 0 for i ≤ m in Claim 1.
Proof.
We first rewrite the linear program as follows.
We introduce new variables xm+1, xm+2, ... , xn−1, defined by xj = Pj i=m+1 ci for j = m + 1, ... , n − 1.
The linear program then becomes: Variables: xm+1, xm+2, ... , xn−1, k Maximize k Subject to: km ≤ (n − m − 1)xm+1 ≤ m km ≤ (m + i)xm+i−1 + (n − m − i)xm+i ≤ m for i = 2, ... , n − m − 1 km ≤ nxn−1 ≤ m xi ≥ 0 for i = m + 1, m + 2, ... , n − 1 We will prove that for any optimal solution to this linear program, k = k∗ .
Moreover, we will prove that when k = k∗ , xj = Pj i=m+1 c∗ i for j = m + 1, ... , n − 1.
This will prove the theorem.
We first make the following observations: (n − m − 1)c∗ m+1 = (n − m − 1) (n−m)(n−1 m−1) (m+1) Pn−1 j=m (n−1 j ) 1 (n−1 m+1) Pn−1 j=m+1 `n−1 j ´ = (n − m − 1) (n−m)(n−1 m−1) (m+1) Pn−1 j=m (n−1 j ) 1 (n−1 m+1) ( Pn−1 j=m `n−1 j ´ − `n−1 m ´ ) = (n − m − 1) m n−m−1 − (n − m − 1) m(n−1 m ) (n−m−1) Pn−1 j=m (n−1 j ) = m − (1 − k∗ )m = k∗ m For i = m + 1, ... , n − 2, ic∗ i + (n − i − 1)c∗ i+1 = i (−1)i+m−1 (n−m)(n−1 m−1) i Pn−1 j=m (n−1 j ) 1 (n−1 i ) Pn−1 j=i `n−1 j ´ + (n − i − 1) (−1)i+m (n−m)(n−1 m−1) (i+1) Pn−1 j=m (n−1 j ) 1 (n−1 i+1 ) Pn−1 j=i+1 `n−1 j ´ = (−1)i+m−1 (n−m)(n−1 m−1) Pn−1 j=m (n−1 j ) 1 (n−1 i ) Pn−1 j=i `n−1 j ´ − (n − i − 1) (−1)i+m−1 (n−m)(n−1 m−1) (i+1) Pn−1 j=m (n−1 j ) i+1 (n−1 i )(n−i−1) Pn−1 j=i+1 `n−1 j ´ = (−1)i+m−1 (n−m)(n−1 m−1) Pn−1 j=m (n−1 j ) = (−1)i+m−1 m(1 − k∗ ) Finally, (n − 1)c∗ n−1 = (n − 1) (−1)n+m (n−m)(n−1 m−1) (n−1) Pn−1 j=m (n−1 j ) 1 (n−1 n−1) Pn−1 j=n−1 `n−1 j ´ = (−1)m+n m(1 − k∗ ) Summarizing the above, we have: (n − m − 1)c∗ m+1 = k∗ m (m + 1)c∗ m+1 + (n − m − 2)c∗ m+2 = m(1 − k∗ ) (m + 2)c∗ m+2 + (n − m − 3)c∗ m+3 = −m(1 − k∗ ) (m + 3)c∗ m+3 + (n − m − 4)c∗ m+4 = m(1 − k∗ ) ... 35 (n − 3)c∗ n−3 + 2c∗ n−2 = (−1)m+n−2 m(1 − k∗ ) (n − 2)c∗ n−2 + c∗ n−1 = (−1)m+n−1 m(1 − k∗ ) (n − 1)c∗ n−1 = (−1)m+n m(1 − k∗ ) Let x∗ j = Pj i=m+1 c∗ i for j = m + 1, m + 2, ... , n − 1, the first equation in the above tells us that (n − m − 1)x∗ m+1 = k∗ m. By adding the first two equations, we get (m + 2)x∗ m+1 + (n − m − 2)x∗ m+2 = m By adding the first three equations, we get (m + 3)x∗ m+2 + (n − m − 3)x∗ m+3 = k∗ m By adding the first i equations, where i = 2, ... , n−m−1, we get (m + i)x∗ m+i−1 + (n − m − i)x∗ m+i = m if i is even (m + i)x∗ m+i−1 + (n − m − i)x∗ m+i = k∗ m if i is odd Finally by adding all the equations, we get nx∗ n−1 = m if n − m is even; nx∗ n−1 = k∗ m if n − m is odd.
Thus, for all of the constraints other than the nonnegativity constraints, we have shown that they are satisfied by setting xj = x∗ j = Pj i=m+1 c∗ i and k = k∗ .
We next show that the nonnegativity constraints are satisfied by these settings as well.
For m + 1 ≤ i, i + 1 ≤ n − 1, we have 1 i Pn−1 j=i (n−1 j ) (n−1 i ) = 1 i Pn−1 j=i i!
(n−1−i)!
j!
(n−1−j)!
≥ 1 i+1 Pn−2 j=i i!
(n−1−i)!
j!
(n−1−j)!
≥ 1 i+1 Pn−2 j=i (i+1)!
(n−1−i−1)!
(j+1)!
(n−1−j−1)!
= 1 i+1 Pn−1 j=i+1 (n−1 j ) (n−1 i+1 ) This implies that the absolute value of c∗ i is decreasing as i increases (if the c∗ contains more than one number).
We further observe that the sign of c∗ i alternates, with the first element c∗ m+1 positive.
So x∗ j = Pj i=m+1 c∗ i ≥ 0 for all j. Thus, we have shown that these xi = x∗ i together with k = k∗ form a feasible solution of the linear program.
We proceed to show that it is in fact the unique optimal solution.
First we prove the following claim: Claim 4.
If ˆk, ˆxi, i = m + 1, m + 2, ... , n − 1 satisfy the following inequalities: ˆkm ≤ (n − m − 1)ˆxm+1 ≤ m ˆkm ≤ (m + i)ˆxm+i−1 + (n − m − i)ˆxm+i ≤ m for i = 2, ... , n − m − 1 ˆkm ≤ nˆxn−1 ≤ m ˆk ≥ k∗ Then we must have that ˆxi = ˆx∗ i and ˆk = k∗ .
Proof of claim.
Consider the first inequality.
We know that (n − m − 1)x∗ m+1 = k∗ m, so (n − m − 1)ˆxm+1 ≥ ˆkm ≥ k∗ m = (n − m − 1)x∗ m+1.
It follows that ˆxm+1 ≥ x∗ m+1 (n − m − 1 = 0).
Now, consider the next inequality for i = 2.
We know that (m + 2)x∗ m+1 + (n − m − 2)x∗ m+2 = m.
It follows that (n−m−2)ˆxm+2 ≤ m−(m+2)ˆxm+1 ≤ m−(m+2)x∗ m+1 = (n − m − 2)x∗ m+2, so ˆxm+2 ≤ x∗ m+2 (i = 2 ≤ n − m − 1 ⇒ n − m − 2 = 0).
Now consider the next inequality for i = 3.
We know that (m + 3)x∗ m+2 + (n − m − 3)x∗ m+3 = m.
It follows that (n−m−3)ˆxm+3 ≥ ˆkm−(m+3)ˆxm+2 ≥ k∗ m−(m+3)x∗ m+2 = (n − m − 3)x∗ m+3, so ˆxm+3 ≥ x∗ m+3 (i = 3 ≤ n − m − 1 ⇒ n − m − 3 = 0).
Proceeding like this all the way up to i = n−m−1, we get that ˆxm+i ≥ x∗ m+i if i is odd and ˆxm+i ≤ x∗ m+i if i is even.
Moreover, if one inequality is strict, then all subsequent inequalities are strict.
Now, if we can prove ˆxn−1 = x∗ n−1, it would follow that the x∗ i are equal to the ˆxi (which also implies that ˆk = k∗ ).
We consider two cases: Case 1: n − m is even.
We have: n − m even ⇒ n − m − 1 odd ⇒ ˆxn−1 ≥ x∗ n−1.
We also have: n−m even ⇒ nx∗ n−1 = m. Combining these two, we get m = nx∗ n−1 ≤ nˆxn−1 ≤ m ⇒ ˆxn−1 = x∗ n−1.
Case 2: n − m is odd.
In this case, we have ˆxn−1 ≤ x∗ n−1, and nx∗ n−1 = k∗ m. Then, we have: k∗ m ≤ ˆkm ≤ nˆxn−1 ≤ nx∗ n−1 = k∗ m ⇒ ˆxn−1 = x∗ n−1.
This completes the proof of the claim.
It follows that if ˆk, ˆxi, i = m + 1, m + 2, ... , n − 1 is a feasible solution and ˆk ≥ k∗ , then since all the inequalities in Claim 4 are satisfied, we must have ˆxi = x∗ i and ˆk = k∗ .
Hence no other feasible solution is as good as the one described in the theorem.
Knowing the analytical characterization of the worst-case optimal mechanism provides us with at least two major benefits.
First, using these formulas is computationally more efficient than solving the linear program using a generalpurpose solver.
Second, we can derive the following corollary.
Corollary 1.
If the number of units m is fixed, then as the number of agents n increases, the worst-case percentage redistributed linearly converges to 1, with a rate of convergence 1 2 .
(That is, limn→∞ 1−k∗ n+1 1−k∗ n = 1 2 .
That is, in the limit, the percentage that is not redistributed halves for every additional agent.)
We note that this is consistent with the experimental data for the single-unit case, where the worst-case remaining percentage roughly halves each time we add another agent.
The worst-case percentage that is redistributed under the Bailey-Cavallo mechanism also converges to 1 as the number of agents goes to infinity, but the convergence is much slower-it does not converge linearly (that is, letting kC n be the percentage redistributed by the Bailey-Cavallo mechanism in the worst case for n agents, limn→∞ 1−kC n+1 1−kC n = limn→∞ n n+1 = 1).
We now present the proof of the corollary.
Proof.
When the number of agents is n, the worst-case percentage redistributed is k∗ n = 1 − (n−1 m ) Pn−1 j=m (n−1 j ) .
When the number of agents is n + 1, the percentage becomes k∗ n+1 = 1 − (n m) Pn j=m (n j ) .
For n sufficiently large, we will have 2n − mnm−1 > 0, and hence 1−k∗ n+1 1−k∗ n = (n m) Pn−1 j=m (n−1 j ) (n−1 m ) Pn j=m (n j ) = n n−m 2n−1 − Pm−1 j=0 (n−1 j ) 2n− Pm−1 j=0 (n j ) , and n n−m 2n−1 −m(n−1)m−1 2n ≤ 1−k∗ n+1 1−k∗ n ≤ n n−m 2n−1 2n−mnm−1 (because `n j ´ ≤ ni if j ≤ i).
Since we have limn→∞ n n−m 2n−1 −m(n−1)m−1 2n = 1 2 , and limn→∞ n n−m 2n−1 2n−mnm−1 = 1 2 , it follows that limn→∞ 1−k∗ n+1 1−k∗ n = 1 2 .
36 8.
WORST-CASE OPTIMALITY OUTSIDE THE FAMILY In this section, we prove that the worst-case optimal redistribution mechanism among linear VCG redistribution mechanisms is in fact optimal (in the worst case) among all redistribution mechanisms that are deterministic, anonymous, strategy-proof, efficient and satisfy the non-deficit constraint.
