1 Introduction
In a twosided market, there are several sellers who hold items for sale and several buyers who consider buying these items. Examples are stock exchanges, usedcar markets, emission trading markets (Godby, 1999; Sturm, 2008), Internet advertisements (Feldman and Gonen, 2016) and markets for spectrum reallocation (LeytonBrown et al., 2017). Each trader has a different valuation to each bundle of items. In contrast to a onesided market, here the valuations of both the buyers and the sellers are their private information, and both sides might act strategically. A double auction is a mechanism for organizing a twosided market — deciding who will buy, who will sell and at what prices.
An important requirement from a doubleauction is efficiency, which is measured by its gainfromtrade (GFT) — the total value gained by the buyers minus the total value contributed by the sellers. As an example, in a usedcar market with a single buyer and a single seller holding a single car, if the seller values the car as and the buyer as , then the potential GFT is .
The most commonly used doubleauction mechanism is the Walrasian mechanism (Rustichini et al., 1994; Babaioff et al., 2014). It computes an equilibrium price — a price at which the supply equals the demand: the total number of units that sellers are interested to sell at this price equals the total number of units that buyers are interested to buy at this price. In a singlegood market, an equilibrium price exists whenever the agents have decreasingmarginalreturns (DMR) — the marginal utility for an agent from having one more unit is weaklydecreasing in his current number of units (Gul and Stacchetti, 2000). Moreover, by the First Welfare Theorem, this mechanism attains the maximum GFT (Nisan et al., 2007, Theorem 11.13). Unfortunately, this mechanism is not incentivecompatible (IC) — agents have an incentive to misreport their valuations in order to manipulate the price.
The problem of designing an IC doubleauction has already been considered by Vickrey. In his seminal paper (Vickrey, 1961) he described a variant of his famous secondprice auction for a twosided market. Like its onesided variant, it is dominantstrategy incentivecompatible (DSIC, aka truthful) — it is a weaklydominant strategy for each agent to reveal its true valuation function. But unlike the onesided variant it is not budgetbalanced (BB) — it has a deficit — it requires the marketmaker to bring money from home. While it is possible to charge entrancefees to cover this cost, this makes the mechanism not individually rational (IR) — some traders might lose from participating.
Myerson and Satterthwaite (1983) proved that, in a twosided market, any mechanism that is IR, BB and IC cannot be efficient . Intuitively, the reason is that it is impossible to truthfully determine prices for trading. Consider any mechanism that charges the buyer and pays the seller . If , then the seller is incentivized to bid to force the price up (the mechanism has to do the deal since it is still efficient). Similarly, if , the buyer is incentivized to force the price down. Setting and leads to a deficit. One way out is to determine takeitorleaveit prices independently of the traders’ valuations, but this might result in a total loss of GFT.
This impossibility result initiated a search for doubleauction mechanisms that are IC, IR and BB, and attain an approximately maximal GFT. We define the competitive ratio of a mechanism as the minimum ratio (over all utility profiles) of its GFT divided by the optimal GFT. The first approximation mechanism was presented by McAfee McAfee (1992) for the case when each seller has a single unit and each buyer wants a single unit. McAfee’s mechanism is truthful — it is a weaklydominant strategy for each agent to report his true valuation for the item. Its competitiveratio is , where is the number of units traded in the optimal situation. (i.e, its GFT is always at least of the maximum GFT). Thus, McAfee’s mechanism is asymptotically efficient — when the marketsize grows to infinity, the GFT approaches the maximum. In addition to being truthful, IR, BB and asymptoticallyefficient, McAfee’s mechanism has a fifth nice property — it is priorfree (PF). This means that it does not require any statistical information about the traders’ valuations; it works well even for worstcase (adversarial) valuations.
