Trade-off between Payoff and Model Rewards in Shapley-Fair Collaborative Machine Learning
Authors: Quoc Phong Nguyen
Bryan Kian Hsiang Low
Patrick Jaillet
Venue: NeurIPS 2022 Accept (2022)
Keywords: collaborative machine learning
model reward
payoff reward
Shapley value

Abstract: This paper investigates the problem of fairly trading off between payoff and model rewards in collaborative machine learning (ML) where parties aggregate their datasets together to obtain improved ML models over that of each party. Supposing parties can afford the optimal model trained on the aggregated dataset, we propose an allocation scheme that distributes the payoff fairly. Notably, the same scheme can be derived from two different approaches based on (a) desirable properties of the parties' payoffs or (b) that of the underlying payoff flows from one party to another. While the former is conceptually simpler, the latter can be used to handle the practical constraint on the budgets of parties. In particular, we propose desirable properties for achieving a fair adjustment of the payoff flows that can trade off between the model reward's performance and the payoff reward. We empirically demonstrate that our proposed scheme is a sensible solution in several scenarios of collaborative ML with different budget constraints.


Trade-off between Payoff and Model Rewards in Shapley-Fair Collaborative Machine Learning
Quoc Phong Nguyen, Bryan Kian Hsiang Low?, and Patrick Jaillet* Institute of Data Science, National University of Singapore, Republic of Singapore Dept. of Computer Science, National University of Singapore, Republic of Singapore Dept. of Electrical Engineering and Computer Science, MIT, USA! qphongmp@gmail com, lowkh@comp... nus. edu. sgå, jaillltllttte faillet@min edu
Abstract
This paper investigates the problem of fairly trading off between payoff and model rewards in collaborative machine learning (ML) where parties aggregate their datasets together to obtain improved ML models over that of each party. Supposing parties can afford the optimal model trained on the aggregated dataset, we propose an allocation scheme that distributes the payoff' fairly. Notably, the same scheme can be derived from two different approaches based on (a) desirable properties of the parties' payoffs or (b) that of the underlying payoff flows from one party to another. While the former is conceptually simpler, the latter can be used to handle the practical constraint on the budgets of parties. In particular, we propose desirable properties for achieving a fair adjustment of the payoff flows that can trade off between the model reward's performance and the payoff reward. We empirically demonstrate that our proposed scheme a sensible solution in several scenarios collaborative ML with different budget constraints.
Introduction
While machine learning (ML) has proven its usefulness in a wide range of applications. significant amount of diverse training data often required to achieve desirable performance. Although this may not be an issue for very large corporations, it can hardly be met by other businesses. In practice, the dataset coming from an individual data owner (e.g., company), called a party, is often limited in size (c.g., due to the size of the customer base) and diversity (e.g., due to the constraints in the geographical location and/or the customer demographics). Hence, there is strong urge for parties to obtain high-quality MI. models by performing collaborative MI.: training models on their aggregated dataset which is often of much larger quantity and higher diversity than those of individual parties.
However, the goal of obtaining improved models may not outweigh the concern of fairness if all parties receive the same model while their contributions differ (due the difference in the size and quality of different parties' datasets). For instance, a party with a larger contribution feels "under-rewarded" if all parties receive models of equal performances without any other componssation/rwwardb for the difference in their contributions. Hence, a "fair" reward allocation scheme is desirable to give all parties enough incentives to join the collaboration. Let us classify existing works on fair reward allocation schemes into 2 groups depending be distributed among parties. This source of reward can thought of as an external buyer of the model who is interested in the model but does not contribute data, so he/she is willing to pay reward to parties in the collaboration to obtain the model.
whether there exists an external source of reward to
As an example, a company distributes monetary payoff's to its users for contributing user data to train a model. Then, one can view users as parties who collaborate train the model and the company as an external buyer of the model. In this case, one can compensate for the difference in users' contributions
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

by fairly distributing the total monetary payoff from the company to users according to the value of each user's data the model. This value can be measured by a data valuation method [[5], e.g.,
the Shapley value S and its variants to incorporate the underlying data distribution [ replication-
robustness 6 20]. the federated learning setting [19], the training-free free setting 18 the online
collaborative learning setting 2 and to speed up the computation 3 3 22].
In contrast, we focus on the other group of works when the obtained model from the collaboration
of several parties is not sold to any external buyer (e.g., due to privacy or the fear of losing the competitive advantage), i.e., there is not any external source of reward to be distributed among parties. In the literature, fairness is maintained by (a) requiring parties to pay participation fee to be distributed back to all parties [ or (b) ensuring that party with less valuable data receives a model with lower performance than those with more valuable data [16]. In brief, parties receive different monetary payoff's (produced by the participation fee) in the former and different models (called model rewards) in the latter. However, there are 2 main drawbacks: the trading-off between monetary payoffs vs. model rewards and the uniqueness of the allocation scheme. First, the former approach [33 prevents parties with an insufficient monetary budget from joining the collaboration regardless of the usefulness of their datasets to other parties. The latter approach [16) prevents parties with low-quality datasets from obtaining good model regardless of their monetary expense capability, Both cases are undesirable. Second, the uniqueness of the allocation schemes in the works of [13] 16 is not discussed. Hence, it raises the question of whether there are other allocation schemes that satisfy the same set of fairness properties in [3 [6]. If such schemes exist, there can be undesirable disagreement among parties in choosing the allocation scheme to use. On the contrary, if an allocation scheme is uniquely defined from a set of fairess properties (e.g., the celebrated Shapley value [14), parties which agree on these properties
certain to adopt the (unique) allocation scheme.
In this work, we resolve the above problems by proposing fairness properties that define not only unique allocation scheme of the monetary payoff but also a unique way of trading off between payoff and model rewards. It is noted that as both payoff and model rewards are allowed, our problem subsumes the two problems in [13 [6]. Nevertheless, our solution is not generalization of existing solutions [3 [6] because even in the special cases of our problem that are equivalent to those
in [13 [6], solution is different from that of [13 [6]. In particular, our solution satisfies the
linearity property while the works of [3 16 do not. The outline of our paper is as follows.
In Sec. 3 we consider parties without budget constraints, which implies that all parties can afford the highest-quality model from the collaboration. Thus, the scheme is reduced to only allocating monetary payoff's to parties. In this case, we propose 4 desirable properties which uniquely determine the fair payoff for each party (in Sec. [.I]. The resulting payofí' fairly compensate for the difference between the contributions (measured by the Shapley value) of parties to the models they receive, which we call Shapley fairness. While these payoffs suffice to characterize a payoff allocation scheme, it does not give us insights into the underlying payoff flows transferred from one party to another. The latter is crucial for us to unravel the unexplored question of trading off between payoff and model rewards when parties have budget constraints. Therefore, in Sec. 3.2 we propose 4 desirable properties for fair payoff flows by drawing inspiration from the Shapley value 11]. These properties uniquely determine new solution called the conditional Shapley value which specifies the payoff flows from one party to another. Interestingly, we arrive at the same set of payoffs as those in Sec. Sec.3.1 Hence, the fairness of our proposed payoff allocation scheme can be consistently explained with both the proposed properties of the payoffs (Sec. 3.1 and those of the underlying payoff flows transferred from one party another (Sec. 3.2
In Sec. 4 we consider parties with budget constraints, i.c., their expense cannot exceed their budgets. Consequently, there may exist parties that cannot afford the highest-quality model if their budgets below the required amount of payment. We propose to trade off between payoff and model rewards such that parties can sacrifice the model reward, i.e., obtaining worse model, to meet the budget. In particular, we construct a set of desirable properties that uniquely define adjusted value functions from which the resulting payoff rewards both meet the budgets and satisfy faimess properties.
In Sec. [5] we empirically demonstrate sensible characteristics of our proposed allocation scheme in several ML problems. For example, even if parties do not have any budget, they are still motivated to join the collaboration because they are able to obtain improved models as long as they have valuable datasets. Furthermore, an increase in the budget of party not only benefits itself but also benefits other parties through improved models and/or higher monetary payoffs.
2

