Beneﬁts of Permutation-Equivariance in Auction Mechanisms
Anonymous Author(s)
AfﬁliationAddress email
Abstract
Designing an incentive-compatible auction mechanism that maximizes the auctioneer’s revenue while minimizes the bidders’ ex-post regret is an importantyet intricate problem in economics.
Remarkable progresses have been achievedthrough learning the optimal auction mechanism by neural networks.
In this paper,we consider the popular additive valuation and symmetric valuation setting; i.e. ,the valuation for a set of items is deﬁned as the sum of all items’ valuations in theset, and the valuation distribution is invariant when the bidders and/or the itemsare permutated.
We prove that permutation-equivariant neural networks have signiﬁcant advantages: the permutation-equivariance decreases the expected ex-postregret, improves the model generalizability, while maintains the expected revenueinvariant.
This implies that the permutation-equivariance helps approach the theoretically optimal dominant strategy incentive compatible condition, and reducesthe required sample complexity for desired generalization.
Extensive experimentsfully support our theory.
To our best knowledge, this is the ﬁrst work towardsunderstanding the beneﬁts of permutation-equivariance in auction mechanisms.
Code will be released publicly.
Introduction
Optimal auction design [26] has wide applications in economics, including online advertising [16],resource allocation [15] and supply chain [4].
In an auction, every bidder has a private valuationproﬁle over all items and accordingly, submits her bid proﬁle.
An auctioneer collects the bids fromall bidders and determines a feasible item allocation to the bidders as well as the prices the biddersneed to pay.
Consequently, every bidder receives her utility.
From the auctioneer’s perspective, theoptimal auction mechanism is required to maximize her revenue , deﬁned as the sum of all bidders’payments.
From the aspect of the bidders, the optimal auction mechanism needs to incentivize everybidder to bid according to their truthful valuation proﬁles (truthful bidding).
This is summarized asthe dominant strategy incentive compatible (DSIC) condition; i.e. , the truthful bidding is always thedominant strategy for every bidder [22].
The optimal auction mechanism can be approximated via neural networks [12, 25, 10].
The “approximation error”, or the “distance” to the DSIC condition, is usually measured by the ex-post regret ,deﬁned as the gap between the bidder’s utility of truthful bidding and the utility when her bid proﬁleis only the best to herself (selﬁsh bidding), while the bid proﬁles of all other bidders are ﬁxed in bothcases [12].
When a bidder’s ex-post regret is always equal to 0 , the truthful bidding is her dominantstrategy.
Therefore, the optimal auction design can be modeled as a linear programming problem,where the object is to maximize the expected revenue subject to the expected ex-post regret being
Another major consideration in learning the optimal mechanism is the generalizability tounseen data, usually measured by the generalization bound , i.e. , the upper bound of the gap betweenthe expected revenue/ex-post regret and their empirical counterparts on the training data [12].
In this paper, we consider the popular setting of additive valuation and symmetric valuation [12,25, 10].
The additive valuation condition deﬁnes the valuation for a set of items as the sum of thevaluations for all items in this set.
The symmetric valuation condition assumes the joint distributionof all bidders’ valuation proﬁles to be invariant when bidders and/or items are permutated.
Thissetting covers many applications in practice.
For example, when items are independent, the additivevaluation condition holds.
Moreover, if the auction is anonymous or the order of the items is notprior-known, the symmetric valuation condition holds.
We demonstrate that permutation-equivariant models have signiﬁcant advantages in learning theoptimal auction mechanism as follows.
(1) We prove that the permutation-equivariance in auctionmechanisms decreases the expected ex-post regret while maintaining the expected revenue invariant.
Conversely, and equivalently, the permutation-equivariance promises a larger expected revenue,when the expected ex-post regret is ﬁxed.
(2) We show that the permutation-equivariance of auctionmechanisms reduces the required sample complexity for a desirable generalizability.
We prove thatthe l 1 , 1 -distance between any two mechanisms in the mechanism space decreases when they areprojected to the permutation-equivariant mechanism (sub-)space.
This smaller distance implies asmaller covering number of the permutation-equivariant mechanism space, which further leads to asmall generalization bound [12].