The main drawback of McAfee’s mechanism is that it works only for singleunit traders. Another potential drawback is that it is only weaklybudgetbalanced (WBB) — the marketmaker need not bring money, but may the have to take money. Moreover, in some cases the marketmaker consumes almost all the GFT, leaving only little GFT to the traders (SegalHalevi et al., 2016b). This may be acceptable when the marketmaker is a monopolist (e.g. a government), but might be problematic when the marketmaker faces competition, e.g. in stock exchange platforms, since a low GFT for the traders might drive them away to the competitors. ^{1}^{1}1Technically it is possible to convert a WBB mechanism to a SBB one: randomly pick an agent before the trade, disallow him to participate in the trade, and give him all the surplus after the trade. However, this might induce a lot of people without any interest in the trade (e.g, “sellers” with no goods or “buyers” with no money) to participate in the auction, in the hope of winning the lottery. We believe this goes against the idea of truthfulness, and might have adverse effects on the market.
This paper presents MUDA — a doubleauction mechanism for traders that buy and sell multiple units. The only assumption required is that all traders have decreasingmarginalreturns; this is the same assumption that guarantees the existence of marketequilibrium.
The main idea of MUDA is to calculate equilibrium prices using random halving. The general scheme is presented below; it is explained in more detail in Section 3.
MUDA
(general scheme): Split the market to two submarkets, left and right, by sending each trader to each side with probability 1/2, independently of the others. Then, in each submarket:

Calculate a marketequilibrium price ( at the right, at the left).

Let the agents trade at the price from the other market ( at the right, at the left).
The randomsampling technique was found useful in onesided markets (Goldberg et al., 2001, 2006; Devanur et al., 2015; Balcan et al., 2008, 2007) and matching markets (Devanur and Hayes, 2009). However, applying it to a twosided market poses a new challenge: material balance — the number of units bought and sold must be exactly equal. This is in contrast to a onesided market, where the seller may leave some units unsold. Randomsampling in twosided markets was used only with oneparametric agents — agents whose valuations are characterized by a single signal Baliga and Vohra (2003); Kojima and Yamashita (2014).
With randomsampling, the price at each submarket is determined exogenously based on the other submarket, so the agents cannot manipulate it by reporting strategically. However, because of the materialbalance requirement, the traders have another reason to report strategically — they might try to manipulate the quantity of trade. Thus, the multiunit twosided setting is more difficult than both the singleunit twosided setting of McAfee and the multiunit onesided setting common in the randomsampling literature. To illustrate this added difficulty, we prove:
Theorem 1.
Suppose a single seller holds units of a good and a single buyer considers buying at most units of this good and the price per unit of the good is determined exogenously. Then, the expected competitive ratio of every DSIC and IR mechanism, whether deterministic or randomized, is:

at most for agents with general valuations, and —

at most for agents with DMR valuations, where is the th harmonic number ().
Both these upper bounds are tight.
The proof is given in SegalHalevi and Hassidim (2017). Theorem 1 can be seen as a dual to the Myerson–Satterthwaite impossibility theorem. In their setting, the quantity is determined exogenously and the traders might manipulate the price; in our setting, the price is determined exogenously and the traders might manipulate the quantity.
We present two variants of MUDA that overcome this impossibility when the market is sufficiently large. We call them LotteryMUDA and VickreyMUDA.
Step 1 is the same in both variants. Step 2 in both variants starts by calculating the aggregate demand and aggregate supply in each submarket, at the prices calculated in the other submarket. Usually, the aggregate demand will be larger or smaller than the aggregate supply, so the submarket will have a long side and a short side (e.g, if there is more demand than supply then the buyers are the long side and the sellers are the short side). The short side can always trade their optimum quantity; i.e, if the buyers are short, we can let each buyer buy the number of units that maximizes his utility given the price. The two variants of MUDA differ in the way they handle the long side: in LotteryMUDA the traders in the long side are selected using a random permutation that ignores their values, while in VickreyMUDA the highvalue traders in the long side are selected. In VickreyMUDA, each selected trader pays the marketmaker a positive tradingfee calculated as in a Vickrey auction; this fee is separate from the money transferred among the traders (the inspiration to this idea came from a recent working paper by Loertscher and Mezzetti (2016)).
Theorem 2.
Both variants of MUDA are priorfree, dominantstrategy incentivecompatible, individuallyrational and budgetbalanced (no deficit). In addition, LotteryMUDA is stronglybudgetbalanced (no surplus).