---

P13
P2
2
P12
P23
Pool
1
3
P21
P52
P3
3
P31
(a) Payoff reward representation.
(b) Payoff flow representation.
Figure 1: Diagrams of the representations of payoff allocation scheme.
2 Background
(1,2,3,.... n). collaborating with each other by aggregating their datasets to train ML models. Any subset of C C N is coalition while A is the grand coalition. To case the notation clutter, we overload the notation i to denote both party and the singleton set {i}. The aggregated training dataset of a coalition C' is denoted as De 2 Uiec where D; denotes the dataset of party i. The performance of model trained on De denoted as s(C). We assume the monotonicity property of s with respect to (w.r.t.) C: AC C c N, s(C") 2 s(C); s(0) O, and s(N) O. In the experiments (in Sec. [5), the model performance measured by the prediction accuracy of models on validation set which is common across different parties. It reflects the assumption that parties solve the same MI. problem. Our approach also works with other performance metrics such as the information gain in the work of [16]
Let us consider a set of n parties, denoted as N &
The value function, denoted v, is mapping from coalition C N its value e(C) such that v00 = 0. In this work, vC)) is considered as the monetary payoff. Adopting the assumption from the work of [13], we define it using an exchange rate v > O: v(C) _ vs(C). From the properties of s, it follows that v is monotonic, v(0) 0, and v(N) > More importantly, a model with the performance sCC) can be exchanged fairly with a payoff vCC) e vs(C) and vice versa.
Given a value function v, the Shapley value is the solution to distributing the payoff v(N) to parties in N that satisfies desirable properties: efficiency, symmetry, linearity, and dummy party (see The Shapley value
App. Hence, it is the foundation of many data valuation methods [3 7
g 21
n n--1 1 [C|
of party i is defined (u) & EcEr'
(i(C) where A. (+|C) & v(CU)) (C)
is the marginal contribution of party i to coalition C. is noted that to use the Shapley value, there should exist payoff v(N) to be distributed to all parties in N, i.e., there exists an external source of reward (see the classification of existing works in Sec. [). In contrast, we assume that such an external source of reward does not exist in this work, which renders the direct use of the Shapley value challenging. Thus, we propose new perspective of distributing a payoff' of value 0 to all parties in A (by allowing parties to receive negative payoff rewards) such that these payoff rewards compensate for the difference in the contributions among parties.
In this work, an allocation scheme specifies both payoff and model rewards that each party receives. We do not distinguish between two models having the same performance. Thus, for a party E N, the scheme specifies the value (equivalently, the performance) of the model and the payoff value receives. To receive negative payoff is to make a payment.
We first consider the case where there are not any budget constraints (i.e., unlimited budgets), so all parties can afford the best model with the performance of s(N) (i.c., the optimal model trained on Dx) while they may receive different payoff rewards to account for the difference in their contributions in Sec. 3 Then, in Sec. 4 we consider the case that parties have budget constraints such that they may not afford the model of performance s(N). In this case, not only do parties receive payoffs constrained by the budgets, but they also receive different model rewards to maintain fairness.
3 Parties without Budget Constraints
In this section, parties have no budget constraints, so they can expend any amount of payoff to receive the optimal model trained on Dy (i.e., with the performance of s(N)). Hence, the problem is reduced
3

---

to an allocation scheme of only the payoff reward. In particular, different parties receive different (and possibly negative) payoff rewards to compensate for the difference their contributions.
Payoff reward vector. Given a value function v, let (i(v) denote the payoff reward that party i receives. Then, a payoff allocation scheme specifies a payoff reward vector p(v) A (p:(t)) While party is rewarded if (v))) > makes a payment of value -Pi(e) if (v()) < O. In practice, it can be implemented as a 2-step process: first, parties with negative payoff's make payments to an intermediate pool; second, parties with positive payoffs take their payoffs from the pool (Fig. [T].
Payoff flow. One may consider defining a payoff allocation scheme by the payoff transferred from one party to another which we call payoff flows (Fig. (Fi However, there can be different ways of transferring payoffs among parties which result in the same payoff reward vector. For example, a payoff reward vector remains unchanged if we increase each payoff flow from party to party r(i) by the same amount where (i) + 1 for all < n and T(n) = 1 (i.e., increasing the payoff along closed loop). Furthermore, regardless of how payoffs are transferred among parties, what matters, in the end, is the payoff reward each party receives which is determined by the payoff reward vector p(v). Therefore, in the next section, we propose desirable properties that uniquely determine a fair payoff reward vector (hence, payoff allocation scheme).
3.1 Fair Payoff Reward Vector
We propose the following 4 desirable properties for the fairness of payoff reward vector p(v). While the symmetry and linearity properties are taken from that of the Shapley value, the efficiency and dummy party properties are modified to fit our problem setting where there is not any external source of reward to be distributed among parties.
Symmetry. If ((CUi) = j) for all C \ {i. then (v = P3 (v). It implies parties with equal marginal contributions any coalitions have the same payoff.
Linearity. Let us consider the value function as a vector in real vector space of dimension 2" (i.e., the number of subsets of A) which is equipped with the usual addition and scalar multiplication. Then, given 2 value functions v and w,
p:(0 + w) = (c)) - p:(w), pi(av) = ap:()), ViEW Va R.
Balance. Recall that there is not any external source that pumps additional payoff's into N (or takes payoffs from N). Therefore, the balance property states that ΣiEN (v(v) O.
Dummy party. Party E N is dummy party if o(Cu,) = v(C) for all C C v. It means that a dummy party can be considered as an external buyer that does not contribute any data and is only interested in buying the model. Therefore, we expect a dummy party to pay the whole value of the model it receives from the collaboratio,, i.c., v(N) (since parties have no budget constraints). In other words, if party is dummy party, then it makes payment of value v(NV), i.c., (v( =v(N).
It is noted that we do not sacrifice the uniqueness of the Shapley value by modifying its properties. The following lemma shows that these prooysproposed properties still characterize a unique payoff reward vector, which is proven in App. B
Lemma 3.1. There exists unique vector (pr(v)) of payoff rewards that satisfies the balance, the symmetry, the linearity, and the dummy party properties:
Vi (u) = - (o,(t)-<(N))
(I)
Corollary 3.2. If we allocate payoff rewards to parties in N according to T then party has
negative payoff reward (i.e.. making a payment) if and only if its Shapley value is less than the average of the Shapley values of all parties, i.e., v(N)/n. /n. This implies that party receives negative payoff if its contribution is less than the average of the contributions of all parties.
We also study another property, namely replication-robustre the proposed payoff allocation scheme is replication- for parties with negative payoffs.
L 6 in App. D where we show that
Corollary 3.3. We observe that pı(u) is a linear function of (x)::
Vii.} C. N1 pi(v) p,(1) ()(p(()) (b)))
(2)