We further provide an explanation for the learning process of non-permutation-equivariant neuralnetworks (NPE-NNs).
In learning the optimal auction mechanism by an NPE-NN, we show thatan extra positive term exists in the quadratic penalty of the ex-post regret based on the result (1) .
This term serves as a regularizer to penalize the “non-permutation-equivariance”.
Moreover, thisregularizer also interferes the revenue maximization, and thus affects the learning performance of
NPE-NNs.
This further explains the advantages of permutation-equivariance in auction design.
Experiments in extensive auction settings are conducted to verify our theory.
We design permutationequivariant versions of RegretNet (RegretNet-test and RegretNet-PE) by projecting the RegretNet [12]to the permutation-equivariant mechanism space in the training and test stage.
The empirical resultsshow that permutation-equivariance helps: (1) signiﬁcantly improve the revenue while maintain thesame ex-post regret; (2) record the same revenue with a signiﬁcantly lower ex-post regret; and (3)narrow the generalization gaps between the training ex-post regret and its test counterpart.
Theseresults fully support our theory.
Related works.
Myerson completely solves the optimal auction design problem in one-item auctions[20].
However, solutions are not clear when the number of bidders/items exceeds one [12].
Initialattempts have been presented on the characterization of optimal auction mechanisms [18, 23, 9]and algorithmic solutions [1, 2, 24].
Remarkable advances have been made on the required samplecomplexity for learning the optimal auction mechanism in various settings, including single-itemauctions [5, 19, 15], multi-item single-bidder auctions [11], combinatorial auctions [3, 29], andallocation mechanisms [21].
Machine learning-based auction design (automated auction design)have obtained considerable progresses [6, 7, 27].
The optimal auction design is modelled as alinear programming problem [6, 7].
However, early works suffer from the scalablility issues thatthe number of the constraints grows exponentically when the bidder number and the item numbersincrease.
To address this issue, recent works propose to learn the optimal auction mechanism by deeplearning.
RegretNet is designed for the multi-bidder and multi-item settings [12].
Then, RegretNet isdeveloped to meet more restrictive constraints, such as the budget condition [14] and the cerifyingstrategyproof condition [8].
Rahme et al. [25] propose the ﬁrst equivariant neural network-basedauction mechanism design method with signiﬁcant empirical advantages. Duan et al. [10] extend theapplicable domain to contextual settings.
Both works make remarkable contributions in designingnew algorithms from the empirical aspect only. However, the theoretical foundations are still elusive. To our best knowledge, our paper is the ﬁrst work on theoretically studying the beneﬁts of permutationequivariance in auction design via deep learning.
Auction.
Suppose n bidders are bidding m items in an auction.
Every bidder i has her biddercontext (feature) x
i 2 X , while every item j is associated with its item-context (feature) y j 2 Y .
The bidder i has a private valuation v ij 2 V ⇢ R  0 for the item j , which is sampled froma conditioned distribution P ( ·| x i , y j ) .
The value proﬁle v i =
( v i 1 , . . .
, v im ) is unknown to theauctioneer.
For the simplicity, we deﬁne x = ( x T 1 , . . .
, x Tn ) T , y =
( y 1 , . . .
y m ) , v = ( v T 1 , . . .
, v Tn ) T ,v  i =
( v T 1 , . . .
, v Ti  1 , v Ti +1 , . . .
, v Tn ) T , and ( v 0 i , v  i )
= ( v T 1 , . . .
, ( v 0 i ) T . . .
, v Tn ) T .
Every bidder submits a bid proﬁle b i to the auctioneer according to her valuation proﬁle.
Then, theauctioneer determines a feasible item allocation g ( b, x, y ) and corresponding payments p ( b, x, y ) asper an auction mechanism ( g, p ) .
Consequently, every bidder receives her utility u i ( v i , b, x, y ) =
P mj =1 g ij ( b, x, y ) · v ij  p i ( b, x, y ) .
The auction mechanism ( g, p ) consists of an allocation ruleg : R n ⇥ m ⇥ X n ⇥ Y m ! R n ⇥ m and a payment rule p : R n ⇥ m ⇥ X n ⇥ Y m ! R n ⇥ m , whereg ij is the probability of allocating item j to the bidder i , and p i =
P mj =1 p ij is the price thatthe bidder i should pay.