Thus, restricting our attention to linear VCG redistribution mechanisms did not come at a loss.
To prove this theorem, we need the following lemma.
This lemma is not new: it was informally stated by Cavallo [3].
For completeness, we present it here with a detailed proof.
Lemma 2.
A VCG redistribution mechanism is deterministic, anonymous and strategy-proof if and only if there exists a function f : Rn−1 → R, so that the redistribution payment zi received by ai satisfies zi = f(ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn) for all i and all bid vectors.
Proof.
First, let us prove the only if direction, that is, if a VCG redistribution mechanism is deterministic, anonymous and strategy-proof then there exists a deterministic function f : Rn−1 → R, which makes zi = f(ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn) for all i and all bid vectors.
If a VCG redistribution mechanism is deterministic and anonymous, then for any bid vector ˆv1 ≥ ˆv2 ≥ ... ≥ ˆvn, the mechanism outputs a unique redistribution payment list: z1, z2, ... , zn.
Let G : Rn → Rn be the function that maps ˆv1, ˆv2, ... , ˆvn to z1, z2, ... , zn for all bid vectors.
Let H(i, x1, x2, ... , xn) be the ith element of G(x1, x2, ... , xn), so that zi = H(i, ˆv1, ˆv2, ... , ˆvn) for all bid vectors and all 1 ≤ i ≤ n. Because the mechanism is anonymous, two agents should receive the same redistribution payment if their bids are the same.
So, if ˆvi = ˆvj, H(i, ˆv1, ˆv2, ... , ˆvn) = H(j, ˆv1, ˆv2, ... , ˆvn).
Hence, if we let j = min{t|ˆvt = ˆvi}, then H(i, ˆv1, ˆv2, ... , ˆvn) = H(j, ˆv1, ˆv2, ... , ˆvn).
Let us define K : Rn → N × Rn as follows: K(y, x1, x2, ... , xn−1) = [j, w1, w2, ... , wn], where w1, w2, ... , wn are y, x1, x2, ... , xn−1 sorted in descending order, and j = min{t|wt = y}.
({t|wt = y} = ∅ because y ∈ {w1, w2, ... , wn}).
Also let us define F : Rn → R by F(ˆvi, ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn) = H ◦ K(ˆvi, ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn) = H(min{t|ˆvt = ˆvi}, ˆv1, ˆv2, ... , ˆvn) = H(i, ˆv1, ˆv2, ... , ˆvn) = zi.
That is, F is the redistribution payment to an agent that bids ˆvi when the other bids are ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn.
Since our mechanism is required to be strategy-proof, and the space of valuations is unrestricted, zi should be independent of ˆvi by Lemma 1 in Cavallo [3].
Hence, we can simply ignore the first variable input to F; let f(x1, x2, ... , xn−1) = F(0, x1, x2, ... , xn−1).
So, we have for all bid vectors and i, zi = f(ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn).
This completes the proof for the only if direction.
For the if direction, if the redistribution payment received by ai satisfies zi = f(ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn) for all bid vectors and i, then this is clearly a deterministic and anonymous mechanism.
To prove strategy-proofness, we observe that because an agent``s redistribution payment is not affected by her own bid, her incentives are the same as in the VCG mechanism, which is strategy-proof.
Now we are ready to introduce the next theorem: Theorem 2.
For any m and n with n ≥ m+2, the worstcase optimal mechanism among the family of linear VCG redistribution mechanisms is worst-case optimal among all mechanisms that are deterministic, anonymous, strategy-proof, efficient and satisfy the non-deficit constraint.
While we needed individual rationality earlier in the paper, this theorem does not mention it, that is, we can not find a mechanism with better worst-case performance even if we sacrifice individual rationality.
(The worst-case optimal linear VCG redistribution mechanism is of course individually rational.)
Proof.
Suppose there is a redistribution mechanism (when the number of units is m and the number of agents is n) that satisfies all of the above properties and has a better worstcase performance than the worst-case optimal linear VCG redistribution mechanism, that is, its worst-case redistribution percentage ˆk is strictly greater than k∗ .
By Lemma 2, for this mechanism, there is a function f : Rn−1 → R so that zi = f(ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn) for all i and all bid vectors.
We first prove that f has the following properties.
Claim 5.
f(1, 1, ... , 1, 0, 0, ... , 0) = 0 if the number of 1s is less than or equal to m. Proof of claim.
We assumed that for this mechanism, the worst-case redistribution percentage satisfies ˆk > k∗ ≥ 0.
If the total VCG payment is x, the total redistribution payment should be in [ˆkx, x] (non-deficit criterion).
Consider the case where all agents bid 0, so that the total VCG payment is also 0.
Hence, the total redistribution payment should be in [ˆk · 0, 0]-that is, it should be 0.
Hence every agent``s redistribution payment f(0, 0, ... , 0) must be 0.
Now, let ti = f(1, 1, ... , 1, 0, 0, ... , 0) where the number of 1s equals i.
We proved t0 = 0.
If tn−1 = 0, consider the bid vector where everyone bids 1.
The total VCG payment is m and the total redistribution payment is nf(1, 1, ... , 1) = ntn−1 = 0.
This corresponds to 0% redistribution, which is contrary to our assumption that ˆk > k∗ ≥ 0.
Now, consider j = min{i|ti = 0} (which is well-defined because tn−1 = 0).
If j > m, the property is satisfied.
If j ≤ m, consider the bid vector where ˆvi = 1 for i ≤ j and ˆvi = 0 for all other i. Under this bid vector, the first j agents each get redistribution payment tj−1 = 0, and the remaining n − j agents each get tj.
Thus, the total redistribution payment is (n − j)tj.
Because the total VCG payment for this bid vector is 0, we must have (n − j)tj = 0.
So tj = 0 (j ≤ m < n).
But this is contrary to the definition of j. Hence f(1, 1, ... , 1, 0, 0, ... , 0) = 0 if the number of 1s is less than or equal to m. Claim 6.
f satisfies the following inequalities: ˆkm ≤ (n − m − 1)tm+1 ≤ m ˆkm ≤ (m + i)tm+i−1 + (n − m − i)tm+i ≤ m for i = 2, 3, ... , n − m − 1 ˆkm ≤ ntn−1 ≤ m Here ti is defined as in the proof of Claim 5.
37 Proof of claim.
For j = m + 1, ... , n, consider the bid vectors where ˆvi = 1 for i ≤ j and ˆvi = 0 for all other i.
These bid vectors together with the non-deficit constraint and worst-case constraint produce the above set of inequalities: for example, when j = m + 1, we consider the bid vector ˆvi = 1 for i ≤ m + 1 and ˆvi = 0 for all other i.
The first m+1 agents each receive a redistribution payment of tm = 0, and all other agents each receive tm+1.
Thus, the total VCG redistribution is (n − m − 1)tm+1.
The nondeficit constraint gives (n − m − 1)tm+1 ≤ m (because the total VCG payment is m).
The worst-case constraint gives (n − m − 1)tm+1 ≥ ˆkm.
Combining these two, we get the first inequality.
The other inequalities can be obtained in the same way.
We now observe that the inequalities in Claim 6, together with ˆk ≥ k∗ , are the same as those in Claim 4 (where the ti are replaced by the ˆxi).
Thus, we can conclude that ˆk = k∗ , which is contrary to our assumption ˆk > k∗ .
Hence no mechanism satisfying all the listed properties has a redistribution percentage greater than k∗ in the worst case.
So far we have only talked about the case where n ≥ m+2.
For the purpose of completeness, we provide the following claim for the n = m + 1 case.
Claim 7.
For any m and n with n = m + 1, the original VCG mechanism (that is, redistributing nothing) is (uniquely) worst-case optimal among all redistribution mechanisms that are deterministic, anonymous, strategy-proof, efficient and satisfy the non-deficit constraint.
We recall that when n = m+1, Claim 1 tells us that the only mechanism inside the family of linear redistribution mechanisms is the original VCG mechanism, so that this mechanism is automatically worst-case optimal inside this family.
However, to prove the above claim, we need to show that it is worst-case optimal among all redistribution mechanisms that have the desired properties.
Proof.
Suppose a redistribution mechanism exists that satisfies all of the above properties and has a worst-case performance as good as the original VCG mechanism, that is, its worst-case redistribution percentage is greater than or equal to 0.
This implies that the total redistribution payment of this mechanism is always nonnegative.
By Lemma 2, for this mechanism, there is a function f : Rn−1 → R so that zi = f(ˆv1, ˆv2, ... , ˆvi−1, ˆvi+1, ... , ˆvn) for all i and all bid vectors.
We will prove that f(x1, x2, ... , xn−1) = 0 for all x1 ≥ x2 ≥ ... ≥ xn−1 ≥ 0.
First, consider the bid vector where ˆvi = 0 for all i. Here, each agent receives a redistribution payment f(0, 0, ... , 0).
The total redistribution payment is then nf(0, 0, ... , 0), which should be both greater than or equal to 0 (by the above observation) as well less than or equal to 0 (using the nondeficit criterion and the fact that the total VCG payment is 0).
It follows that f(0, 0, ... , 0) = 0.
Now, let us consider the bid vector where ˆv1 = x1 ≥ 0 and ˆvi = 0 for all other i. For this bid vector, the agent with the highest bid receives a redistribution payment of f(0, 0, ... , 0) = 0, and the other n − 1 agents each receive f(x1, 0, ... , 0).
By the same reasoning as above, the total redistribution payment should be both greater than or equal to 0 and less than or equal to 0, hence f(x1, 0, ... , 0) = 0 for all x1 ≥ 0.
Proceeding by induction, let us assume f(x1, x2, ... , xk, 0, ... , 0) = 0 for all x1 ≥ x2 ≥ ... ≥ xk ≥ 0, for some k < n − 1.
Consider the bid vector where ˆvi = xi for i ≤ k + 1, and ˆvi = 0 for all other i, where the xi are arbitrary numbers satisfying x1 ≥ x2 ≥ ... ≥ xk ≥ xk+1 ≥ 0.
For the agents with the highest k + 1 bids, their redistribution payment is specified by f acting on an input with only k non-zero variables.
Hence they all receive 0 by induction assumption.
The other n − k − 1 agents each receive f(x1, x2, ... , xk, xk+1, 0, ... , 0).
The total redistribution payment is then (n−k−1)f(x1, x2, ... , xk, xk+1, 0, ... , 0), which should be both greater than or equal to 0, and less than or equal to the total VCG payment.
Now, in this bid vector, the lowest bid is 0 because k + 1 < n.
But since n = m + 1, the total VCG payment is mˆvn = 0.
So we have f(x1, x2, ... , xk, xk+1, 0, ... , 0) = 0 for all x1 ≥ x2 ≥ ... ≥ xk ≥ xk+1 ≥ 0.
By induction, this statement holds for all k < n − 1; when k + 1 = n − 1, we have f(x1, x2, ... , xn−2, xn−1) = 0 for all x1 ≥ x2 ≥ ... ≥ xn−2 ≥ xn−1 ≥ 0.
Hence, in this mechanism, the redistribution payment is always 0; that is, the mechanism is just the original VCG mechanism.
Incidentally, we obtain the following corollary: Corollary 2.