---

While the Shapley values are often scaled to ensure all of the rewards are distributed (i.e., being efficient) [T3 [6] the scaling transformation violates not only the linearity property but also the above balance property: if all parties have positive Shapley values, any payoff rewards proportional to the Shapley values (by scaling) are either all positive or all negative. Unlike the sealing transformation, the linear transformation in Corollary 3.3 allows a party with positive Shapley value to have a negative payoff reward. Furthermore, it realizes our intuition of fairness: the payoff rewards compensate the difference in the contributions (measured by the Shapley value) between parties, which we call Shapley fairness. In particular, larger difference in the contributions of two parties leads to larger difference in their payoff rewards, and parties with higher contributions receive higher payoff rewards.
Remark 3.4. While the Shapley value represents the contribution of a party to the optimal model trained on Dyr, there should be n such models distributed to n parties. Thus, the contribution of party to n models should be (v) which explains the factor n in 2 The fact that data is freely replicable (e.g., n parties) is special property of data that differentiates itself from the usual commodity studied in cooperative game theory [1
Remark 3.5. From (T). one interpretation of the allocation of pc(u) is as follows: first, every party deposits v(N) to common pool as if they were dummy parties (i.e., they have not made any contribution); second, parties contribute their datasets to form the aggregated dataset Dx from which the optimal model is trained; third, parties receive the optimal model and also parts of the pool of value (eV (deposited by all parties in the first step) according to their contributions to the optimal model measured by the Shapley values, i.e., (v).
Remark 3.6. We note that the above modification of the efficiency and dummy party properties can be generalized while maintaining the uniqueness of the payoff reward vector. This can be of indepedent interest to readers. particular, there exists a unique payoff reward vector that satisfies the symmetry property, the linearity property, the generalized efficiency property: ΣiEN Pi(u) = B, and the generalized dummy party property: pr(vv) = ( (-areN) + $)/n /n for any dummy party i' and scalars a, 3.. It is specified by (u) = (p))(1) (B av(N)) /n. For example, p(v) is the Shapley value for a = 1 and | = v(N), while p(u) is our proposed payoff reward vector in Lemma 3.1 O. When a = O, every party receives the same payoff of value B/n.
for a n and
Although a payoff allocation scheme is fully determined by the payoff reward vector p(v), it remains an open question how one fairly adjusts the payoff reward under budget constraints. Thus, the next section describes particular way specify payoff flows from one party to another such that it gives rise to the same desirable payoff vector as the one in Lemma 3.1 The notion of payoff flow allows us to handle budget constraints as explained later in Sec. 4
3.2 Fair Payoff Flows
Let us decompose the payoff reward ps (v) E TR into the total cost flow (v) 2 (from party to others) that party needs to pay to obtain the optimal model trained on Dxx and the total revenue flow (v) 2 0 (from others to party i) that party obtains by contributing its dataset to the models other parties receive (the revenue does not deduct any cost):
p;(v)= Pt.+ (v) (v).
Since P.,- is the total cost that party incurs to acquire the additional performance of the model from the datasets of other parties N i i, the total cost flow can be decomposed into
(u)
Σjery Pij
where the payoff flow Pıj denotes the cost that party pays to party j for party j's contribution. Equivalently, it is the revenue that party j obtains from party i. Hence, the total revenue flow is decomposed into
Pi (v) = EjEN
Note that the payoff flow Piy(e) (v) can be viewed as either a cost of party or a revenue of party j. Therefore, to construct all payoff flows between parties, choose to only examine the costs of all parties because it is easier to argue about the total cost of a party (see the fair cost property below) than the total revenue of a party.
In particular, for a party E N, we are interested in the (cost) payoff flows are many different ways of transferring payoff flows from one party to another that result in the same
As there
5