Proof.
Priorfreeness holds by design: MUDA does not use any statistical information on the valuations.
DSIC and IR will be proved in Section 4 after presenting the mechanism details.
LotteryMUDA is stronglybudgetbalanced (SBB) since all money goes from buyers to sellers; the marketmaker neither brings nor takes any money. VickreyMUDA is only weakly budgetbalanced (WBB) — the marketmaker does not lose money, but may make profit from tradingfees. ∎
We emphasize that the properties are true expost — for every outcome of the randomization in the mechanism.
We distinguish between the totalGFT, which includes the gain of both the agents and the marketmaker, and the agentsGFT, which includes only the gain of the agents. Since LotteryMUDA is SBB, the agentsGFT equals the totalGFT. In VickreyMUDA, the totalGFT is always higher since the most profitable deals are selected. However, the agentsGFT might be much lower. In a particular example shown in the arXiv version, the totalGFT is high but the agentsGFT is near 0. Thus, VickreyMUDA may be preferred when the marketmaker is a monopolist (e.g, a government), while LotteryMUDA may be preferred when the marketmaker faces competition (e.g, a stockexchange).
The competitiveratio of MUDA depends on — the number of units traded in the optimal situation (the used by McAfee 1992), and — the maximum number of units offered(demanded) by a single seller(buyer). represents the market concentration — how much market power is held by a single trader.
Theorem 3.
The expected totalGFT of both LotteryMUDA and VickreyMUDA is at least a fraction of the maximum totalGFT.
The proof is given in Section 5.
While VickreyMUDA attains more totalGFT than LotteryMUDA, their asymptotic behavior is similar — both of them approach the optimal totalGFT when the market size () is large, as long as the marketconcentration factor (represented by ) is kept constant. In contrast, if is very large () the market effectively has a single buyer and a single seller, and the impossibility result of (Myerson and Satterthwaite, 1983) implies that no positive approximation of the GFT is possible. The arXiv version shows an example in which the agentsGFT of VickreyMUDA is close to 0 (the agentsGFT of LotteryMUDA always equals its totalGFT).
While we focus on approximating the maximum GFT (buyers’ values minus trading sellers’ values), other mechanisms in the literature approximate the maximum social welfare (buyers’ values plus nontrading sellers’ values). But any mechanism that attains a fraction of the optimal GFT also attains a fraction of at least of the optimal social welfare (Brustle et al., 2017). Hence, MUDA is asymptoticallyoptimal with respect to the socialwelfare too.
The lower bound of Theorem 3 depends on — the optimal trade size. Ideally, we would like a bound that depends on the number of traders that come to the market (say, ). However, we cannot attain such bound theoretically in a worstcase analysis, even for the socialwelfare. As an example, consider a singlegood singleunit market having sellers with value , buyers with value and one buyer with value . Here as there is only one relevant trade. The competitive ratio of any mechanism depends only on the probability with which it performs this single trade; this probability does not change even when .
We complement our worstcase analysis that depends on with simulations of both variants of MUDA on agents drawn from both synthetic and realistic distributions. The simulations show that, when valuations are random (and not worstcase), the competitiveratio of MUDA increases with the number of traders. These are presented in Section 6.
1.1 Related Work
Most existing mechanisms for multiunit doubleauction are not truthful, e.g. Plott and Gray (1990).
The research on truthful double auction has made many advancements since McAfee (1992). There are variants of McAfee’s mechanism for maximizing the auctioneer’s surplus (Deshmukh et al., 2002), handling spatiallydistributed markets (Babaioff et al., 2004), transaction costs (Chu and Shen, 2006), supply chains (Babaioff and Walsh, 2006), constraints on the set of traders that can trade simultaneously (Yao et al., 2011; Dütting et al., 2017; Brustle et al., 2017), online arrival of buyers (Wang et al., 2010) and strong budgetbalance (ColiniBaldeschi et al., 2016). All these advancements are for singleunit agents.
