Bazaar economics

Journal of Economic Behavior & Organization 119 (2015) 163–181

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Bazaar economics John H. Miller a,b,∗ , Michele Tumminello a,c a b c

Department of Social and Decision Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA Santa Fe Institute, Santa Fe, NM 87501, USA Dipartimento di Scienze Economiche, Aziendali e Statistiche, University of Palermo, Palermo 90128, Italy

a r t i c l e

i n f o

Article history: Received 29 October 2014 Received in revised form 12 June 2015 Accepted 9 August 2015 Available online 18 August 2015 Keywords: Competitive Equilibrium Disequilibrium Supply and demand Stochastic processes

a b s t r a c t Competitive Equilibrium theory has been a widely accepted and extensively used cornerstone in economics for over a century. Here, we suggest a complementary model—motivated by the haggling in a bazaar—that offers a useful, first-principle account of market behavior that better accounts for the observed outcomes in forty market experiments. The Bazaar model uses simple stochastic processes to drive the matching of traders and the determination of price. We show that as agents become more impatient, the system tends toward more Competitive-Equilibrium-like outcomes. © 2015 Elsevier B.V. All rights reserved.

Our current understanding of how markets function emerged slowly starting in the late 1600s and progressed on into the next two centuries. With the development of Competitive Equilibrium (CE) theory, and the associated analytic tools of supply and demand curves (begun by Jenkin, 1870 (see Brownlie and Lloyd Prichard, 1963) and refined and popularized by Marshall, 1890), a lovely picture emerged by which the apparent chaos of the marketplace resulted in a nicely ordered economic universe. While studying CE has been quite useful, the notion that CE theory might not fully explain market behavior is not new. Chamberlin’s (1948) path breaking experimental work was motivated in no small part by his feelings that markets may not, in fact, seek out CE, as he explicitly noted his “failure, upon reflection stimulated by the problem, to find any reason why it should do so.” Early experiments with computation-based markets composed of algorithmic traders (Rust et al., 1994) allowed the process of price formation to be explicitly explored. This work made it clear that even unsophisticated traders can result in observed behavior that apparently aligns with the gross predictions of CE under some circumstances (Gode and Sunder, 1993, 1997; Rust et al., 1994). More recently, Bergstrom and Kwok (2005) and Parendo (2010) have analyzed data from classroom experiments, and observed that the predictions of CE are not realized in any exact way and are only marginally supported if one has a generous interpretation of the data. Here we develop some coherent analytic results that add to this new line of inquiry. Along with the potential difficulty of CE to truly explain experimental observations, another challenge it confronts is its lack of a first-principle derivation. For example, the notion of a Walrasian auctioneer has been invoked to intuitively justify CE theory, yet we know that Walrasian auctioneers are like unicorns, perhaps possible but never observed in the wild. While

∗ Corresponding author at: Department of Social and Decision Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Tel.: +1 4122683229. E-mail addresses: [email protected] (J.H. Miller), [email protected] (M. Tumminello). http://dx.doi.org/10.1016/j.jebo.2015.08.005 0167-2681/© 2015 Elsevier B.V. All rights reserved.

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a first-principle derivation is not necessarily needed for scientific advance, the ability to grow a theory from first principles is desirable nonetheless. Hayek (1945), over seventy years ago, recognized this issue in the context of markets: The problem is in no way solved if we can show that all the facts, if they were known to a single mind, would uniquely determine the solution; instead we must show how a solution is produced by the interactions of people each of whom possesses only partial knowledge. To assume that all the knowledge to be given to a single mind in the same manner in which we assume it to be given to us as the explaining economists is to assume the problem away and to disregard everything that is important and significant in the real world. The approach pursued in this paper represents an interesting branch of economic theory that in many ways was the road not taken. The work of Walras (1877) initiated a long arc of economic theorizing about Competitive Equilibrium that predominates to this day. While Walras did note the importance of dynamics in any tatonnement process, this emphasis is only present in vestigial forms like the Walrasian auctioneer. Moreover, more recent work, for example, Sonnenschein (1974), Mantel (1974), Debreu (1974), and Saari and Simon (1978) has raised serious concerns about the ability of these systems to converge to an equilibrium. There is also a branch of work about non-tatonnement process, whereby some trades can occur at out-of-equilibrium prices. Yet, in both tatonnement and non-tatonnement processes, it is assumed that there is a given price vector known to everyone and that prices are adjusted by a (fictitious) central authority. An alternative branch of market theory began in a series of disagreements between Walras and Edgeworth (Kirman, 2010) and focused on the market implications of processes that were driven by decentralized haggling. Hayek (1945) conjectured that such processes would arrive at an efficient outcome. This alternative modeling approach has been pursued, to various degrees, in the more modern economic literature as well, for example, Fisher’s (1983) work on disequilibirum outcomes and the models of Diamond (1982), Myerson and Satterthwaite (1983), and Foley (1994). This paper also pursues this road not taken, the one endorsed by Edgeworth, by considering the behavior of a decentralized market starting from a non-equilibrium state. There is also a well developed literature on dynamic matching and bargaining games (see, for example, Rubinstein and Wolinsky, 1990; Gale, 2000; Gale and Sabourian, 2006), wherein market behavior is explored by bringing together search theory (buyers and sellers dynamically matching with one another) and bargaining theory (once matched, agents attempt to reach a deal and make a trade based on a game-theoretic bargaining solution). This literature arose in response to some of the theoretical issues, mentioned above, confronting the standard model of CE, and it is an attempt to establish firmer theoretical foundations from which market behavior, and potentially CE, could be understood. The dynamic matching and bargaining games research has advanced our theoretical understanding of how prices can arise in markets, and whether CE is a reasonable description of the potential outcome. This work considered highly simplified markets, for example, Rubinstein and Wolinsky (1985) focused on a market with a large number of homogeneous buyers and sellers (with the inflow of agents just equal to the outflow) who meet, bargain, and if an agreeable price is reached (using an equilibrium refinement from game theory), leave the market. Using this framework, the authors demonstrated the existence, and uniqueness, of a market equilibrium, which turns out to be different than the CE. While latter work loosened some of these restrictions, for example, Gale (1987) generalized this model to heterogeneous agents and showed that under certain frictionless conditions the game theoretic equilibrium coincides with CE, these models still require some intense assumptions about, say, each agent’s ability to have full knowledge of prior matches, offers, and acceptances and declines. (See Osborne and Rubinstein, 1990, for a more extensive history.) The work here departs from these prior efforts by having agent bargaining behavior be governed by a simple, more behaviorally realistic rule, rather than seeking an outcome dictated by an equilibrium refinement (which typically requires extraordinarily capable and informed agents). This alternative approach has been embraced before, for example, Gode and Sunder (1993) used Zero-Intelligence traders to show how simple, double-auction market rules can funnel even unskilled traders toward CE-like outcomes.1 While the choice of such an assumption may be a matter of taste (or paradigm), ultimately we are driven here by the theoretical leverage and empirical possibilities one gains from this assumption. First, it allows us to explore markets with a variety of configurations of heterogeneous agents, including markets amenable to actual experiments. Second, the resulting model makes exact probabilistic predictions about the outcomes that will arise (both those that embrace CE and those that do not), versus the equilibrium predictions of the prior models that do not provide a sense of how likely any such outcome might be—a particular problem when these prior models predict multiple equilibria.2 Finally, along with specifying an interesting theoretical model of markets in its own right, our approach has the advantage of being testable against experimental outcomes, something we pursue based on data from forty classroom experiments. This empirical test provides new insights into both our model and prior conclusions about observing CE in experiments.

1 Other researchers have also focused on the impact of market conditions and rules on allocative efficiency, including Feldman (1973, 1974), Albin and Foley (1992), Gode and Sunder (1997), Easley and Ledyard (1993), Cason and Friedman (1993), and Brewer et al. (1999). 2 Kirman (2010) argues that Walras’ own notion of equilibrium “could either be thought of as a rest point of a dynamic process, or as a static solution to the problem of finding a price vector at which all markets would clear,” the latter of which took hold with the work of Arrow and Debreu. Our work uses a dynamic process to create a probability distribution over the possible outcomes (configuration of trades) that will arise in the market. This approach is akin to a statistical mechanics model in physics, where the inherent random nature of the system is explicitly recognized (versus the deterministic approach pursued in classical mechanics). Sometimes, the importance of randomness in the statistical model disappears, and the outcomes of the two approaches align. As discussed later, we find a similar phenomenon here when the spread between the buyer values and seller costs is large.