---

payoff reward vector p(r), it is an ill- ill-posed problem to reconstruct payoff flows solely from p(v) in Lemma 3.1 Therefore, we propose the following desirable properties of fair payoff flows that are able to not only uniquely determine (P:j(T)) (we assume pa (v) O in our construction), but also result in the same payoff reward vector as that in Lemma 3.1
Fair cost. As explained above, (v) is the cost that party incurs to acquire the additional performance from the datasets of other parties N i. This additional performance is the difference between the performance s(V) of the obtained model trained on DN and the performance s(i) of the model trained on the dataset of party í. Equivalently. the increase in the value of the model is v(s(NN - s(i)) = v(N) (i). Therefore, we suggest the fair cost property: the total cost flow Pt, (v) of party should be equal the value of the additional performance obtained by party i:
Ps (v) v vi). equivalently, ΣjENY, Pty(t) v(N) u(i).
(4)
Conditional symmetry. two parties j. k N\ have the same marginal contribution to coalition CU i for any C C {i.j.k] k), then the cost party i pays to party j is the same as that party pays to party k. i.e., Piy = Pix (v). It also implies that the cost of party does not depend on the marginal contributions of other parties to coalitions that do not contain i.
Linearity. Let us consider the value function vector of 2" real numbers with the usual addition and scalar multiplication operations, then for value functions v, w,
(v + w) = (t) + Pij(w), Piy(aw) = ap)s(v), Vi,j C.N, Va E..
Conditional dummy party. A conditional dummy party j W \ given party i satisfies vCC (i,j)) = ((Cuă) for all C C N ji For a conditional dummy party j given party i, (v) 0. It means that party i does not pay any payoff to a party that does not contribute to any coalition containing i.
The above 4 properties uniquely determine the payoff flows according to the following lemma which is proved in App.
Lemma 3.7. There exists unique solution (ptj (E)) that satisfies the above 4 properties. We call this unique solution the conditional Shapley value given party i and denote it as (1))(())))ENT)) is defined as follows:
1 ()((V)
Σ (EA)
("1C)
A.UiCCu).
(5)
Like the Shapley value, the conditional Shaple value given party only depends on the marginal contribution A, i) for C c C i (5), so it is translation invariant as elaborated in the following corollary.
Corollary 3.8. (CU()) = =(Cui)+C AC CNV V where C E IR is constant, then (p)((()) = P))(')) for all E
Remark 3.9. Since the formulation of the conditional Shapley value is similar to that of the Shapley to the conditional
value, we can apply existing approximation methods of the Shapley value 3 5 Shapley value.
Remark 3.10. The payoff flow (y)((v) from party i to party is interpreted as the amount of payoff that party i needs to pay party j for its contribution to the model recevied by party i. It is also viewed as the amount of contribution that party j contributes party through any coalitions that containing party i. Besides, fj (v) is independent of the marginal contributions of party j to any coalitions that do not contain party i. It can be clearly observed from the alternative formulation of the conditional Shapley value:
(u)(())
1
A.!! HElL
Σ
(n
where π denotes a permutation of M, Pl denotes parties preceding j in n, and II, denotes the set of permutations that start with
Decomposition of the Shapley value. The Shapley value of party can be decomposed into the conditional Shapley values and its value as shown in App. F
(v) = mii) Ejera (v)

---

We recall in Remark 3.4 that a party contributes to n models (each of which is distributed to one party in M). Hence, the total contribution of party nep((e) which is the LHS of (6). On the other hand, the RHS of (6) is the sum of the contributions of party i to other parties j E .N measured by the conditional Shapley values Pi1s (v) and its own value v(i). The latter is interpreted as the contribution of party to the model it receives from the collaboration. It is v(i) because party can always obtain model of u(i) from its dataset. In brief, the contribution of party i (to n models for parties) measured by the Shapley value can be decomposed into the contributions of party to the models that other parties receive measured by the conditional Shapley values and that to the model party receives.
We set Pai = 0. so by setting (v) = (s)a(()) for i + j. all payoff flows are fully specified. Then, for all i E N, the payoff reward p:(1)) is
pc(v) - (v) Pi (v) = np:(()) v(N)
where the total cost flow is Pt. (v) 2 E PiENP(((U) = v(N) - v(i) (from the fair cost property) and
the total revenue flow is (v) & EjEN Pfi(1)= - (u) v(i) (from (6)) The payoff rewards
specified by (7) are exactly the same that in Lemma [.1 Thus, they also satisfy the 4 desirable properties in Sec. 3.1 symmetry, linearity, balance, and dummy party.
Remark 3.1 (Omnipotent party). Furthermore, if we define an omnipotent party as a party i that satisfies (C U i) rii) for all C . Then, party i does not pay to any other parties, i.e., Pi.- = 0. It is because other parties j EW \ i are conditional dummy parties given party i.
In brief, when parties have no budget constraints, we present a fair payoff allocation scheme that arises from either a set of desirable properties of the payoff rewards (Sec. or that of the underlying payoff flows (Sec. 3.2). While the former is conceptually simpler, the latter can be utilized in the next section address the budget constraints of parties.
4 Parties with Budget Constraints
In the previous section, we assume all parties have no budget constraints, so they can afford the optimal model with the performance s(N) (trained on D2). However, in practice, parties often have budget constraints. Let b, 2 O denote the budget of party i. When (w) > b; party i cannot afford the optimal model trained on Dxv To allow party to join the collaboration without sacrificing fairness, we would like to reduce the performance of the model it receives such that its corresponding payoff reward satisfies the budget constraint. Intuitively, a "'ight-budget" party has to buy a "cheaper" model (i.e.,& model of lower quality). This implies that different parties receive models of different performances. We call it the heterogeneous model rewards. As we discuss in the following paragraph, while the reward vector representation based on the Shapley value is not suitable, the payoff flows based on our proposed conditional Shapley value are able to handle this heterogeneity.
We recall in the Sec. 3 where there are not any budget constraints, all parties receive the same model of value v(N). In contrast, from the above diseussion, the heterogeneous model rewards imply that different parties may receive models of different values, e.g., party i and party j receive models of different values, denoted as v; (N)) and v, respectively. It means a party may contribute to different parties differently. As the payoff reward vector in Sec. 3. relies on the Shapley value that does not differentiate the contributions of a party to different parties, it is challenging to devise principled adjustment of the payoff reward vector meet the budget constraint. On the other hand, by investigating the payoff flows from one party to another through our proposed conditional Shapley value (which is also viewed the contribution in Remark 3.10). we can adjust the amount of contribution from one party to different parties differently.
Let us consider party which cannot afford the optimal model of value v(N) due to its budget, i... b; < <p) For party: i to meet its budget, we need to increase (v()) (equivalently, decreas- ing (v) such that the adjusted payoff reward is at least -b. From the decomposition of payoff rewards into payoff flows in Sec. 3.2 the payoff reward is px(v) = Pi.+ (v) - pi.- (v) 7 It is not pos- sible for party to increase its total revenue flow Ps, i (v) since it comes from other parties. Hence, the only way for party i to increase (v) is by decreasing its total cost flow ps, (v) to other parties N \ i.
We recall that this total cost flow is pi (v) _ Ejen, (v)(v) = Ejen,
Hence, another interpretation is that party reduces the contributions ())(()) of other parties j e N.) i to itself (in
7

