1 Introduction

Contrasting the conventional requirement of infinitely many traders for perfect competition, laboratory experiments using the double auction mechanism observe the convergence to competitive equilibrium and attainment of full efficiency with as few as eight traders (Smith Reference Smith1962, Reference Smith1982). Theories on strategic market games, which organize the market using a static double auction mechanism, have also verified the competitiveness of Nash equilibria outcomes with few traders in certain scenarios (Dubey Reference Dubey1982; Simon Reference Simon1984; Benassy Reference Benassy1986, DSB hereafter). This paper reconciles the laboratory results with theories on strategic market games. We propose a strategic market game applicable to markets with indivisible commodities as in most double auction experiments, we derive conditions for the equivalence between Nash and competitive equilibrium that require as few as four traders, and we test the equivalence in the laboratory.

In a strategic market game, buyers and sellers submit price-quantity pairs to a clearing house, which acts as a profit-maximizing middleman, and allocates trades accordingly. Like in a double auction, priorities are given to higher bids and lower asks in a strategic market game, inducing a price competition between traders. In line with Bertrand’s argument, DSB prove that having two active sellers and two active buyers in a Nash equilibrium is sufficient to make the outcome competitive in certain strategic market games.

This paper addresses a few differences between the settings of the aforementioned strategic market games and the double auction mechanism. Firstly, the divisibility of the commodities. Proofs in DSB rely on the divisibility of commodities; in a conventional double auction market setting, there is an indivisible commodity for trade with divisible money. Analogous to what Mas-Colell (Reference Mas-Colell1977) finds when all but one commodities in the general equilibrium model become indivisible, we show that the “main results ...with divisible commodities and convex preferences remain valid” in the double auction setting. Unlike Mas-Colell (Reference Mas-Colell1977), we do not assume a continuum of traders.

The condition DSB provide is a sufficient one for the competitiveness of outcomes from a Nash equilibrium, thus is not informative on the Nash equilibria where it is not met. We provide a necessary and sufficient condition for equivalence between the outcomes of the Nash equilibrium and the competitive equilibrium with indivisible commodities. Essentially, our condition requires that on each side of the market there are two inframarginal traders, in the sense that they are willing to trade at every competitive price.Footnote ¹ Unlike previous work, our condition relies on the characteristics of the set of competitive equilibria, and place no requirement on Nash equilibria other than the occurrence of trade. Notably, our equivalence result includes contestable markets, in which a single active seller sells in the market at the competitive price.Footnote ²

Different from the common measures for market power such as residual demand elasticity or market concentration, the condition in DSB, which we show also holds in the indivisible setting, uses the minimum number of active traders as an indicator for the attainment of a competitive outcome. Our condition further differentiates the market structures that admit of monopolistic outcomes in the Nash equilibria and the ones that do not. When our condition is not met, i.e. a market structure allows for exploitation of market power in the equilibrium, it does not rule out the possibility of achieving a competitive outcome in a different equilibrium. Indeed, there always exists a Nash equilibrium with competitive outcome. Hence, when there is a monopoly in the market, whether the outcome is competitive can be a result of equilibrium selection. This helps to explain the convergence to competitive outcomes in some monopolistic markets in Smith and Williams (Reference Smith, Williams, Green and Kagel1990). Our condition is general and met in double auction markets such as in Smith (Reference Smith1962, Reference Smith and Amihud1976, Reference Smith1982), as well as the stress-tests that yield supportive evidence for the institution’s robust convergence to competitive outcomes [for example, duopoly markets, swastika design and box design in Smith and Williams (Reference Smith, Williams, Green and Kagel1990), and swastika design in Smith (Reference Smith1965) and Holt et al. (Reference Holt, Langan and Villamil1986)].Footnote ³ Parallel to the asymptote competitiveness in Cournot competition, when traders’ reservation values are independently drawn from certain distributions, our condition is more likely to be met as the number of traders grows.Footnote ⁴ While an exact model for the dynamic double auction institution remains absent, in our experiment, we examine the effectiveness of our condition in providing reference for the double auction markets.Footnote ⁵