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Overall, our approach here is to take seriously the theoretical and empirical concerns that have arisen in CE theory, and develop a simple alternative model that allows a much more prominent role for the randomness inherent in the matching of potential traders and the making of offers than is normally assumed. To capture these notions, we consider agents milling about a bazaar, randomly encountering one another, making unsophisticated offers perhaps driven by behavioral considerations, and accepting deals that are mutually profitable. We call this approach the Bazaar model, at the obvious risk of easy wordplay. We will show that the impatience of agents seeking to make deals introduces a selection mechanism into our model that favors transactions with higher gains to trade and pushes the system toward CE-like outcomes. Thus, under our model, the impatience of agents increasingly favors the conditions needed to reach global market efficiency. 1. Bazaar economics in a simple market We wish to model a simple market in a simple way. For the market fundamentals, we assume a finite set of potential traders. Some of the potential traders are demanders of goods, and if they are able to negotiate a trade they will receive a buyer value (v) that pays the monetary equivalent of acquiring a good. Other potential traders are suppliers of goods, and if they are able to trade, they will need to pay a seller cost (c) to produce the good. The set of buyer values and seller costs induce the underlying forces of supply and demand in the market. For much of the discussion below, both for rhetorical ease and linkage to our empirical data, we assume that demander’s buyer values are either $40 or $20 and that supplier’s seller costs are either $10 or $30. If a supplier and demander negotiate a price of p, then the profit to the seller (supplier) will be p − ci and the consumer surplus to the buyer (demander) will be vj − p, where i and j index the particular traders involved in the transaction. Consider a simple market with two traders on each side of the market, demanders with buyer values of $40 and $20 and suppliers with seller costs of $10 and $30. The CE of this market is one trade at a price in [$ 20, $ 30],3 with the high-value demander trading with the low-cost supplier. For our alternative market mechanism, we assume that potential traders are milling about the bazaar and randomly encounter one another. When two agents meet, if the buyer value of the demander is greater than or equal to the seller cost of the supplier, a trade occurs at a random price on [cj , vi ] and the two traders leave the bazaar.4 This process continues until no remaining demander has a buyer value that is greater than or equal to the lowest seller cost among the remaining suppliers. Once the system reaches this final state, there are no mutually profitable trades (even with 0 profit) remaining in the market, and we assume that the market ends. To hone some intuitions on how the Bazaar model compares to CE, consider the above market with two traders on each side of the market. If there is an initial encounter between the high-value demander and low-cost supplier, then under Bazaar a trade will occur at a price distributed on [$ 10, $ 40]5 and the market will end given the impossibility of a mutually profitable trade between the remaining low-value demander and high-cost supplier. In this case, the CE and Bazaar predictions align, as both models predict one trade between the high-value demander and low-cost seller, at an expected price of $25, which is the midpoint of the [$ 20, $ 30] range predicted by CE and the [$ 10, $ 40] range predicted by Bazaar. Of course, while the expected prices are identical, the observed price under Bazaar may be outside of the range predicted by CE. The Bazaar model also admits a second possible market outcome. If the high-value demander matches with the highcost supplier, they will trade at a price on [$ 30, $ 40], leaving the low-value demander and low-cost supplier to trade on [$ 10, $ 20]. In this case, we will observe two trades, one at an expected price of $35 and the other at $15. This alternative configuration of trades6 results in an outcome that is quite different from the CE prediction, namely, we get two trades (at expected prices of $35 and $15) instead of one (at an expected price of $25), and a total surplus of $20 rather than $30. If we randomly form a single pair of traders (consisting of one demander and one supplier) at each time step, and we assume that paired traders with a buyer value larger then the seller cost will always end up agreeing on a transaction prices, then 2/3 of the time this alternative outcome (to CE) will emerge, since two of the three possible initial pairings put the system on a path resulting in the alternative outcome. The above example shows how a simple, and we claim reasonable, alternative model could lead to predictions that are fundamentally different from those arising in CE theory. While the two models lead to different predictions, the example above hints at how empirically it might be difficult to distinguish between them. For example, if Bazaar holds, we may observe a configuration of trades that is consistent with the CE predictions,7 though in this case there is a chance of seeing some “outlier” prices. Even if we end up in the second configuration of trades admitted by Bazaar in the case above—where the quantity traded is one higher than expected—the

3 CE theory does not specify how the price is chosen in this interval, only that any price in this interval is possible. We will often assume that the mean price (assuming uniform priors over the interval) is a good CE predictor in such cases. 4 Specifically, each agent produces a random offer subject to the proviso that if the offer were accepted the agent would not lose money, and transactions occur when the offers cross. We explore the impact of this mechanism later. 5 We assume a symmetric distribution of the transaction price around the mean value of $25. A formal mathematical model to obtain the distribution of the transaction price will be discussed later. 6 For rhetorical ease, we will use the term “trader configuration” to designate which traders are paired for transactions in a market, and “configuration of trades” to describe both those pairings and resulting prices. 7 As will be discussed, the CE trader configuration is always possible under Bazaar.

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expected mean price is still $25, which is consistent with the CE prediction. Thus, one may have some difficulty distinguishing between the two models using casual empirics. The previous paragraph raises an important issue as to what it means to have an “error” in CE. CE theory typically makes very stark predictions about the configuration of trades (who trades, prices, quantity, and the total surplus obtained) that we should observe in a market. Even in carefully controlled experimental markets, these stark predictions are violated all of the time. Such violations may not be too disturbing, as we typically are willing to admit some reasonable error process tied to a theory as driving differences between observations and theoretical predictions. Of course, one needs to be explicit as to the actual error process involved. In the analysis of CE experiments, one often sees a fairly casual approach with support for the theory being tied to whether the means of the observed prices and quantities are “close” to the CE prediction. It is not entirely clear what underlying error process would be responsible for such an outcome—presumably the errors occur at the level of trades and traders, rather than at the aggregate outcome—and without being explicit here, such results are difficult to interpret. To be explicit about such a process in CE, one might assume that errant demanders and suppliers occasionally meet and make “random” deals with one another, rather than being driven by the aggregate forces of supply and demand and optimal behavior. If one views this as a reasonable accommodation to CE theory, then the foundations of the Bazaar model are not too far afield, as they just extend this notion to having the entire market governed by such behavior. Below, we will formalize the intuitions developed above and derive, and empirically test, the Bazaar model. At the heart of the Bazaar theory is the outcome of a stochastic matching process that pairs together the demanders and suppliers for potential trades in the market. For any given trader configuration, we can predict the key market outcomes concerning the number of trades, prices, who will trade, and total surplus. Thus, our analytic approach is driven by identifying the probabilities of the various trade configurations that could arise given the market fundamentals (tied to the underlying buyer values and supply costs of the potential traders). Armed with these analytics, we empirically test both the Bazaar and CE theories using a large number of classroom experiments. 2. The Asynchronous Bazaar model In this section, we provide a formal description of the Asynchronous Bazaar (AB) model. The theory is presented by considering the distribution of possible trader configurations for a given market. We assume that there are |b| unique buyer values and |s| unique seller costs in the market, and thus a trade configuration is characterized by |s| × |b| numbers, nij , indicating the number of trades that occurred between suppliers with seller cost ci and demanders with buyer value vj . Once we know the trader configuration, we can predict the number of trades, the market efficiency, and the distribution of prices. The distribution of trader configurations depends on the mechanism of pairing. Here we assume an asynchronous mechanism. Under asynchronous pairing, only one pair of agents is randomly formed at each time step. If the selected pair of agents trades, they leave the market, and the process is iterated anew with any remaining agents.8 The underlying market parameters of the Bazaar model are the number, Bj , j = 1, . . ., |b|, of demanders with buyer value vj and the number, Si , i = 1, . . ., |s|, of suppliers with seller cost ci . We assume, without loss of generality, that c1 < c2 < . . . < c|s| , and v1 < v2 < . . . < v|b| . We also assume that (1) each supplier (demander) can sell (buy) at most one good, (2) both suppliers and demanders do not trade at a loss, and (3) trade ends when all of the remaining suppliers have seller costs that are above the buyer values of the remaining demanders. The only constraint on transaction prices imposed by the model is (2), and it implies that the distribution of the transaction price between a demander with buyer value v and a supplier with seller cost c, where v ≥ c, will be constrained to the range [c, v]. In the AB model, suppliers and demanders are paired asynchronously, one pair at a time. To solve this system we use recursive equations to calculate the trader configuration probabilities. The recursive equations will be constructed in the space of trades, in order to obtain all of the configuration probabilities in a finite number of iterations.9 A trader configuration C is described by a |s| × |b| matrix, with entries nij designating the number of trades between suppliers with seller cost ci and demanders with buyer value vj (hereafter, we will use (ci , vj ) to designate such pairs). In the following, we will indicate the probability of observing a given trader configuration after t transactions as P(Ct ), where the sum of all the elements of Ct equals t. At the core of the Bazaar model is a prediction about the trader configuration probabilities. These probabilities depend on the probability that any two given types of agents randomly match and agree to trade. This probability depends on the specific buyer value and seller cost of the two agents and their trading patience (see below). Let the probability that two paired agents (ci , vj ) will trade, once paired, be given by pij . After the first transaction, |s| × |b| different trader configurations are possible, namely, all the matrices C1 with one entry equal to 1 and all the others equal to 0.10

8 One can also assume a synchronous mechanism of pairing. Such a mechanism may give different configuration probabilities depending on the distribution of buyer values and seller costs. An application of the synchronous mechanism is discussed in Appendix A, while a more general presentation is reported in Miller and Tumminello (2012). 9 Under AB, there is an infinitesimal probability that the market will never end, for example, suppose that two traders with a buyer value less than the seller cost are continually paired. 10 The probability of some of these configurations can be zero, in particular, when the buyer value is lower than the seller cost for the two types of traders in question.

