1 Introduction
Twosided 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, usedcar markets, and preowned 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 twosided market that maximize the broker’s profit. For convenience, we refer to the submarket between the sellers and the broker the sellerside market and to the submarket between the broker and the buyers the buyerside market.
If the broker had all the items, then we would only have the buyerside 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 sellerside 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 twosided 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 dominantstrategy 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 productioncost markets into mechanisms for twosided markets. In a productioncost 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 productioncost markets is costmonotone if, when the cost of an item increases, the likelihood that it is sold does not increase. We show that any costmonotone mechanism for productioncost markets can be converted into a mechanism for twosided markets via a blackbox approach. This reduction holds for general combinatorial valuation functions of buyers.
Theorem 1 (Informal). Any costmonotone DSIC mechanism that is an approximation for productioncost markets, can be converted into a DSIC mechanism that is an approximation for twosided markets.
Next, we use costmonotonicity as a guideline in constructing concrete mechanisms for twosided markets. When the buyers have additive valuations, we generalize the duality framework of [Cai et al., 2016] and the mechanism there to design a costmonotone mechanism for productioncost markets. Following our reduction, we immediately obtain a mechanism for twosided markets.
Theorem 3 (Informal). When the buyers have additive valuations, there exists a DSIC mechanism for twosided markets which is an 8approximation to the optimal profit.
1.2 Related Work
Bayesian auctions have been extensively studied since the seminal work of [Myerson, 1981]. For singleparameter settings, Myerson’s mechanism is optimal. The problem becomes more complicated in multiparameter settings [Hart and Nisan, 2017]. Although optimal Bayesian incentivecompatible (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]
Twosided 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 twosided 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 singleparameter 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 twosided 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 multiparameter settings with a general number of sellers and buyers.
Finally, we briefly discuss the efficiency of twosided markets, which is measured by gainfromtrade (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, [SegalHalevi et al., 2018a] and [SegalHalevi et al., 2018b] study the asymptotically efficient mechanisms instead of constant approximations. For maximizing social welfare, [ColiniBaldeschi et al., 2016, ColiniBaldeschi et al., 2017] provide constantapproximation mechanisms.
2 Preliminaries
A twosided 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 productioncost and twosided 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 twosided market instance.A mechanism for twosided 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.^{1}^{1}1 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 quasilinear. 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 dominantstrategy incentivecompatible (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 incentivecompatible (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 twosided markets is productioncost 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 productioncost market instance and a productioncost 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 productioncost 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 productioncost 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 Twosided markets to ProductionCost Markets
Note that the sellers are singleparameter in the twosided markets under consideration. Thus, each seller is truthful in a mechanism if and only if the selling probability of her item is nonincreasing 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 singlevalue 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 singleparameter 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 productioncost markets into a twosided market’s mechanism. The main idea is to use the sellers’ virtual values in twosided markets as costs, and run the mechanism for productioncost markets.
Definition 1.
A mechanism for productioncost markets is costmonotone 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 twosided market instance. For any valuation profile of the sellers, denote by the sellers’ virtualvalue vector, and let be a productioncost market instance.
We first show that the optimal profit of the twosided market is no more than the optimal profit generated by the corresponding productioncost markets in expectation.
Lemma 1.
For any twosided market instance , .
Proof.
It suffices to show that for any DSIC mechanism for twosided markets, there exists a DSIC mechanism for productioncost 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) preimage 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 lowerbound 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 productioncost markets is costmonotone, then it can be converted into a mechanism for twosided markets.
Lemma 2.
Given any DSIC costmonotone mechanism for productioncost markets, there exists a DSIC mechanism for twosided 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 productioncost 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 costmonotone and each (ironed) virtual value function is nondecreasing in , the allocation is nonincreasing in . As the payments to the sellers are the threshold payments, the sellers are truthful as well.
Theorem 1.
Given any DSIC mechanism for productioncost markets, if is costmonotone and is an approximation to the optimal profit, then there exists a DSIC mechanism for twosided markets that is an approximation to the optimal profit.
4 A Mechanism for TwoSided Markets with Additive Valuations
4.1 Broker’s Profit in ProductionCost Markets
We first design a mechanism for productioncost markets which is an 8approximation 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 twosided 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 productioncost markets (denoted by ) and runs it with probability .
Essentially, Mechanism runs a secondprice 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 singleparameter valuations. Mechanism is well studied in auctions [Yao, 2015, Cai et al., 2016], and we describe it in Mechanism 1 for productioncost markets . Essentially, it is a VCG mechanism with peritem reserve prices and peragent entry fees.
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 upperbound the optimal profit for any productioncost 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 2sided market. But their goal is to maximize the gain from trade and the buyers have unitdemand valuations.
Theorem 2 ([Brustle et al., 2017]).
When the buyers have additive valuations, Mechanism is DSIC and is an 8approximation to the optimal profit for productioncost markets.
4.2 Converting to Twosided Markets
Next we prove the costmonotonicity for Mechanism . First, we start with Mechanism .
Lemma 3.
is costmonotone.
Proof.
For any two productioncost 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 costmonotonicity. ∎
Next we show is costmonotone. 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 costmonotone.
Proof.
Similarly, for any two productioncost 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
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