Another important feature of the double auction is its demonstration of “the economy of knowledge with which it operates, or how little the individual participants need to know in order to be able to take the right action” (Hayek Reference Hayek1945, p. 527). To resemble the limited information each trader possesses in real markets, double auction experiments provide traders with private information of their own values of the commodity. However, when incomplete information solution concepts such as Bayesian Nash equilibrium are used in another strand of models for the double auction, the k-double auction, the attainment of full efficiency requires infinitely many traders (for literature reviews, see Satterthwaite and Williams Reference Satterthwaite, Williams, Friedman and Rust1993; Zheng Reference Zheng2020). As suggested by previous results from the experiments, “although traders’ information ...is far from complete, it is possible for them to learn to use the relevant ‘complete information’ strategies” (Friedman and Ostroy Reference Friedman and Ostroy1995, p. 23).

To check whether the Nash equilibrium provides a good description of outcomes under the standard double auction settings, we run a laboratory experiment in which traders are informed about their own valuations but not about the valuations of other traders. We consider two market institutions: the clearing house institution, which is exactly the strategic market game we model, and a double auction following the rules of Smith (Reference Smith1962). These two institutions are static/dynamic versions of each other—the clearing house institution is indeed a call market with a pay-as-bid feature.Footnote ⁶ The double auction institution is known to facilitate learning of the relevant information for traders when compared to call markets (see e.g. Smith Reference Smith1982; Smith et al. Reference Smith, Williams, Bratton and Vannoni1982). Under the clearing house institution, we provide traders with feedback on the trading prices and quantities at the end of each round, to check whether this information is enough for them to learn to act as-if there is complete information in the market as Hayek hypothesized.

Each market in our experiment consists of two buyers and two sellers—the minimal size allowing for the equivalence of Nash equilibrium and competitive equilibrium outcomes, and thus adequate for a stringent test. Under each institution, we consider two market environments: one in which the two buyers and the two sellers are inframarginal, so that all Nash equilibrium outcomes are competitive, and one in which the two buyers but only one of the sellers are inframarginal (i.e. there is monopoly power) so that some Nash equilibrium outcomes are non-competitive.

In the absence of monopoly power, the results from our experiment confirm the double auction institution’s convergence to competitive outcomes, though we have fewer traders than previous experiments.Footnote ⁷ Efficiency under the clearing house institution remains below efficiency under the double auction, but seemingly converges over time, in line with the results obtained by Smith et al. (Reference Smith, Williams, Bratton and Vannoni1982) and Friedman and Ostroy (Reference Friedman and Ostroy1995) for larger numbers of traders. Under both institutions, trading prices lay mostly in the competitive range in the absence of monopoly power, consistent with equilibrium predictions.

When monopoly power exists, higher trading price, lower trading volume and an efficiency loss can be observed under the double auction compared to the environment without monopoly, as expected. Under the clearing house institution, trading volume is lower compared to the environment without monopoly, but the efficiency loss is not significant, and prices seem to converge to competitive levels over time. This surprising result may be either a consequence of the inability of the monopolist to gather enough information about the other side of the market to exploit monopoly power under the clearing house institution, or a consequence of coordination on a low-price outcome, which remains a Nash equilibrium outcome under monopolistic conditions. It is an interesting and open question whether the convergence to competitive outcomes for the clearing house institution even in the presence of monopoly power is robust to learning with a longer horizon and to variations in the parameters describing the economy.

Gode and Sunder (Reference Gode and Sunder1993) show the attainment of high efficiency when the double auction market is populated by simulated zero-intelligence traders who submit random non-loss-incurring bids and asks.Footnote ⁸ Our model helps to complete the story by focusing on the factor they omitted, the strategic behavior in the market, and how it may lead to the competitive outcome in double auction with a small number of traders. The convergence to competitive price by budget-constrained zero-intelligent agents was helped by a “Marshallian path,” i.e. trading occurring first to those who have more to gain from trade (Gode and Sunder Reference Gode and Sunder1993; Brewer et al. Reference Brewer, Huang, Nelson and Plott2002). Given the static nature of our clearing house institution, the Marshallian path cannot have an impact within a round. However, there may be a similar effect through learning across rounds. Low cost and high value units are more frequently traded.Footnote ⁹ Based on previous transaction prices, traders may improve coordination and further narrow down the price range (Fig. 8).

