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Convergence of outcomes and evolution of strategic behavior in double auctions

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Abstract

We study the emergence of strategic behavior in double auctions with an equal number of buyers and sellers, under the distinct assumptions that orders are cleared simultaneously or asynchronously. The evolution of strategic behavior is modeled as a learning process driven by a genetic algorithm. We find that, as the size of the market grows, allocative inefficiency tends to zero and performance converges to the competitive outcome, regardless of the order-clearing rule. The main result concerns the evolution of strategic behavior as the size of the market gets larger. Under simultaneous order-clearing, only marginal traders learn to be price takers and make offers equal to their valuations/costs. Under asynchronous order-clearing, all intramarginal traders learn to be price makers and make offers equal to the competitive equilibrium price. The nature of the order-clearing rule affects in a fundamental way what kind of strategic behavior we should expect to emerge.

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Notes

  1. We use a stronger notion of role-symmetry later in the paper.

  2. With obvious modifications, Results 1–3 listed in Section 5 for the continuous double auction hold also for the call market.

  3. For brevity, we leave this qualification implicit in the statement of similar following results.

  4. They look at two cases: cartels and single players. We consider only this latter, as it is more appropriate.

  5. In the language of mechanism design, ZF’s trading strategies from Eq. 3 are not incentive-compatible for the sellers. The convex markup rule in Cervone et al. (2009) is incentive-compatible.

References

  • Anufriev M, Arifovic J, Ledyard J, Panchenko V (2011) Efficiency of continuous double auctions under individual evolutionary learning with full or limited information. J Evol Econ. doi:10.1007/s00191-011-0230-8

    Google Scholar 

  • Ashlock D (2006) Evolutionary computation for modeling and optimization. Springer

  • Aumann R (1987) Game theory. In: Eatwell J, Milgate M, Newman P (eds) The New Palgrave: a dictionary of economics, vol 2. Macmillan, pp 460–482

  • Cervone R, Galavotti S, LiCalzi M (2009) Symmetric equilibria in double auctions with markdown buyers and markup sellers. In: Hernández C, Posada M, López-Paredes A (eds) Artificial economics. Springer, pp 81–92

  • Chatterjee K, Samuelson W (1983) Bargaining under incomplete information. Oper Res 31:835–851

    Article  Google Scholar 

  • Dawid H (1999) On the convergence of genetic learning in a double auction market. J Econ Dyn Control 23:1545–1567

    Article  Google Scholar 

  • Fano S (2010) Comportamento strategico nell’asta doppia continua. Master’s thesis, Università Ca’ Foscari Venezia

  • Fano S, LiCalzi M, Pellizzari P (2010) Convergence of outcomes and evolution of strategic behavior in double auctions. Working paper 196, Department of Applied Mathematics, Università Ca’ Foscari Venezia, February. Available at: http://ideas.repec.org/p/vnm/wpaper/196.html

  • Friedman D (1993) The double auction institution: a survey. In: Friedman D, Rust J (eds) The double auction market: institutions, theories and evidence. Perseus, Cambridge, pp 3–25

    Google Scholar 

  • Gode DK, Sunder S (1993) Allocative efficiency of markets with zero intelligence traders: market as a partial substitute for individual rationality. J Polit Econ 101:119–137

    Article  Google Scholar 

  • Leininger W, Linhart PB, Radner R (1989) Equilibria of the sealed-bid mechanism for bargaining with incomplete information. J Econ Theory 48:63–106

    Article  Google Scholar 

  • Lettau M (1997) Explaining the facts with adaptive agents: the case of mutual fund flows. J Econ Dyn Control 21:1117–1147

    Article  Google Scholar 

  • MacKie-Mason JK, Wellman MP (2006) Automated markets and trading agents. In: Tesfatsion L, Judd KL (eds) Handbook of computational economics, vol 2. Elsevier, pp 1381–1431

  • Mendelson H (1985) Random competitive exchange: price distributions and gains from trade. J Econ Theory 37:254–280

    Article  Google Scholar 

  • Myerson RB, Satterthwaite MA (1983) Efficient mechanisms for bilateral trading. J Econ Theory 29:265–281

    Article  Google Scholar 

  • Phelps S, Parsons S, McBurney P (2006) An evolutionary game-theoretic comparison of two double-auction market designs. In: Faratin P, Rodriguez-Aguílar JA (eds) Agent-mediated electronic commerce VI. Springer, pp 101–114

  • Rustichini A, Satterthwaite MA, Williams SR (1994) Convergence to efficiency in a simple market with incomplete information. Econometrica 62:1041–1063