No VCG redistribution mechanism satisfies all of the following: determinism, anonymity, strategyproofness, efficiency, and (strong) budget balance.
This holds for any n ≥ m + 1.
Proof.
For the case n ≥ m + 2: If such a mechanism exists, its worst-case performance would be better than that of the worst-case optimal linear VCG redistribution mechanism, which by Theorem 1 obtains a redistribution percentage strictly less than 1.
But Theorem 2 shows that it is impossible to outperform this mechanism in the worst case.
For the case n = m + 1: If such a mechanism exists, it would perform as well as the original VCG mechanism in the worst case, which implies that it is identical to the VCG mechanism by Claim 7.
But the VCG mechanism is not (strongly) budget balanced.
9.
CONCLUSIONS For allocation problems with one or more items, the wellknown Vickrey-Clarke-Groves (VCG) mechanism is efficient, strategy-proof, individually rational, and does not incur a deficit.
However, the VCG mechanism is not (strongly) budget balanced: generally, the agents'' payments will sum to more than 0.
If there is an auctioneer who is selling the items, this may be desirable, because the surplus payment corresponds to revenue for the auctioneer.
However, if the items do not have an owner and the agents are merely interested in allocating the items efficiently among themselves, any surplus payment is undesirable, because it will have to flow out of the system of agents.
In 2006, Cavallo [3] proposed a mechanism that redistributes some of the VCG payment back to the agents, while maintaining efficiency, strategy-proofness, individual rationality, and the non-deficit property.
In this paper, we extended this result in a restricted setting.
We studied allocation settings where there are multiple indistinguishable units of a single good, and agents have unit demand.
(For this specific setting, Cavallo``s mechanism coincides with a mechanism proposed by Bailey in 1997 [2].)
Here we proposed a family of mechanisms that redistribute some of the VCG payment 38 back to the agents.
All mechanisms in the family are efficient, strategy-proof, individually rational, and never incur a deficit.
The family includes the Bailey-Cavallo mechanism as a special case.
We then provided an optimization model for finding the optimal mechanism-that is, the mechanism that maximizes redistribution in the worst case-inside the family, and showed how to cast this model as a linear program.
We gave both numerical and analytical solutions of this linear program, and the (unique) resulting mechanism shows significant improvement over the Bailey-Cavallo mechanism (in the worst case).
Finally, we proved that the obtained mechanism is optimal among all anonymous deterministic mechanisms that satisfy the above properties.
One important direction for future research is to try to extend these results beyond multi-unit auctions with unit demand.
However, it turns out that in sufficiently general settings, the worst-case optimal redistribution percentage is 0.
In such settings, the worst-case criterion provides no guidance in determining a good redistribution mechanism (even redistributing nothing achieves the optimal worst-case percentage), so it becomes necessary to pursue other criteria.
Alternatively, one can try to identify other special settings in which positive redistribution in the worst case is possible.
Another direction for future research is to consider whether this mechanism has applications to collusion.
For example, in a typical collusive scheme, there is a bidding ring consisting of a number of colluders, who submit only a single bid [10, 17].
If this bid wins, the colluders must allocate the item amongst themselves, perhaps using payments-but of course they do not want payments to flow out of the ring.
This work is part of a growing literature on designing mechanisms that obtain good results in the worst case.
Traditionally, economists have mostly focused either on designing mechanisms that always obtain certain properties (such as the VCG mechanism), or on designing mechanisms that are optimal with respect to some prior distribution over the agents'' preferences (such as the Myerson auction [20] and the Maskin-Riley auction [18] for maximizing expected revenue).
Some more recent papers have focused on designing mechanisms for profit maximization using worst-case competitive analysis (e.g. [9, 1, 15, 8]).
There has also been growing interest in the design of online mechanisms [7] where the agents arrive over time and decisions must be taken before all the agents have arrived.
Such work often also takes a worst-case competitive analysis approach [14, 13].
It does not appear that there are direct connections between our work and these other works that focus on designing mechanisms that perform well in the worst case.
Nevertheless, it seems likely that future research will continue to investigate mechanism design for the worst case, and hopefully a coherent framework will emerge.
10.
REFERENCES [1] G. Aggarwal, A. Fiat, A. Goldberg, J. Hartline, N. Immorlica, and M. Sudan.
Derandomization of auctions.
STOC, 619-625, 2005.
[2] M. J. Bailey.
The demand revealing process: to distribute the surplus.
Public Choice, 91:107-126, 1997.
[3] R. Cavallo.
Optimal decision-making with minimal waste: Strategyproof redistribution of VCG payments.
AAMAS, 882-889, 2006.
[4] E. H. Clarke.
Multipart pricing of public goods.
Public Choice, 11:17-33, 1971.
[5] B. Faltings.
A budget-balanced, incentive-compatible scheme for social choice.
AMEC, 30-43, 2005.
[6] J. Feigenbaum, C. Papadimitriou, and S. Shenker.
Sharing the cost of muliticast transmissions.
JCSS, 63:21-41, 2001.
[7] E. Friedman and D. Parkes.
Pricing WiFi at Starbucks - Issues in online mechanism design.
EC, 240-241, 2003.
[8] A. Goldberg, J. Hartline, A. Karlin, M. Saks, and A. Wright.
Competitive auctions.
Games and Economic Behavior, 2006.
[9] A. Goldberg, J. Hartline, and A. Wright.
Competitive auctions and digital goods.
SODA, 735-744, 2001.
[10] D. A. Graham and R. C. Marshall.
Collusive bidder behavior at single-object second-price and English auctions.
Journal of Political Economy, 95(6):1217-1239, 1987.
[11] J. Green and J.-J.
Laffont.
Characterization of satisfactory mechanisms for the revelation of preferences for public goods.
Econometrica, 45:427-438, 1977.
[12] T. Groves.
Incentives in teams.
Econometrica, 41:617-631, 1973.
[13] M. T. Hajiaghayi, R. Kleinberg, M. Mahdian, and D. C. Parkes.
Online auctions with re-usable goods.
EC, 165-174, 2005.
[14] M. T. Hajiaghayi, R. Kleinberg, and D. C. Parkes.
Adaptive limited-supply online auctions.
EC, 71-80, 2004.
[15] J. Hartline and R. McGrew.
From optimal limited to unlimited supply auctions.
EC, 175-182, 2005.
[16] L. Hurwicz.
On the existence of allocation systems whose manipulative Nash equilibria are Pareto optimal, 1975.
Presented at the 3rd World Congress of the Econometric Society.
[17] K. Leyton-Brown, Y. Shoham, and M. Tennenholtz.
Bidding clubs in first-price auctions.
AAAI, 373-378, 2002.
[18] E. Maskin and J. Riley.
Optimal multi-unit auctions.
In F. Hahn, editor, The Economics of Missing Markets, Information, and Games, chapter 14, 312-335.
Clarendon Press, Oxford, 1989.
[19] H. Moulin.
Efficient and strategy-proof assignment with a cheap residual claimant.
Working paper, March 2007.
[20] R. Myerson.
Optimal auction design.
Mathematics of Operations Research, 6:58-73, 1981.
[21] R. Myerson and M. Satterthwaite.
Efficient mechanisms for bilateral trading.
Journal of Economic Theory, 28:265-281, 1983.
[22] D. Parkes, J. Kalagnanam, and M. Eso.
Achieving budget-balance with Vickrey-based payment schemes in exchanges.
IJCAI, 1161-1168, 2001.
[23] W. Vickrey.
Counterspeculation, auctions, and competitive sealed tenders.
Journal of Finance, 16:8-37, 1961.
39

Worst-Case Optimal Redistribution of VCG Payments in Heterogeneous-Item Auctions with Unit Demand
ABSTRACT
Many important problems in multiagent systems involve the allocation of multiple resources among the agents .
For resource allocation problems , the well-known VCG mechanism satisfies a list of desired properties , including efficiency , strategy-proofness , individual rationality , and the non-deficit property .
However , VCG is generally not budget-balanced .
Under VCG , agents pay the VCG payments , which reduces social welfare .
To offset the loss of social welfare due to the VCG payments , VCG redistribution mechanisms were introduced .
These mechanisms aim to redistribute as much VCG payments back to the agents as possible , while maintaining the aforementioned desired properties of the VCG mechanism .
We continue the search for worst-case optimal VCG redistribution mechanisms -- mechanisms that maximize the fraction of total VCG payment redistributed in the worst case .
Previously , a worst-case optimal VCG redistribution mechanism ( denoted by WCO ) was characterized for multi-unit auctions with nonincreasing marginal values [ 7 ] .
Later , WCO was generalized to settings involving heterogeneous items [ 4 ] , resulting in the HETERO mechanism .
[ 4 ] conjectured that HETERO is feasible and worst-case optimal for heterogeneous-item auctions with unit demand .
In this paper , we propose a more natural way to generalize the WCO mechanism .
We prove that our generalized mechanism , though represented differently , actually coincides with HETERO .
Based on this new representation of HETERO , we prove that HETERO is indeed feasible and worst-case optimal in heterogeneous-item auctions with unit demand .
Finally , we conjecture that HETERO remains feasible and worst-case optimal in the even more general setting of combinatorial auctions with gross substitutes .
1 .
INTRODUCTION
1.1 VCG Redistribution Mechanisms
Many important problems in multiagent systems involve the allocation of multiple resources among the agents .
For resource allocation problems , the well-known VCG mechanism satisfies the following list of desired properties :
• Efficiency : the allocation maximizes the agents ' total valuation ( without considering payments ) .
• Strategy-proofness : for any agent , reporting truthfully is a dominant strategy , regardless of the other agents ' types .
• ( Ex post ) individual rationality : Every agent 's final utility ( after deducting her payment ) is always nonnegative .
• Non-deficit : the total paymentfrom the agents is nonnegative .
However , VCG is generally not budget-balanced .
Under VCG , agents pay the VCG payments , which reduces social welfare .
To offset the loss of social welfare due to the VCG payments , VCG redistribution mechanisms were introduced .
These mechanisms still allocate the resources using VCG .
On top of VCG , these mechanisms try to redistribute as much VCG payments back to the agents as possible .
We require that an agent 's redistribution be independent of her own type .
This is sufficient for maintaining strategyproofness and efficiency ( an agent has no control over her own redistribution ) .
For smoothly connected domains ( including multiunit auctions with nonincreasing marginal values and heterogeneousitem auctions with unit demand ) , the above requirement is also necessary for maintaining strategy-proofness and efficiency [ 8 ] .
A VCG redistribution mechanism is feasible if it maintains all the desired properties of the VCG mechanism .
That is , we also require that the redistribution process maintains individual rationality and the non-deficit property .
Let n be the number of agents .
Since all VCG redistribution mechanisms start by allocating according to the VCG mechanism , a VCG redistribution mechanism is characterized by its redistribution scheme r ~ = ( r1 , r2 , ... , rn ) .
Under VCG redistribution mechanism ~ r , agent i 's redistribution equals ri ( 01 , ... , 0i − 1 , 0i +1 , ... , 0n ) , where 0j is agent j 's type .
( We do not have to differentiate between an agent 's true type and her reported type , since all VCG redistribution mechanisms are strategy-proof . )
For the mechanism design objective studied in this paper , it is without loss of generality to only consider VCG redistribution mechanisms that are anonymous ( we defer the proof of this claim to the appendix ) .