---

order to decrease the cost party needs to pay to other parties). The amount of reduction in (p)((() is different for different parties since they may have different budget constraints. Therefore, for each party E NN, we define an adjusted value function, denoted v/. from which the adjusted payoff flows [T It is noted that
flows
Pij (v() are uniquely defined as the conditional Shapley values Py)a(o() (Sec. 3.2
instead of adjusting the underlying value function, existing works [3] 16 17 D) directly modify the Shapley value. However, this invalidates its fairness properties which is undesirable.
Let us denote the set of these adjusted value functions as J a (v))=1- Then, by defining Pty Pts( (7') & Piy (4,), Pt. (J")= Pt, (v²) = (A) (i), and pt, (T)) = -,, P))(e))), the 4 fairesss properties
in Sec. 3.2 uniquely define the payoff flow as the conditional Shapley value Piş(v() - Pyls (t))
Let 8' ( 1=1 (C( v (C) = vCC) for all j, and C CN (v is the un-modified value function), = ())(() is the payoff flow when all parties have no budget constraint (Sec. (31)
then Pyje( 1)
Given v and the budget b - (b.) we would like to find adjusted value functions T" & (v))=_ such that the payoff flows 4pjaa( Th)) Pyje(t() meet the budget constraints, i.c., (p) - b,, Vi E W. To achieve this goal, we propose the following fairness properties of the adjustment.
Feasibility & Individual Rationality. Party is interested in reducing its cost due to the increase in the value v(N) vi). Therefore, a rational party should receive a model with the (adjusted) value (AV) that is (Feasibility) at most v\V); and (individual rationality) at least v(i) (i.e. not worse than the value v(i) of the model it can obtain by itself). The same argument should apply to any coalition containing i:
W.W, VC C v(CU)) 2 ((CU)) 2 v11)
(8)
Marginal contribution consistency. From an alternative perspective, party reduces its cost by reducing the contribution of other parties to the increase in the performance of the model it receives. This can be measured by the conditional Shapley values of other parties given i which consist of the marginal contributions of other parties any coalition containing Therefore, the marginal contribution consistency states that the marginal contributions of a party j A i to any coalition containing should be consistently reduced by the same factor:
Vi E N, 'YE > 0, Vj E N (i, VC c NV\ {i,s} A., (ICU.) (iCu... (9)
It means that the adjustment should not unfairly bias the marginal contribution of party to coalition.
Budget efficiency. The budget efficiency encourages every party to obtain the best model that it can afford by expending as much of its budget as possible. On one hand, if party i can afford the optimal model of value v(N) (i.e., (v) < bi), then it receives the optimal model under the adjusted payoff flows, i.e., (A" = v(N). On the other hand, if party i cannot afford the optimal model of value v(N) due to its budget (i.e.. (p: > b;), then the adjusted payoff exactly -b;, i.e., ps (x()() b, It means that the budget is fully consumed.
The above properties and the budget constraints uniquely define fair adjustment of the value function as shown in the following lemma (proven in App. G
Lemma 4.1. Let X & (2i)in be the solution to the following linear programming problem:
maximize subject to
EjinioN (e)))(())
A4 P3)1()))) EueNss ARPa)(()) & bi Vi VIEN (0
(P)
Vi N.
The feasibility & individual rationality, the marginal contribution consistency, the budget efficiency, and the budget constraints uniquely determine the following adjusted value functions:
ViE VC CN vf(C = Xv(CUi) + (L-X))v(1))
(10)
Then, from the fairness properties in Sec. 3.2 the payoff flow Piç(e() is uniquely defined as the conditional Shapley value ()((())) which can be directly computed from P))()):
Vj E N 11, (1) (p)) = ()):(A; = \1)((1).
(11)
As the conditional Shapley value given i only depends on the value function at coalitions containing i, we are only concerned with defining of on the restricted domain (CuiC \..
8

---

We note that the LHS of the first set of constraints in (P) are the negative values of the payoffs, i.c.,
-P; (p,)"" = (3") Pi.+ (²)), Vi N, so they are the budget constraints.
Given the adjusted payoff flows, the total revenue flow and the total cost flow are computed as
(3') = Σ Ejen (1)) and p=' = x EjENY Ps)(t), respectively.
The value of the model obtained by party according to the above adjusted value function is (N) XFv(N) + (1 - (t)) = v(t) + Pt,,- (v) = uni)= - Pt.- (vz). Hence, the value of the model obtained by party the sum of the value v(t) of model trained on party i's dataset and the value of
the contribution (rf( of other parties to party i. This also demonstrates the feasibility & individual rationality property (since A, E [0, 1). To produce model with a pre-specified performance (e.g., the validation accuracy), we suggest using early stopping. Starting from the optimal model trained on the dataset D. of party i (with the performance v(i)/v p = s(i)), we continue to train the model with the aggregated dataset Dx and stop the training when the performance of the model reaches (o(1) + A.Pt,-()))
Remark 4.2. There always exists party in N that receives the optimal model trained on Dx (of value v(N)). Unlike the work of [16) enforcing this property, namely weak efficiency, from the construction of the allocation scheme, this property naturally arises from our above fairness properties as shown in App. H implies that all of the aggregated dataset is utilized by least a party.
Remark 4.3. Our proposed framework can be extended to handle the performance constraint. In particular, while the budget constraint prevents party's 's expense from exceeding its budget, the performance constraint implies that a party is not interested in obtainng a model with a performance larger than a desirable model performance. The details are in App.
5 Experiments
This section empirically illustrates the proposed allocation scheme of both payoff and model rewards in collaborative ML. To easily interpret the results, we choose the MNIST dataset [10] which contains 70, 000 images of handwritten digits. The dataset is partitioned based on the digit labels. A
party, denoted as [s-e] (s S e). owns a subset of the MNIST training dataset (of size 60, 000) that are labeled with digits s, + .,.... There are 5 parties in N: [0] (i.e., [0-0]). [1-2], [0-3], [3-5]. and [6-9]. For simplicity, we set v/ = 1. The model performance is measured by the prediction accuracy on separate dataset of size 10, 000.
Additional experiments on the CIFAR-10 p9 dataset and the IMDB movie reviews dataset dataset are in A All experiments share the following 4 desirable observations. First, when there are not any budget constraints, all parties receive models with the highest performance, and parties with more valuable datasets receive higher payoff rewards. Second, even if all parties" budgets are O, they still benefit from the collaboration (hence, motivated to join the collaboration) by obtaining better models than their own. Third, increasing the budget of a party not only increases the model performance of the party but can also benefit other parties through improved model performance and/or higher payoffs. Fourth, there is always party receiving the best model of value v(N), i.c., the weak efficiency property.
Figs. 2 and show the payoff and model rewards when there are not any budget constraints (i.e., b = o01), respectively. In Fig. 2 party [6-9] receives the largest payoff reward because party [6-9] owns the dataset with the largest number of digits which is not overlapped with any other parties' datasets. Party [0-3] receives a larger payoff reward than that of parties [0] and [1-2] because party [0-3] 's dataset contains the datasets of both parties [0] and [1-2]. In Fig. Fig.2b. all parties receive models with the highest performance as there are not any budget constraints. While the work of [33 can address the
case of no budget constraints, it results in payoff vector [-0.41, -0.26, 0.002 0.12, 0.58) which is different from ours. It is noted that their solution does not satisfy the linear property while ours does.
Figs. 2 and 2 show the payoff and model rewards when all parties are not allowed to spend any payoff (i.e,
0), respectively. From the balance property. the payoff rewards are for all parties (in Fig. p However, in Fig. D1. they still benefit from the collaboration by obtaining better models (of value v(i) pt. (A.r)) than their own models (of value vi)). In particular, we observe that party (6-9) receives the optimal model trained on the aggregated dataset Dxx (of value v(N)), as stated by the weak efficiency property. Thus, when monetary payoff's are not allowed, our allocation scheme is still able to incentivize parties to fairly collaborate using their data. Compared to the p phapley