Other doubleauction mechanisms either assume that the agents’ valuations are additive (Huang et al., 2002; Xu et al., 2010; Feldman and Gonen, 2016; Goel et al., 2016; Hirai and Sato, 2017) or assume that their valuations are represented by a single parameter Gonen et al. (2007); Kojima and Yamashita (2014). The DMR valuations handled by MUDA are multiparametric and include additive valuations as a special case.
Several recent doubleauction mechanisms work in a Bayesian setting — they assume that the traders’ valuations are drawn from some probability distribution that is public knowledge. Such knowledge allows the mechanism designer to attain approximate efficiency without relying on the agents’ reports. Examples are
Yoon (2008); ColiniBaldeschi et al. (2017). Some other mechanisms require partial prior knowledge on the valuations, such as their median (Blumrosen and Dobzinski, 2014) or their maximum and minimum value Gonen and Egri (2017). (Baliga and Vohra, 2003) assume that the agents’ valuations are drawn from some unknown distribution (they handle singleunit traders) . In contrast, MUDA needs no prior information on the agents’ valuations and does not even assume that they are drawn from some distribution.We are aware of two truthful mechanisms that handle multiparametric agents with DMR valuations in a priorfree way. The first is by Blumrosen and Dobzinski (2014): their competitive ratio is 1/48 — it is not asymptoticallyefficient. The second is by Loertscher and Mezzetti (2016), which is being developed simultaneously to our work. Their mechanism (that does not use markethalving) is asymptoticallyefficient when the valuations of the traders are drawn from probability distributions satisfying certain conditions. In contrast, our convergence theorem does not assume that agents’ valuations are drawn from a distribution at all.
2 Model
Agents and valuations
We consider a market for a single good. Some agents, the “sellers”, are endowed with at most units of that good, and other agents, the “buyers”, are endowed with an unlimited supply of money. Each agent has a valuationfunction that returns, for every integer between and , the agent’s value for owning units. The valuations are normalized such that .
All agents have decreasing marginal returns (DMR). Formally, , for every agent and .
We will often represent a multiunit agent as singleunit virtual agents; the value of virtualagent of agent is the agent’s marginal value for having the th unit: for (a similar idea was used by Chawla et al. (2010)).
We assume that the agents’ valuations are generic, i.e, all marginal values of different traders are different and linearlyindependent over the integers (no linear combination with integer coefficients equals zero). This assumption can be dropped if we use centralized tiebreaking; see Hsu et al. (2016) for other ways to handle ties in markets.
Given a price per unit of the good, the gain of a buyer from buying units is , and the gain of a seller from selling units is , where is the number of units initially held by seller , .
A mechanism is a (randomized) function that takes the agents’ valuations and returns (1) a tradingprice , (2) for each buyer(seller) , the amount of units he should buy(sell) at price , and possibly a tradingfee paid to the marketmaker. A mechanism is materiallybalanced if the number of units bought equals the number of units sold: . It is budgetbalanced (BB) if it is materiallybalanced and . It is stronglybudgetbalanced (SBB) if .
A mechanism is individuallyrational (IR) if every agent has a weaklypositive gain: with probability 1. It is DSIC (= truthful) if an agent can never increase his gain by pretending to have different valuations.
The demand of buyer is the number of units that maximizes the gain: . The genericity assumption implies that the maximum is unique. When has DMR, equals the number of virtualbuyers with . ^{2}^{2}2 This is true only for a DMR agent. For example, suppose buyer values one unit as 3 and two units as 4. Then and . If the price is 2, then there is one virtual buyer with value above the price, and indeed the agent’s demand is 1. However, if the buyer values one unit as 1 and two units as 4, then and . If the price is 2, then there is still one virtual buyer with value above the price, but the agent’s demand is 0.
The aggregatedemand at price is the sum of demands of all buyers. Equivalently, this is the number of virtualbuyers with . The supply and aggregatesupply are defined analogously.
The totalgainfromtrade is the sum of gains of all buyers and sellers: . Note that in a materiallybalanced mechanism, the totalGFT does not depend on the tradingprice. The agentsGFT is the total GFT minus the total fees, .
3 Mechanism Details
This section explains the details of the MUDA mechanism presented in the introduction. The steps are done in each submarket separately. For convenience we describe step 1 in the right market and step 2 in the left market; the execution in the opposite direction is entirely analogous.