To avoid allocating an item over once, the allocation rule is constrainedsuch that
[-=
-]P
[Equation] 1 g ij ( b, x, y )  1 for all j 2 [ m ] .
Every v i in our notations can be replaced byb i .
Thus, we can deﬁne the similar notations: b  i =
( b T 1 , . . .
, b Ti  1 , b Ti +1 , . . .
, b Tn ) T , ( b i , v  i ) =( v T 1 , . . .
, ( b i ) T . .
. , v Tn ) T and ( v i , b  i ) =
( b T 1 , . . .
, ( v i ) T . . .
, b Tn ) T .
Optimal auction mechanism.
An auction mechanism ( g, p ) is deﬁned to be dominant strategyincentive compatible (DSIC), if truthful bidding is always a dominant strategy of every bidder; i.e. ,u i ( v i , ( v i , b  i ) , x, y )  u
i ( v i , b, x, y ) for all i 2 [ n ] , v, b 2 V n ⇥ m , x 2 X n and y 2 Y m .
Inaddition, an auction mechanism ( g, p ) is called individually rational (IR), if for any bidder-contextsx 2 X n , any item-contexts y 2 Y m , and any bidder i 2 [ n ] x 2 X n , valuation proﬁle and bid proﬁlev, b 2 V n ⇥ m , truthful bidding always leads to a non-negative utility; i.e. , u i ( v i , ( v i , b  i ) , x, y )  0 .
If an auction mechanism is DSIC and IR, a rational bidder with an obvious dominant strategy willplay it (bidding truthfully).
Moreover, an optimal auction mechanism is required to maximize theauctioneer’s expected revenue rev = E ( v,x,y )
[Equation]
Auction design.
The ex-post regret reg i ( v, x, y ) for the bidder i is deﬁned as
[Equation]
, x, y )  u i ( v i , v, x, y ) .
An auction mechanism ( g, p ) is DSIC, if andonly if
P ni =1 reg i ( v, x, y ) = {+
+}
for any value proﬁle v 2 V n ⇥ m , bidder-context x 2 X n , and itemcontext y 2 Y m .
Suppose the payment rule p satisfy p
i ( b, x, y ) 
P mj =1 g ij ( b, x, y ) b ij .
Then, theauction design can be modeled as a linear programming problem that maximizes the expected revenue
E ( v,x,y ) [ P ni =1 p i ( v, x, y )] subject to the expected ex-post regret E ( v,x,y ) [
P ni =1 reg i ( v, x, y )]
= 0 .
Without loss of generality, the ex-post regret may refer to the sum of all bidders’ ex-post regrets.
Deﬁnition 2.1.
Suppose the network’s parameter is ! , and the bidder i ’s empirical payment and expost regret are deﬁned as 1 L
[Equation] and
[Equation] ) ,where the sample set { ( v ( l ) , x ( l ) , y ( l ) ) } Ll =1 is i.i.d.
sampled from the following prior distribution,
P ( v, x, y ) =
Q n,mi,j =1 P ( v ij | x i , y j ) P X
i ( x i ) P Y j ( y j ) .
Equivariant mapping.
We deﬁne a mapping f as G - equivariant if g  f = f  ⇢ g for two chosengroup linear representations ⇢ and and any g in group G .
Deﬁnition 2.2.
(Permutaion-Equivariant Mapping)
A permutation-equivariant mapping is deﬁned tobe f : R n ⇥ m ! R n ⇥ m that for any instance x 2 R n ⇥ m , and permutation matrices  n 2 R n ⇥ n and m 2 R m ⇥ m , we have f (  n x  m ) =
n f ( x )  m .
In this paper, we consider the bidder-permutation  n 2 R n ⇥ n and item-permutation  m 2 R m ⇥ m .
Speciﬁcally, we deﬁne a mapping f is bidder-symmetric or item-symmetric, if f (  n x ) =  n f ( x )or f ( x  m ) =
f ( x )  m , respectively.