In the competitive environment under the double auction institution, we observe the Marshallian path to hold better in earlier rounds, as corresponds to a phenomenon linked to market learning out of equilibrium. Under the double auction institution, Coasian dynamics would prevent the monopolist seller from selling to the highest valuation buyer at above competitive prices since a buyer would anticipate that the monopolist would be willing to lower the price afterwards to trade with the other buyer. We do find some evidence of the monopolist occasionally lowering the price to sell a second unit. However, trading volume remains on average below competitive levels, and estimated price asymptotes suggest higher than competitive prices in the monopolistic environment under the double auction.

In previous experimental work on market games, Duffy et al. (Reference Duffy, Matros and Temzelides2011) explore a quantity strategic market game with divisible commodities, where traders retain market power. They compare outcomes with two and with ten traders per side, and obtain higher efficiency and more coherence to competitive behavior if there are more traders. Dufwenberg and Gneezy (Reference Dufwenberg and Gneezy2000) also obtain—somewhat surprisingly—a beneficial effect of the number of traders in an experiment on Bertrand competition, comparing outcomes with two, three, and four traders per side. We differ from both as we explore the boundary between competitive and noncompetitive environments.

This paper is also closely related to Friedman and Ostroy (Reference Friedman and Ostroy1995), which compares a strategic market game institution with double auction under various structural parameters with divisible commodities and larger number of traders, and draws a connection between DSB and results from double auction experiments. With the conditions we obtain for the equivalence of Nash and competitive equilibria outcomes, we are able to carry out a stringent test on their as-if complete information theory in the lab.

The rest of the paper is organized as follows. Section 2 gives a formal description of the economy. Section 3 gives a detailed explanation of the strategic market mechanism. Section 4 contains the theorems of coincidence of Nash equilibrium and competitive equilibrium. Section 5 presents the experimental design and hypotheses. Section 6 describes the results. Section 7 concludes. Proofs are collected in “Online Appendix 1”, and experimental instructions and quizzes are collected in “Online Appendix 2”.

2 The economy

We describe a general equilibrium model related to laboratory experiments. Our notation follows Friedman and Ostroy (Reference Friedman and Ostroy1995). There are two goods, a divisible ‘money’ and a traded good that can only be traded in indivisible units. Let $I=B\cup S$ be the set of individuals, classified as either buyers (B) or sellers (S). Each $i\in I$ is defined by a vector $(r_{i1},\dots ,r_{\mathrm{ik}})$ , where $r_{\mathrm{ij}}$ indicates the reservation value for the $j_{\mathrm{th}}$ unit of the traded good. The parameter $k\ge 1$ indicates the maximum number of units of the traded good that an individual can buy or sell. For each $i\in B$ , reservation values decrease with the quantity demanded: $r_{i1}\ge \cdots \ge r_{\mathrm{ik}}\ge 0$ . For each $i\in S$ , reservation values increase with the quantity supplied $0\le r_{i1}\le \cdots \le r_{\mathrm{ik}}$ .

Each trader’s utility is given by

$\begin{array}{c}{u}_i(q_i,m_i)=\left(\begin{array}{cc}{\delta }_i{\sum }_{j=1}^{|q_i|}{r}_{\mathrm{ij}}+m_i& \text{if}{q}_i\ne 0\\ m_i& \text{if}{q}_i=0\end{array}\right),\text{with}{\delta }_i=\left(\begin{array}{cc}1& \text{if}i\in B\\ -1& \text{if}i\in S\end{array}\right),\end{array}$

where $q_i\in Q_i$ is the quantity of the good traded by i and $m_i\in \Re$ are the money holdings of i. We let $Q_i=\{0,1,\dots ,k\}$ if $i\in B$ and $Q_i=\{0,-1,\dots ,-k\}$ if $i\in S$ , so that supply is described as negative demand. We assume that initial endowment of money of each individual is equal to 0; note that individuals are allowed negative money holdings.