Step 1: Price calculation.
We calculate a price that is an equilibrium price at the right market — a price for which . Such a price exists even in more general (multigood) settings. It can be found, for example, by simulating an English auction (Gul and Stacchetti, 2000), or by binary search.
Step 2: Postedprice trade.
For each buyer in the left market, calculate . For each seller in the left market, calculate . Let be the sum of demands and the sum of supplies. If , then we can let the traders trade freely at price and the market will clear. Usually, however, we will not be so lucky: there will be either excess demand () or excess supply (). These two cases are handled analogously; henceforth we describe how to handle excess supply. First, we ask each buyer to pay in advance for the optimal number of units he wants to buy at price , so we have money for units. Since , we will have more than enough units to give to all these buyers for their money. The problem is how to select the sellers that will supply these units. We present two solutions.
(a) Lottery: Order the sellers randomly; let each seller in turn sell at price as many units as he likes, while there is money (i.e, while at most units are sold).
(b) Vickreystyle auction: Order the virtual sellers in increasing order of their value. From the virtual sellers whose value is below , pick the virtualsellers with the lowest values, and have each of them sell an item at price .
The Vickreystyle auction is followed by a 4th step: tradingfees. For each seller , let be the number of virtualsellers of this seller that are picked. Note that . The fee paid by seller is determined by the potential gains of the virtualsellers that are “pushed” out of the market because of seller . Specifically, consider the set of virtualsellers who want to trade in the leftmarket, except the virtualsellers of . From this set, pick the (at most) lowvalue ones. In this set, there are (at most) virtualsellers that do not trade when seller is present. Seller pays to the marketmaker, the gain (price minus value) of these virtual sellers. Note that the tradingfee is positive if and zero if .
Example 1.
Suppose . The virtualbuyers in the left market have values , so the aggregate demand is 4. There are two sellers: Alice’s values are and Bob’s values are , so the aggregate supply is 7 and there is excess supply. The four highvalue virtualbuyers (with values 100,90,80,60) each pays 50 and is guaranteed a unit.
In LotteryMUDA, the two sellers are ordered randomly; if Alice is first then she sells 3 units and gains , and Bob sells 1 unit and gains , so the GFT is 115. If Bob is first then he sells 4 units and gains , and Alice sells nothing. The price is 50 per unit, all money collected from the buyers is given to the sellers, and no money is given to the marketmaker.
In VickreyMUDA, the four lowvalue virtualsellers are picked: they are the virtualsellers with values 10,15,20,25. So each each real seller sells two units for 50 per unit. At this point Alice’s gain is and Bob’s gain is , so the totalGFT is 130. Now, Alice pays a fee of 20, since her presence prevents Bob from selling two units worth for him 35 and 45, so her net gain is 50. Bob pays 10, since his presence prevents Alice from selling a unit worth for her 40 (the unit worth 60 would not have been sold anyway), so his net gain is 50 too, and the agentsGFT is 100. The marketmaker gains 30. ∎
4 Strategic Properties of MUDA
In both variants of MUDA, the traders cannot affect their tradingprice. Moreover, in both variants, the traders in the short side of each submarket trade the amount of units that maximizes their gain given the price, so for them the mechanism is obviously IR and DSIC. As for the longside traders:
(a) in LotteryMUDA they play random serial dictatorship: the first agents in the line trade their optimal quantity given the price and the last agents in the line cannot trade at all. There is at most a single agent, in the middle of the line, who trades less than his optimal quantity. Because of the DMR assumption, it is always optimal to trade as many units as possible up to the optimal quantity (since the highest gain comes from trading the first units). Therefore, the mechanism is IR and DSIC for all traders.
(b) In VickreyMUDA, longside virtualtraders trade only if they have positive gain. In this case, a trader with participating virtualtraders pays a fee that is equal to the gain of at most nonparticipating virtualtraders. Since the mechanism always selects the virtualtraders with the highest gain, the total fee paid by any trader is lower than his gain, so the net gain remains positive and the mechanism is IR. The mechanism is DSIC since it is effectively a multiunit Vickreyauction with a reserveprice. It is known that such a mechanism is DSIC; we omit the proof.