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First, focus on a generic configuration matrix C1 with element nij = 1, indicating a single trade between a pair (ci , vj ), and all other elements equal to 0. The probability, P(C1 ), of observing such a configuration is given by the probability that the two indicated agents transacted at the first iteration. This probability is proportional to the product of two terms: the probability that a ci supplier and a vj demander are randomly and independently chosen to form a pair and the probability of making a transaction given the pairing, that is, pij introduced above. The first probability is given by si · bj , where si = is the proportion of suppliers with seller cost ci in a market with M|s| = of demanders with buyer value vj in a market with M|b| = P(C1 ) =

|b|

B j=1 j

|s|

S i=1 i

total suppliers and bj =

Bj M|b|

Si M|s|

is the proportion

total demanders. Therefore,

1 ·s ·b ·p , K0 i j ij

(1)

|s| |b|

where K0 is a normalization constant11 in the space of trades. Specifically, K0 = s · bj · pij if that sum is different i=1 j=1 i from 0, and 1 otherwise. We are now ready to construct the recursive equations for the system. Suppose we know the probability of every possible configuration with exactly t transactions. Consider a new configuration Ct+1 . This configuration can only occur if one of the |b| × |s| possible configurations Ctn −1 (which is identical to Ct+1 , but with entry nij less one trade) occurred after t trades for ij

all i and j.12 Thus |s| |b|  

P(Ct+1 ) =

P(Ct+1 |Ctn

ij −1

) · P(Ctn

ij −1

),

(2)

i=1 j=1

where P(X|Y) represents the conditional probability that configuration x is observed given the occurrence of configuration Y. Eq. (2) provides the recursive relation for calculating the configuration probabilities. The probability P(Ct+1 |Ctn −1 ) is proportional to the probability of selecting one of the remaining Si − the M|s| − t suppliers and one of the Bj −

|s|

n p=1 pj

|b|

ij

n suppliers with seller cost ci from among q=1 iq

remaining demanders with buyer value vj among the M|b| − t remaining

demanders after t transactions (thus, the sums are over the elements of the Ctn

ij −1

matrix), and to the probability that the

selected demander and supplier transact, which is given by pij . Thus P(Ct+1 |Ctn

ij −1

)=

1 Kt (Ctn

|b|

where sit (Ctn

ij

−1 ) =

Si −

ij −1

n j=1 ij

M|s| −t

)

· sit (Ctn

ij −1

) · btj (Ctn

ij −1

) · pij ,

(3)

|s|

is the proportion of the remaining suppliers with seller cost ci , btj (Cnt

ij

)= −1

Bj −

n i=1 ij

M|b| −t

the

proportion of the remaining demanders with buyer value vj , and Kt (Ctn −1 ) is a normalization factor that only depends on the ij configuration Ctn −1 . If t = Min[M|s| , M|b| ] some of the proportions above become indeterminate. However, the corresponding ij configuration will be a complete configuration, in which further trading is not possible, and we set the indeterminate proportions equal to 0. Thus, analogous to K0 , the quantity Kt (Ctn −1 ) is a normalization constant accounting for the different possible configurations in which configuration Ctn to calculate K0 , Kt (Ctn

ij −1

)=

|s| |b| p=1

ij −1

ij

can evolve after the next trade. By following the same reasoning used

st (C ) · btq (Cnij −1 ) · ppq q=1 p nij −1

if the summation is different than 0 and 1 otherwise.

Eqs. (1)–(3) allow one to calculate the exact probability of each trader configuration that can occur under the AB model, and therefore to evaluate the expected number of transactions and the efficiency of the market. 2.1. An example To illustrate the above equations, consider a simple market of three demanders and three suppliers, with NB = 1 demander with buyer value vB and Nb = 2 demanders with buyer value vB , and Ns = 2 suppliers with seller cost cs and NS = 1 supplier with seller cost cs . We assume that vB > cS > vb > cs like in the experiments discussed later. Both the CE and AB models predict that no trades occur at a loss, which implies that no transactions between high-cost suppliers and low-value demanders will occur. CE theory would predict that the high-value demander and one of the two low-value demanders would trade with the two low-cost sellers at a price equal to the low buyer value. We call this configuration C2CE .13 Another trader configuration is possible according to the AB model. In this configuration, the two low-cost suppliers trade with the two

The notation K0 is used to stress that the normalization constant only depends on the configuration observed after t − 1 transactions. The probability of any trade configuration with any negative entries is zero. 13 Note that, under CE theory, it is possible to have multiple trader configurations (though not in the example presented here). CE predicts which demanders and suppliers will trade, but it is agnostic as to how these traders match up, and under some conditions this can lead to multiple trader configurations. 11 12

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low-value demanders, and the high-cost seller trades with the high-value demander. We call this configuration C3∼CE . We can calculate P(C3∼CE ) using the above recursive equations. Since, in this case, P(C3∼CE ) = 1 − P(C2CE ), we will calculate P(C2CE ). After the first trade, only three configurations can be observed, specifically C1sB with a single trade between a low-cost supplier and high-value demander, C1sB with a single trade between a low-cost supplier and low-value demander, and C1sB with a single trade between a high-cost supplier and high-value demander. Since C1sB cannot lead to the CE configuration, we focus on the first two configurations. Here, the normalization constant, K0 in Eq. (1), is equal to Nb Ns p =(2psB + 4psb + pSB )/9. From Eq. (1), P(C1sB ) 3 3 sb (2), P(C2CE ) = P(C2CE |C1sB ) P(C1sB ) + P(C2CE |C1sb ) P(C1sb ),

= 2psB /(2psB + 4psb + pSB ) and

P(C1sb )

NB Ns p 3 3 sB

+

NB NS 3 3

pSB +

= 4psb /(2psB + 4psb + pSB ). Using Eq.

where the conditional probabilities are calculated using Eq. (3). Since C1sB can only evolve into the CE configuration (as no high-value demanders are left on the market, and thus the only feasible trade is between a low-cost supplier and low-cost demander), P(C2CE |C1sB ) = 1.14 To calculate P(C2CE |C1sb ) we first evaluate the normalization constant K1 (C1sb ) which is equal to (psB + pSB + psb )/4. Configuration C1sB will evolve into the CE configuration if a trade occurs between the high-value demander and the remaining low-cost supplier. Thus, by applying Eq. (3), we get P(C2CE |C1sb ) = psB /(psB + pSB + psb ). Therefore, the probability of the CE configuration is P(C2CE ) =

2 psB 2 psB + 4 psb + pSB



1+



2 psb . psb + psB + pSB

(4)

Note that this CE-configuration probability depends on the probability that the different types of suppliers and demanders agree on a transaction price. This, in turn, depends on the hypothesized transaction mechanism. A possible transaction mechanism is discussed later. However, in Eq. (4) we see that the likelihood of observing the CE configuration increases, and eventually tends to 1, as psB becomes much larger than the trading probability between the other types of agents. We will show later that this outcome is favored as traders become more impatient.