An anonymous VCG redistribution mechanism is characterized by a single function r. Under ( anonymous ) VCG redistribution mechanism r , agent i 's redistribution equals r ( 0 − i ) , where 0 − i is the multiset of the types of the agents other than i.
We use θ ~ to denote the type profile .
Let V CG ( ~ θ ) be the total We organize existing results by their settings .
VCG payment for this type profile .
A VCG redistribution mechanism r satisfies the non-deficit property if the total redistribution never exceeds the total VCG payment .
That is , for any type profile ~ θ , Ei r ( θ − i ) ≤ V CG ( ~ θ ) .
A VCG redistribution mechanism r is ( ex post ) individually rational if every agent 's final utility is always nonnegative .
Since VCG is individually rational , we have that a sufficient condition for r to be individually rational is for any θ ~ and any i , r ( θ − i ) ≥ 0 ( on top of VCG , every agent also receives a redistribution amount that is always nonnegative ) .
On the other hand , when agent i is not interested in any item ( her valuation on any item bundle equals 0 ) , under VCG , i 's utility always equals 0 .
After redistribution , agent i 's utility is exactly her redistribution r ( θ − i ) .
That is , r ( θ − i ) ≥ 0 for all θ − i ( hence for all θ ~ and all i ) is also necessary for individual rationality .
We want to find VCG redistribution mechanisms that maximize the fraction of total VCG payment redistributed in the worst-case .
This mechanism design problem is equivalent to the following functional optimization model :
In this paper , we will analytically characterize one worst-case optimal VCG redistribution mechanism for heterogeneous-item auctions with unit demand .1 We conclude this subsection with an example VCG redistribution mechanism in the simplest setting of single-item auctions .
In a single-item auction , an agent 's type is a nonnegative real number representing her utility for winning the item .
Without loss of generality , we assume that θ1 ≥ θ2 ≥ ... ≥ θn ≥ 0 .
In single-item auctions , the Bailey-Cavallo VCG redistribution mechanism [ 2 , 3 ] works as follows :
• Allocate the item according to VCG : Agent 1 wins the item and pays θ2 .
The other agents win nothing and do not pay .
• Every agent receives a redistribution that equals n1 times the
second highest other type : Agent 1 and 2 each receives n1 θ3 .
The other agents each receives n1θ2 .
The above mechanism obviously maintains strategy-proofness and efficiency ( an agent 's redistribution does not depend on her own type ) .
It also maintains individual rationality because all redistributions are nonnegative .
The total redistribution equals 2n θ3 +
the above mechanism maintains the non-deficit property .
Finally , the total redistribution 2 n item auctions , this example mechanism 's worst-case redistribution fraction is n − 2 n
1.2 Previous Research on Worst-Case Optimal VCG Redistribution Mechanisms
In this subsection , we review existing results on worst-case optimal VCG redistribution mechanisms .
Besides high-level discussions , we also choose to include a certain level of technical details , as they are needed for later sections .
Worst-Case Optimal Redistribution in Multi-Unit Auctions with Unit Demand [ 7 , 12 ] : In multi-unit auctions with unit demand , the items for sale are identical .
Each agent wants at most one copy of the item .
( Single-item auctions are special cases of multi-unit auctions with unit demand . )
Let m be the number of items .
Throughout this paper , we only consider cases where m ≤ n − 2.2 Here , an agent 's type is a nonnegative real number representing her valuation for winning one copy of the item .
It is without loss of generality to assume that θ1 ≥ θ2 ≥ ... ≥ θn ≥ 0 .
[ 7 ] showed that for multi-unit auctions with unit demand , any VCG redistribution mechanism 's worst-case redistribution fraction is at most If we switch to a more general setting , then α ∗ is still an upper bound : if there exists a VCG redistribution mechanism whose worst-case redistribution fraction is strictly larger than α ∗ in a more general setting , then this mechanism , when applied to multi-unit auctions with unit demand , has a worst-case redistribution fraction that is strictly larger than α ∗ , which contradicts with the meaning of α ∗ .
[ 7 ] also characterized a VCG redistribution mechanism for multiunit auctions with unit demand , called the WCO mechanism .3 WCO 's worst-case redistribution fraction is exactly α ∗ .
That is , it is worst-case optimal .
WCO was obtained by optimizing within the family of linear VCG redistribution mechanisms .
A linear VCG redistribution mechanism r takes the following form : Here , the ci are constants .
( We only consider the ci that correspond to feasible VCG redistribution mechanisms . )
[ θ − i ] j is the j-th highest type among θ − i. Linear mechanism r is characterized by the values of the ci .
The optimal values the ci are as follows :
for i = m + 1 , ... , n − 1 , and c ∗ i = 0 for i = 1 , 2 , ... , m .
The characterization of WCO then follows :
Worst-Case Optimal Redistribution in Multi-Unit Auctions with
Nonincreasing Marginal Values [ 7 ] : Multi-unit auctions with non2 [ 7 ] showed that for multi-unit auctions with unit demand , when m = n − 1 , the worst-case redistribution fraction ( of any feasible VCG redistribution mechanism ) is at most 0 .
Since the setting studied in this paper is more general ( heterogeneous-item auctions with unit demand ) , we also have that the worst-case redistribution fraction is at most 0 when m = n − 1 .
Since heterogeneous-item auctions with x units are special cases of heterogeneous-item auctions with x + 1 units , we have that for our setting the worst-case redistribution fraction is at most 0 when m ≥ n − 1 .
That is , not redistributing anything is worst-case optimal when m ≥ n − 1 .
3WCO has also been independently derived in [ 12 ] , under a slightly different objective of maximizing worst-case efficiency ratio .
Also , for [ 12 ] 's objective , the optimal mechanism coincides with WCO only when the individual rationality constraint is enforced .
increasing marginal values are more general than multi-unit auctions with unit demand .
In this more general setting , the items are still identical , but an agent may demand more than one copy of the item .
An agent 's valuation for winning the first copy of the item is called her initial/first marginal value .
Similarly , an agent 's additional valuation for winning the i-th copy of the item is called her i-th marginal value .
An agent 's type contains m nonnegative real numbers ( i-th marginal value for i = 1 , ... , m ) .
In this setting , it is further assumed that the marginal values are nonincreasing .
As discussed earlier , in this more general setting , any VCG redistribution mechanism 's worst-case redistribution fraction is still bounded above by α * .
[ 7 ] generalized WCO to this setting , and proved that its worst-case redistribution fraction remains the same .
Therefore , WCO ( after generalization ) is also worst-case optimal for multi-unit auctions with nonincreasing marginal values .
The original definition of WCO does not directly generalize to multi-unit auctions with nonincreasing marginal values .
When it comes to multi-unit auctions with nonincreasing marginal values , an agent 's type is no longer a single value , which means that there is no such thing as `` the j-th highest type among θ_i '' .
To address this , [ 7 ] replaced [ θ_i ] j bym1R ( θ_i , j − m − 1 ) for j = m +1 , ... , n − 1 .
Basically , R ( θ_i , j − m − 1 ) is the generalization of [ θ_i ] j : it is identical to [ θ_i ] j in the unit demand setting , and it remains well-defined for multi-unit auctions with nonincreasing marginal values .
We abuse notation by not differentiating the agents and their types .
For example , θ_i is equivalent to the set of agents other than i. Let S be a set of agents .
R ( S , i ) is formally defined as follows ( this definition is included for completeness ; we will not use it anywhere ) :
• R ( S , 0 ) = V CG ( S ) ( the total VCG payment when only those in S participate in the auction ) .
Pm + i • For i = 1 , ... , | S | − m − 1 , R ( S , i ) = 1 j = 1 R ( U ( S , j ) , m + i
i − 1 ) .
Here , U ( S , j ) is the new set of agents , after removing the agent with the j-th highest initial marginal value in S from S .
The general form of WCO is as follows :
Worst-Case Optimal Redistribution in Heterogeneous-Item Auctions with Unit Demand [ 4 ] : In heterogeneous-item auctions with unit demand , the items for sale are different .
Each agent demands at most one item .
Here , an agent 's type consists of m nonnegative real numbers ( her valuation for winning item i for i = 1 , ... , m ) .
Heterogeneous-item auctions with unit demand is the main focus of this paper .
Since heterogeneous-item auctions with unit demand is more general than multi-unit auctions with unit demand , α * is still an upper bound on the worst-case redistribution fraction .
[ 4 ] proposed the HETERO mechanism , by generalizing WCO .
The authors conjectured that HETERO is feasible and has a worst-case redistribution fraction that equals α * .
That is , the authors conjectured that HETERO is worst-case optimal in this setting .
The main contribution of this paper is a proof of this conjecture .
Redistribution in Combinatorial Auctions with Gross Substitutes [ 6 ] : The gross substitutes condition was first proposed in [ 9 ] .
Like unit demand , the gross substitutes condition is a condition on an agent 's type ( does not depend on the mechanism under discussion ) .
In words , an agent 's type satisfies the gross substitutes condition if her demand for an item does not decrease when the prices of the other items increase .
Both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand are special cases of combinatorial auctions with gross substitutes [ 5 , 9 ] .
[ 6 ] showed that for this setting , the worst-case redistribution fraction of the Bailey-Cavallo mechanism [ 2 , 3 ] is exactly n_m_1 n ( when n ≥ m + 1 ) , and it is possible to construct mechanisms with even higher worst-case redistribution fractions .
The authors did not find a worst-case optimal mechanism for this setting .
At the end of this paper , we conjecture that HETERO is optimal for combinatorial auctions with gross substitutes .
Finally , Naroditskiy et al. [ 13 ] proposed a numerical technique for designing worst-case optimal redistribution mechanisms .
The proposed technique only works for single-parameter domains .
It does not apply to our setting ( multi-parameter domain ) .
1.3 Our contribution
We generalize WCO to heterogeneous-item auctions with unit demand .
We prove that the generalized mechanism , though represented differently , coincides with the HETERO mechanism proposed in [ 4 ] .
That is , what we proposed is not a new mechanism , but a new representation of an existing mechanism .
Based on our new representation of HETERO , we prove that HETERO is indeed feasible and worst-case optimal when applied to heterogeneousitem auctions with unit demand , thus confirming the conjecture raised in [ 4 ] .
We conclude with a new conjecture that HETERO remains feasible and worst-case optimal in the even more general setting of combinatorial auctions with gross substitutes .
2 .
NEW REPRESENTATION OF HETERO
3 .
FEASIBILITY AND WORST-CASE OPTIMALITY OF HETERO
4 .
CONCLUSION
We conclude our paper with the following conjecture : CONJECTURE 1 .
Gross substitutes implies redistribution monotonicity .
That is , HETERO remainsfeasible and worst-case optimal in combinatorial auctions with gross substitutes .
The idea is that both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand satisfy redistribution monotonicity .
A natural conjecture is that the `` most restrictive joint '' of these two settings also satisfies redistribution monotonicity .
There are many well-studied auction settings that contain both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand ( a list of which can be found in [ 10 ] ) .
Among these well-studied settings , combinatorial auctions with gross substitutes is the most restrictive .
