Optimal Procurement Auction for Cooperative Production of Virtual Products: Vickrey-Clarke-Groves Meet Cremer-McLean

We set up a supply-side game-theoretic model for the cooperative production of virtual products. In our model, a group of producers collaboratively produce a virtual product by contributing costly input resources to a production coalition. Producers are capacitated, i.e., they cannot contribute more resources than their capacity limits. Our model is an abstraction of emerging internet-based business models such as federated learning and crowd computing. To maintain an efficient and stable production coalition, the coordinator should share with producers the income brought by the virtual product. Besides the demand-side information asymmetry, another two sources of supply-side information asymmetry intertwined in this problem: 1) the capacity limit of each producer and 2) the cost incurred to each producer. In this paper, we rigorously prove that a supply-side mechanism from the VCG family, PVCG, can overcome such multiple information asymmetry and guarantee truthfulness. Furthermore, with some reasonable assumptions, PVCG simultaneously attains truthfulness, ex-post allocative efficiency, ex-post individual rationality, and ex-post weak budget balancedness on the supply side, easing the well-known tension between these four objectives in the mechanism design literature.

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Introduction

On this flat earth created by the internet, hierarchies of companies are falling away. Production activities are no longer a process confined within the border of an enterprise. Independent businesses can cooperate seamlessly through distributed computing systems to produce valuable virtual products. For example, a fast-developing technology, federated learning (FL), enables businesses to train (produce) artificial intelligence models collaboratively

[31]. Another example is crowd computing, where participants contribute their redundant computing capacity to distributed computing tasks (e.g., protein folding simulation in the Folding@home project [3]). These production models differ from classical ones in at least two respects: First, collaborative production creates synergies, i.e., participants working together generate more value than they working separately; Second, the output is virtual products, which is non-tangible and non-rivalrous, i.e., the consumption of the virtual products by one participant does not reduce the amount available for others. While the first feature encourages cooperation, the second feature results in the free rider problem, i.e., participants lack the incentives to ally to attain the socially optimal result. Currently, cooperative production applications are mostly run as small-scale not-for-profit projects, heavily relying on volunteers’ participation, thus hindering their popularization. A profitable business model is necessary for cooperative production to develop faster in the economic world.

The design of such a business model is challenging due to two reasons: First, there are multiple dimensions of asymmetric information, i.e., the demand side is privately informed about its valuation of cooperative production while the supply side is privately informed about its cost and capacity constraint of supplying input resources; Second, in many realistic applications, an agent can both demand and supply cooperative production, e.g., businesses may both demand and supply data under federated learning. In this paper, we show that if the demand side’s valuation satisfies a correlation condition, then we can use the combination of Crémer-McLean mechanism [12, 20] and VCG mechanism to separately solve the asymmetric information problem for the demand side and supply side: on the demand side, the Crémer-McLean mechanism enables us to extract full consumer surplus for their usage of the virtual products and maximize the revenue distributable to the supply side; on the supply side, the VCG mechanism enables us to overcome the free-rider problem by compensating producers for their contribution of input resources and achieve socially efficient allocation.

In this paper, our contributions include: 1) We set up a game-theoretical model for cooperative production of virtual products, which takes into consideration twofold supply-side information asymmetry in addition to the demand-side information asymmetry; 2) We propose a procurement auction that incentivizes participants to report their private information honestly (truthfulness) and maximizes social welfare ex post (allocative efficiency); 3) We apply neural network methods to optimizing for ex-post individual rationality and ex-post weak budget balancedness; 4) We prove with some reasonable assumptions, truthfulness, ex-post allocative efficiency, ex-post individual rationality and ex-post weak budget balancedness can be attained simultaneously.

Related Works

Our game-theoretic model is an abstraction of emerging internet-based cooperative production practices. Although underlying technologies of cooperative production are fast developing, existing studies on these new business models are quite limited. From a more general perspective, our model is a special type of cooperative games [23, 4, 7]

, which have attracted great attention from the algorithmic game theory community

[26, 11, 6, 14, 2]. Designing an optimal sharing rule is a key concern when studying cooperative games [21, 22, 29].

We decouples the cooperative production game into a demand-side game and a supply-side game. On the demand side, we use the Crémer-McLean mechanism [12, 20, 1] to extract consumer surplus. On the supply side, we proposed the PVCG mechanism, which borrows the idea of procurement auction from procurement games [9, 8, 17, 13]. Compared to existing research, our model takes into account three sources of information asymmetry, one on the demand side and two on the supply side, yet we simultaneously achieve truthfulness, allocative efficiency, individual rationality, and ex-post weak budget balancedness, easing the well-known tension between these objectives [18]. We also follow the recent trend of neural-network-based mechanism design [27].