Moreover, we deﬁne an auction mechanism ( g, p ) as biddersymmetric or item-symmetric, if the allocation rule g and the payment rule p are both biddersymmetric or item-symmetric.
Orbit averaging.
For any feature mapping f : F ! G , the orbit averaging Q on f is deﬁned as
Q f = 1 | G | P g 2 G  1 g  f  ⇢ g , where ⇢ and are two chosen group representations acting on thefeature spaces F and G , respectively.
Orbit averaging can project any mapping to be equivariant:
Proposition 2.3.
Orbit averaging Q is a projection to the equivariant mapping space { f :  f = 137 f  ⇢ } , i.e.,  Q f = Q f  ⇢ and Q 2 = Q .
In particular, if f is already equivariant, then Q f = f .
Moreover, Q u and Q reg refer to the utility and the ex-post regret induced by Q g and Q p .
Forthe simplicity, we denote the orbit averagings that modify the auction mechanism to be biddersymmetric, item-symmetric, and bidder/item-symmetric by bidder averaging Q 1 , item averaging Q 2 ,and bidder-item aggregated averaging Q 3 .
The proof of the feasibility can be found in Appendix A.1.
Hypothesis complexity.
The generalizatbility to unseen data is usually measured by the generalizationbound, which depends on the hypothesis set’s complexity.
To characterize the complexity of thehypothesis set, we introduce the following deﬁnitions of covering number N 1 , 1 and its correspondingdistance l 1 , 1 .
Based on the covering number, we can obtain a generalization bound in Theorem 3.6.
Deﬁnition 2.4.
( l 1 , 1 -distance)
Let X be a feature space and F a space of some functions from X to R n .
The l 1 , 1 -distance on the space F is deﬁned as l 1 , 1 ( f, g ) = max x 2 X ( P n i =1 | f i ( x )  g i ( x ) | ) .
Deﬁnition 2.5.
(Covering number) Covering number N 1 , 1 ( F , r ) is the minimum number of ballswith radius r that can cover F under l 1 , 1 -distance.
Theoretical Results
This section presents the theoretical results.
For simplicity, we view p
=
( p 1 , . . .
, p n ) T as a n ⇥ 1matrix to present the prices the bidders should pay.
We ﬁrst prove that the permutation-equivarianceinduces a same expected revenue and a smaller expected ex-post regret in Section 3.1.
Next in Section3.2, we prove that the permutation-equivariant mechanism space has a smaller covering number,which promises a smaller required sample complexity and a better generalization.
Detailed proofs areomitted from the main text and given in supplementary materials due to space limitation.
3.1 Beneﬁts for Revenue and Ex-Post Regret
In this section, we discuss the beneﬁts for the revenue and the ex-post regret in the conditions ofbidder-symmetry and item-symmetry separately, and then discuss the beneﬁts when both of themhold.
Based on these results, we also study the learning process of non-permutation-equivariantneural networks for auction design.
3.1.1 Beneﬁts in the Bidder/Item-Symmetry Condition
When the bidders come from the same distribution, the joint valuation distribution f is invariantunder bidder-permutation, i.e. f (  n v,  n x, y ) =
f ( v, x, y ) for any  n 2 S n .
Meanwhile, whenthe items are indistinguishable, the joint distribution f is invariant under item-permutation, i.e. ,f ( v  m , x, y  m ) =
f ( v, x, y ) for any  m 2 S m .
Both conditions do not always hold simultaneously.
In this section, we study them separately.
To measure the “non-permutation-equivariance” of the mechanism, we introduce the conception ofregret gap between the projected mechanism and the original mechanism for as below,
[Equation]
where v is the valuation proﬁles, v i is valuation proﬁle of bidder i , x is the bidder-context, y is theitem-context, and the orbit averaging Q · can be the bidder averaging Q 1 or the item averaging Q 2 .
The bidder averaging Q 1 and the item averaging Q 2 acting on the allocation rule g and the paymentrule p , respectively, are as below,
[Equation]
We thus can prove the following theorem that characterizes the beneﬁts of permutation-equivariancefor revenue and ex-post regret in the condition of bidder/item-symmetry.
Theorem 3.1 (Beneﬁts for revenue and ex-post regret in the condition of bidder/item-symmetry) .