---

1.00
0.25
0.00
0.00
(e
(ea)
[0)
[12]
10-3/
[3-5]
(e-9)
[221
103))
(a-)
(6-9)
(a) Payoff
(b) Model reward
(c) Payof
(d) Model reward
Revenue flows (x10)
[o]-
0.00 0:36
0.39
Revenue flows (x10)
11:21-
9.12
0.00
0.24
0.56
0.92
[1-21
0.15
0.00
0.31
0.73
1.1
0.75
[03)
0.50
[3
0.29
45
0.92
0.00
2.20
[3
2.38
0.85
1.21
0.00
2.99
[6-]]
0:48
[69]
0.00
[0]
(2:2)
[e:3]
[2.5]
16-91
(0)
[1:21
[0-3]
[3:5]
[6.9]
flows (x10)
flos (x10)
(e)
Payoff
(f) Model reward
(g) Payoff flow
(h) Payoff flow
Figure 2: Plots of the payoff's, model rewards, and payoff flows in the MNIST experiments with budget constraints: (a.b) b 00 (c.d.g) b and (e.f.h) b, 0.31;
value [16] (which only works in this special case), the model reward allocation in Fig. 2 cannot be obtained from the p-Shapley value (by varying its parameters). In fact, unlike our proposed allocation scheme, when P + 1, the p-Shapley value does not satisfy the linear property.
Figs. 2 and 2 show the payoff and model rewards when only party [0] has a positive budget of 0.3 (other parties have budgets of 0), i.e. b; = 0.31;-1- While it is intuitive that with a positive budget, party [0] improves the performance
its obtained model in comparison to the previous case of no budget, it is interesting that parties [1-2], [0-3]. and (3-5] also have the performance of their models
improved (by comparing Figs. 2 and 2t. The reason is that revenues parties [1-2], [0-3), and [3-5]
from party [0] increase (comparing the first row in Figs. 2 2h) These increased revenues
used to further improve the models of [1-2]. [0-3]. and [3-5]. c.g., by getting more contributions from party [6-9] (comparing the last column in Figs. 2 and 2h). Furthermore, not only does party [6-9]
obtain the best model of value v(N) as that in Fig. 2H, but also obtains an additional payoff reward for its contributions to other parties in Fig. 2p. As seen in Figs. D and 2h. the cost flows of party [6-9] (the last rows) are the same between the two cases. Thus, according to our allocation
scheme, increasing the budget of party benefits not only itself but also other parties.
6
Conclusion
This paper presents an allocation scheme for both payoff and model rewards in fair collaborative ML. When there are not any budget constraints, all parties receive the same model trained on the aggregated dataset. On the other hand, they receive different payoff rewards account for the difference in the contributions of different parties. We construct such a payoff allocation scheme that is fair according to (i) desirable properties of the parties' payoff's or (ii) that of the underlying payoff flows from one party to another. The latter is useful to handle the practical constraint on the budgets of parties. In particular, when there exist budget constraints, we propose a solution to trade off between payoff and model rewards through fair adjustments of the payoff flows. The experiments empirically show that our scheme is reasonable in several collaborative MI. scenarios with different budget constraints.
Acknowledgments and Disclosure of Funding
This research/project is supported by the National Research Foundation Singapore and DSO National Laboratories under the Al Singapore Programme (AISG Award No:
10

---

References
[1] A. Agarwal, M. Dahleh, and T. Sarkar. A marketplace for data: An algorithmic solution. In Proceedings of the 2019 ACM Conference Economics and Computation, pages 701-726, 2019.
[2] I.. Agussurji, Xu, and B. K. H. Low, the convergence of the Shapley value in parametric Bayesian learning games. In Proc. ICMI,, pages 180-196, 2022.
[3] J. Castro, D. Gómez, and J. Tejada. Polynomial calculation of the Shapley value based on sampling. Computers operations research, 3665): 2009.
[4] A. Ghorbani, M. Kim, and J. Zou. A distributional framework for data valuation. In Proc. ICML, pages
PMLR, 2020.
[5] A. Ghorbani and J. Zou. Data Shapley: Equitable valuation of data for machine learning. In Proc. ICML, pages 2242-2251, 2019.
[6] D. Han, S. Tople, A. Rogers, M. Wooldridge, O. Ohrimenko, and Tschiatschek. Replication- robust payoff-allocation with applications in machine learning marketplaces. arXiv preprint arXiv:2006. 14583, 2020.
[7] R. Jia, D. Dao, B. Wang, F. A. Hubis, N. M. Gurel, B. Li, C. Zhang, C. J. Spanos, and D. Song. Efficient tpsc-sp data valuation for nearest neighbor algorithms. In Proc. VLDB Endowment, volume 12, pages 1610-1623, 2019.
[8] R. Jia, D. Dao, B. Wang, R. A. Hubis, N. Hynes, N. M. Gürel, B. Li, C. Zhang, D. Song, and C. J. Spanos. Towards efficient data valuation based on the Shapley value. In AISTATS, pages 1167-1176. PMLR, 2019.
[9] A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. 2009.
[10] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient- based learning applied document recognition. Proceedings of the IEEE, 86( 11:2218:2722333 November 1998.
[11] A. L.. Maas, R. F. Daly, P. T. Pham, D. Huang, A. Ng. and C. Potts. Learning word vectors for sentiment analysis. In Proceedings of the 49th annual meeting of the association for computational linguistics: Human language technologies, pages 142-150. Association for Computational Linguistics, June 2011.
[12] S. Maleki, L. Tran-Thanh, G. Hines, T. Rahwan, and A. Rogers. Bounding the estimation error of sampling-basee Shapley value approximation. arXiv preprint arXiv: 1306. 4265,, 2013.
[13] O. Ohrimenko, Tople, and Tschiatschek. Collaborative machine learning markets with nata-repliiattinn-robss payments. arXiv preprint
2019.
[14] L. S. Shapley. A value for n-person games. Contributions to the theory of games, 2(28):307-317, 1953.
[15] R. H. L. Sim, X. Xu, and B. K. H. Low. Data valuation in machine learning: "Ingredients", strategies, and open challenges. In Proc. IJCAI, pages 5607-5614, 2022.
[16] R. H. L. Sim, Y. Zhang, M. C. Chan, and B. K. H. Low. Collaborative machine learning with incentive-aware model rewards. In Proc. ICML, pages 8927-8936. PMLR, 2020.
[17] S. Tay, X. Xu, S. Foo, and B. K. H. Low. Incentivizing collaboration in machine learning via synthetic data rewards. In Proc. AAAI, 2022.
[18] Z. Wu, Y. Shu, and B. K. H. Low. DAVINZ: Data valuation using deep neural networks at initialization. In Proc. ICML, pages 14150-24176 2022.
[19] X. Xu, L. Lyu, X. Ma, C. Miao, C. S. Foo, and B. K. H. Low. Gradient driven rewards to guarantee fairness in collaborative machine learning. In Proc. NeurIPS, volume 34, 2021.
[20] X. Xu, Z. Wu, S. Foo, and B. K. Low. Validation free and replication robust colume-based data valuation. In Proc. NeurIPS, pages 10837-10848, 2021.