Federated learning [31, 32] and crowd computing [3, larson2009Folding]

are examples of the cooperative production game under study. Therefore, our work is also related to the literature on training machine learning models with strategic participants

[19, 28, 5, 25, 30, 33] and crowd computing from the angle of game theory [10].

Game Settings

We study a cooperative production game where a set of producers, denoted by , cooperatively produce a valuable virtual product, of which copies are dilivered to consumers, denoted by . A participant may be both a producer and a consumer, but we assume that its behaviors as a producer and as a consumer are independent. On the supply side, producers contribute input resources to the production coalition, e.g., labor, raw materials, equipments, etc. On the demand side, consumers are granted access to utilize the output virtual product. We introduce a parameter to measure the input resources contributed by producer .

may be a vector when multiple input resources are involved. We use another parameter

to measure the usefulness of the output product, which is determined by the contributed input resources from all producers, i.e., is a function of .

The usefulness of the output product determines the value it brings to consumers. We use the parameter to denote the valuation of participant on the output product. is a function of and a type parameter (called valuation type) that reflects the heterogeneity among consumers. The type space is of cardinality and we assume that for all . We denote this function by and call the composite function of and the individual valuation function, denoted by , i.e., . For convenience, is such chosen that when . We denote . Contributing resources to the cooperative production process incurs costs to participants. Producer ’s cost is a function (called individual cost function) of and another type parameter that reflects the heterogeneity of producer . We denote . In our game, producers are assumed to be capacitated, i.e., producer cannot contribute more resources than its capacity limit , i.e., . Both the type parameters and the capacity limit are private information unknown to the coalition coordinator a priori. This coordinator makes the transfer payment to producer and to consumer . Participants’ preference is represented by quasi-linear utilities and . We use social surplus to measure the social effect of the production coalition. Social surplus maximization implies Pareto efficiency (or allocative efficiency in the language of mechanism design).

Parameters in our model have concrete meanings in scenarios such as federated learning (FL) and crowd computing. In FL, the input resources are data. measures the size and quality of the contributed dataset. (Measuring the quality of datasets is a separate problem. We do not go deeper here.) The capacity limit is the best dataset owned by participant . is the cost of collecting and cleaning data. is the value of the federated model to each participant (e.g., for the use case that banks use FL to train AI models to predict credit risk, is calculated as the reduced bad debt rate times the principle of loans). In crowd computing, is the contributed computing power, costs of computing power (e.g., electricity costs, hardware costs, etc), and is the value of the computation result (e.g., the value of protein folding result to drug development).

We put forward five assumptions on the individual valuation function and the individual cost function . Assumption 1-2 are used for proving truthfulness and alloactive efficiency. Assumption 3-5 are used for proving individual rationality and ex-post weak budget balancedness.

Assumption 1 (Smoothness and monotocity).

The individual valuation function is a smooth and monotonic increasing function of and . The individual cost function is a smooth and monotonic increasing function of and .

Assumption 2 (Zero input resource makes no difference).

If , then: (1) participant makes no difference to the value of the output product, i.e., ; (2) participant bears no cost, i.e., .

Assumption 3 (Super additivity).

The individual valuation function is super additive with respect to , i.e.,

(1)
Assumption 4 (Decreasing cross marginal returns).

The marginal return of one producer’s input resources decreases when other producers contribute more input resources, i.e., for all and , , the following inequality holds:

(2)
Assumption 5 (Correlated and identifiable valuation types).

The prior belief on valuation types, Prior(), is identifiable and correlated, i.e., the identifiability condition [20] and Crémer-McLean condition [12] hold for all consumers.

The economic meaning of super additivity is that cooperative production brings synergies, i.e., the value created by the production coalition is higher than the total value created by independent producers. The rationality of Assumption 4 comes from the law of diminishing marginal returns: when many resources have already been involved in a production process, the marginal return brought by an additional unit of input resources decreases. As an example, the following individual valuation function satisfies all these four assumptions:

(3)

where is an arbitrary smooth and increasing function (e.g., the Cobb-Douglas function). Assumption 5 is introduced to guarantee full consumer surplus, so that we can derive the income of the coalition from individual valuation functions, i.e., , because we have the following theorem:

Theorem 1 (Crémer-McLean Theorem).

There exists an interim individually rational and ex-post budget balanced Bayesian mechanism that extracts full consumer surplus if Prior() is identifiable and Crémer-McLean condition holds for all consumers.

Proofs of Theorem 1 can be found in [12] and [20]. We can construct such a demand-side Crémer-McLean mechanism by following the constructive proof of Lemma A3 in [20] or by automated mechanism design techniques [1]. We will not go into further detail here because we focus on the supply-side game. For theoretical analyses in this paper, we take as a given that the production coalition uses Crémer-McLean mechanism to extract full consumer surplus. This also guarantees the reported valuation type equals the true valuation type for all consumer . Theorem 1 transfers the ex-post weak budget balance constraint to the following inequality:

(4)

This decouples the supply-side mechanism design problem from the demand-side problem.

The Procurement Auction

As a counterpart of Crémer-McLean mechanism which is optimal on the demand side, we introduce an optimal supply-side procurement auction in this section. This proposed procurement auction, accompnied by the demand-side Crémer-McLean mechanism, maximizes social surplus by incentivizing producers to truthfully report their capacity limits and type parameters. This procurement auction consists of four steps, of which the soundness will be proved in the next section.

Step 1. Producers bid on capacity limits and cost types

As the first step, every producer submits a sealed bid for their respective capacity limits and cost types. The reported capacity limit is the maximum resources that producer is willing to offer to the coalition. It may differ from the true capacity limit . Similarly, the reported cost type may differ from the true cost type .

Step 2. The coordinator chooses the optimal acceptance ratios

Then, the coalition coordinator decides how many input resources to accept from each producer. It chooses that maximize the social surplus, constrained by reported capacity limits and based on reported type parameters. Equivalently, the coordinator calculates the optimal acceptance ratio such that , where and denote the element-wise multiplication and division respectively, and denotes the interval between and . The economic meaning of is the ratio of input resources accepted by the coalition to those offered by the producer.

The optimal acceptance ratios are calculated according to the following formula:

(5)

Because different results in different , is written as . Correspondingly, the maximum social surplus is denoted by . It is worth noting that although and both represent social surplus, they are different functions. The first parameter in is the accepted input resources, whereas the first parameter in is the reported capacity limits. and are related by .

Step 3. Producers contribute accepted input resources to the production coalition

In this step, producers are required to contribute units of input resources to the production coalition. Since in the first step, producer has promised to offer at most units of input resources, if it cannot contribute , we impose a high punishment on it. With the contributed input resources, producers collaboratively produce the output virtual product, bringing value to consumer .

Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule

In this final step, the coordinator pays producers according to the PVCG sharing rule. The PVCG payment

(6)

is composed of two parts, the VCG payment and the optimal adjustment payment . The VCG payment is designed to induce truthfulness, i.e., the reported capacity limits and reported cost type are equal to the true capacity limits and true cost type . The adjustment payment is optimized so that ex-post individual rationality and ex-post weak budget balancedness can also be attained.

With calculated in Step 2, the VCG payment to producer is:

(7)

where denotes the reported capacity limits and the reported cost types excluding producer . and are the corresponding optimal acceptance ratios and maximum social surplus. Note that is different from : the former maximizes , whereas the latter is the component of that maximizes . is a function of , written as .

The adjustment payment is a function of . The optimal adjustment payments are determined by solving the following functional equation (a type of equation in which the unknowns are functions instead of variables; refer to [24] for more details):

(8)

where is the support of the prior distribution of the true parameters estimated by the coalition coordinator. Support is a terminology from measure theory, defined by . In general, there is no closed-form solution to Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule, so we employ neural network techniques to learn the solution. We will go into more details later.

Theoretical Analyses

Theoretical analyses in this section are organized through the following strand:

  1. First (in Proposition 1 and Proposition 2), we prove that for arbitrary , , the PVCG payments guarantee supply-side truthfulness (dominant strategy incentive compatibility) and maximize social surplus (allocative efficiency).

  2. Second (in Proposition 3 and Proposition 4), given that Crémer-McLean Theorem (Theorem 1) holds on the demand side, we derive two ineqaulity cosntraints on the adjustment payment , which are sufficient and necessary conditions for ex-post individual rationality and ex-post weak budget balancedness. We show that these constraints can be transformed into Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule, which is equivalent to a minimization problem that can be solved by neural network methods.

  3. Lastly (in Theorem 2 and Corollary 4), we prove the existence of at least one solution to Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule. The PVCG payment corresponding to this solution attains truthfulness, ex-post allocative efficiency, ex-post individual rationality, and ex-post weak budget balancedness simultaneously on the supply side.