When the valuation distribution is invariant under permutations of bidders/items, the projectedmechanism has the same expected revenue and a smaller expected ex-post regret, that is,
[Equation]
x
[Equation]
where p is the payment rule, reg is the ex-post regret, and Q · is the bidder/item averaging.
[-=
-]
A smaller expected ex-post regret implies this mechanism is closer to the dominant strategy incentivecompatible condition.
Conversely, and equivalently, when the expected ex-post regrets are ﬁxed,the projected auction mechanism has a larger expected revenue.
For any auction mechanism, in thebidder/item-symmetry condition, we can project it through the bidder/item averaging.
Remark 3.2.
The mechanism space can be decompose into the direct sum of the permutationequivariant mechanism space {M : QM =
M} and the complementary space {N : QN = 0 } 186 [13].
Thus, an mechanism M has an unique decomposition: M = QM + N .
The pure permutationequivariant part QM contains all and only the “permutation-equivariance” of the mechanism M .
The pure non-permutation-equivariant part N is independent from the permutation-equivairance.
Inthis way, we may study the inﬂuence of permutation-equivariance by comparing the mechanism Mand its permutation equivariant part QM .
3.1.2 Interplay between Bidder-Symmetry and Item-Symmetry.
If the valuation distribution is invariant under both bidder-permutation and item-permutation, wecan project the mechanism to be permutation-equivariant with respect to both bidder and item intwo steps (by mapping Q 1  Q 2 or mapping Q 2  Q 1 ).
Consequently, the projected mechanism hasthe same expected revenue and a smaller expected ex-post regret.
Equivalently, we can also projectan auction mechanism to be bidder-symmetric and item-symmetric immediately by the bidder-itemaggregated averaging Q 3 as below,
[Equation]
(  n v  m ,  n x, y  m ) .
We can prove that the bidder-item aggregated averaging Q 3 is the composition of the orbit averagingoperators Q 1 and Q 2 , as shown in the following lemma.
This lemma shows that the order of Q 1 and
Q 2 would not inﬂuence their composition.
Lemma 3.3.
The bidder-item aggregated averaging is the composition of bidder averaging and itemaveraging: Q 3 = Q 1  Q 2 = Q 2  Q 1 .
Based on this lemma, we can prove the following theorem on the beneﬁts of permutation-equivariancefor revenue and ex-post regret in the condition of both bidder-symmetry and item-symmetry.
Theorem 3.4 (Beneﬁts for revenue and ex-post regret in the condition of both bidder-symmetryand item-symmetry) .
When the valuation distribution is invariant under both item-permutation anditem-permutation, then the projected mechanism has a same expected revenue and a smaller expectedex-post regret, that is,
[Equation]
and
[Equation]
where p is the payment rule, reg is the ex-post regret, and Q 3 is the bidder-item aggregated averaging.
The difference between bidder-symmetry and item-symmetry is signiﬁcant in practice.
For example,for a symmetric valuation distribution, when the mechanism is already bidder-symmetric but notitem-symmetric, we can project it to be item-symmetric to gain an extra beneﬁt from item-symmetry.
That means, the two regret gaps induced by Q 1 and Q 2 are “additive” as below,
[Equation]
In general, E [  2 ( g, p ; v, x, y )] 6 = E [  2 ( Q 1 g, Q 1 p ; v, x, y )] and thus E [  3 ( g, p ; v, x, y )]
6 =
E [  1 ( g, p ; v, x, y )]
+ E [  2 ( g, p ; v, x, y )] .
That means, the beneﬁts from bidder-symmetry and itemsymmetry are “additive” but not “independent”.
3.1.3 Insights on Training Non-Permutation-Equivariant Mechanism
Because the expected revenue is always same for the original mechanism and the projectedpermutation-equivariant mechanism, we only consider the gredient caused by the expected expost regret.
We can decompose the original expected ex-post regret into the sum of the expectedex-post regret of the projected mechanism and the expectation of the regret gap as below,
[Equation]
i .
The regret gap  3 ( g, p ; · ) follows from the “non-permutation-equivariance” of the mechanism M .