To prove the conjecture , we need to prove that gross substitutes implies that for any set of agents S , R ( S , 0 ) > R ( S , 1 ) > ... > R ( S , ISI -- m -- 1 ) > 0 .
So far , we have only proved R ( S , 0 ) > R ( S , 1 ) > 0 .

Worst-Case Optimal Redistribution of VCG Payments in Heterogeneous-Item Auctions with Unit Demand
ABSTRACT
Many important problems in multiagent systems involve the allocation of multiple resources among the agents .
For resource allocation problems , the well-known VCG mechanism satisfies a list of desired properties , including efficiency , strategy-proofness , individual rationality , and the non-deficit property .
However , VCG is generally not budget-balanced .
Under VCG , agents pay the VCG payments , which reduces social welfare .
To offset the loss of social welfare due to the VCG payments , VCG redistribution mechanisms were introduced .
These mechanisms aim to redistribute as much VCG payments back to the agents as possible , while maintaining the aforementioned desired properties of the VCG mechanism .
We continue the search for worst-case optimal VCG redistribution mechanisms -- mechanisms that maximize the fraction of total VCG payment redistributed in the worst case .
Previously , a worst-case optimal VCG redistribution mechanism ( denoted by WCO ) was characterized for multi-unit auctions with nonincreasing marginal values [ 7 ] .
Later , WCO was generalized to settings involving heterogeneous items [ 4 ] , resulting in the HETERO mechanism .
[ 4 ] conjectured that HETERO is feasible and worst-case optimal for heterogeneous-item auctions with unit demand .
In this paper , we propose a more natural way to generalize the WCO mechanism .
We prove that our generalized mechanism , though represented differently , actually coincides with HETERO .
Based on this new representation of HETERO , we prove that HETERO is indeed feasible and worst-case optimal in heterogeneous-item auctions with unit demand .
Finally , we conjecture that HETERO remains feasible and worst-case optimal in the even more general setting of combinatorial auctions with gross substitutes .
1 .
INTRODUCTION
1.1 VCG Redistribution Mechanisms
Many important problems in multiagent systems involve the allocation of multiple resources among the agents .
For resource allocation problems , the well-known VCG mechanism satisfies the following list of desired properties :
• Efficiency : the allocation maximizes the agents ' total valuation ( without considering payments ) .
• Strategy-proofness : for any agent , reporting truthfully is a dominant strategy , regardless of the other agents ' types .
• ( Ex post ) individual rationality : Every agent 's final utility ( after deducting her payment ) is always nonnegative .
• Non-deficit : the total paymentfrom the agents is nonnegative .
However , VCG is generally not budget-balanced .
Under VCG , agents pay the VCG payments , which reduces social welfare .
To offset the loss of social welfare due to the VCG payments , VCG redistribution mechanisms were introduced .
These mechanisms still allocate the resources using VCG .
On top of VCG , these mechanisms try to redistribute as much VCG payments back to the agents as possible .
We require that an agent 's redistribution be independent of her own type .
This is sufficient for maintaining strategyproofness and efficiency ( an agent has no control over her own redistribution ) .
For smoothly connected domains ( including multiunit auctions with nonincreasing marginal values and heterogeneousitem auctions with unit demand ) , the above requirement is also necessary for maintaining strategy-proofness and efficiency [ 8 ] .
A VCG redistribution mechanism is feasible if it maintains all the desired properties of the VCG mechanism .
That is , we also require that the redistribution process maintains individual rationality and the non-deficit property .
Let n be the number of agents .
Since all VCG redistribution mechanisms start by allocating according to the VCG mechanism , a VCG redistribution mechanism is characterized by its redistribution scheme r ~ = ( r1 , r2 , ... , rn ) .
Under VCG redistribution mechanism ~ r , agent i 's redistribution equals ri ( 01 , ... , 0i − 1 , 0i +1 , ... , 0n ) , where 0j is agent j 's type .
( We do not have to differentiate between an agent 's true type and her reported type , since all VCG redistribution mechanisms are strategy-proof . )
An anonymous VCG redistribution mechanism is characterized by a single function r. Under ( anonymous ) VCG redistribution mechanism r , agent i 's redistribution equals r ( 0 − i ) , where 0 − i is the multiset of the types of the agents other than i.
We use θ ~ to denote the type profile .
Let V CG ( ~ θ ) be the total We organize existing results by their settings .
VCG payment for this type profile .
A VCG redistribution mechanism r satisfies the non-deficit property if the total redistribution never exceeds the total VCG payment .
A VCG redistribution mechanism r is ( ex post ) individually rational if every agent 's final utility is always nonnegative .
After redistribution , agent i 's utility is exactly her redistribution r ( θ − i ) .
We want to find VCG redistribution mechanisms that maximize the fraction of total VCG payment redistributed in the worst-case .
This mechanism design problem is equivalent to the following functional optimization model :
In this paper , we will analytically characterize one worst-case optimal VCG redistribution mechanism for heterogeneous-item auctions with unit demand .1 We conclude this subsection with an example VCG redistribution mechanism in the simplest setting of single-item auctions .
In a single-item auction , an agent 's type is a nonnegative real number representing her utility for winning the item .
In single-item auctions , the Bailey-Cavallo VCG redistribution mechanism [ 2 , 3 ] works as follows :
• Allocate the item according to VCG : Agent 1 wins the item and pays θ2 .
The other agents win nothing and do not pay .
• Every agent receives a redistribution that equals n1 times the
second highest other type : Agent 1 and 2 each receives n1 θ3 .
The other agents each receives n1θ2 .
The above mechanism obviously maintains strategy-proofness and efficiency ( an agent 's redistribution does not depend on her own type ) .
It also maintains individual rationality because all redistributions are nonnegative .
The total redistribution equals 2n θ3 +
the above mechanism maintains the non-deficit property .
Finally , the total redistribution 2 n item auctions , this example mechanism 's worst-case redistribution fraction is n − 2 n
1.2 Previous Research on Worst-Case Optimal VCG Redistribution Mechanisms
In this subsection , we review existing results on worst-case optimal VCG redistribution mechanisms .
Worst-Case Optimal Redistribution in Multi-Unit Auctions with Unit Demand [ 7 , 12 ] : In multi-unit auctions with unit demand , the items for sale are identical .
Each agent wants at most one copy of the item .
( Single-item auctions are special cases of multi-unit auctions with unit demand . )
Let m be the number of items .
Throughout this paper , we only consider cases where m ≤ n − 2.2 Here , an agent 's type is a nonnegative real number representing her valuation for winning one copy of the item .
[ 7 ] also characterized a VCG redistribution mechanism for multiunit auctions with unit demand , called the WCO mechanism .3 WCO 's worst-case redistribution fraction is exactly α ∗ .
That is , it is worst-case optimal .
WCO was obtained by optimizing within the family of linear VCG redistribution mechanisms .
A linear VCG redistribution mechanism r takes the following form : Here , the ci are constants .
( We only consider the ci that correspond to feasible VCG redistribution mechanisms . )
[ θ − i ] j is the j-th highest type among θ − i. Linear mechanism r is characterized by the values of the ci .
The optimal values the ci are as follows :
The characterization of WCO then follows :
Worst-Case Optimal Redistribution in Multi-Unit Auctions with
Nonincreasing Marginal Values [ 7 ] : Multi-unit auctions with non2 [ 7 ] showed that for multi-unit auctions with unit demand , when m = n − 1 , the worst-case redistribution fraction ( of any feasible VCG redistribution mechanism ) is at most 0 .
Since the setting studied in this paper is more general ( heterogeneous-item auctions with unit demand ) , we also have that the worst-case redistribution fraction is at most 0 when m = n − 1 .
Since heterogeneous-item auctions with x units are special cases of heterogeneous-item auctions with x + 1 units , we have that for our setting the worst-case redistribution fraction is at most 0 when m ≥ n − 1 .
That is , not redistributing anything is worst-case optimal when m ≥ n − 1 .
Also , for [ 12 ] 's objective , the optimal mechanism coincides with WCO only when the individual rationality constraint is enforced .
increasing marginal values are more general than multi-unit auctions with unit demand .
In this more general setting , the items are still identical , but an agent may demand more than one copy of the item .
An agent 's valuation for winning the first copy of the item is called her initial/first marginal value .
Similarly , an agent 's additional valuation for winning the i-th copy of the item is called her i-th marginal value .
An agent 's type contains m nonnegative real numbers ( i-th marginal value for i = 1 , ... , m ) .
In this setting , it is further assumed that the marginal values are nonincreasing .
As discussed earlier , in this more general setting , any VCG redistribution mechanism 's worst-case redistribution fraction is still bounded above by α * .
[ 7 ] generalized WCO to this setting , and proved that its worst-case redistribution fraction remains the same .
Therefore , WCO ( after generalization ) is also worst-case optimal for multi-unit auctions with nonincreasing marginal values .
The original definition of WCO does not directly generalize to multi-unit auctions with nonincreasing marginal values .
When it comes to multi-unit auctions with nonincreasing marginal values , an agent 's type is no longer a single value , which means that there is no such thing as `` the j-th highest type among θ_i '' .
We abuse notation by not differentiating the agents and their types .
For example , θ_i is equivalent to the set of agents other than i. Let S be a set of agents .
i − 1 ) .
Here , U ( S , j ) is the new set of agents , after removing the agent with the j-th highest initial marginal value in S from S .
The general form of WCO is as follows :
Worst-Case Optimal Redistribution in Heterogeneous-Item Auctions with Unit Demand [ 4 ] : In heterogeneous-item auctions with unit demand , the items for sale are different .
Each agent demands at most one item .
Here , an agent 's type consists of m nonnegative real numbers ( her valuation for winning item i for i = 1 , ... , m ) .
Heterogeneous-item auctions with unit demand is the main focus of this paper .
Since heterogeneous-item auctions with unit demand is more general than multi-unit auctions with unit demand , α * is still an upper bound on the worst-case redistribution fraction .
[ 4 ] proposed the HETERO mechanism , by generalizing WCO .
The authors conjectured that HETERO is feasible and has a worst-case redistribution fraction that equals α * .
That is , the authors conjectured that HETERO is worst-case optimal in this setting .
The main contribution of this paper is a proof of this conjecture .
Redistribution in Combinatorial Auctions with Gross Substitutes [ 6 ] : The gross substitutes condition was first proposed in [ 9 ] .
Like unit demand , the gross substitutes condition is a condition on an agent 's type ( does not depend on the mechanism under discussion ) .
In words , an agent 's type satisfies the gross substitutes condition if her demand for an item does not decrease when the prices of the other items increase .
Both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand are special cases of combinatorial auctions with gross substitutes [ 5 , 9 ] .
The authors did not find a worst-case optimal mechanism for this setting .
At the end of this paper , we conjecture that HETERO is optimal for combinatorial auctions with gross substitutes .
Finally , Naroditskiy et al. [ 13 ] proposed a numerical technique for designing worst-case optimal redistribution mechanisms .
The proposed technique only works for single-parameter domains .
It does not apply to our setting ( multi-parameter domain ) .
1.3 Our contribution
We generalize WCO to heterogeneous-item auctions with unit demand .
We prove that the generalized mechanism , though represented differently , coincides with the HETERO mechanism proposed in [ 4 ] .
That is , what we proposed is not a new mechanism , but a new representation of an existing mechanism .
Based on our new representation of HETERO , we prove that HETERO is indeed feasible and worst-case optimal when applied to heterogeneousitem auctions with unit demand , thus confirming the conjecture raised in [ 4 ] .
We conclude with a new conjecture that HETERO remains feasible and worst-case optimal in the even more general setting of combinatorial auctions with gross substitutes .
4 .
CONCLUSION
We conclude our paper with the following conjecture : CONJECTURE 1 .
Gross substitutes implies redistribution monotonicity .
That is , HETERO remainsfeasible and worst-case optimal in combinatorial auctions with gross substitutes .
The idea is that both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand satisfy redistribution monotonicity .
A natural conjecture is that the `` most restrictive joint '' of these two settings also satisfies redistribution monotonicity .
There are many well-studied auction settings that contain both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand ( a list of which can be found in [ 10 ] ) .
Among these well-studied settings , combinatorial auctions with gross substitutes is the most restrictive .

Worst-Case Optimal Redistribution of VCG Payments in Heterogeneous-Item Auctions with Unit Demand
ABSTRACT
Many important problems in multiagent systems involve the allocation of multiple resources among the agents .
For resource allocation problems , the well-known VCG mechanism satisfies a list of desired properties , including efficiency , strategy-proofness , individual rationality , and the non-deficit property .
However , VCG is generally not budget-balanced .
Under VCG , agents pay the VCG payments , which reduces social welfare .
To offset the loss of social welfare due to the VCG payments , VCG redistribution mechanisms were introduced .
These mechanisms aim to redistribute as much VCG payments back to the agents as possible , while maintaining the aforementioned desired properties of the VCG mechanism .
We continue the search for worst-case optimal VCG redistribution mechanisms -- mechanisms that maximize the fraction of total VCG payment redistributed in the worst case .
Previously , a worst-case optimal VCG redistribution mechanism ( denoted by WCO ) was characterized for multi-unit auctions with nonincreasing marginal values [ 7 ] .
Later , WCO was generalized to settings involving heterogeneous items [ 4 ] , resulting in the HETERO mechanism .
[ 4 ] conjectured that HETERO is feasible and worst-case optimal for heterogeneous-item auctions with unit demand .
In this paper , we propose a more natural way to generalize the WCO mechanism .
We prove that our generalized mechanism , though represented differently , actually coincides with HETERO .
Based on this new representation of HETERO , we prove that HETERO is indeed feasible and worst-case optimal in heterogeneous-item auctions with unit demand .
Finally , we conjecture that HETERO remains feasible and worst-case optimal in the even more general setting of combinatorial auctions with gross substitutes .
1 .
INTRODUCTION
1.1 VCG Redistribution Mechanisms
Many important problems in multiagent systems involve the allocation of multiple resources among the agents .
For resource allocation problems , the well-known VCG mechanism satisfies the following list of desired properties :
• Efficiency : the allocation maximizes the agents ' total valuation ( without considering payments ) .
• Strategy-proofness : for any agent , reporting truthfully is a dominant strategy , regardless of the other agents ' types .
• ( Ex post ) individual rationality : Every agent 's final utility ( after deducting her payment ) is always nonnegative .
• Non-deficit : the total paymentfrom the agents is nonnegative .
However , VCG is generally not budget-balanced .
Under VCG , agents pay the VCG payments , which reduces social welfare .
To offset the loss of social welfare due to the VCG payments , VCG redistribution mechanisms were introduced .
These mechanisms still allocate the resources using VCG .
On top of VCG , these mechanisms try to redistribute as much VCG payments back to the agents as possible .
We require that an agent 's redistribution be independent of her own type .
This is sufficient for maintaining strategyproofness and efficiency ( an agent has no control over her own redistribution ) .
For smoothly connected domains ( including multiunit auctions with nonincreasing marginal values and heterogeneousitem auctions with unit demand ) , the above requirement is also necessary for maintaining strategy-proofness and efficiency [ 8 ] .
A VCG redistribution mechanism is feasible if it maintains all the desired properties of the VCG mechanism .
That is , we also require that the redistribution process maintains individual rationality and the non-deficit property .
Let n be the number of agents .
Since all VCG redistribution mechanisms start by allocating according to the VCG mechanism , a VCG redistribution mechanism is characterized by its redistribution scheme r ~ = ( r1 , r2 , ... , rn ) .
Under VCG redistribution mechanism ~ r , agent i 's redistribution equals ri ( 01 , ... , 0i − 1 , 0i +1 , ... , 0n ) , where 0j is agent j 's type .
( We do not have to differentiate between an agent 's true type and her reported type , since all VCG redistribution mechanisms are strategy-proof . )
For the mechanism design objective studied in this paper , it is without loss of generality to only consider VCG redistribution mechanisms that are anonymous ( we defer the proof of this claim to the appendix ) .
An anonymous VCG redistribution mechanism is characterized by a single function r. Under ( anonymous ) VCG redistribution mechanism r , agent i 's redistribution equals r ( 0 − i ) , where 0 − i is the multiset of the types of the agents other than i.
We use θ ~ to denote the type profile .
Let V CG ( ~ θ ) be the total We organize existing results by their settings .
VCG payment for this type profile .
A VCG redistribution mechanism r satisfies the non-deficit property if the total redistribution never exceeds the total VCG payment .
That is , for any type profile ~ θ , Ei r ( θ − i ) ≤ V CG ( ~ θ ) .
A VCG redistribution mechanism r is ( ex post ) individually rational if every agent 's final utility is always nonnegative .
Since VCG is individually rational , we have that a sufficient condition for r to be individually rational is for any θ ~ and any i , r ( θ − i ) ≥ 0 ( on top of VCG , every agent also receives a redistribution amount that is always nonnegative ) .
On the other hand , when agent i is not interested in any item ( her valuation on any item bundle equals 0 ) , under VCG , i 's utility always equals 0 .
After redistribution , agent i 's utility is exactly her redistribution r ( θ − i ) .
That is , r ( θ − i ) ≥ 0 for all θ − i ( hence for all θ ~ and all i ) is also necessary for individual rationality .
We want to find VCG redistribution mechanisms that maximize the fraction of total VCG payment redistributed in the worst-case .
This mechanism design problem is equivalent to the following functional optimization model :
In this paper , we will analytically characterize one worst-case optimal VCG redistribution mechanism for heterogeneous-item auctions with unit demand .1 We conclude this subsection with an example VCG redistribution mechanism in the simplest setting of single-item auctions .
In a single-item auction , an agent 's type is a nonnegative real number representing her utility for winning the item .
Without loss of generality , we assume that θ1 ≥ θ2 ≥ ... ≥ θn ≥ 0 .
In single-item auctions , the Bailey-Cavallo VCG redistribution mechanism [ 2 , 3 ] works as follows :
• Allocate the item according to VCG : Agent 1 wins the item and pays θ2 .
The other agents win nothing and do not pay .
• Every agent receives a redistribution that equals n1 times the
second highest other type : Agent 1 and 2 each receives n1 θ3 .
The other agents each receives n1θ2 .
The above mechanism obviously maintains strategy-proofness and efficiency ( an agent 's redistribution does not depend on her own type ) .
It also maintains individual rationality because all redistributions are nonnegative .
The total redistribution equals 2n θ3 +
the above mechanism maintains the non-deficit property .
Finally , the total redistribution 2 n item auctions , this example mechanism 's worst-case redistribution fraction is n − 2 n
1.2 Previous Research on Worst-Case Optimal VCG Redistribution Mechanisms
In this subsection , we review existing results on worst-case optimal VCG redistribution mechanisms .
Besides high-level discussions , we also choose to include a certain level of technical details , as they are needed for later sections .
Worst-Case Optimal Redistribution in Multi-Unit Auctions with Unit Demand [ 7 , 12 ] : In multi-unit auctions with unit demand , the items for sale are identical .
Each agent wants at most one copy of the item .
( Single-item auctions are special cases of multi-unit auctions with unit demand . )
Let m be the number of items .
Throughout this paper , we only consider cases where m ≤ n − 2.2 Here , an agent 's type is a nonnegative real number representing her valuation for winning one copy of the item .
It is without loss of generality to assume that θ1 ≥ θ2 ≥ ... ≥ θn ≥ 0 .
[ 7 ] showed that for multi-unit auctions with unit demand , any VCG redistribution mechanism 's worst-case redistribution fraction is at most If we switch to a more general setting , then α ∗ is still an upper bound : if there exists a VCG redistribution mechanism whose worst-case redistribution fraction is strictly larger than α ∗ in a more general setting , then this mechanism , when applied to multi-unit auctions with unit demand , has a worst-case redistribution fraction that is strictly larger than α ∗ , which contradicts with the meaning of α ∗ .
[ 7 ] also characterized a VCG redistribution mechanism for multiunit auctions with unit demand , called the WCO mechanism .3 WCO 's worst-case redistribution fraction is exactly α ∗ .
That is , it is worst-case optimal .
WCO was obtained by optimizing within the family of linear VCG redistribution mechanisms .
A linear VCG redistribution mechanism r takes the following form : Here , the ci are constants .
( We only consider the ci that correspond to feasible VCG redistribution mechanisms . )
[ θ − i ] j is the j-th highest type among θ − i. Linear mechanism r is characterized by the values of the ci .
The optimal values the ci are as follows :
for i = m + 1 , ... , n − 1 , and c ∗ i = 0 for i = 1 , 2 , ... , m .
The characterization of WCO then follows :
Worst-Case Optimal Redistribution in Multi-Unit Auctions with
Nonincreasing Marginal Values [ 7 ] : Multi-unit auctions with non2 [ 7 ] showed that for multi-unit auctions with unit demand , when m = n − 1 , the worst-case redistribution fraction ( of any feasible VCG redistribution mechanism ) is at most 0 .
Since the setting studied in this paper is more general ( heterogeneous-item auctions with unit demand ) , we also have that the worst-case redistribution fraction is at most 0 when m = n − 1 .
Since heterogeneous-item auctions with x units are special cases of heterogeneous-item auctions with x + 1 units , we have that for our setting the worst-case redistribution fraction is at most 0 when m ≥ n − 1 .
That is , not redistributing anything is worst-case optimal when m ≥ n − 1 .
3WCO has also been independently derived in [ 12 ] , under a slightly different objective of maximizing worst-case efficiency ratio .
Also , for [ 12 ] 's objective , the optimal mechanism coincides with WCO only when the individual rationality constraint is enforced .
increasing marginal values are more general than multi-unit auctions with unit demand .
In this more general setting , the items are still identical , but an agent may demand more than one copy of the item .
An agent 's valuation for winning the first copy of the item is called her initial/first marginal value .
Similarly , an agent 's additional valuation for winning the i-th copy of the item is called her i-th marginal value .
An agent 's type contains m nonnegative real numbers ( i-th marginal value for i = 1 , ... , m ) .
In this setting , it is further assumed that the marginal values are nonincreasing .
As discussed earlier , in this more general setting , any VCG redistribution mechanism 's worst-case redistribution fraction is still bounded above by α * .
[ 7 ] generalized WCO to this setting , and proved that its worst-case redistribution fraction remains the same .
Therefore , WCO ( after generalization ) is also worst-case optimal for multi-unit auctions with nonincreasing marginal values .