5 CompetitiveRatio Analysis
In this section we prove Theorem 3. In fact, we prove a more general claim that depends on a third parameter, — the minimum number of units offered(demanded) by a single seller(buyer) at same value. We assume that the virtualtraders of each trader are divided to groups of size at least , such that the values in each group are the same. So each agent values units as , units as , etc., with for all and . Theorem 3 follows from the expression at the end of this section by setting .
We first analyze the optimal trade, then the right submarket and finally the left submarket.
5.1 Optimal trade
In the optimal trade, there is a set of virtualbuyers who buy goods from a set of virtualsellers. By materialbalance the numbers of virtualagents in both groups are the same; this is the number we denoted by :
(1) 
We call these buyers and sellers the efficient traders. We make the pessimistic assumption that all GFT in the submarkets comes from these efficient traders. Therefore, the GFT of our mechanism depends on the numbers of efficient traders that trade in each submarket.
The reduction in GFT has two reasons: one is the sampling error — efficient buyers and sellers land in different submarkets, so they do not meet and cannot trade. This error is easy to bound using standard tail bounds. The second reason is the pricing error — the price at the submarket might be too high or too low, which might create imbalance in the demand and supply. Analyzing this error requires careful analysis of the equilibrium in the optimal situation vs. the equilibrium in each submarket.
5.2 Right submarket
In the right submarket, MUDA calculates an equilibrium price . We define four sets of virtualtraders:

is the set of efficient virtualbuyers (members of ) whose value is below (so they won’t buy at price ).

is the set of efficient virtualsellers (members of ) whose value is above (so they won’t sell at price ).

is the set of inefficient virtualbuyers whose value is above (so they want to buy at price ).

is the set of inefficient virtualsellers whose value is below (so they want to sell at price ).
These sets represent the pricing error, so we want to upperbound their sizes.
For any set of agents, denote by the subset of that is sampled to the right market and by the subset of sampled to the left market. By definition of the equilibrium price :
(2) 
In order to relate (1) and (2), we have to relate to . This is be done using the following lemma.
Lemma.
For every set of virtualtraders and for every integer :
(3) 
(“w.p. ” is a shorthand to “with probability at least ”).
The lemma is proved using Hoeffding’s inequality. The proof is standard and we omit it.
We apply this lemma twice, to and , and combine the outcomes using the union bound. This gives, :
(4) 
Combining equations 1,2,4 gives, :
(5) 
Of the two sets in the lefthand side, at least one must be empty: if is too high (relative to some optimal equilibrium price ) then efficient buyers quit and inefficient sellers join, but no inefficient buyers join and no efficient sellers quit, so . This situation is illustrated in Figure 1. Analogously, if is too low then .
From now on we assume that the situation is like in Figure 1 (the other situation is analogous). So (5) implies:
(6) 
Our goal is now to derive an upper bound on and . They are entirely analogous; we focus on . Note that we cannot apply (3) directly to , since is a randomset — it depends on the randomsampling through . (3) does not apply to sets that depend on the randomsampling; as an illustration, suppose the set is selected such that it contains only virtualagents from the right market. Then so (3) is obviously not satisfied. Fortunately, is a special randomset — it is onedimensional: for every integer , it has only a single possible value with cardinality , which is the set of virtualbuyers with the lowest value in (see Figure 1, where ). Denote these sets by . For every , is independent of the randomsampling, so it is eligible to (3). Substituting there gives:
(7)  
If and , then necessarily . So by combining (6) and (7) with the union bound we get, :
(8) 
To simplify the expression we choose ; assuming , this implies , so (8) simplifies to:
(9) 
5.3 Left submarket
Denote by () the set of efficient buyers(sellers) who want to buy(sell) in the left submarket at price :
(Recall that we assume the case in which and are empty; the other case is analogous).