---

[21] J. Yoon, S. Arik, and T. Pfister. Data valuation using reinforcement learning, In Proc. ICML, pages 10842-10857) PMLR, 2020.
[22] Z. Zhou, X. Xu, R. H. L. Sim, C. S. Foo, and Low. Probably approximate Shapley fairness with applications in machine learning. In Proc. AAAI, 2023.
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes]
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] We include the proofs in the App..
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main ex- perimental results (either in the supplemental material or as URL)? [Yes]] https
//github. caaccuu..oo/qppaaaae
saaneel-pyyof--lloaattiooo allocation
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [N/A]
(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]
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]
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any new assets either the supplemental material or as URL? [No]
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
12


Reviews:
[{"review_id": "ZaRSOYc_9A", "reviewer": "Reviewer_GnPz", "summary": "The paper studies the problem of fair payoff allocation of compensation/cost among a group of collaborating parties which pool their data to jointly train a machine learning model. Specifically, the authors considered several fairness properties (including a linearity property which helps to uniquely determines an allocation scheme). The authors then presented a unique allocation scheme which satisfies these properties. From there, the authors considered and resolved a more general setting where parties may not have sufficient budget by balancing their budget and model performances.", "questions": "Can the authors comment on the payoff allocation schemes that do not satisfy the linearity property?", "limitations": "I do not observe any negative societal impact.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- the collaborative machine learning setting is interesting and important.\n- the considered fairness axioms are reasonable, for example, the budge balance that keeps the total payoff/cost to zero among the group of parties.\n- the proposed payoff allocation scheme is unique and supported with theoretical fairness axioms\n\nWeaknesses:\n- there is limited empircal comparison and analysis with prior works.\n\nSome minor comments:\n- the uniqueness of the allocation scheme is largely a result from the linearity axiom, that draws a connection to the Shapley value. I can see that the advantage of imposing the linearity is that it utilises the fairness of the Shapley value (a classic solution concept), which is a reasonable constraint for finding a unique solution. However, I do not see why allocation schemes that do not exhibit the linearity dependency on the Shapley value are inferior in the particular setting. This may be beyond the scope of the current paper, but a more comprehensive analysis on the class of payoff allocation schemes (without the linearity property) may be explored further in future works.  I also acknowledge the motivation and overall contributions of the present paper.\n- As far as I know, there are quite a few papers on payoff allocation with external funds, but only a few on fully collaborative settings. I find this topic to be interesting for the community. I would not argue for acceptance weighing the strengths/weaknesses, but I am slightly positive hence a weak accept.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "T9LFx1MNJ3Z", "reviewer": "Reviewer_iwL1", "summary": "This paper considered the setting where there exists a tradeoff between payoff and model rewards in collaborative machine learning. Inspired by the observation that the difference between the contributions of parties should take into consideration during reward allocation, they proposed a new solution concept \\emph{conditional Shapley value}, and proved their payoff function is unique for reasonable properties. Additionally, they decide their allocation based on the payoff function, and consider the settings that the parties have budget constraints, or without budget constraints, and empirically demonstrate the performance of their allocation scheme. ", "questions": "1. Is there any rationale/literature on whether the party with more contributions should receive higher payoff rewards? From the reviewer's perspective, it's not a standard Fairness notion to assume this. ", "limitations": "See weakness section. \n", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper has many strengths:\n1. They consider a more general setting when the given model should of different payments based on the parties' contribution.  \n2. Their characterization of the desirable properties for a payoff reward vector produces a provable \\emph{unique} payoff reward vector. \n3. Their payoff allocation also satisfies replication-robustness that is studied in the previous literature.\n4. The paper includes experiments on MNIST, CIFAR-10, and IMDB datasets. \n\nWeaknesses:\n1. The paper doesn’t compare against the existing benchmark either in theoretical statements or empirically. \n2. No contribution section in the introduction. ([line 65-90] has an overview of the sections, but doesn't include the theoretical results of this paper. \n3. The descriptions of the payoff functions are hard to follow. \n4. No running time analysis. \n\n=====================================================================\nAfter review rebuttal period, the reviewer would like to add more weakness as follows:\n\n1. Results for similar settings, though not explicitly stated as \"Fair Allocation\", has been published in 2019, see [1]. \n\n2. Many of the concepts in this paper are stated without appropriate references, for example, \"Individual Rationality\", \"Marginal contribution consistency\", \"balance\", \"linear Value\", \"dummy party\" etc. These concepts are widely used metrics for mechanism design and beyond. \n3. The conditional Shapley value seems to be similar to Shapley value, as in stated in Remark 3.5. Additionally, for the properties introduced in this paper, i.e., balance, linear Value, dummy party, similar properties are already been proved as in [2] for Shapley value, which makes the theoretical contributions for providing such metric incremental. \n\n3. This paper doesn't have comprehensive references, at least existing literature on collaborative learning/ allocations rules. From the reviewer's perspective, while there are total 17 references, many of them are not directly related to collaborative learning. Additionally, the current references are informal, i.e., without the conference/ journal name, e,g. [15], [16], [8] in the paper. \n\nThe reviewer is from a more theoretical background, so my evaluations are more on the theoretical contributions of this paper, i.e., the novelty of the proof for theorem statements, and how the results are different from existing metrics. From the reviewer's perspective, the writing of this paper itself (references, connection to existing results ) needs a major revision. \n\n[1] Ghorbani, A., & Zou, J. (2019, May). Data Shapley: Equitable valuation of data for machine learning. In International Conference on Machine Learning (pp. 2242-2251). PMLR.\n\n[2] Agarwal, A., Dahleh, M., & Sarkar, T. (2019, June). A marketplace for data: An algorithmic solution. In Proceedings of the 2019 ACM Conference on Economics and Computation (pp. 701-726).\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "SxvVzE-89Ne", "reviewer": "Reviewer_nJsn", "summary": "The paper studies collaborative machine learning, where data from multiple parties is aggregated to train a machine learning model with better performance. It is concerned with fairness in the sense that parties should receive a reward that accounts for how much they contributed to the quality of the model, as measured by the Shapley value. The paper notably presents a set of desirable properties of the payoff flows between parties from which it derives a fair allocation scheme. This approach allows to address settings with or without budget constraints. The method is evaluated on a simulation with segments from the MNIST dataset.", "questions": "* What is the difference between this work and [Sim et al., 2020] ([14]), in the use of Shapley values to measure the contribution of a party? \n\n* Sec. 4: why should we consider fair to reduce the performance of the model a party receive, in the presence of budget constraints? \n", "limitations": "The limitations and societal impact are not discussed in this work. In particular, the authors did not discuss the privacy issues involved when different parties share their data.