The original definition of WCO does not directly generalize to multi-unit auctions with nonincreasing marginal values .
When it comes to multi-unit auctions with nonincreasing marginal values , an agent 's type is no longer a single value , which means that there is no such thing as `` the j-th highest type among θ_i '' .
To address this , [ 7 ] replaced [ θ_i ] j bym1R ( θ_i , j − m − 1 ) for j = m +1 , ... , n − 1 .
Basically , R ( θ_i , j − m − 1 ) is the generalization of [ θ_i ] j : it is identical to [ θ_i ] j in the unit demand setting , and it remains well-defined for multi-unit auctions with nonincreasing marginal values .
We abuse notation by not differentiating the agents and their types .
For example , θ_i is equivalent to the set of agents other than i. Let S be a set of agents .
R ( S , i ) is formally defined as follows ( this definition is included for completeness ; we will not use it anywhere ) :
• R ( S , 0 ) = V CG ( S ) ( the total VCG payment when only those in S participate in the auction ) .
Pm + i • For i = 1 , ... , | S | − m − 1 , R ( S , i ) = 1 j = 1 R ( U ( S , j ) , m + i
i − 1 ) .
Here , U ( S , j ) is the new set of agents , after removing the agent with the j-th highest initial marginal value in S from S .
The general form of WCO is as follows :
Worst-Case Optimal Redistribution in Heterogeneous-Item Auctions with Unit Demand [ 4 ] : In heterogeneous-item auctions with unit demand , the items for sale are different .
Each agent demands at most one item .
Here , an agent 's type consists of m nonnegative real numbers ( her valuation for winning item i for i = 1 , ... , m ) .
Heterogeneous-item auctions with unit demand is the main focus of this paper .
Since heterogeneous-item auctions with unit demand is more general than multi-unit auctions with unit demand , α * is still an upper bound on the worst-case redistribution fraction .
[ 4 ] proposed the HETERO mechanism , by generalizing WCO .
The authors conjectured that HETERO is feasible and has a worst-case redistribution fraction that equals α * .
That is , the authors conjectured that HETERO is worst-case optimal in this setting .
The main contribution of this paper is a proof of this conjecture .
Redistribution in Combinatorial Auctions with Gross Substitutes [ 6 ] : The gross substitutes condition was first proposed in [ 9 ] .
Like unit demand , the gross substitutes condition is a condition on an agent 's type ( does not depend on the mechanism under discussion ) .
In words , an agent 's type satisfies the gross substitutes condition if her demand for an item does not decrease when the prices of the other items increase .
Both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand are special cases of combinatorial auctions with gross substitutes [ 5 , 9 ] .
[ 6 ] showed that for this setting , the worst-case redistribution fraction of the Bailey-Cavallo mechanism [ 2 , 3 ] is exactly n_m_1 n ( when n ≥ m + 1 ) , and it is possible to construct mechanisms with even higher worst-case redistribution fractions .
The authors did not find a worst-case optimal mechanism for this setting .
At the end of this paper , we conjecture that HETERO is optimal for combinatorial auctions with gross substitutes .
Finally , Naroditskiy et al. [ 13 ] proposed a numerical technique for designing worst-case optimal redistribution mechanisms .
The proposed technique only works for single-parameter domains .
It does not apply to our setting ( multi-parameter domain ) .
1.3 Our contribution
We generalize WCO to heterogeneous-item auctions with unit demand .
We prove that the generalized mechanism , though represented differently , coincides with the HETERO mechanism proposed in [ 4 ] .
That is , what we proposed is not a new mechanism , but a new representation of an existing mechanism .
Based on our new representation of HETERO , we prove that HETERO is indeed feasible and worst-case optimal when applied to heterogeneousitem auctions with unit demand , thus confirming the conjecture raised in [ 4 ] .
We conclude with a new conjecture that HETERO remains feasible and worst-case optimal in the even more general setting of combinatorial auctions with gross substitutes .
2 .
NEW REPRESENTATION OF HETERO
We recall that WCO was obtained by optimizing within the family of linear VCG redistribution mechanisms .
The original representation of HETERO was obtained using a similar approach [ 4 ] .
The authors focused on the following family of mechanisms :
Here , the βi are constants .
t ( S , j ) is the expected total VCG payment when we remove j agents uniformly at random from S , and allocate all the items to the remaining agents .
It is easy to see that all member mechanisms of the above family are well-defined for general combinatorial auctions .
Not every member mechanism is feasible though .
[ 4 ] did not attempt optimizing over the family .
Instead , the βi are chosen so that the corresponding mechanism coincides with WCO when it comes to multi-unit auctions with unit demand .
It turns out that the choice is unique , and the corresponding mechanism is called HETERO .
[ 4 ] conjectured that HETERO is feasible and worst-case optimal for heterogeneous-item auctions with unit demand .
In this section , we propose another way to generalize WCO .
We will show that the generalized WCO actually coincides with HETERO .
That is , what we derive is a new representation of HETERO .
This new representation will prove itself useful in later discussions .
We recall that the characterization of WCO for multi-unit auctions with nonincreasing marginal values is based on a series of functions R ( S , i ) .
These functions do not directly generalize to settings involving heterogeneous items , because , for i > 0 , R ( S , i ) is defined explicitly based on the agents ' initial marginal values .
Fortunately , there is an easy way to rewrite R ( S , i ) , so that it becomes well-defined for settings involving heterogeneous items .
Based on Equation 1 , WCO can be rewritten into the following form ( the only changes are that for i > 0 , R ( S , i ) 's definition no longer mentions `` initial marginal values '' ) : Definition 1 .
Heterogeneous WCO ( new representation of HETERO ) :
c ∗ j R ( θ − i , j − m − 1 ) • R ( S , 0 ) = V CG ( S ) • For i = 1 , ... , | S | − m − 1 , R ( S , i ) equals :
Heterogeneous WCO is well-defined for general combinatorial auctions , so we can directly apply it to heterogeneous-item auctions with unit demand .
Of course , we still have the burden to prove that it remains feasible and worst-case optimal .
We will do so in the next section .
Heterogeneous WCO is not a new mechanism .
It turns out that it coincides with HETERO for general combinatorial auctions .
That is , Definition 1 is a new representation of the existing mechanism HETERO .
Similarly , R ( S , 2 ) = [ S ] 4 , R ( S , 3 ) = [ S ] 5 , ... , and finally R ( S , | S | − m − 1 ) = R ( S , | S | − 2 ) = [ S ] | S | ( lowest type from the agents in S ) .
It is clear that redistribution monotonicity holds here .
More generally , redistribution monotonicity holds for multi-unit auctions with nonincreasing marginal values : Claim 17 of [ 7 ] proved that R ( S , i ) is nonincreasing in i for multi-unit auctions with nonincreasing marginal values ; R ( S , i ) 's original definition as described in Subsection 1.2 makes it clear that the R ( S , i ) are nonnegative .
The following proposition greatly simplifies our task :
PROPOSITION 2 .
Ifthe setting satisfies redistribution monotonicity , then HETERO is feasible ( strategy-proof , efficient , individually rational , and non-deficit ) , and its worst-case redistributionfraction is at least α ∗ .
If the setting is also more general than multi-unit auctions with unit demand , then HETERO is worst-case optimal .
PROOF .
We first prove that HETERO is feasible given redistribution monotonicity .
According to Definition 1 , under HETERO , an agent 's redistribution does not depend on her own type .
That is , HETERO is strategy-proof and efficient in all settings .
We only need to prove that HETERO is individually rational and non-deficit given redistribution monotonicity .
Individual rationality : As discussed in Subsection 1.1 , individual rationality is equivalent to redistributions being nonnegative .
We recall that for multi-unit auctions with unit demand , under WCO , agent i 's redistribution equals
WCO is known to be individually rational .
That is , for all θ − i , PROPOSITION 1 .
Heterogeneous WCO coincides with HETERO n − 1X cj [ θ − i ] j ≥ 0 for general combinatorial auctions .
j = m +1 ∗
Proof omitted since it is based on pure algebraic manipulation .
3 .
FEASIBILITY AND WORST-CASE OPTIMALITY OF HETERO
In this section , we prove that HETERO , as represented in Definition 1 , is feasible and worst-case optimal for heterogeneous-item auctions with unit demand .
We first define the redistribution monotonicity condition : Definition 2 .
An auction setting satisfies redistribution monotonicity if for any set of agents S , we have that R ( S , 0 ) ≥ R ( S , 1 ) ≥ ... ≥ R ( S , | S | − m − 1 ) ≥ 0 R was defined in Definition 1 .
That is , R ( S , 0 ) = VCG ( S ) , and for i = 1 , ... , | S | − m − 1 , R ( S , i ) equals !
R ( S − a , i − 1 ) − ( | S | − m − i ) R ( S , i − 1 ) .
For example , the setting of single-item auctions satisfies redistribution monotonicity .
In a single-item auction , R ( S , 0 ) = V CG ( S ) = [ S ] 2 ( [ S ] i is the i-th highest type from the agents in S ) .
This is equivalent to for all x0 ≥ ... ≥ xn − m − 2 ≥ 0 ,
Based on ( 2 ) and ( 4 ) ( substituting R ( θ − i , j ) for xj for all j ) , we have that ( 3 ) is nonnegative .
Therefore , redistribution monotonicity implies individual rationality .
Non-deficit and worst-case optimality : For multi-unit auctions with unit demand , under WCO , the total VCG payment is mθm +1 .
The total redistribution is
Under HETERO , the total redistribution is The total VCG payment equals V CG ( ~ θ ) = R ( ~ θ , 0 ) .
Redistribution monotonicity implies that
Given ( 5 ) and ( 7 ) ( substituting R ( ~ θ , j ) for xj for all j ) , we have that ( 6 ) is between α ∗ times the total VCG payment and the total VCG payment .
Therefore , redistribution monotonicity implies the non-deficit property and also worst-case optimality .
We want to prove that for heterogeneous-item auctions with unit demand , the following redistribution monotonicity condition holds .
R ( S , 0 ) ≥ R ( S , 1 ) ≥ ... ≥ R ( S , | S | − m − 1 ) ≥ 0 By Proposition 3 , it suffices to prove that for all j , Rj ( S , 0 ) ≥ Rj ( S , 1 ) ≥ ... ≥ Rj ( S , | S | − m − 1 ) ≥ 0 .
Without loss of generality , we will prove
To prove the above inequality , we need the following definitions and propositions .
From now on to the end ofthis section , the setting by default is heterogeneous-item auctions with unit demand , unless specified .
We use E ( T , S ) to denote the efficient total valuation when we allocate all the items in T to the agents in S.
In the remaining of this section , we prove that heterogeneousitem auctions with unit demand satisfies redistribution monotonicity , which would then imply that HETERO is feasible and worstcase optimal for heterogeneous-item auctions with unit demand .
We define Rj ( S , i ) by modifying the definition of R ( S , i ) in Definition 1 .
• Rj ( S , 0 ) = V CGj ( S ) .
V CGj ( S ) is the VCG price of item j ( the VCG payment from the agent winning item j ) when we allocate all the items to the agents in S using VCG .
• For i = 1 , ... , | S | − m − 1 , Rj ( S , i ) equals
[ 14 ] showed that the proposition is true when gross substitutes condition holds .