When the distance l ( M , QM ) tends to 0 , the regret gap converges to 0 .
When the auction mechanismhas a negligible ex-post regret, the expectation of the regret gap is also close to 0 .
That means, themechanism is closing to being permutation-equivariant.
However, even using symmetric dataset oradopting data augmentation in training, the learned mechanism will not be permutation-equivariantin general [17].
As a result, to achieve negligible ex-post regret, the non-permutation-equivariantmodels need to learn more samples to approach permutation-equivariance.
That is because thenon-permutation-equivariant part (expected regret gap) would mislead the gradient of the expectedregret but have no beneﬁt to the expected revenue and the expected ex-post regret.
On the other hand, the regret gap can be viewed as a regularizer in the ex-post regret to penalizethe “non-permutation-equivariance” of the mechanism.
When the optimizer tries to minimize theex-post regret, the auction mechanism approaches to be permutation-equivariant.
Therefore, if themechanism achieves a negligible ex-post regret, it is almost to be permutationn-equivariant.
Thisresult can explain why RegretNet struggles to ﬁnd permutation-equivariant auction mechanisms[25].
However, in complex settings, it will be harder for non-permutation-equivariant models toapproach the negligible ex-post regret.
It can explain why the permutation-equivariant models show asigniﬁcant improvement in complex setting, compared with that they have similar performances insimple settings [10], which shows the great importance of adopting permutation-equivariant modelsin complex settings.
3.2 Beneﬁts for Generalization
In this section, we studies permutation-equivariance from the aspect of generalizability [28], whichcharacterizes the performance gap of a learned mechanism on collected training data and unseen data.
We ﬁrst study the covering number of the permutation-equivariant mechanism space.
Let U ={ u ! : ! 2 ⌦ } and P = { p ! : ! 2 ⌦ } be the sets of all possible utilities and payment rules, and
Q · U = {Q · u : u 2 U} and Q · P = {Q · p : p 2 P} the spaces of all projected utilities and payment 247 rules.
In addition, let N 1 , 1 ( U , r ) and N 1 , 1 ( P , r ) be the minimum numbers of balls with radius 248 r that can cover U and P under l 1 , 1 -distance, respectively.
We obtain the following result, whichindicates the projected permutation-equivariant mechanism space has smaller covering numbers.
Theorem 3.5 (Covering number of the permutation-equivariant mechanism space) .
The space of allprojected bidder-symmetric mechanisms has smaller covering numbers, that is,
N 1 , 1 ( Q 1 U , r )  N 1 , 1 ( U , r ) and N 1 , 1 ( Q 1 P , r )  N 1 , 1 ( P , r ) .
The space of all projected item-symmetric mechanisms has smaller covering numbers, that is,
N 1 , 1 ( Q 2 U , r )  N 1 , 1 ( U , r ) and N 1 , 1 ( Q 2 P , r )  N 1 , 1 ( P , r ) .
Intuitively, the orbit averaging Q narrows the distance between two mechanisms: l ( QM , QM 0 ) l ( M , M 0 ) , for any two mechanisms.
Then, any r -cover r > 0 A for space U or space P induces anr -cover QA for space QU or space QP .
Combining with Lemma 3.3, we have the following results,
[Equation]
[Equation]
Q 2 U , r )  N
[Equation]
1 ( U , r ) and
[Equation]
We then prove that two generalization bounds of permutation-equivariant mechanisms which characterize the gap between the expected revenue/ex-post regret and their empirical counterparts.
Theorem 3.6 (Generalization bounds of permutation-equivariant mechanisms) .
If for any bidder, hervaluation satisﬁes that v i ( S )  1 for any S ⇢ [ m ] , then with probability at least 1   , we have thefollowing inequalities with ✏  [Equation]
[Equation]
where L is the number of samples, U and P are the spaces of all possible utilities and payment rules.
Equivalently, we can rewrite this result in the form of the sample complexity,
Corollary 3.7.
For any ✏ > 0 ,  2 (0 , 1) , and mechanism parameter ! , when the sample complexity
L
[Equation]
o⌘ , with probability at least 1   , thegeneralization bounds, eqs.
(3) and (4), hold.
Remark 3.8.