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: The paper is overall well written. The fairness properties of payoff flows for fair allocation are interesting and allow to address both budgeted and non-budgeted settings. The scheme based on payoff rewards is limited to non-constrained problems but it is simpler to understand. The presentation of both types of allocation schemes allows to understand well the advantages and drawbacks of either approaches.\n\nWeaknesses: The contribution of the paper is not very clear: How do the properties defined in the paper differ from classical axioms of cooperative game theory? This is important because the paper is not technical, and its main interest thus lies in the conceptual framework studied. \n\nTypos: L60 et -> It, L388 collaboration -> collaborative\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "7ajrktZd66", "reviewer": "Reviewer_Bdvn", "summary": "The paper considers collaborative machine learners where multiple vendors combine their data so as the generate more accurate models than either of them can achieve on their own. The authors focus on tradeoffs between payoff and model rewards, including for budget restricted vendors. The method proposed can be derived from two methods,  based on desirable properties of the parties’ payoffs or based on the underlying payoff flows from one party to another (the second allows handling constraints on the budgets of vendors). The authors propose desiderata for buiding a fair adjustment of the payoff flows that trades off between the model reward’s performance and the payoff reward. The authors also support their claims on several collaborative ML scenarios. \n\nThansk for the detailed author comments. ", "questions": "Are there any guarantees on the shared data? Can one vendor \"poison\" the data given to others so as to achieve its goals? See discussion in: Meir, Reshef, Ariel D. Procaccia, and Jeffrey S. Rosenschein. \"Algorithms for strategyproof classification.\" Artificial Intelligence 186 (2012): 123-156.\n\nFurther, can other foundations of mechanism design (such as auctions or the Vickery-Clarke-Groves mechanism) be used here for similar applications (e.g. auction off the data by a vendor, rather than redistributing payments). \n\nCould there be other constraints (e.g. computational ones) that be tackled here, rather than the budget constraints. \n\nFinally, you show that your solutions are \"sensible\" and result with reasonable outcomes. What kind of additional metrics could quantify that? What could one show in future work to demonstrate an even \"more fair\" outcome? ", "limitations": "This builds heavily on earlier work and foundations from cooperative game theory. The empirical analysis is illuminating but limited (few domains, few metrics). The authors should do more to justify their choices of desiderata (and surey relevant game theory work on how different desiderata yield different solutions). \n\nWhile the question of whether budget constraints are reasonable ones for ML vendors is subjective, the authors focus on this one constraint. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I find the topic of the paper very interesting. There have recently been multiple works on mechanisms for allowing vendors to share their data, focusing on how to incentivize them to do so by fairly sharing the proceeds of the resulting model (or by yielding smaller models where the quality of a model a party receives depends on its contribution). The key strength of the paper is by combining key methodology from cooperative game theory, such as the Shapley value and the desiderata yielding this value, and applying them to this domain. \n\nHowever, given that there are already multiple papers on this topic, I believe the bar for additional work here is higher. The paper has some weaknesses. First, while taking an axiomatic (desiderata) approach, I'm not sure I fully by into the proposed desiderata here. Specifically, for the Shapley value which is a common theme here, one of the axioms is linearity across games. Other powre indices are triggered by alternative approaches. See, for e.g. (and I'd encourage the authors to discuss these, in more depth):\n\nDubey, Pradeep. \"On the uniqueness of the Shapley value.\" International Journal of Game Theory 4.3 (1975): 131-139.\nDubey, Pradeep, and Lloyd S. Shapley. \"Mathematical properties of the Banzhaf power index.\" Mathematics of Operations Research 4.2 (1979): 99-131.Feltkamp, Vincent. \"Alternative axiomatic characterizations of the Shapley and Banzhaf values.\" International Journal of Game Theory 24.2 (1995): 179-186.\nMonderer, Dov, and Dov Samet. \"Variations on the Shapley value.\" Handbook of game theory with economic applications 3 (2002): 2055-2076.\n(and see the very related discussion in Torra, Vicenç, and Yasuo Narukawa. Modeling decisions: information fusion and aggregation operators. Springer Science & Business Media, 2007.)\n\nFurther, regarding writing, I feel like a more detailed introduction in cooperative game theory is warranted. You build on these foundations, but the discussion is short. For example, what are the key differences between such power indices and other known methods such as the Core or Nucleolus? Why are these less applicable here? \n\nFinally, I suggest addressing computational constraints. What is the exact time (and memory) complexity of the proposed approaches? Are you computing Shapley values exacly, or approximating? Can alternative computational mechanisms be used? \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "7ATU6ta97Qx", "reviewer": "Reviewer_xYwo", "summary": "The authors considered an interesting and practical problem where groups of data owner contribute their data to collaborative build an ML model that is better than models they build on their own. The key contribution comes into designing a unique solution concept that simultaneously consider how monetary payoffs and model rewards should be assigned to each contributor using (Conditional) Shapley values.", "questions": "On the contrary, if a party has a constraint or minimal requirement on model performance instead of monetary budget, is your proposed optimisation framework applicable in this case? More concretely, if a party only requires the model to be as good as 90% of the best performing model in order to obtain some discounted payoffs, is your method applicable in this situation?\n\nAnother question: is the monotonicity assumption of s needed in your formulation? It seems to be needed to ensure joining the cooperation will be better than not joining, but in terms of Shapley values computation there is no such restriction needed when defining a value function? Can you comment a bit on this please? ", "limitations": "NA", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- proposing a solution that handles monetary payoff + model rewards simultaneously is not something the literature has considered before. \n- Communication of ideas are mostly clear besides the use of payoff flows (see below).\n\nWeaknesses:\n- The introduction of payoff flow and conditional SVs weren't apparent to readers until getting to Sec.4, and it seemed to complicate the payoff allocation scheme. It was only made clear later in section 4 that we can handle the budget constraints by modifying the value functions of all parties in the Conditional SV. Maybe this intuition should be made a bit earlier to make it easier to follow.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept