Approximately Maximizing the Broker's Profit in a Two-sided Market

We study how to maximize the broker's (expected) profit in a two-sided market, where she buys items from a set of sellers and resells them to a set of buyers. Each seller has a single item to sell and holds a private value on her item, and each buyer has a valuation function over the bundles of the sellers' items. We consider the Bayesian setting where the agents' values are independently drawn from prior distributions, and aim at designing dominant-strategy incentive-compatible (DSIC) mechanisms that are approximately optimal. Production-cost markets, where each item has a publicly-known cost to be produced, provide a platform for us to study two-sided markets. Briefly, we show how to covert a mechanism for production-cost markets into a mechanism for the broker, whenever the former satisfies cost-monotonicity. This reduction holds even when buyers have general combinatorial valuation functions. When the buyers' valuations are additive, we generalize an existing mechanism to production-cost markets in an approximation-preserving way. We then show that the resulting mechanism is cost-monotone and thus can be converted into an 8-approximation mechanism for two-sided markets.

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1 Introduction

Two-sided markets are widely studied markets in economics [Myerson and Satterthwaite, 1983, McAfee, 1992, McAfee, 2008], where a number of buyers and a number of sellers are connected by an intermediary, such as antique markets, used-car markets, and pre-owned house markets. Here each seller has a single item to trade for money and holds a private value for her owned item, while each buyer’s private information is a general combinatorial valuation function over the bundles of the sellers’ items. A common feature in these situations is that the intermediary keeps the difference between the payments made by the buyers and the payments made to the sellers —that is, the intermediary’s profit. We call such an intermediary a broker. The objective of the broker is to acquire the items from the sellers and resell them to the buyers to maximize her profit. The problem studied in our paper is to design the mechanism in the two-sided market that maximize the broker’s profit. For convenience, we refer to the sub-market between the sellers and the broker the seller-side market and to the sub-market between the broker and the buyers the buyer-side market.

If the broker had all the items, then we would only have the buyer-side market, which is an auction where the broker tries to maximize her revenue. Auctions have been well studied in the literature following the seminal work of Myerson [Myerson, 1981]. In Section 1.2

, we will briefly recall the most relevant literature on auctions. If the broker would keep the items, then we only have the seller-side market, which is a procurement game. Budget feasible procurement has been studied by many in the Algorithmic Game Theory literature

[Singer, 2010, Dobzinski et al., 2011, Chen et al., 2011, Chan and Chen, 2014]. The broker wants to maximize her value for the items she buys, subject to a budget constraint.

Although auctions and procurements are closely related to the broker’s problem, they cannot be dealt with separately in two-sided markets. Indeed, the difficulty of the broker’s problem is to simultaneously and truthfully elicit both the sellers’ and the buyers’ valuations, so as to generate a good profit.

1.1 Main Results and Techniques

In this paper we assume the values of the sellers and buyers are independently distributed, and we study simple dominant-strategy incentive compatible (DSIC) mechanisms. To approximately maximize the (expected) profit of the broker, we first develop a reduction, through which we can directly convert mechanisms for production-cost markets into mechanisms for two-sided markets. In a production-cost market, the broker is able to produce all the items, each item has a cost to be produced and the costs are publicly known. Roughly speaking, we say a mechanism for production-cost markets is cost-monotone if, when the cost of an item increases, the likelihood that it is sold does not increase. We show that any cost-monotone mechanism for production-cost markets can be converted into a mechanism for two-sided markets via a black-box approach. This reduction holds for general combinatorial valuation functions of buyers.

Theorem 1 (Informal). Any cost-monotone DSIC mechanism that is an -approximation for production-cost markets, can be converted into a DSIC mechanism that is an -approximation for two-sided markets.

Next, we use cost-monotonicity as a guideline in constructing concrete mechanisms for two-sided markets. When the buyers have additive valuations, we generalize the duality framework of [Cai et al., 2016] and the mechanism there to design a cost-monotone mechanism for production-cost markets. Following our reduction, we immediately obtain a mechanism for two-sided markets.

Theorem 3 (Informal). When the buyers have additive valuations, there exists a DSIC mechanism for two-sided markets which is an 8-approximation to the optimal profit.

1.2 Related Work

Bayesian auctions have been extensively studied since the seminal work of [Myerson, 1981]. For single-parameter settings, Myerson’s mechanism is optimal. The problem becomes more complicated in multi-parameter settings [Hart and Nisan, 2017]. Although optimal Bayesian incentive-compatible (BIC) mechanisms have been characterized [Cai et al., 2012b, Cai et al., 2012a], they are too complex to be practical. Also, optimal DSIC mechanisms remain unknown. Thus, simple DSIC mechanisms that are approximately optimal have been studied in the literature, such as [Kleinberg and Weinberg, 2012, Yao, 2015, Cai et al., 2016]

Two-sided markets are also called double auctions [McAfee, 1992], bilateral trading [Myerson and Satterthwaite, 1983] or market intermediation [Jain and Wilkens, 2012] in the literature. Maximizing the broker’s profit is an important objective for two-sided market. The seminal paper [Myerson and Satterthwaite, 1983] characterized the optimal mechanism for one seller and one buyer, which is further generalized by [Deng et al., 2014] to multiple single-parameter sellers and buyers. Unlike our work, [Deng et al., 2014] studies the Bayesian Incentive Compatible (BIC) mechanisms. DSIC mechanisms are also studied in the literature, but only for some special cases: [Jain and Wilkens, 2012] studies the case of a single buyer and multiple sellers, [Balseiro et al., 2019] studies the case of a single seller and multiple buyers, and [Gerstgrasser et al., 2016] studies the optimal mechanism when the numbers of sellers and buyers are both constants. Although [Chan and Chen, 2016] studies two-sided markets with multiple buyers and multiple sellers, the dealer there has a fixed budget and their mechanism guarantees that the payment to sellers is within the budget. Before our work, it remained unknown how to design a (simple) DSIC mechanism that approximates the optimal profit in multi-parameter settings with a general number of sellers and buyers.

Finally, we briefly discuss the efficiency of two-sided markets, which is measured by gain-from-trade (GFT), i.e., the total value gained by the buyers minus the value contributed by the sellers. [McAfee, 1992] gave the first approximation mechanism for the one seller and one buyer case, and [Brustle et al., 2017] gives approximation mechanisms for multiple buyers with unit demand valuations. Recently, [Segal-Halevi et al., 2018a] and [Segal-Halevi et al., 2018b] study the asymptotically efficient mechanisms instead of constant approximations. For maximizing social welfare, [Colini-Baldeschi et al., 2016, Colini-Baldeschi et al., 2017] provide constant-approximation mechanisms.

2 Preliminaries

A two-sided market includes a set of sellers, and a set of buyers. We consider the setting where each seller has one item to sell, so we may refer to items and sellers interchangeably. The total payment made by the buyers is the broker’s revenue, and her profit is the revenue minus the total payment to the sellers.

Each buyer has valuation with . The function is monotone: for any , . In our reduction between production-cost and two-sided markets, we consider combinatorial valuations and do not impose any restriction on .

Each function is independently drawn from a distribution over the set of all possible valuation functions, with density function

and cumulative probability

. Let , and . Each seller ’s value on her item, , is independently drawn from a distribution , with density function and cumulative probability . Let , and . Let the supports of distributions and be and , respectively. and are called the valuation spaces of buyer and seller . Let and . Finally, denote by a two-sided market instance.

A mechanism for two-sided markets is represented by . Given a valuation profile ,

  • is the allocation of the buyers, where with , representing the probability that buyer  gets the item set , under valuation profile and . Moreover, .

  • is the allocation of the sellers with , representing the probability that seller ’s item is sold under .

  • is the payment of the buyers, where .

  • is the payment to the sellers, where .

A feasible mechanism is such that

for any item and any valuation profile . In principle, the above condition may allow a mechanism to sell an item that it didn’t buy or to buy an item without selling it. However, these cases never happen in the mechanisms in this paper.111 Note that our feasibility constraint only requires “feasible in expectation” which is weaker than ex post feasibility. All of our results still hold if we change the requirement to be ex post feasible. The expected profit of mechanism for instance is

The utilities of the agents are quasi-linear. That is, for each buyer , for any valuation subprofile of the buyers and any valuation profile of the sellers, when reports her true valuation function , her utility under mechanism is

For each seller , for any valuation subprofile and , when reports her true value , her utility is

Mechanism is dominant-strategy incentive-compatible (DSIC) if: (1) for any buyer , , , and , ,

and (2) for any seller , , and ,

Mechanism is individually rational (IR) if: (1) for any buyer , , and , and (2) for any seller , , and ,

Mechanism is Bayesian incentive-compatible (BIC) if (1) for any buyer and valuation functions , ,

and (2) for any seller and values , ,

Mechanism is Bayesian individually rational (BIR) if (1) for any buyer and valuation function , ; and (2) for any seller and value , .

Finally, we denote by the (expected) profit generated by the optimal DSIC mechanism for instance .

A special case of two-sided markets is production-cost markets, where the broker can produce the items by himself and each item has a publicly known production cost . Therefore we do not need to consider the sellers’ incentives. Letting , we use to denote a production-cost market instance and a production-cost market mechanism, where the input of and is the buyers’ valuation profile. Then the broker’s profit is the revenue minus the total production cost , which is

Auctions are production-cost markets with cost 0. We use to denote an auction instance and an mechanism. The expected revenue is . When there is no ambiguity, the superscript is omitted in auctions and production-cost markets.

In Section 4, we will consider additive valuations for the buyers. In this case, for any buyer

, there exists a valuation vector

such that is ’s value on each item . Then, is additive if for any . To simplify the notation, in this case we use to denote the vector instead of the corresponding function. Each is independently drawn from a distribution , and . Finally, when buyers have additive valuations, their allocation is simplified as , where with , representing the probability that buyer  gets the item , when the valuations are and .

3 A Reduction from Two-sided markets to Production-Cost Markets

Note that the sellers are single-parameter in the two-sided markets under consideration. Thus, each seller is truthful in a mechanism if and only if the selling probability of her item is non-increasing with respect to her value and the payment to her is the threshold payment, i.e., the highest value such that her item can still be sold. More precisely, for any single-value distribution with density function and cumulative probability , if is a seller’s value distribution, then the virtual value function is . In addition, if is not regular then is the ironed virtual value. Following [Myerson and Satterthwaite, 1983], for single-parameter sellers and any DSIC mechanism , the total payment to the sellers is the virtual social welfare of them, i.e.,

(1)

for any valuation profile of the buyers.

We now show how to convert a mechanism for production-cost markets into a two-sided market’s mechanism. The main idea is to use the sellers’ virtual values in two-sided markets as costs, and run the mechanism for production-cost markets.

Definition 1.

A mechanism for production-cost markets is cost-monotone if for any two instances and , where and differ only at an item  and , for any buyers’ valuation profile , the probabilities of item being sold under the two instances, and , satisfy .

Reduction. Let be a two-sided market instance. For any valuation profile of the sellers, denote by the sellers’ virtual-value vector, and let be a production-cost market instance.

We first show that the optimal profit of the two-sided market is no more than the optimal profit generated by the corresponding production-cost markets in expectation.

Lemma 1.

For any two-sided market instance , .

Proof.

It suffices to show that for any DSIC mechanism for two-sided markets, there exists a DSIC mechanism for production-cost markets such that . Indeed, this would imply for any , and thus .

Given and , we define mechanism as follows. For any instance , first computes , the (randomized) pre-image of with respect to . In particular, if for some seller , the (ironed) virtual value corresponds to a value interval in the support of , then is randomly sampled from conditional on it belongs to this interval.

For any reported valuation profile and buyer ,

for any , and

It is easy to see that, given any and , for any true valuation , buyer has the same utility in and by reporting the same . Thus is DSIC whenever is DSIC. Next, we lower-bound the profit of for each instance .

The inequality above is because holds for any feasible mechanism, any and any valuation profiles . Thus,

as desired. Here is the distribution of virtual values induced by , and the second equality is by Equation 1. ∎

In the following, we show that if a mechanism for production-cost markets is cost-monotone, then it can be converted into a mechanism for two-sided markets.

Lemma 2.

Given any DSIC cost-monotone mechanism for production-cost markets, there exists a DSIC mechanism for two-sided markets such that

Proof.

Given mechanism , the mechanism is defined as follows: first collects and reported by the buyers and the sellers, and then run on the production-cost instance to obtain and . Then for each buyer , let

for any and

For each seller , let

and let be the threshold payment for : namely, the highest reported value of seller such that the probability that item  is bought by the broker is .

We claim that is DSIC. First, the buyers will truthfully report their valuations because is DSIC and each buyer has the same allocation and payment in and . For the sellers, since is cost-monotone and each (ironed) virtual value function is non-decreasing in , the allocation is non-increasing in . As the payments to the sellers are the threshold payments, the sellers are truthful as well.

Next we show that

Thus Lemma 2 holds. ∎

Combining Lemmas 1 and 2, we get our first main result.

Theorem 1.

Given any DSIC mechanism for production-cost markets, if is cost-monotone and is an -approximation to the optimal profit, then there exists a DSIC mechanism for two-sided markets that is an -approximation to the optimal profit.

Proof.

Mechanism is defined as in Lemma 2. For any two-sided market instance ,

where the equality is by Lemma 2 and the last inequality is by Lemma 1. ∎

4 A Mechanism for Two-Sided Markets with Additive Valuations

4.1 Broker’s Profit in Production-Cost Markets

We first design a mechanism for production-cost markets which is an 8-approximation of the optimal profit. Our mechanism is inspired by the mechanism in [Yao, 2015] and the duality framework in [Cai et al., 2016] for auctions. In particular, with probability , runs the mechanism of [Myerson and Satterthwaite, 1983] for two-sided markets for each item separately, denoted by . The mechanism of [Myerson and Satterthwaite, 1983] is for a single buyer and a single seller, but can be generalized to multiple buyers and a single seller as shown in [Deng et al., 2014]. Furthermore, generalizes the bundling VCG mechanism of [Yao, 2015] to production-cost markets (denoted by ) and runs it with probability .

Essentially, Mechanism runs a second-price auction on the buyers’ virtual values, with a reserve price which is the production cost of the item. As shown in [Myerson and Satterthwaite, 1983, Deng et al., 2014], this mechanism is optimal for the broker’s profit when the buyers have single-parameter valuations. Mechanism is well studied in auctions [Yao, 2015, Cai et al., 2016], and we describe it in Mechanism 1 for production-cost markets . Essentially, it is a VCG mechanism with per-item reserve prices and per-agent entry fees.

1:  Collect the valuation profile from the buyers.
2:  For any buyer and item , let and .
3:  For any buyer , set the reserve price for item to be . Set the entry fee

to be the median of the random variable

, where and for any .
4:  Each buyer is considered to accept her entry fee if and only if .
5:  If a buyer accepts her entry fee, then she gets the set of items with , and her price is . If does not accept her entry fee, then she gets no item and pays 0.
Mechanism 1 for Production-Cost Markets

It is not hard to see that both and are DSIC and IR. Indeed, the mechanism of [Myerson and Satterthwaite, 1983] is DSIC and IR, directly applies it to each item, and the buyers have additive valuations across the items. Moreover, is DSIC and IR with respect to any reserve prices that do not depend on , and Mechanism 1 simply incorporates the production costs into reserve prices.

In Theorem 2 we use to upper-bound the optimal profit for any production-cost instance , with proof provided in the full version. In fact, this proof is similar to the proof in [Cai et al., 2016] with modifications to incorporate the production costs into consideration. Note that [Brustle et al., 2017] also adapts the framework of [Cai et al., 2016] to the 2-sided market. But their goal is to maximize the gain from trade and the buyers have unit-demand valuations.

Theorem 2 ([Brustle et al., 2017]).

When the buyers have additive valuations, Mechanism is DSIC and is an 8-approximation to the optimal profit for production-cost markets.

4.2 Converting to Two-sided Markets

Next we prove the cost-monotonicity for Mechanism . First, we start with Mechanism .

Lemma 3.

is cost-monotone.

Proof.

For any two production-cost instances and , where there exists an item such that and for any , we show that in Mechanism , when buyers’ valuation profile is , if item is not sold in , then item is not sold in . Since all buyers’ valuation functions are additive and sells each item individually, the result of selling one item does not effect any other item. In the mechanism of [Myerson and Satterthwaite, 1983], given the reported valuation profile , the potential winner of item is the buyer who has highest virtual value on it, denoted by . If her virtual value is at least the cost of item , buyer takes item . Otherwise, item is kept unsold. Therefore, if item is not sold in , then which implies and item cannot be sold in . Thus satisfies cost-monotonicity. ∎

Next we show is cost-monotone. Since we need to apply to different instances with different cost vectors and , we explicitly write and in Steps 2 and 3 of Mechanism 1.

Lemma 4.

is cost-monotone.

Proof.

Similarly, for any two production-cost instances and , where there exists an item such that and for any , we show that in Mechanism , when buyers’ valuation profile is , if item is sold in , then item is also sold in .

In Mechanism , given the valuation profile , the potential winner of item is the buyer who has highest value on item , denoted by . When the cost vector is , item  is sold to if and only if accepts the entry fee and . Otherwise item is unsold. Note that given different production cost , the potential winner of item remains unchanged.

Note that the entry fee is selected such that the probability that buyer accepts it is exactly . Let be the increase of the item reserve in for buyer . Then

(2)

Indeed the equality holds in Inequality 2 if .

Let . When the cost vector is , buyer ’s utility is

When the cost is , if the entry fee is , buyer ’s utility is