2.2. Distribution of transaction prices and trading probability The analytics above have focused on identifying the possible trader configurations that arise in the Bazaar model. Our only condition on admitting a pair of traders to the configuration was that they could each make a non-negative profit at some price. This condition requires only that the buyer value, v, of the demander is at least as big as the seller cost, c, of the supplier, and that the price is in [c, v]. Here we wish to explicitly explore the price setting process. We assume that the transactions are independent of one another and that our paired traders continue to propose potential transaction prices to one another until they either agree on a trade or exhaust their patience. We use only a uniform proposal mechanism, however other mechanisms could be considered. Under uniform proposals, we assume that the two traders propose a potential price—each based on a uniform, random distribution—and execute a transaction when the proposed prices cross. The supplier randomly proposes a price ps in the range [c, vmax ], where vmax = v|b| is the maximum of the buyer values of the demanders, while the demander proposes a price pb in the range[cmin , v], where cmin = c1 is the minimum of the seller costs of the suppliers. The corresponding probability density functions are fs (ps ) = fb (pb ) =

1

vmax − c 1

v − cmin

∀ps ∈ [c, vmax ] and 0 otherwise, ∀pb ∈ [cmin , v] and 0 otherwise, p +p

for the supplier and demander respectively.15 If ps < pb then the trade occurs at the average of the two offers p = s 2 b . The probability density function of the transaction price, if a transaction occurs, is obtained by renormalizing the integral



Min[p−c,v−p]

fs (ps = p − t) × fb (pb = p + t)dt = 0

Min[p − c, v − p] (vmax − c) (v − cmin )

(5)

in the range [c, v]. So, the probability density function of the transaction price is fs,b (p) =

4 Min[p − c, v − p] (v − c)2

∀p ∈ [c, v] and 0 otherwise.

(6)

This result can also be obtained by applying Eq. (3) to configuration C1sB . For the sake of simplicity, continuous distributions are assumed. However, qualitatively identical results are obtained by considering discrete distributions of proposed prices. 14 15

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According to this distribution, the expected transaction price between a supplier with seller cost c and a demander with buyer value v is

 E(p) =



v

p · fs,b (p) dp = c

v

p·

4 Min[p − c, v − p] (v − c)

c

2

dp =

v+c 2

.

(7)

Note that, unlike CE, this model predicts that the expected transaction price depends on the seller cost and buyer value of the two traders. What is the probability that the prices proposed by the two traders match and a transaction occurs? This probability is given by the probability that ps < pb , which is 0 if c > v and

 P(ps < pb ) =



v

fb (pb )dpb · c

pb

fs (ps )dps =

c

(v − c)2 2(v − cmin ) · (vmax − c)

(8)

otherwise. This probability can be used to calculate the probability that a buyer and a seller with limited “patience” end up trading. Specifically, assume that the maximum number of proposals attempted before giving up is . Then the probability that they will trade is given by: P = 1 − [1 − P(ps < pb ) s(v − c)] ,

(9)

where the step function, s(x) = 1 if x ≥ 0 and s(x) = 0 otherwise, is introduced here to include the case where c > v. This probability is a non-decreasing function of . If the two traders are infinitely patient then the probability they trade reaches the maximum value, s(v − c), indicating that the two agents will always agree on a transaction price, unless the seller cost of the supplier is higher than the buyer value of the demander. We designate this case as AB∞ . On the contrary, if the two traders are impatient and make only one offer, that is = 1, then the probability takes its minimum value, P(ps < pb ) · s(v − c). We designate this case as AB1 . We focus on these two extremes in our comparisons of the Bazaar model’s predictions and the experimental outcomes. To illustrate the above equations, consider again the simple market of three demanders and three suppliers, with NB = 1 demander with vB and Nb = 2 demanders with vB , and Ns = 2 suppliers with cs and NS = 1 supplier with cs . To simplify things, v −c suppose that vB − cS = vb − cs = d and set vB − cs = ˛d, with ˛ = vB −css ≥ 1. Given these parameters, the probability that a b

attempts is, according to Eq. (9), high-value demander and a low-cost supplier will reach an agreement and trade in psB = 1 − (1/2) , while the probability that a low-value demander (high-value demander) and a low-cost supplier (high-cost supplier) will reach an agreement and trade in



attempts is psb = pSB = 1 − 1 −

1 2˛



.

If agents are infinitely patient (AB∞ ), that is → ∞, then psB = psb = pSB = 1, and P(C2CE ) = 10/21 < 1/2, according to Eq. (4). So, if agents are infinitely patient then the CE trader configuration is not the most likely one to arise. On the contrary, if ˛(˛+4) 1 , and P(C2CE ) = (2 2˛+5)(˛+2) ≥ 12 for any value of we assume that agents are impatient (AB1 ), = 1 and psB = 21 , psb = pSB = 2˛ ˛ ≥ 1.09. Thus, even for small differences between the buyer values of the two types of buyers and between the seller costs of the two types of sellers (˛ > 1), trades between high-value demanders and low-cost sellers are favored by more impatient agents, and the CE trader configuration is more likely to occur. Thus, the impatience of agents, together with an increasing spread among the buyer values of demanders and among the seller costs of suppliers, introduces a selection mechanism into the bazaar that favors socially efficient outcomes and, eventually, leads to CE. A more general insight about the relationship between the AB1 model and CE can be obtained by considering the specific set of configuration branches leading to the CE trader configuration according to Eq. (2). Consider a simple market as above, but relax the constraint on the total number of both suppliers and demanders. The CE trader configuration for this market will be CCE = (nsB , nsb , nSB , nSb ) = (Min[Ns , NB ], Ns − Min[Ns , NB ], NB − Min[Ns , NB ], 0). Now, consider the branch of trader configurations in which the Min[Ns , NB ] trades between high-value demanders and low-cost suppliers occur first. If the first Min[Ns , NB ] trades occur between high-value demanders and low-cost suppliers, then the final configuration can only be the CE configuration, regardless of the order and nature of the last transactions, since the configuration (Min[Ns , NB ], Ns − Min[Ns , NB ], NB − Min[Ns , NB ], 0) is completely determined by the value of nsB = Min[Ns , NB ]. Given Eq. (2), we know that



Q −1

PQ (Q, 0, 0, 0) =

nsB =0

(Ns − nsB )(NB − nsB )˛ , (Ns − nsB )(NB − nsB )˛ + (Ns − nsB )Nb + (NB − nsB )NS

(10)

where Q = Min[Ns , NB ]. We know that P(CCE ) is at least as big as PQ (Q, 0, 0, 0), since this is only one possible path to this end. So, P(CCE ) ≥ PQ (Q, 0, 0, 0). In the limit as ˛→ ∞, PQ (Q, 0, 0, 0) → 1 according to Eq. (10), and therefore P(CCE ) will also tend to one. Thus, as we bias the expectations of the agents toward more constrained price intervals and increase the spread between the two buyer values, we find that the AB1 and CE models converge.

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2.3. An approximate formula for the expected configuration Though Eqs. (1)–(3) completely determine the trader configuration probabilities in the AB model, it is useful to have an approximate formula for the expected configuration. In particular, this approximation dramatically reduces the computation needed for markets with large numbers of traders. The expected value of the number of trades, n¯ ij , between (si , vj ) can be approximated by n¯ ij = lim n¯ tij , where n¯ tij is defined t→∞

through the following recursive equation:



n¯ t+1 ij

=

n¯ tij

+

Bj − M|b| −

|s|

n¯ t q=1 qj |s| |b| n¯ t r=1 q=1 rq







·

Si − M|s| −

|b|

n¯ t r=1 ir |s| |b| n¯ t r=1 q=1 rq





 pij ,

(11)

and initial condition n¯ 0ij = 0, where the terms in parentheses represent the expected proportion of demanders with vj (suppliers with ci ) remaining in the market after t attempts to trade. This equation can then be used to obtain an approximation |s| |b| |s| |b| of (1) the expected number of trades, E(T ) ∼ n¯ ij , (2) the expected surplus E(S) ∼ n¯ ij (vj − ci ), and (3) the = =





i=1

j=1

i=1

j=1

|s| |b| 1 expected average price E(p) ∼ n¯ (v + ci )/2 that will be observed in the AB model. = E(T i=1 j=1 ij j ) To demonstrate the use of Eq. (11), consider a market with 30 suppliers and 30 demanders. More specifically, assume that we have NB = 10 demanders with vB = $40 and Nb = 20 demanders with vb = $20, and Ns = 20 suppliers with cs = $10 and NS = 10 suppliers with cS = $30. The proportions of the different types of suppliers and demanders are the same as those considered in our prior example, while the buyer values and seller costs are drawn from the classroom experiments discussed v −c below. Given these parameters, ˛ = vB −css = 3. The (exact) expected configuration under AB1 is (nsB = 7.559, nsb = 12.441, b nSB = 2.441, nSb = 0), while the modal configuration, occurring 32.6% of the time, is (nsB = 8, nsb = 12, nSB = 2, nSb = 0). Estimating the expected trader configuration using Eq. (11) we obtain (nsB = 7.571, nsb = 12.429, nSB = 2.429, nSb = 0), which is close to the exact expected configuration.