Keeping fixed the sets of buyers and sellers and k, an economy $r\in {\Re }_{+}^{k\left|I\right|}$ is described by a set of vectors of reservation values that are weakly decreasing for each buyer and weakly increasing for each seller, as described above. Given an economy r, an allocation (of the indivisible good) is a vector $q=\left(q_i\right)\in {\times }_{i\in I}{Q}_i$ and an outcome is a vector (q, m) where q is an allocation and $m\in {\Re }^{\left|I\right|}$ .

Denote by $\xi \left(r\right)$ the set of competitive equilibria for an economy r. A competitive equilibrium $(p,q)\in \xi \left(r\right)$ is a price $p\in {\Re }_{+}$ and an allocation q such that

1. 1. (utility maximization) for each i, $u_i(q_i,-p{q}_i)\ge u_i(q_i^{\prime },-p{q}_i^{\prime })$ for all $q_i^{\prime }\in Q_i$ .

2. 2. (market clearance) ${\sum }_{i\in I}{q}_i=0$ . By utility maximization, if (p, q) is a competitive equilibrium for economy r, then

   • • for every $i\in B$ , either $q_i=0$ and $r_{i1}\le p$ , or $0<q_i<k$ and $r_{i{q}_i}\ge p\ge r_{i,q_i+1}$ , or $q_i=k$ and $r_{\mathrm{ik}}\ge p$ .

   • • for every $i\in S$ , either $q_i=0$ and $r_{i1}\ge p$ , or $-k<q_i<0$ and $r_{i|q_i|}\le p\le r_{i,|q_i|+1}$ , or $q_i=-k$ and $r_{\mathrm{ik}}\le p$ .

Note that (p, q) induces the outcome $(q,m)=(q,(-p{q}_i))$ .

It is easy to prove that for any economy r, there is a competitive equilibrium. We can order the units that sellers can supply in ascending order according to their reservation values, and the units that buyers can demand in descending order according to their reservation values, to obtain the familiar supply and demand curves. Equilibrium prices and allocations can be obtained by the crossing of the supply and demand curves. As it is well-known for economies with quasi-linear preferences, the set of competitive allocations is the set of solutions to the problem of maximizing social surplus, that is

$\begin{array}{c}\underset{q\in Q}{max}\sum _{i\in I}\sum _{0\le j\le |q_i|}{\delta }_i{r}_{\mathrm{ij}},\end{array}$

where $Q=\{q:q_i\in Q_i,{\sum }_i{q}_i=0\}$ is the set of feasible allocations.

Trade is positive in every competitive equilibrium if and only if

$\begin{array}{c}\underset{i\in S}{min}{r}_{i1}<\underset{i\in B}{max}{r}_{i1}.\left(\text{A}\right)\end{array}$

As we will see, an important condition for the equivalence between competitive equilibrium outcomes and the outcomes of a strategic game is that there are at least two trading individuals on each side of the market.

Related works on price-quantity strategic market games feature divisible commodities under the usual assumptions of continuous, increasing marginal costs for each seller, and continuous, decreasing marginal utility of consumption for each buyer. Note that in economies with divisible units active traders compete “at the margin,” in the sense that in a competitive equilibrium the marginal utility of consumption and the marginal cost of production for the last unit are equated to the price for all active traders. Our results illustrate that competition at the margin is unnecessary for the equivalence between competitive and strategic outcomes.

3 The strategic market game

Each individual submits a price-quantity offer $({\tilde{p}}_i,{\tilde{q}}_i)$ to the clearing house, where ${\tilde{p}}_i\ge 0$ and ${\tilde{q}}_i\in Q_i$ . Intuitively, each individual offers to trade up to $|{\tilde{q}}_i|$ units of the traded good at the price ${\tilde{p}}_i$ . Denote the set of feasible offers for individual i by

$\begin{array}{c}{W}_i=\{({\tilde{p}}_i,{\tilde{q}}_i):{\tilde{p}}_i\ge 0;{\tilde{q}}_i\in Q_i\}.\end{array}$

Given an offer profile $w\in W={\times }_{i\in I}{W}_i$ , the set of feasible allocation vectors for the clearing house is

(3.1) $\begin{array}{cc}Y\left(w\right)=\{(y_1,...y_n)\text{:}& 0\le y_i\le {\tilde{q}}_i,\text{if}i\in B;\end{array}$