Using (4) and (9) with the same gives, for every :
In VickreyMUDA, the most efficient traders in each side trade with each other. Therefore, the mechanism makes at least the most efficient deals in the left submarket. Similar considerations are true in the right submarket. All in all, VickreyMUDA does at least the most efficient out of the efficient deals. Therefore, w.p. , the competitive ratio is at least .
In LotteryMUDA, the efficient sellers in have to compete with the inefficient sellers in in the random lottery. The expected number of efficient deals carried out is thus:
Therefore, w.p. , the expected competitive ratio is at least . So w.p. 1, the expected competitive ratio is at least .
All our analysis so far holds simultaneously for every . Now, we select to maximize the expected competitive ratio. With some tedious calculations we find that, when is sufficiently large, we can select such that the competitive ratio is at least: ^{3}^{3}3 Define . We look for an that minimizes . The firstorder condition is:
The analysis for VickreyMUDA is obtained by replacing 36 with 18; the asymptotic behavior is the same.
6 Simulations
To complement our theoretic analysis, we simulated MUDA on traders with valuations sampled both from a synthetic distribution and an empirical distribution based on real stockexchange data.
6.1 Uniform distribution
In the first experiment, for each agent, we sampled
values from a uniform distribution with support
. We considered each of these values as the marginalvalue of virtualtraders, so that each agent has virtualtraders. We ordered the values in decreasing order to get DMR valuations. In the experiments, we took and and varied the noiseamplitude between and . Here we show the results for ; varying did not have much effect on the results. We repeated each experiment 100 times and averaged the results.In the first subexperiment, we kept constant (at ) and varied the number of real traders between 0 and 1000. The results are shown in Figure 2/Left. The competitive ratios of all variants of MUDA increase towards when the number of traders grows.
In the second subexperiment, we kept constant the total number of units held by all sellers. We increased from 100 to , and decreased the number of traders accordingly so that the total number of units remains constant. The results are shown in Figure 2/Right. The competitive ratios decrease when increases and the same number of units are concentrated in the hand of fewer traders. When is very large we effectively have one buyer and one seller, and then the impossibility result of (Myerson and Satterthwaite, 1983) implies that no positive competitive ratio is possible.
Both plots show that the GFT of LotteryMUDA is approximately middleway between the totalGFT and the agentsGFT of VickreyMUDA. This fact has a practical implication regarding the choice of mechanism: if the market is managed by a monopolist (e.g. the government), then VickreyMUDA is better since its totalGFT is higher; but if the market competes with other markets (e.g. stock trading platforms), then LotteryMUDA is better since its agentsGFT is higher.
We found that, when the number of units per trader is fixed, the performance is determined by the number of traders in the market. Therefore, the plot at the left of Figure 2 remains the same even when we set e.g. and . As for the plot at the right of Figure 2, if we set the total number of units to e.g. , then the plot approaches zero already when , since in this case there is a single buyer and a single seller.
6.2 Empirical stockexchange distribution
In the second experiment, we used the TORQ database (Hasbrouck, 1992; Lee and Radhakrishna, 2000)
. It contains buy and sell orders given for a sample of 144 NYSE stocks for the three months 199011 through 199101. In the NYSE, each day before the continuous trade begins, there is a phase of “startofday auctions”. For each stock, a separate multiunit doubleauction is conducted in the following way. All buy and sell orders given for that stock before the start of day are collected. An equilibrium price is calculated. All buyorders above the price and all sellorders below the price are executed. As explained in the introduction, this mechanism is efficient but not truthful. Therefore, the orders may not represent the true values of the traders. While there are econometric methods for estimating the true values from the reported values, these are beyond the scope of the present paper, since we do not need to know the true value of every trader — all we need is a distribution of values. Our results are valid as long as the empirical distribution of reported values resembles the distribution of values in the real world.
In the present paper we assume that the empirical distribution of the orders is similar to the empirical distribution of the true values.
For the experiment, we considered only the startofday orders. These orders are given in the format (Symbol, Date, Order date, Side, Price, Quantity) where

Symbol is the threeletter acronym of the stock;

Date is the day of the auction.

Order date is the day the order was made, which may be earlier than Date;

Side is BUY or SELL; and

Price and Quantity are the actual bid.