3. An empirical test of the models We test the Bazaar and CE models against data from twenty-four experiments conducted by Ted Bergstrom and coworkers at different universities, and from sixteen experiments conducted by Jon Sonstelie at the University of California, Santa Barbara. The classroom experiments follow Bergstrom and Miller (2000). Students were first privately assigned buyer values and seller costs. The students were then allowed to wander in the classroom and haggle with one another. Anytime a deal was made, the students reported the transaction to the experimenter, and the agreed upon price was posted in a central location.16 Suppliers have seller costs of either $10 (cs ) or $30 (cs ), and demanders have buyer values of either $20 (vB ) or $40 (vB ). Each experiment consists of two sessions differing in their proportion of the two types of demanders and suppliers, with the number of low-value demanders (low-cost suppliers) about twice the number of high-value demanders (high-cost suppliers) in Session 1, and the number of high-value demanders (high-cost suppliers) about twice the number of low-value demanders (low-cost suppliers) in Session 2. These parameters imply a CE price of $20 in Session 1 and $30 in Session 2. Each session consists of two rounds, involving repetitions with the same group of participants each with the identical buyer values and seller costs. Thus, in the second round, participants might exploit the experience gained in the first round in their attempts to trade. The Bergstrom data includes both the first and second round of each session, while Sonstelie only reports the second round. We compare the predictions of the models across four dimensions: (1) transaction prices, (2) number of transactions, (3) market efficiency, and (4) the observed trader configuration. Consistent with the notation used in the previous section, a trader configuration in these experiments is described by four numbers (nsB , nsb , nSb , nSB ), where nij gives the number of trades between sellers of type i and buyers of type j, with lower-case letters representing either low-cost suppliers or low-value demanders and upper-case letters representing either high-cost suppliers or high-value demanders, respectively. CE provides precise predictions across all four of the empirical dimensions. In general, under these experimental conditions, CE predicts a single price and quantity, 100% market efficiency, and a single trader configuration. The Bazaar models are innately probabilistic, and thus they make a number of alternative predictions: prices will vary (and will be conditional on the underlying seller cost and buyer value of the two traders), quantity will equal or exceed that observed in CE (Bergstrom and Kwok, 2005), market efficiency may not be 100%, and different trader configurations can arise with known probabilities. 3.1. Transaction prices Fig. 1 shows the mean price observed in the experiments and the various theoretical predictions. Note that the data on the x-axis are arrayed nominally. The top panel of Fig. 1 contains Session 1 while the bottom contains Session 2. We find that the mean prices observed are typically above the CE prediction in Session 1 and below it in Session 2. Both AB1

16

The full experimental design and instructions to participants are provide in the supplementary materials.

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171

Fig. 1. Mean transaction prices Session 1 (top) and 2 (bottom panel). The error bar associated with the empirical averages corresponds to a one standard error of the mean given the sample size. The x-axis is nominal, with Bergstrom’s experiments labeled with “B” and Sonstelie’s with “S” followed by the experiment number. The mean expected price is given for the Asynchronous Bazaar (AB∞ ), Inpatient Asynchronous Bazaar (AB1 ), and Competitive Equilibrium (CE) models. R-1 is Round 1 (Bergstrom only) and R-2 is Round 2 observations.

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and AB∞ make price predictions that are typically closer to the observations than the CE predictions, for both sessions and datasets.17 One dramatic difference between the CE and Bazaar models is that the Bazaar models predict very different price distributions conditional on the two traders involved in the transaction, while the CE prediction is indifferent to such conditions. Figs. 2 and 3 present the observed price distributions for Sessions 1 and 2, respectively, of Bergstrom’s (top panels, both rounds) and Sonstelie’s (bottom panels, Round 2 only) experiments. The distributions are conditional on the seller cost and buyer value of the two agents involved in the transaction.18 Red and green lines indicate the expected prices according to the Bazaar and CE models, respectively. Both of the figures indicate that the observed price distribution strongly varies depending on the type of agents involved in the transaction, and that the predictions of the Bazaar models typically outperform those of CE. In the experiments, CE predicts no trade between high-value demanders and high-cost suppliers (S-B in the figures) in Session 1, nor between low-value demanders and low-cost suppliers (s-b in the figures) in Session 2. The outcomes reported in Figs. 2 and 3 deviate from this prediction, as several trades occur between these types of agents in both sessions of both sets of experiments. Furthermore, major deviations from the CE theory are observed in transactions between high-value demanders and low-cost suppliers (s-B in the figures) in both sessions of the experiments. Indeed, the average transaction price over all the transactions between low-cost suppliers and high-value demanders is around $25 in both sessions of both sets of experiments,19 and the shape of the distribution does not change significantly across the two sessions. This average value of the transaction price is in agreement with the predictions of the Bazaar models, and is at odds with those of CE, not only because the expected prices are significantly different (in terms of standard errors) than the observed ones, but also because CE predicts different expected prices for the two sessions ($20 in Session 1 and $30 in Session 2), while the observed distributions are not remarkably different from those predicted by the Bazaar model.20 3.2. Market efficiency Market efficiency is given by the relative proportion of total surplus earned by the market participants at the conclusion of trading to the maximum total surplus available in the market. CE guarantees that total surplus is maximized under the conditions of the experiments, and thus it predicts a market efficiency of 100%. Fig. 4 shows the attained efficiency for Session 1 (top panel) and 2 (bottom panel) for both Bergstrom’s and Sonstelie’s experiments. In Session 1, maximum efficiency is attained in just 7 of 48 rounds in Bergstrom’s experiments and 3 of 16 rounds in Sonstelie’s. In Session 2, 14 of Bergstrom’s rounds achieve 100% market efficiency while only 1 of Sonstelie’s does so. Thus, CE typically overestimates the level of market efficiency observed in the experiments. Alas, the Bazaar models tend to underestimate market efficiency (see Fig. 4), though the observed efficiency is closer, on average, to that predicted by AB1 than CE. The AB∞ model predicts a lower efficiency than AB1 , due to the increased probability of extra-marginal trading in AB∞ (AB1 predicts the highest efficiency of the Bazaar models, and slightly overestimates the market efficiency in Session 1).21 3.3. Number of transactions Fig. 5 reports the number of trades observed in the experiments. CE theory tends to slightly underestimate the total number of transactions observed in the experiments (Chamberlin, 1948; Bergstrom and Kwok, 2005), while the two Bazaar models tend to overestimate this value. Both the Bazaar models and CE theory predict that the final trader configuration will

17 The divergence of expected prices from those observed can be captured by the mean of the values D(model) and the mean of the absolute values DA (model) of the difference between the mean transaction price observed in the experiment and that predicted by the various models. For Session 1, Round 1, we find D(CE) = $1.6, D(AB∞ ) = − $0.02, D(AB1 ) = $0.4, DA (CE) = $1.9, DA (AB∞ ) = $1.7, and DA (AB1 ) = $1.7; and for Round 2 (which includes Sonstelie’s experiments) D(CE) = $1.0, D(AB∞ ) = − $0.5, D(AB1 ) = − $0.1, DA (CE) = $1.5, DA (AB∞ ) = $1.2, and DA (AB1 ) = $1.1. For Session 2, Round 1, we find D(CE) = − $3.0, D(AB∞ ) = − $1.6, D(AB1 ) = − $2.0, DA (CE) = $3.0, DA (AB∞ ) = $1.6, and DA (AB1 ) = $2.0; and for Round 2 (which includes Sonstelie’s experiments) D(CE) = − $1.9, D(AB∞ ) = − $0.6, D(AB1 ) = − $1.0, DA (CE) = $2.0, DA (AB∞ ) = $1.2, and DA (AB1 ) = $1.4. These results indicate that, on average, Bazaar models outperform CE in predicting the mean transaction price observed in the experiments. 18 Specifically, s-B includes transactions between low-cost suppliers and high-value demanders, s-b includes transactions between low-cost suppliers and low-value demanders, S-B includes transactions between high-cost suppliers and high-value demanders, and S-b includes transactions between high-cost suppliers and low-value demanders. 19 Average price and standard error are reported in each panel of Figs. 2 and 3. 20 A two-sample (exponentially tilted) bootstrap test (Efron and Tibshirani, 1993), based on 106 bootstrap replicates of data, testing the null hypothesis that the population mean is the same for the distributions of transaction prices (s-B) in Session 1 and Session 2, gives a p-value of 0.004 for Bergstrom’s experiments and of 0.002 for Sonstelie’s experiments. If we exclude from the analysis all the transaction prices observed in the experiments presenting some inconsistency in the outcome, that is those marked by  =“-” in Table 1 and 2, the p-value for Bergstrom’s experiments becomes 0.017. Some of the empirical results from Bergstrom’s and Sonstelie’s experiments show statistically significant differences, in particular, in the distribution of transaction prices between low-cost suppliers and high-value demanders (s-B) in Session 1. A bootstrap test, testing the null hypothesis that the population mean is the same for the two datasets, gives a p-value smaller than 0.001; for the same types of agents in Session 2, the p-value is 0.035. The bootstrap approach has been used here in place of more standard Student’s t-tests because the empirical distributions of transaction prices show significant deviations from the Normal distribution. However, t-tests results qualitatively agree with the numbers reported here. 21 The means of the values of the difference, D(model), and of the absolute values of the difference, DA (model), between the observed market efficiency and the prediction of the models are for Session 1, Round 1, D(CE) = −0.09, D(AB∞ ) = 0.05, D(AB1 ) = −0.02, DA (CE) = 0.09, DA (AB∞ ) = 0.07, and DA (AB1 ) = 0.05; and for Round 2 (which includes Sonstelie’s experiments), D(CE) = −0.09, D(AB∞ ) = 0.05, D(AB1 ) = −0.02, DA (CE) = 0.09, DA (AB∞ ) = 0.09, and DA (AB1 ) = 0.07. The difference between the Rounds 1 and 2 experiments is marginal. For Session 2, Round 1, we find D(CE) = −0.04, D(AB∞ ) = 0.09, D(AB1 ) = 0.03, DA (CE) = 0.04, DA (AB∞ ) = 0.10, and DA (AB1 ) = 0.05; and for Round 2 (which includes Sonstelie’s experiments), D(CE) = −0.06, D(AB∞ ) = 0.07, D(AB1 ) = 0.002, DA (CE) = 0.06, DA (AB∞ ) = 0.09, and DA (AB1 ) = 0.05.