(3.2) $\begin{array}{cc}& 0\ge y_i\ge {\tilde{q}}_i,\text{if}i\in S;\end{array}$

(3.3) $\begin{array}{cc}& {\sum }_i{y}_i=0;\end{array}$

(3.4) $\begin{array}{cc}& y_i\in Z\}.\end{array}$

Note that Y(w) is a finite set. Conditions (3.1) and (3.2) guarantee that trade is voluntary, i.e. individuals do not end up trading more than what they offered. Condition (3.3) ensures that the market clears and the clearing house keeps no inventory. Condition (3.4) conveys the assumption that the good is indivisible.

After the clearing house chooses an allocation $y=(y_1,\dots ,y_n)\in Y\left(w\right)$ , individual i receives $y_i$ units of the traded good and earns an amount of money equal to $-{\tilde{p}}_i{y}_i$ . We assume that the clearing house allocates trade to maximize the arbitrage profit, ${\sum }_{i\in I}{y}_i{\tilde{p}}_i$ , as if the clearing house buys units from the sellers and sells them to buyers at the agents’ proposed prices. Thus, given an offer profile w, the resulting allocation y must satisfy

$\begin{array}{c}y\in \Pi \left(w\right)=\{y\in Y\left(w\right):\sum _{i\in I}{y}_i{\tilde{p}}_i\ge \sum _{i\in I}{y}_i^{\prime }{\tilde{p}}_i\text{for all}{y}^{\prime }\in Y\left(w\right)\}.\end{array}$

Intuitively, as in Dubey (Reference Dubey1982), buying offers are ranked in a descending order by price while the quantities offered are accumulated to form the demand curve, and selling offers are ranked in an ascending order by price while the quantities offered are accumulated to form the supply curve. Since goods are traded at their proposed prices, the clearing house extracts the surplus between the supply and demand, as Fig. 1 illustrates. That is, the clearing house chooses a competitive equilibrium allocation for a fictitious economy $\tilde{r}$ given by

$\begin{array}{c}{\tilde{r}}_{\mathrm{ij}}=\left(\begin{array}{cc}{\tilde{p}}_i& \text{if}1\le j\le |{\tilde{q}}_i|\\ 0& \text{if}|{\tilde{q}}_i|<j\le k\text{and}i\in B\\ +\infty & \text{if}|{\tilde{q}}_i|<j\le k\text{and}i\in S\end{array}\right),\end{array}$

and appropriates the revealed social surplus.

Fig. 1

Arbitrage profit for the clearing house

In scenario (a) of Fig. 1, $\Pi \left(w\right)$ is a singleton set. To maximize the arbitrage profit, the clearing house would fulfill all demand and supply to the left of the dashed line. The dotted area is the profit for the clearing house, and the profit is positive in this case. In scenario (b), $\Pi \left(w\right)$ contains two allocations if units A and B are offered by different sellers, depending on which of the two sellers is allowed to sell the last unit. In scenario (c), the clearing house gets the same profit allocating $q_1$ or $q_2>q_1$ units. Similarly, in scenario (d), buying and selling q units gives the same profit for the clearing house as making no trade.

To make trade happen whenever possible, following Simon (Reference Simon1984), we assume that the clearing house chooses an allocation from the set

$\begin{array}{c}F\left(w\right)=\{y\in \Pi (w):\text{there}\text{is}\text{no}\varphi \in \Pi (w)\\ \text{such}\text{that}\varphi \ne y\text{and}|{\varphi }_i|\ge |y_i|\text{for}\text{all}i\in I\}.\end{array}$

That is, the clearing house does not choose allocations that are ray-dominated. Then in scenario (c), $q_2$ units will be bought and sold, and in scenario (d), q units will be traded. We still have two allocations in F(w) in scenario (b) if units A and B are offered by different sellers. We assume that the clearing house chooses randomly according to the distribution ${\mu }_w$ that gives probability ${\mu }_w\left(y\right)>0$ to each allocation $y\in F\left(w\right)$ and probability ${\mu }_w\left(y\right)=0$ to every other allocation in Y(w) such that ${\sum }_{y\in F\left(w\right)}{\mu }_w\left(y\right)=1$ . Propositions Footnote ¹–6 in the Online Appendix provide a characterization on F(w).