The dataset contains 144 symbols, 7914 different symboldate combinations and orders.
The dataset does not contain the identities of the traders. We made two experiments: in the first we treated each order as a separate trader, so that all traders are additive (each trader values a certain quantity of units for the same priceperunit). In the second experiment, we simulated traders with nonadditive valuations by merging bids based on the orderdate. I.e, we heuristically assumed that two bids with the same symbol, date, orderdate and side belong to the same trader. We ordered the bids of the same trader in descending order to create DMR valuations, similarly to the uniform experiment. In average, each merged bidder contained 10 different orders. The number of units per order ranges between 100 and 99000, so
. The number of units per trader, after combining orders from the same orderdate, ranges between 5000 and 12000000 with median 148000, so .For each symbol, we created a collection of all traders in all days to represent the empirical distribution of this symbol. Then, in each experiment we sampled traders, where . All in all we simulated auctions and averaged the auctions for each . The results are shown in Figure 3.
6.3 Discussion
It is interesting that the plots in Figure 3 look almost the same, i.e, the nonadditivity had little effect on the results. The reason might be that stocktraders have many different bids with very similar prices, so that they are almost additive.
The competitive ratio in the TORQ experiment is somewhat lower than in the uniform case. We attribute this to the large variation in the number of units: while some traders hold only 5000 units, others hold as many as 12 million.
The source code used for the experiments is available at the following URL:
7 Future Work
It is interesting whether MUDA can be extended to agents with general valuations (not DMR). This poses two challenges. First, in step 1, a priceequilibrium might not exist (the existence of a priceequilibrium is guaranteed only when the agents have DMR). Second, in step 2, material balance might fail. For example, if there is one buyer who only wants to buy 2 units and one seller who only wants to sell 3 units, then with DMR we can assume that both agree to trade 2 units, but without DMR this assumption might be false — the seller might have negative utility from selling 3 units.
Another challenge is handling double auctions with multiple types of items. Preliminary results in this direction were presented by SegalHalevi et al. (2016a).
We will also be happy to compare MUDA to other doubleauction mechanisms. In particular, an interesting practical question is when would MUDA’s truthfulness actually matter — how does its GFT compare to the GFT in Nash equilibrium of the standard (nontruthful) market mechanism? As far as we know, the Nashequilibrium GFT of the nontruthful market mechanism (i.e, its “priceofanarchy”) is still an open question. The closest reference we found is by Babaioff et al. (2014), who consider the priceofanarchy of the market mechanism in a singlesided market. We are not aware of analogous results for doublesided markets.
But nontruthful market mechanisms have disadvantages beyond the priceofanarchy. First, the players need to spend resources to obtain information about other players and calculate their best response. Second, the players may not even play an equilibrium. It is possible that players do not know the correct state of the market and hence play bestresponse to a fictitious state. This can bring social welfare to zero. Consider any mechanism that sets a single price clearing the market. For every active buyer(seller), it is a best response to bid below(above) the market price in order to push the price down(up). If all agents play these best responses, the outcome is that nothing gets sold. Such market failures are prevented by the truthfulness of MUDA.
8 Acknowledgments
Assaf Romm helped us access and analyze the TORQ dataset and provided many helpful comments. Ron Adin, Simcha Haber, Ron Peretz, Tom van der Zanden, Jack D’Aurizio, Andre Nicolas, Brian Tung, Robert Israel, Clement C. and C. Rose helped us with the probability calculations. The participants of the gametheory seminar in BarIlan University, computational economics and economic theory seminars in the Hebrew university of Jerusalem, algorithms seminar in TelAviv university and computer science seminar in Ariel university provided helpful comments on various aspects of the paper. The anonymous reviewers in Theoretical Economics, SODA 2016, EC 2016, SODA 2017, EC 2017, and, of course, AAAI 2018, gave us detailed and helpful feedback.
Erel was supported by ISF grant 1083/13, Doctoral Fellowships of Excellence Program and Mordecai and Monique Katz Graduate Fellowship Program at BarIlan University. Avinatan Hassidim is supported by ISF grant 1394/16.
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