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Fig. 2. Transaction price distributions in Session 1 conditional on trader types, for Bergstrom’s (top panels, both rounds) and Sonstelie’s (bottom panels, Round 2 only) experiments. Red (dark) lines indicate the expected transaction price according to the Bazaar models, assuming a symmetric distribution of transaction prices, and green (light) lines indicate the expected transaction price according to CE. The mean and standard error of transaction prices are reported in each panel. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

be complete, in the sense that at the end of the trading activity, no further mutually profitable trades exist, that is, nothing is left on the table. Moreover, the final trader configurations admissible in the Bazaar models always involve either the same number, or more, trades than CE theory. While CE theory better fits the empirical observations on total trades relative to the Bazaar models,22 this result is driven by observations tied to markets that end with incomplete trader configurations,

22 The means of the values of the difference, D(model), and of the absolute values of the difference, DA (model), between the number of trades observed and those predicted by the models, DA (model), for Session 1, Round 1, are D(CE) = 0.8, D(AB∞ ) = −3.1, D(AB1 ) = −1.3, DA (CE) = 1.0, DA (AB∞ ) = 5.3, and DA (AB1 ) = 4.1; and for Round 2 (which includes Sonstelie’s experiments), D(CE) = 0.9, D(AB∞ ) = −2.9, D(AB1 ) = −1.0, DA (CE) = 1.1, DA (AB∞ ) = 2.9, and DA (AB1 ) = 1.3. For Session 2, Round 1, D(CE) = 0.4, D(AB∞ ) = −3.1, D(AB1 ) = −1.4, DA (CE) = 0.8, DA (AB∞ ) = 3.1, and DA (AB1 ) = 1.4; and for Round 2 (which includes Sonstelie’s experiments), D(CE) = 0.6, D(AB∞ ) = −3.1, D(AB1 ) = −1.2, DA (CE) = 1.0, DA (AB∞ ) = 3.1, and DA (AB1 ) = 1.4.

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Fig. 3. Transaction price distributions in Session 2 conditional on trader types, for Bergstrom’s (top panels, both rounds) and Sonstelie’s (bottom panels, Round 2 only) experiments. Red (dark) lines indicate the expected transaction price according to the Bazaar models, assuming a symmetric distribution of transaction prices, and green (light) lines indicate the expected transaction price according to CE. The mean and standard error of transaction prices are reported in each panel. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

that is, markets that end with some mutually beneficial trades left undone. Thus, some participants in the experiments decide not to trade even thought they could earn a mutual profit. Indeed, in some experiments, we find residual highvalue demanders or low-cost suppliers, and either of these agent types could make mutually beneficial trades with any counterpart in the market. This violation of the assumption of complete trading—an assumption underlying both the Bazaar and CE models—will tend to favor CE predictions about quantity, as the Bazaar models always have the CE quantity as a lower bound.

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175

Fig. 4. Market efficiency in Session 1 (top) and 2 (bottom). The figure shows the mean difference between the observed market efficiency and the predictions of the models under Competitive Equilibrium (CE), Asynchronous Bazaar (AB∞ ), and Inpatient Asynchronous Bazaar (AB1 ). R-1 is Round 1 (Bergstrom only) and R-2 is Round 2 observations.

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Fig. 5. Number of trades in Session 1 (top) and 2 (bottom). The figure shows the mean difference between the observed number of trades and the predictions under Competitive Equilibrium (CE), Asynchronous Bazaar (AB∞ ), and Inpatient Asynchronous Bazaar (AB1 ). R-1 is Round 1 (Bergstrom only) and R-2 is Round 2 observations.

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177

3.4. Configurations The models explored here make varying predictions about the trader configurations we will observe in the world. Tables 1 (for Session 1) and 2 (for Session 2) provide the final trader configurations (columns 7–10) observed in the experiments. Here we investigate how the final trader configurations align with the various models. At the outset, all of the models predict that no agent would ever voluntarily trade at a loss. One implication of this assumption, given the parameters of the experiments, is that nSb = 0. However, we do observe some markets that violate this constraint. In Session 1, six of the forty markets show such transactions (five in Round 1 and one in Round 2), while in Session 2, eight of the forty markets have trades where at least one of the traders takes a loss (five in Round 1 and three in Round 2). These latter observations, linked with our earlier observation that markets often end with possible unrealized gains to trade, provide some interesting fodder for the development of future theories. Are these actions due to mistakes and laziness on the part of our participants, or do they reflect more insidious forces driven by behavioral dynamics or information flows? We note that in these types of experimental markets, if it becomes known that gains remain, participants do tend to make the additional trades and realize the gains. Modulo the issues raised in the previous paragraph, how well do the final trader configurations match the theoretical predictions? To facilitate this comparison, we start with the final trader configuration observed in the experiment (columns 7–10 in Tables 1 and 2). If the values in columns 7–9 represent a complete configuration, then we use these values as the final trader configuration (shown in columns 11–13, as we implicitly set nSb to 0). If not, we use the complete trader configurations that could arise from the realized trader configuration. That is, assuming we start at the realized trader configuration, what additional profitable trades are possible that will result in a complete trader configuration. We call this new configuration the associated trader configuration. In most cases, there is only one such associated configuration, but in a few cases (designated by m), multiple configurations are possible. Columns 11–13 list the associated complete trader configuration that we use in the analysis below. The minimum number of configuration changes needed to obtain a complete configuration from the observed one, is reported in column 14 (labeled ).23 How often do the associated trader configurations conform to the theoretical predictions? Using CE as a benchmark, we find that (see column 15) fifteen of the sixty-four configurations in Session 1 and twenty-seven of the sixty-four in Session 2 are consistent with the CE prediction. For the two Bazaar models, we derive one-sided p-values (columns 16–18) based on the null hypothesis that the trading configuration observed would arise in the respective Bazaar model. The p-value is calculated as the probability of observing a configuration with equal or higher efficiency than the one observed. In general, the p-values tend to be ordered with AB1 > AB∞ . As is evident in the tables, many of the trading configurations that arose are quite likely under the Bazaar models (assuming we reject the null at a 1% confidence level). Indeed, all of the p-values for the AB1 model are larger than 0.01 in both sessions, suggesting non-trivial agreement between this model and the data. Typically, the observed trader configurations that are not all that likely under the Bazaar models, are those minority of cases where the CE is attained. In some of these cases, six in Session 1 and two in Session 2 of the experiments, the observed CE trader configuration is not that likely of an outcome according to the Bazaar models, in the sense that all of the p-values are smaller than 0.05. For instance, the CE trader configuration observed in Round 2 of Session 1 of Experiment S06 is significantly unlikely under AB∞ (p-value = 0.0003) and moderately unlikely under the AB1 model (p-value = 0.039). When the number of agents is small (for example, in Session 1, Round 1 of B18), it becomes very difficult to discriminate between CE and Bazaar models, as the configuration predictions converge between the two models. As an overall measure of agreement between model predictions and experiments, we calculate the log-likelihood of the observed configurations separately for each round of each session of the two sets of experiments. The results, reported in Table 3, show that the likelihood of the observed outcomes according to AB1 is always higher than the likelihood according to AB∞ . We can associate a p-value with each value of the likelihood, given by the probability of obtaining a smaller value of the likelihood according to the AB∞ and AB1 models. This probability is based on the analysis of 106 independent realizations of each set of experiments according to the two Bazaar models. The calculated p-values indicate that the AB1 hypothesis cannot be rejected at the 1% level of statistical significance in all sets of experiments, except for Session 1 of Sonstelie’s experiments. 3.5. Summary of the empirical results In summary, the analysis of experiments indicates that (1) the distribution of transaction prices strongly depends on the agent types involved in the transaction, which is a result predicted by the Bazaar models and not by CE, (2) most of the observed trader configurations have a non-negligible probability of occurrence according to the Bazaar models and many are inconsistent with the predictions of CE, (3) typically, the CE predictions of market efficiency overestimate while