Given this market mechanism, define an active trader given offer profile w as a trader that has positive probability to trade. In other words, agent i is an active trader given offer profile w if there exists $y\in F\left(w\right)$ such that $y_i\ne 0$ . Furthermore, denote by AS(w) the set of active sellers, and AB(w) the set of active buyers given offer profile w.

4 Nash equilibrium and competitive outcomes

Note that each offer profile $w\in W$ induces a lottery over outcomes. Each outcome $(y,(-\widetilde{p_i}{y}_i))$ is realized with probability ${\mu }_w\left(y\right)>0$ if $y\in F\left(w\right)$ , and ${\mu }_w\left(y\right)=0$ if not. Given an offer profile $w\in W$ , the expected utility for each trader is,

$\begin{array}{c}E{u}_i\left(w\right)=\sum _{y\in F\left(w\right)}{\mu }_w\left(y\right)u_i(y_i,-\widetilde{p_i}{y}_i).\end{array}$

A (pure strategy) Nash equilibrium for an economy r is an offer profile $w^{\ast }\in W$ such that for every $i\in I$ ,

$\begin{array}{c}E{u}_i(w_i^{\ast },w_{-i}^{\ast })\ge E{u}_i(w_i^{\prime },w_{-i}^{\ast })\text{for all}{w}_i^{\prime }\in W_i.\end{array}$

As in other price-quantity strategic market games, every competitive equilibrium outcome can be reached with probability one by at least one Nash equilibrium offer profile, and all the positive probability outcomes of a Nash equilibrium are competitive as long as in the Nash equilibrium there are at least two active traders on each side of the market.

In what follows, we provide necessary and sufficient conditions for all the outcomes of every Nash equilibrium with trade to be competitive. Define the buyers’ marginal value, $v_b$ , as the maximum of the lowest reservation value for buyers’ units traded in competitive equilibria, that is,

$\begin{array}{c}{v}_b=\underset{(p,q)\in \xi \left(r\right)}{max}\underset{q_i>0}{min}{r}_{i,q_i}.\end{array}$

Similarly, define the sellers’ marginal value, $v_s$ , as the minimum of the highest reservation value for sellers’ units traded in competitive equilibria, that is,

$\begin{array}{c}{v}_s=\underset{(p,q)\in \xi \left(r\right)}{min}\underset{q_i<0}{max}{r}_{i,|q_i|}.\end{array}$

In economies such that (A) is satisfied, $v_b$ and $v_s$ are well-defined, since in every competitive equilibrium at least some $i^{\prime }\in S$ with the minimum cost (i.e. $r_{i^{\prime }1}={min}_{i\in S}{r}_{i1}$ ) must have $q_{i^{\prime }}<0$ , and at least some $i^{\prime \prime }\in B$ with the maximum reservation value (i.e. $r_{i^{\prime \prime }1}={max}_{i\in B}{r}_{i1}$ ) must have $q_{i^{\prime \prime }}>0$ . As shown in the Online Appendix, $v_b$ and $v_s$ are equal, respectively, to the lowest reservation value of buyers’ traded unit(s) and the highest reservation value of sellers’ traded unit(s) in any competitive equilibrium with the smallest number of transactions. Moreover, if (A) is satisfied, we must have $v_b>v_s$ , because if there is a competitive equilibrium such that both the marginal buyer and the marginal seller are indifferent (i.e. ${min}_{q_i>0}{r}_{i,q_i}=p={max}_{q_i<0}{r}_{i,|q_i|}$ ), there is another competitive equilibrium in which one fewer unit is traded.

Denote by $\overline{p}$ and $\underline{p}$ the highest and lowest competitive price respectively. It is easy to check that

$\begin{array}{c}{v}_s\le \underline{p}\le \overline{p}\le v_b.\end{array}$

The first and third inequalities above follow from the fact that for every equilibrium $(p,q)\in \xi \left(r\right)$ we must have ${max}_{q_i<0}{r}_{i,|q_i|}\le p\le {min}_{q_i>0}{r}_{i,q_i}$ .