Combining Theorem 3.5, we have proved that the permutation-equivariance canimprove the generalizability.
Experiments
This section presents our experimental results.
More details and results are presented in the supplementary materails.
Code will be released publicly.
Model architecture.
We project RegretNet [12] to the permutation-equivariant mechanism spacevia orbit averaging: (1) using bidder averaging for the bidder-symmetry condition, (2) applying item [Table] [Table] RegretNet
[Table]
RegretNet [Table] PE [Table]
averaging for the item-symmetry condition, and (3) employing bidder-item aggregated averaging forthe bidder-symmetry and item-symmetry condition.
The projected model is called RegretNet-PE.
Speciﬁcally, RegretNet is an auction mechanism deﬁned as ( g ! , p ! ) , in which both of the allocationrule g ! and the payment rule p ! are neural networks that consist of three fully-connected layers, and! is the overall model parameter of the auction mechanism.
The detailed architecture is given in thesupplementary materials.
Comparison with EquivariantNet.
RegretNet uses two feed-forward full-connected networksto learn the allocation rule and payment rule, respectively.
We denote the weight matrix in thelayer ` as W ( ` ) .
Both EquivariantNet and RegretNet-PE inherit the architecture of RegretNet(with some modiﬁcations), but utilize different approaches to realize the permutation-equivariance. EquivariantNet applies parameter-sharing in every layer during training, to constrain W ( ` ) to beequivariant. In contrast, RegretNet-PE employs orbit averaging to be permutation-equivariant. Speciﬁcally, RegretNet-PE adopts a weight matrix
I K ⌦ W
( ` ) ( ⇢ Tg 1 . . . ⇢ Tg K ) T in the ﬁrst layer, weightmatrices I K ⌦ W ( ` ) in the following layers, and multiples a matrix ( ⇢  1 g 1 , . .
, ⇢  1 g K ) to the outputlayer, where K is the scale of the group G = { g 1 , . .
. , g K } , ⇢ g k represents the permutation operatoron bidders and items, I K is an identity matrix, and ⌦ is the Kronecker product.
Auction settings.
We adopt two families of auction settings in our experiments: (1) single-itemauctions, including two-bidder, three-bidder and ﬁve-bidder settings; and (2) multi-item auctions,including two-bidder and two-item, two-bidder and ﬁve-item, and ﬁve-bidder and three-item settings.
The optimal auction mechanism for any single-item auction is known [20].
We thus compare ourmethod with the optimal auction mechanisms in the single-item settings.
Also, we compare RegretNetPE and EquivariantNet in the one-bidder, two-item setting, and the two-bidder, two-item setting.
Besides, we employ a multivariate uniform distribution U [0 , 1] m to model the bidder valuation proﬁles.
We sample 5,000 data points for training and 100 data batches of size 100 (totally 10,000data points) for test in every setting.
Model training.
We optimize the auction mechanism model via solving the following optimizationproblem, following the standard settings [12, 25, 10],
[Equation]
where  2 R n is the Lagrange multiplier and ⇢ > 0 is the factor of the quadratic regularization term.
During the training process, the objective function L ⇢ ( ! ,  ) is minimized via Adam with learningrate 0.001 with respect to the model parameter ! and the Largrange multiplier  is updated via SGDonce in every 100 iterations, until the ex-post regret is smaller than 0 .
The regularization factor ⇢is is set to 1000 and 5 initially for single-item settings and multiple and gradually increased along thetraining process.
In calculating the best bid proﬁle v 0 i of every bidder i , we ﬁrst randomly initializethe bid proﬁles for 1,000 times, optimize each of them individually via Adam with the same settings,and take the best one as the approximated best bidding.
Evaluation.
We leverage three metrics to evaluate the performance of the auction mechanism, whichare: (1) the empirical revenue d rev , (2) the empirical ex-post regret across all bidders, i.e. , P ni = d reg i ,and (3) the generalization error deﬁned on top of regrets, i.e. , GE = | d reg test  d reg train | .
Computing resource.
The experiments are conducted on 1 GPU (NVIDIA ® Tesla ® V100 16GB)and 10 CPU cores (Intel ® Xeon ® Processor E5-2650 v4 @
2.20GHz).