23 For a few of the experiments, the reported data is inconsistent, for example, the number of transactions involving agents of a certain type exceeds the number of that type of agent in the experiment (for example, in Session 1 of B05, ten high-value demanders are involved in trades, yet only eight such demanders exist in the market). Such cases are designated by “-” and are not used in the analysis. More generally, 39 out of 64 experimental configurations were either incomplete or inconsistent in Session 1 of the experiments, and 29 out of 64 in Session 2.

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Table 1 Final trading configurations, session 1. Exp

R

Ns

nSB

nSb

n∗sB

n∗sb

n∗SB



CE

p-val AB∞

p-val AB1

B01 B01 B02 B02 B03 B03 B04 B04 B05 B05 B06 B06 B07 B07 B08 B08 B09 B09 B10 B10 B11 B11 B12 B12 B13 B13 B14 B14 B15 B15 B16 B16 B17 B17 B18 B18 B19 B19 B20 B20 B21 B21 B22 B22 B23 B23 B24 B24

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

16 16 16 16 20 20 20 20 16 16 16 16 17 17 15 15 16 16 15 15 8 8 12 12 18 18 6 6 8 8 22 22 8 8 3 3 8 8 6 6 8 8 6 6 7 7 13 13

8 8 8 8 9 9 10 10 8 8 8 8 8 8 7 7 8 8 9 9 4 4 6 6 9 9 3 3 4 4 11 11 4 4 1 1 4 4 3 3 4 4 3 3 3 3 6 6

9 9 9 9 10 10 11 11 8 8 10 10 8 8 8 8 9 9 8 8 4 4 7 7 10 10 3 3 5 5 12 12 4 4 2 2 5 5 4 4 5 5 5 5 4 4 8 8

18 18 18 18 19 19 21 21 17 17 17 17 19 19 16 16 17 17 16 16 8 8 14 14 19 19 6 6 10 10 23 23 8 8 4 4 10 10 6 6 10 10 4 4 8 8 12 12

9 8 7 7 9 7 11 10 10 6 6 8 6 7 7 8 9 9 7 8 5 3 7 6 8 6 2 2 3 4 11 10 4 4 2 1 3 2 3 4 3 5 3 3 4 4 7 7

7 8 8 8 9 12 9 9 6 10 9 6 9 10 8 7 7 6 9 8 3 4 6 4 9 11 4 4 5 4 11 12 3 4 1 1 5 6 3 2 5 2 3 3 2 3 5 6

0 1 2 2 1 3 0 1 0 2 2 4 2 1 1 0 0 0 1 0 0 3 1 1 1 4 0 0 1 1 1 2 0 0 0 0 2 3 1 0 1 2 1 1 0 0 1 1

0 0 0 0 0 0 2 0 1 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0

9 8 7 7 9 7 11 10 – 6 m – 6 7 7 8 9 9 – – – – – 6 m 6 2 2 3 4 11 10 4 4 2 m 3 2 3 4 3 – 3 3 4 4 7 7

7 8 9 9 11 13 9 10 – 10 m – 11 10 8 7 7 7 – – – – – 6 m 12 4 4 5 4 11 12 4 4 1 m 5 6 3 2 5 – 3 3 3 3 6 6

0 1 2 2 1 3 0 1 – 2 m – 2 1 1 0 0 0 – – – – – 1 m 4 1 1 2 1 1 2 0 0 0 m 2 3 1 0 2 – 2 2 0 0 1 1

0 0 1 1 2 1 2 1 – 0 2–3 – 4 0 0 0 0 1 – – – – – 2 1–2 1 1 1 1 0 0 0 1 0 0 1–2 0 0 0 0 1 – 1 1 1 0 3 0

Y N N N N N Y N N N N N N N N Y Y Y N N N N N N N N N N N N N N Y Y Y N/Y N N N Y N N N N Y Y N N

0.0001* 0.003* 0.028 0.028 0.004* 0.161 0.00002* 0.001* – 0.081 0.262 – 0.083 0.012 0.011 0.001* 0.0001* 0.0001* – – – – – 0.015 0.017 0.292 0.49 0.49 0.352 0.069 0.0003* 0.004* 0.03 0.03 0.138 0.743 0.352 0.77 0.221 0.018 0.352 – 0.603 0.603 0.022 0.022 0.013 0.013

0.028 0.191 0.523 0.523 0.258 0.871 0.016 0.124 – 0.689 0.934 – 0.703 0.336 0.317 0.059 0.032 0.032 – – – – – 0.315 0.467 0.947 0.887 0.887 0.876 0.497 0.094 0.321 0.273 0.273 0.45 0.954 0.876 0.99 0.714 0.196 0.876 – 0.956 0.956 0.232 0.232 0.32 0.32

S01 S02 S03 S04 S05 S06 S07 S08 S09 S10 S11 S12 S13 S14 S15 S16

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

14 14 12 13 14 14 15 13 14 15 15 14 13 14 14 14

7 7 6 7 7 7 8 7 7 7 7 7 7 7 7 7

8 5 7 7 7 8 8 7 7 8 8 8 8 8 8 8

14 16 14 14 14 16 16 14 14 16 16 14 16 15 16 14

4 4 7 6 6 8 7 5 4 5 5 7 8 3 4 5

8 10 5 6 7 6 7 8 10 10 9 7 5 10 10 7

3 1 0 1 1 0 1 1 2 3 2 1 0 3 4 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

m 4 7 6 6 8 7 5 4 5 m 7 8 m 4 m

m 10 5 7 8 6 8 8 10 10 m 7 5 m 10 m

m 1 0 1 1 0 1 2 3 3 m 1 0 m 4 m

2–3 0 0 1 1 0 1 1 1 0 1–2 0 0 2–3 0 2–4

N N Y N N Y N N N N N N Y N N N

0.649 0.133 0.001* 0.019 0.03 0.0003* 0.008* 0.128 0.494 0.303 0.303 0.01 0.0001* 0.88 0.563 0.303

0.991 0.665 0.054 0.355 0.431 0.039 0.278 0.739 0.966 0.925 0.925 0.296 0.023 0.999 0.984 0.921

*

p-Value < 0.01.