    Article  Google Scholar 

  • Satterthwaite MA, Williams SR (1993) The bayesian theory of the k-double auction. In: Friedman D, Rust J (eds) The double auction market: institutions, theories and evidence. Perseus, Cambridge, pp 99–123

    Google Scholar 

  • Satterthwaite MA, Williams SR (2002) The optimality of a simple market mechanism. Econometrica 70:1841–1863

    Article  Google Scholar 

  • Zhan W, Friedman D (2007) Markups in double auction markets. J Econ Dyn Control 31:2984–3005

    Article  Google Scholar 

Download references

Acknowledgements

We thank the audiences at Scuola Superiore Sant’Anna, Université de Cergy-Pontoise, Université Paris 1–Sorbonne, Universität Innsbruck, Katholieke Universiteit Leuven, ESHIA 2010, CEF 2010, as well as two referees for their comments. Impronta Café generously hosted the final drafting of the paper. The second and third author acknowledge financial support from MIUR under grants 2007EENEAX and 2007TKLTSR.

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Correspondence to Marco LiCalzi.

Appendix: The bilateral trading model

Appendix: The bilateral trading model

The bilateral trading model (with simultaneous order clearing) was introduced in Chatterjee and Samuelson (1983) and studied in Myerson and Satterthwaite (1983), spawning a long and still flourishing literature. It describes a situation where one buyer and one seller are engaged in the trade of a single object. The environment is the same as in Section 2.1, but there are only one buyer and one seller so n = 1. Viewed as a game with incomplete information, it is equivalent to the k-double auction described in Section 2.2 for k = 1/2 and n = 1.

An equilibrium profile (β, α) of bidding and asking functions for the bilateral trading model requires that a buyer with valuation v offers a bid b = β(v) that solves

$$ \max\limits_b \int^1_0 \left[ v - \frac{\alpha(c)+b}{2} \right] \mathbf{1} \! \left\{b \ge \alpha(c)\right\} \mbox{d} G(c) $$
(4)

and a seller with cost c submits an ask a = α(c) that solves

$$ \max\limits_a \int^1_0 \left[ \frac{a+\beta (v)}{2} - c \right] \mathbf{1} \! \left\{a \le \beta (v)\right\} \mbox{d} F(v), $$
(5)

where \(\mathbf{1}\! \left\{ \cdot \right\}\) is the indicator function.

The formal description of the bilateral trading model under asynchronous order clearing requires only the following changes. There are two equally likely queues: the buyer arrives first, or the seller arrives first. If a buyer with valuation v arrives first, he finds no outstanding ask and hence records his bid b = β(v) on the buying book and, if trade occurs, the price is b. Similarly, if a seller with cost c arrives first, she writes her ask a = α(c) on the selling book and, if trade occurs, the price is a. Therefore, if trade occurs, it takes place at price b with probability 1/2 and at price a with probability 1/2. Recall that, under simultaneous order clearing, the transaction price is p = (a + b)/2. Therefore, roughly speaking, the expected value of the trading price is the same under simultaneous or asynchronous order clearing but the latter one adds some variability around it. (Formally speaking, the distribution of the trading price is a mean-preserving spread.)

Given that traders are risk neutral, this immediately translates in the strategic equivalence of the equilibria for the two models. We prove this claim by showing that the expected payoffs under any strategy profile (β, α) are the same for the two models. In the bilateral trading model under asynchronous order clearing, a buyer with valuation v who offers a bid b = β(v) obtains a payoff

$$ \max\limits_b \frac{1}{2} \int^1_0 \left[ \strut v - \alpha (c) \right] {\bf{1}} \! \left\{b \ge \alpha(c)\right\} {\rm{d}}G(c) + \frac{1}{2} \int^1_0 \left[ \strut v - b \right] {\bf{1}} \! \left\{b \ge \alpha(c)\right\} {\rm{d}}G(c). $$
(6)

In the bilateral trading model under simultaneous order clearing, the payoff to a buyer with valuation v who offers a bid b = β(v) is given in Eq. 4. Clearly, Eqs. 4 and 6 are identical. A similar argument applies for the seller. Therefore, the set of equilibria under arbitrary priors for the two models is the same. More generally, a similar argument applies to show the strategic equivalence of the equilibria for a k-double auction and a continuous double auction where the buyer arrives before the seller with probability k. Note that, while strategic equivalence holds, equilibrium payoffs coincide only in expectation.

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Fano, S., LiCalzi, M. & Pellizzari, P. Convergence of outcomes and evolution of strategic behavior in double auctions. J Evol Econ 23, 513–538 (2013). https://doi.org/10.1007/s00191-011-0226-4

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