5 Experimental design and hypotheses

5.1 Experimental design

We test the predictive ability of our market game model in laboratory experiments. We consider two markets with indivisible commodities. Each market has two buyers $B=\{B1,B2\}$ and two sellers $S=\{S1,S2\}$ , and each trader can either buy or sell two units. We assign the first and third highest demand reservation values to one buyer, and the second and fourth to the other buyer. By assigning the units to sellers in different ways, we create a market that satisfies the condition in Theorem Footnote ³, and a market that does not. This design is similar to one implemented by Davis and Holt (Reference Davis and Holt1994).

In our competitive market, the two supply units that can be traded in competitive equilibrium are assigned each to each one of the two sellers. Thus, as shown in the left part of Fig. 2, there are two inframarginal traders on each side of the market. By Theorem Footnote ³, the Nash equilibrium outcomes with trading of the strategic market mechanism coincide with competitive equilibrium outcomes. That is, both units with lowers costs should be traded, and the price should be in the competitive price range, $15–$19. Correspondingly, efficiency (as a percentage of the maximum possible surplus) should be $100\%$ .

In our monopoly market, instead, the two low cost units are assigned to the same seller, as shown in the right part of Fig. 2. The set of Nash equilibrium outcomes with trade includes the set of competitive equilibria just described, as well as monopolistic market equilibria in which only the unit with the lowest cost is traded, and the price is between $19 and $30. Efficiency of monopolistic equilibria is $(32-2)/(32-2+19-15)$ , that is approximately $88\%$ .

Fig. 2

Competitive and monopolistic markets

In the experiment, we compare the performance of our strategic market mechanism (Clearing House, or CH hereafter) with the continuous time double auction (Double Auction, or DA hereafter), in the two markets.

In the clearing house institution, each trader submits a price-quantity pair to the clearing house. The clearing house then decides trade by the rules described in Sect. 3, and reports the trader’s own transaction price and quantity, together with the price and quantity traded in the market. We let ${\mu }_w\left(y\right)=1/\left|F\left(w\right)\right|$ for all $y\in F\left(w\right)$ in the experiment; that is, the clearing house assigns equal probability to all arbitrage-maximizing allocations. When making decisions, traders are given their own values, but not other traders’ values or offers.

In the double auction institution, the traders buy/sell the good unit by unit, i.e. after a trader has traded his/her first unit, he/she can submit offers for the second unit. Valid offers are listed on the screen in real time as public information for all traders in the market. Bids are ranked from high to low, and asks are ranked from low to high. Each offer is a limit order for one unit. We implement the bid/ask improvement rule such that each offer has to reduce the bid-ask spread to be valid. A transaction happens automatically if a valid bid is no lower than a valid ask. In each transaction, a bid will always be matched with the highest-ranked ask, and an ask will be traded with the highest-ranked valid bid. In case of unequal bid and ask in a transaction, the trading price will be the price in the pair that was submitted later.Footnote ¹¹ The double auction institution follows a soft-close rule. A round ends if no new valid offer shows up in 20 seconds.

The experiment was conducted in the Interdisciplinary Center for Economic Science (ICES) lab in George Mason University. In total, 240 subjects participated in the 18 sessions, and each session lasted for no more than 100 minutes.Footnote ¹² Each subject participated in only one treatment, playing the same role (B1, B2, S1 or S2) in the same market for 20 rounds. The final payoffs ranged from $5 to $36. The average payoff is $11.25 including a $5 show-up fee.

The experiment was computerized, and programmed in oTree (Chen et al. Reference Chen, Schonger and Wickens2016).Footnote ¹³ At the beginning of the session, the participants were seated at partitioned computer stations and allowed 10 minutes to read the instructions on their own. Then an experimenter read the instructions out loud to all participants. Afterward, a quiz was handed out, and the experiment began after each participant gave correct answers to all the questions in the quiz. Then the role a participant had in the experiment was revealed to him/her, and the participants were given a practice round before the formal rounds began. There were 20 formal rounds, one of which was randomly chosen for payment. After the 20 formal rounds, each participant was informed of the round chosen for payment and his/her own payoff. The payment was made privately.