NS

NB

Nb

nsB

nsb

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179

Table 2 Final trading configurations, session 2. Exp

R

NS

NB

nSb

n∗sB

n∗sb

n∗SB



CE

p-val AB∞

p-val AB1

B01 B01 B02 B02 B03 B03 B04 B04 B05 B05 B06 B06 B07 B07 B08 B08 B09 B09 B10 B10 B11 B11 B12 B12 B13 B13 B14 B14 B15 B15 B16 B16 B17 B17 B18 B18 B19 B19 B20 B20 B21 B21 B22 B22 B23 B23 B24 B24

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

9 9 9 9 10 10 11 11 9 9 9 9 10 10 8 8 9 9 8 8 4 4 7 7 10 10 3 3 5 5 12 12 4 4 2 2 5 5 3 3 5 5 3 3 4 4 7 7

16 16 17 17 19 19 21 21 16 16 18 18 16 16 15 15 17 17 15 15 8 8 13 13 19 19 6 6 9 9 23 23 8 8 3 3 9 9 7 7 9 9 6 6 7 7 13 13

17 17 17 17 18 18 20 20 16 16 16 16 17 17 15 15 16 16 17 17 8 8 13 13 18 18 6 6 9 9 22 22 8 8 3 3 9 9 6 6 9 9 6 6 7 7 12 12

9 9 8 8 11 11 10 10 8 8 8 8 9 9 8 8 8 8 8 8 4 4 6 6 9 9 3 3 4 4 11 11 4 4 2 2 4 4 3 3 4 4 3 3 4 4 7 7

7 8 8 9 10 9 11 10 7 8 7 8 9 9 8 7 7 8 8 7 4 4 6 7 9 9 1 1 5 5 9 10 4 4 2 1 4 5 3 3 5 5 3 2 3 4 7 6

1 1 1 0 0 1 1 1 2 1 2 1 0 0 0 1 2 1 0 0 0 0 4 0 1 1 0 0 0 0 3 2 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1

11 9 9 8 8 9 9 10 8 8 5 8 7 7 6 8 10 8 8 9 4 4 1 6 10 8 3 4 4 4 13 12 4 3 1 2 5 4 3 3 4 3 3 4 4 3 5 6

1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

– 8 8 9 10 9 – 10 7 8 7 8 m m 8 7 – 8 8 m 4 4 – 7 – 9 m m 5 5 9 10 4 4 2 1 4 5 3 3 5 5 3 2 3 4 7 6

– 1 1 0 0 1 – 1 2 1 2 1 m m 0 1 – 1 0 m 0 0 – 0 – 1 m m 0 0 3 2 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1

– 9 9 8 8 9 – 10 9 8 9 8 m m 7 8 – 8 9 m 4 4 – 6 – 9 m m 4 4 13 12 4 4 1 2 5 4 3 3 4 4 3 4 4 3 5 6

– 0 0 0 1 1 – 0 1 0 4 0 1–2 1–2 1 0 – 0 1 1–2 0 0 – 0 – 2 3–5 2–3 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0

N N N Y Y N N N N N N N N/Y N/Y Y N N N Y N/Y Y Y N Y N N N/Y N Y Y N N Y Y Y N N Y Y Y Y Y Y N N Y Y N

– 0.006* 0.006* 0.0003* 0.00003* 0.001* – 0.001* 0.04 0.004* 0.028 0.003* 0.002* 0.002* 0.0005* 0.01 – 0.004* 0.002* 0.025 0.03 0.03 – 0.002* – 0.002* 0.927 0.927 0.011 0.011 0.027 0.004* 0.03 0.03 0.133 0.742 0.137 0.011 0.06 0.06 0.011 0.011 0.076 0.49 0.221 0.021 0.001* 0.014

– 0.253 0.269 0.047 0.015 0.12 – 0.124 0.583 0.23 0.523 0.191 0.163 0.163 0.054 0.301 – 0.21 0.1 0.428 0.273 0.273 – 0.094 – 0.162 0.997 0.997 0.189 0.189 0.632 0.321 0.273 0.273 0.43 0.951 0.658 0.189 0.346 0.346 0.189 0.189 0.386 0.887 0.719 0.217 0.051 0.302

S01 S02 S03 S04 S05 S06 S07 S08 S09 S10 S11 S12 S13 S14 S15 S16

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

7 8 7 7 7 8 8 7 7 8 8 7 8 8 8 7

15 12 13 13 14 15 15 13 14 15 15 15 15 15 15 13

14 15 13 14 14 15 16 14 14 15 15 14 15 14 15 14

7 7 6 7 7 7 8 7 7 8 8 7 6 7 7 7

5 7 7 6 5 8 7 7 5 4 7 5 5 5 7 6

2 1 0 1 2 0 1 0 2 3 1 2 2 3 0 1

9 7 6 8 8 7 9 6 8 11 5 9 7 9 8 8

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

5 7 7 6 5 8 7 7 5 4 7 5 m 5 7 6

2 1 0 1 2 0 1 0 2 4 1 2 m 3 1 1

9 8 6 8 9 7 9 7 9 11 8 9 m 9 8 8

0 1 0 0 1 1 0 1 1 1 3 0 3–4 0 1 0

N N Y N N Y N Y N N N N N N N N

0.152 0.026 0.002* 0.036 0.173 0.001* 0.016 0.002* 0.173 0.632 0.01 0.152 0.39 0.268 0.014 0.036

0.778 0.425 0.094 0.458 0.801 0.067 0.366 0.107 0.801 0.989 0.301 0.778 0.956 0.906 0.343 0.458

*

Ns

p-Value < 0.01.

Nb

nsB

nsb

nSB

180

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Table 3 Likelihood of associated configurations. Session

Exp.

Round



Log-Likelihood

p-value

AB∞

AB1

AB∞

AB1

1 1 1

B B S

1 2 2

Any Any Any

−80.0 −75.6 −55.1

−31.7 −30.9 −32.5

<10−3 <10−3 <10−3

0.054 0.085 0.002

1 1 1

B B S

1 2 2

0 0 0

−35.6 −45.6 −33.9

−13.3 −17.3 −15.9

<10−3 <10−3 <10−3

0.024 0.122 0.004

2 2 2

B B S

1 2 2

Any Any Any

−82.1 −111.8 −55.4

−31.6 −39.7 −24.0

<10−3 <10−3 <10−3

0.010 0.013 0.25

2 2 2

B B S

1 2 2

0 0 0

−39.9 −79.3 −23.0

−14.6 −27.2 −9.0

<10−3 <10−3 <10−3

0.213 0.046 0.635

the Bazaar-model predictions underestimate empirical outcomes, (4) typically, the total number of trades predicted by CE underestimates the observed number of trades while the Bazaar-model predictions tend to overestimate it, (5) a number of incomplete trader configurations are observed in the experiments, which is something unexpected according to all of the models, and (6) some CE trader configurations that were observed (six in Session 1 and two in Session 2) were unlikely according to the Asynchronous Bazaar model (p-values smaller than 5%).24 4. Conclusions The development of the CE model was an amazing accomplishment in the annals of social science, and it has been a long-lasting and useful framework for describing market behavior. Notwithstanding this vitality, its theoretical foundations have occasionally been questioned (for example, Hayek, 1945) and experimental data have not always aligned easily with its predictions (most recently, see Bergstrom and Kwok, 2005; Parendo, 2010). Admittedly, our experimental data—with profit-seeking students motivated by pride or extra credit, under a market mechanism that did not involve a centralized auctioneer (but did publicly post transaction prices)—pose a bit more of a challenge to the idealized notion of CE than previous experiments involving some form of a double auction. Yet, it is precisely such challenges that truly test a theory. Here we propose a simple, first-principle approach to the problem of market behavior inspired by the chaos embodied in a trading bazaar. At the core of this alternative model is the notion that traders randomly meet (asynchronously) and that trade may occur when mutual gains are possible. Such a model puts a premium on the configuration of traders and at the heart of our analytic results is a probability distribution across these possible configurations. In the Bazaar model, the expected prices are tied to the characteristics of the specific traders involved in the trade, perhaps reflecting behavioral considerations, but regardless a clearly observable feature of the experiments. CE also makes a prediction about the traders involved in its hypothesized transactions, but the explicit notion of a probability distribution across these configurations has not been emphasized, since all trades occur at the equilibrium price and the expected surplus is not contingent on the exact pairings. Moreover, the models diverge on the expected quantity of trades (with the Bazaar model predicting at least as many trades as CE, but perhaps more) and on the expected efficiency of the market (with CE’s guaranteed 100% serving only as an upper bound for the Bazaar model). While the principles underlying the Bazaar model are rather simple, deriving precise analytics is less so. However, we were able to obtain recursive equations to calculate the exact trader configuration probability. Perhaps there is an easier analytic approach here that can be derived—in the case of CE, the non-trivial mathematical and mechanistic problems posed by the theory were resolved in one of Science’s most elegant graphical solutions: supply and demand curves. Regardless, even without simple analytic solutions, the predictions of the Bazaar model are easily derived using straight-forward computational methods. The CE and Bazaar model are closely related. Indeed, one can think of the Bazaar model as a way to introduce a reasonable error process into CE. The relationship between the two models becomes most apparent when we introduce into the Bazaar model a simple mechanism for preferential trading based on the impatience of traders. As traders become more impatient, the system favors trades that generate more surplus, and CE-like outcomes become more likely. The CE model has certainly stood the test of time. Nonetheless, a lot has changed in our ability to experiment on, and model, complex social systems since CE was created. The Bazaar model provides an interesting class of market models, with an inherent simplicity in first-principle design and an apparent ability to capture some observations that are anomalous under CE, while also providing a micro-foundation from which CE-like outcomes can emerge.

24

The synchronous trading model (Miller and Tumminello, 2012) results in an increased probability of such configurations.

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