First, we prove that for arbitrary , the payment encourages all producers to report their capacity limits and cost types truthfully.

Proposition 1 (Dominant strategy incentive compatibility).

For every producer , truthfully reporting its capacity limit and cost type is its dominant strategy, i.e.,

(9)
Proof.

We differentiate the cases where with those where .

(1) When ,
it is impossible for producer to contribute the accepted amount of input resources because this exceeds its capacity limit. In this case, producer suffers a high punishment, i.e., is a large negative number. Hence, the right side of Eq. 1 becomes extremely negative and Eq. 1 holds. This shows a rational producer will not choose such that .

(2) When ,
we aim to prove that by truthfully reporting , , and , the utility of producer is at least the same as before.

By definition,
. We substitute with and get

(10)

In particular, for , Eq. Theoretical Analyses holds. Therefore,

(11)

Adding to both sides of Eq. Theoretical Analyses and substituting , we get Eq. 1. ∎

From Proposition 1, we know reported parameters equal true parameters, i.e., . Therefore, we can use these two sets of parameters interchangeably in the remaining parts of this paper.

Proposition 2 (Ex-post social surplus maximization / allocative efficiency).

PVCG maximizes social surplus ex post.

Proof.

Suppose and . We aim to prove that PVCG results in ex-post social surplus no less than .

By definition of ,

(12)

Incentive compatibility guarantees , , and . Therefore,

(13)

Particularly, Eq. 13 holds for , i.e.,

(14)

The left side of Eq. Theoretical Analyses is the ex-post social surplus achieved by PVCG, while the right side is the maximum social surplus across all possible , given . Hence, the right side is also no less than the left side. Therefore, the left side equals the right side. ∎

Furthermore, given that incentive compatibility has been proved in Proposition 1 and Theorem 1, the following two propositions provide sufficient and necessary conditions for ex-post individual rationality and ex-post weak budget balancedness.

Proposition 3 (Condition for ex-post individual rationality).

PVCG is ex-post individual rational (IR) for all producers i.f.f. the true capacity limits , the true type parameters , and the adjustment payments satisfy

(15)
Proof.

According to truthfulness proved in Proposition 1 and Theorem 1, we use to substitute in Eq. 6 and Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule. Then, the ex-post utility of producer becomes

(16)

Ex-post IR requires , which is equivalent to the inequality in Eq. 15. ∎

Proposition 4 (Condition for ex-post weak budget balancedness).

PVCG is ex-post weakly budget balanced (WBB) on the supply side i.f.f. the true capacity limits , the true type parameters , and the adjustment payments satisfy

(17)
Proof.

According to truthfulness proved in Proposition 1 and Theorem 1, we use to substitute in Eq. 6 and Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule. Then, the ex-post total payment to all producers is

(18)

As explained before, Theorem 1 transforms the WBB condition to . This inequality, together with the definition , transforms Eq. Theoretical Analyses to Eq. 4. ∎

Then, we prove Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule is a sufficient and necessary confition for Proposition 3 and Proposition 4.

Corollary 1.

Under PVCG, a sufficient and necessary condition for ex-post IR and ex-post WBB to coexist on the supply side is

(19)

where

(20)
(21)
Proof.

. i.f.f. Eq. 15 holds; i.f.f. Eq. 4 holds. ∎

From Corollary 1, we know that if the adjustment payments are such chosen that Eq. 19 holds for true capacity limits and true type parameters , then PVCG attains IR and WBB simultaneously. However, these true parameters are unknown to the coalition coordinator a priori, we need to find the optimal adjustment payment function such that for all possible and drawn from their respective prior distributions, Eq. 19 holds. This is equivalent to solve the functional equation in Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule. When such a functional solution exists, it minimizes the expected value of .

Corollary 2.

The solution to Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule, if existing, is also a solution to the following minimization problem:

(22)

where the expectation is over the prior distribution of .

Proof.

For the solution to Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule,
. Therefore, .

For all other function , since , we have . ∎

From Corollary 2, we can find the functional solution to Eq. Step 4. The coordinator makes transfer payments to participants according to the PVCG sharing rule by minimizing the expected . The remaining problem is to prove the existence of such a solution. The following theorem holds.

Theorem 2.

The following inequality is a sufficient and necessary condition for the existence of such that the PVCG payments satisfy WBB and IR for all .

(23)

where and are the extreme values of and on their support set .

In order to prove Theorem 2, we introduce the following lemma first.

Lemma 1.

The maximum social surplus monotonically increases with and monotonically decreases with for every producer .

Proof.

Suppose ; then, by definition,