Experimental results.
We train a RegretNet and a RegretNet-PE on the training data.
The welltrained RegretNet is then projected to be permutation-equivariant, denoted as “RegretNet-Test”.
Theresults are collected in Tables 1 and 2.
From Tables 1 and 2, we observe that (1) compared to RegretNet, RegretNet-PE has a signiﬁcantlyhigher revenue with a lower ex-post regret, and narrows the generalization gap between the trainingex-post regret and its test counterpart; (2) compared to RegretNet, RegretNet-Test receives the samerevenue with a signiﬁcantly lower ex-post regret; and (3) under comparable ex-post regrets, RegretNetPE has a considerably higher revenue than EquivariantNet, while all permutation-equivariant models(EquivariantNet, RegretNet-Test, and RegretNet-PE) can outperform RegretNet.
These results showsigniﬁcant beneﬁts of permutation-equivariance on revenue, ex-post regret, and generalizability,which fully supports our theoretical ﬁndings in Theorems 3.1, 3.4, and 3.5.
Conclusion and future works
In this paper, under additive valuation and symmetric valuation setting, we study the beneﬁts ofpermutation-equivariance in auction mechanisms in two aspects: a better performance and a bettergeneralization.
First, we prove a smaller expected ex-post regret and a same expected revenue whenprojecting an mechanism to be permutation-equivariant.
Next, we propose the permutation-equivariantmechanism space has a smaller covering number, which promises the permutation-equivariant modelsa better generalization.
Extensive experiments are conducted to verify our theoretical results.
Ourresults help understand the optimal auction mechanisms’ characterization and the learning processes’difference between non-equivariant models and equivaraint models.
Beyond the additive valuation setting, an interesting direction is to extend our results to otherconditions, including the combinatorial and the unit-demand auctions.
Meanwhile, the understandingof the difference in the aspect of the training process between non-equivariant models and equivariantmodels is still elusive.
Social impact.
Our results can help understand and design optimal auction mechanisms for symmetricvaluation distribution.
As a result, our work could inspire more near-optimal auction mechanismsand promote economic growth.
No potential negative social impact is identiﬁed.
Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’scontributions and scope?
[Yes] Our contributions are stated in the 4th-7th sentences ofthe abstract, and the 4th-7th paragraphs of the introduction.
The scope is described inthe related work section.(b)
Did you describe the limitations of your work?
[Yes] We describes the problem setting
we considered in the 3rd paragraph of the introduction.
Our contributions are for this
problem setting.
(c) Did you discuss any potential negative societal impacts of your work?
[Yes] Our workis committed to learn the optimal auction mechanism.
No negative societal impact isidentiﬁed.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to
them?
[Yes] We have read the ethics review guidelines.
We ensure that our paperconforms all of them.
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes] We stateall of our assumptions in the third paragraph of introduction, the ﬁrst paragraph ofpreliminary, Theorem 3.1, and Theorem 3.2.
Wherever we use an assumption, we
always state it again.
(b) Did you include complete proofs of all theoretical results?
[Yes] We give a proofsketch in Section 3 due to space limitation.
Full proofs are given in the supplementalmaterial.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either in the supplemental material or as a URL)?
[No] We promise to
release our code publicly.
We are associated in a industrial organization.
The code is
currently under review of our organization.
(b) Did you specify all the training details (e
g., data splits, hyperparameters,
how they
were chosen)?
[Yes] We mostly follow the setting of the RegretNet.
Other details are given in Section 4.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
[No]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
[Yes] They are given in Section 4.
If you are using existing assets (e
g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[Yes] They are given inSection 4.
(b) Did you mention the license of the assets?
[Yes] They are given in Section 4.
(c) Did you include any new assets either in the supplemental material or as a URL?
[No]
We promise to release our code publicly.
We are associated in a industrial organization.
The code is currently under review of our organization.
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[Yes] The data is synthetic in our data.
(e) Did you discuss whether the data you are using/curating contains personally identiﬁable
information or offensive content?
[Yes] The data is synthetic in our data.
No personally
identiﬁable information or offensive content is contained.
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
[N/A]