Heterogeneous-item auctions with unit demand satisfies gross substitutes .
We use { 1 } ⊕ { 1 , ... , m } to denote the item set that contains not only item 1 to m , but also an additional duplicate of item 1 .
PROPOSITION 5 .
Let S be any set of agents .
Let a be the agent who wins item 1 when we allocate the items { 1 , ... , m } to the agents in S .
We have that E ( { 1 } ⊕ { 1 , ... , m } , S ) = E ( { 1 } , a ) + E ( { 1 , ... , m } , S − a ) .
That is , after we add an additional duplicate of item 1 to the auction , there exists an efficient allocation under which agent a still wins item 1 .
The above proposition was proved in [ 11 ] .
PROOF .
We prove by induction .
When i = 0 , by definition , for any S ,
PROOF .
Let w1 be the winner of item 1 when we allocate the items { 1 , ... , m } to the agents in S using VCG .
V CG1 ( S ) = E ( { 1 , ... , m } , S − w1 ) − E ( { 2 , ... , m } , S − w1 ) .
a could be either w1 or some other agent .
We discuss case by case .
Case a = w1 : Let w ′ 1 be the new winner of item 1 when we allocate the items { 1 , ... , m } to the agents in S − w1 using VCG .
V CG1 ( S − w1 ) = E ( { 1 , ... , m } , S − w1 − w ′ 1 ) − E ( { 2 , ... , m } , S − w1 − w ′ 1 ) .
We need to prove that E ( { 1 , ... , m } , S − w1 ) − E ( { 2 , ... , m } , S − w1 ) ≥ E ( { 1 , ... , m } , S − w1 − w ′ 1 ) − E ( { 2 , ... , m } , S − w1 − w ′ 1 ) .
We construct a new agent x. Let x 's valuation for item 1 be extremely high so that she wins item 1 .
The above inequality can be rewritten as E ( { 1 , ... , m } , S − w1 ) − E ( { 2 , ... , m } , S − w1 ) − E ( { 1 } , x ) ≥ E ( { 1 , ... , m } , S − w1 − w ′ 1 ) − E ( { 2 , ... , m } , S − w1 − w ′ 1 ) − E ( { 1 } , x ) .
This
is , E ( { 1 , ... , m } , S − w1 ) − E ( { 1 , ... , m } , S − w1 + x ) > E ( { 1 , ... , m } , S − w1 − w ′ 1 ) − E ( { 1 , ... , m } , S − w1 − w ′ 1 + x ) .
We rearrange the terms , and get E ( { 1 , ... , m } , S − w1 ) +
Case a = 6 w1 : Let w ′ 1 be the new winner of item 1 when we allocate all the items { 1 , ... , m } to the agents in S − a using VCG .
PROPOSITION 7 .
Winners still win after we remove some other agents [ 4 , 6 ] :4 For any set of agents S and any set of items T , we use W to denote the set of winners when we allocate the items in T to the agents in S using VCG .
After we remove some agents in S , those in W that have not been removed remain to be winners , provided that a consistent tie-breaking rule exists .
It should be noted that there may not exist a consistent tie-breaking rule that satisfies the above proposition .
Fortunately , we are able to prove that tie-breaking is irrelevant for the goal of proving redistribution monotonicity .
We say that a type profile is tie-free if it satisfies the following : Let T1 = { 1 } ® { 1 , ... , m } .
Let T2 = { 1 , ... , m } .
Basically , T1 and T2 are the only item sets that we will ever mention .
A type profile is tie-free if for any set of agents S , when we allocate the items in T1 ( or T2 ) to S , the set of VCG winners is unique .
If we only consider tie-free type profiles , then we do not need to be bothered by tie-breaking .
We notice that the set of tie-free type profiles is a dense subset of the set of all type profiles -- any type profile can be perturbed infinitesimally to become a tie-free type profile .
Our ultimate goal is to prove that for any set of agents S , R ( S , 0 ) > R ( S , 1 ) > ... > R ( S , | S | − m − 1 ) > 0 We notice that the R ( S , j ) are continuous in the agents ' types .
Therefore , it suffices to prove the above inequality for tie-free type profiles only .
From now on , we simply assume that the set of VCG winners is always unique .
Definition 3 .
For any set of agents S with | S | > m +1 , let D ( S ) be the set of m + 1 winners when we allocate { 1 } ® { 1 , ... , m } to the agents in S. D ( S ) is called the determination set of S.
The above proposition says that for the purpose of calculating item 1 's VCG price , only those agents in D ( S ) are relevant .
4The proposition was originally introduced in [ 4 ] .
A more rigorous proof of a more general claim was also given in [ 6 ] .
Let w1 be the winner of item 1 when we allocate { 1 , ... , m } to the agents in S. V CG1 ( S ) = E ( { 1 , ... , m } , S − w1 ) − E ( { 2 , ... , m } , S − w1 ) = E ( { 1 , ... , m } , S − w1 ) + E ( { 1 } , w1 ) − E ( { 1 , ... , m } , S ) = E ( { 1 } ® { 1 , ... , m } , S ) − E ( { 1 , ... , m } , S ) ( the last step is due to Proposition 5 ) .
The first term only depends on those in D ( S ) .
The second term also only depends on those in D ( S ) for the following reason : Let S ′ be the set of VCG winners when we allocate { 1 , ... , m } to the agents in S .
The second term only depends on those in S ′ .
We introduce an agent x whose valuation for item 1 is extremely high so that she wins item 1 .
When we allocate { 1 } ® { 1 , ... , m } to the agents in S + x , the set of VCG winners are then x + S ′ .
D ( S ) are the new set of VCG winners after we remove x. By Proposition 7 , those in S ′ must still remain in D ( S ) .
Overall , V CG1 ( S ) only depends on those agents in D ( S ) .
Similarly , V CG1 ( S − a ) only depends on those agents in D ( S − a ) .
For a ∈ S − D ( S ) , D ( S ) = D ( S − a ) .
Therefore , we must have VCG1 ( S ) = VCG1 ( S − a ) .
Definition 4 .
Let S be any set of agents .
Let k be any integer from 1 to | S | .
Let a1 ≺ a2 ≺ ... ≺ ak be a sequence of k distinct agents in S .
We say these k agents form a winner sequence with respect to S if
Let S ′ be a subset of S of size k .
We say that S ′ forms a winner sequence with respect to S if there exists an ordering of the agents in S ′ that forms a winner sequence with respect to S .
When S ′ forms a winner sequence with respect to S , we call S ′ a size - | S ′ | winner sequence set with respect to S. Let H ( S ′ , S ) = 1 if S ′ forms a winner sequence with respect to S , and let H ( S ′ , S ) = 0 otherwise .
For presentation purpose , we say that the empty set forms a winner sequence ( of size 0 ) with respect to any set S .
That is , H ( ∅ , S ) = 1 .
Now we are ready to prove that heterogeneous-item auctions with unit demand satisfies redistribution monotonicity .
We recall that it suffices to prove that for any set of agents S ,
Now let us analyze the expression Pa ∈ S R1 ( S − a , k ) .
By induction assumption , it can be rewritten as
The following lemmas are needed for the proof of the above proposition .
All these lemmas are implications of `` winners still win after we remove some other agents '' .
The proofs are omitted due to space constraints .
Now we are ready to prove the proposition .
PROOF .
We prove by induction .
Initial step : We have R1 ( S , 0 ) = VCG1 ( S ) .
When k = 0 ,
We need to prove that the results hold for k + 1 .
That is , for any S ( | S | ≥ k + m + 2 ) , | S | m + k By induction assumption , the above expression is the sum of ~ terms .
Each term corresponds to one choice of a among S and one choice of S ′ among S − a .
We divide these | S | m + k
terms into two groups : Group A , terms with a ∈ / D ( S − S ′ ) : By Lemma 2 , S ′ must also form a winner sequence with respect to S .
That is , there are at most m + k ~ choices of S ′ .
For each choice of S ′ , there are at k most | S − S ′ − D ( S − S ′ ) | = | S | − k − m − 1 choices of a. Overall , there are at most m + k ~ ( | S | − k − m − 1 ) terms in Group k A. On the other hand , for any S ′ that forms a winner sequence with respect to S , S ′ must also form a winner sequence with respect to S − a by Lemma 1 .
For any a ∈ / D ( S − S ′ ) , there must be a term in Group A that is characterized by a and S ′ .
That is , there are at least m + k ~ ( | S | − k − m − 1 ) terms in Group A. Hence , there are exactly m + k
a ∈ / D ( S − S ′ ) , we have that V CG1 ( S − a − S ′ ) = V CG1 ( S − S ′ ) by Proposition 8 .
Therefore , the sum of all the terms in Group A equals
( k + 1 ) winner sequence sets with respect to S .
According to Lemma 3 and Lemma 4 , every term in Group B must corresponds to an element in X , and every element in X must correspond to exactly k + 1 terms in Group B ( e.g. , a size - ( k + 1 ) winner sequence set Y = { x1 , ... , xk +1 } corresponds to the following k + 1 terms : a = xi and S ′ = Y − xi for all i ) .
Therefore , the total number of elements in X must be m + k +1 ~ .
m The sum of the terms in Group B equals
Proposition 9 implies that function R1 is always nonnegative .
We still need to prove that R1 ( S , 0 ) > R1 ( S , 1 ) > ... > R1 ( S , ISI -- m -- 1 ) .
Due to space constraint , we only present an outline of the proof of R1 ( S , 3 ) > R1 ( S , 4 ) , which highlights the main idea behind the full proof .
a , 3 ) -- ( ISI -- m -- 4 ) R1 ( S , 3 ) ) .
To prove that R1 ( S , 4 ) < R1 ( S , 3 ) , it suffices to prove that R1 ( S , 3 ) > R1 ( S -- a , 3 ) for any a E S. Let a be an arbitrary agent in S .
According to Proposition 9 , we need to prove
sition 9 ) .
Every term is characterized by a size-3 winner sequence set S ′ .
.
For every term on the right-hand side , we map it to a corresponding term on the left-hand side .
The corresponding term on the left-hand side is larger or the same .
.
We prove that the mapping is injective .
That is , different terms on the right-hand side are mapped to different terms on the left-hand side .
.
Therefore , the left-hand side must be greater than or equal to the right-hand side .
4 .
CONCLUSION
We conclude our paper with the following conjecture : CONJECTURE 1 .
Gross substitutes implies redistribution monotonicity .
That is , HETERO remainsfeasible and worst-case optimal in combinatorial auctions with gross substitutes .
The idea is that both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand satisfy redistribution monotonicity .
A natural conjecture is that the `` most restrictive joint '' of these two settings also satisfies redistribution monotonicity .
There are many well-studied auction settings that contain both multi-unit auctions with nonincreasing marginal values and heterogeneous-item auctions with unit demand ( a list of which can be found in [ 10 ] ) .
Among these well-studied settings , combinatorial auctions with gross substitutes is the most restrictive .
To prove the conjecture , we need to prove that gross substitutes implies that for any set of agents S , R ( S , 0 ) > R ( S , 1 ) > ... > R ( S , ISI -- m -- 1 ) > 0 .
So far , we have only proved R ( S , 0 ) > R ( S , 1 ) > 0 .