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Theory and Decision

Erschienen in:

02.06.2015

Optimal stealing time

verfasst von: Andrea Gallice

Erschienen in: Theory and Decision | Ausgabe 3/2016

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Abstract

We study a dynamic game in which players can steal parts of a homogeneous and perfectly divisible pie from each other. The effectiveness of a player’s theft is a random function which is stochastically increasing in the share of the pie the agent currently owns. We show how the incentives to preempt or to follow the rivals change with the number of players involved in the game and investigate the conditions that lead to the occurrence of symmetric or asymmetric equilibria.

Vorheriger Artikel Richter–Peleg multi-utility representations of preorders

Nächster Artikel Make-up and suspicion in bargaining with cheap talk: An experiment controlling for gender and gender constellation

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As an example of a situation that matches some of the key features of the game, consider the case of electoral competition among political candidates. By campaigning on specific topics, a candidate may target a particular opponent and thus “steal” a portion of his voters. Moreover, larger players (i.e., candidates with many supporters) are usually able to raise more funds, so they can afford more expensive campaigns, which are in turn expected to be more effective.

More recent literature on timing games has focused on generalizing former results (Bulow and Klemperer 1999), in providing a unified framework to study preemption games and wars of attrition (Park and Smith 2008), or in experimentally testing some of the theoretical results (Brunnermeier and Morgan 2010).

There are a number of things to notice here. First, we set \(K<T\) because we are interested in studying a situation in which stealing opportunities are a scarce resource, and players must decide when to use them. Second, an agent can freely change the rival he targets across periods: agent a may steal from b in a certain period and then from c in a subsequent period. Finally, we assume for simplicity that there are no explicit monetary costs associated with the act of stealing. Such an assumption implies little loss of generality since all the results would remain valid as long as stealing costs do not exceed a certain threshold.

Consider, for instance, a stealing game with \(n=5\). Proposition 3 implies that preempting equilibria can emerge only in partitions (5) and (3, 2). The number of preempting equilibria is thus 44: there exist \(4!=24\) equilibrium outcomes in partition (5) and 20 equilibrium outcomes in partition (3, 2) (ten couples can be drawn from a set of 5 elements; for any of these couples there are two possible circles that can emerge in the part that involves 3 players). On the contrary, Proposition 4 states that postponing equilibria can emerge only in partition (5) since players must necessarily belong to a circle \(C_{5}^{2}\). It follows that there are only 24 postponing equilibria.

Let \(N=\left\{ a,b,c,d,e\right\} \). The outcome \(O_{1}=\left( (b,\emptyset ),(a,\emptyset ),(\emptyset ,d),(\emptyset ,e)(\emptyset ,c)\right) \) is an example of an asymmetric equilibrium: a and b belong to the circle \( C_{2}^{1}\) and move in \(t=1\) while c, d, and e belong to a circle \( C_{3}^{2}\) and move in \(t=2\).

Borel, E. (1921). La theorie du jeu les equations integrales a noyau symetrique. Comptes Rendus de l’Academie 173, 1304–1308. English translation by Savage, L., 1953. The theory of play and integral equations with Skew Symmetric Kernels. Econometrica, 21, 97–100.

Brunnermeier, M. K., & Morgan, J. (2010). Clock games: Theory and experiments. Games and Economic Behavior, 68, 532–550.CrossRef

Bulow, J., & Klemperer, P. (1999). The generalized war of attrition. American Economic Review, 89, 175–189.CrossRef

Dubovik, A., & Parakhonyak, A. (2014). Drugs, guns, and targeted competition. Games and Economic Behavior, 87, 497–507.CrossRef

Harsanyi, J., & Selten, R. (1988). A general theory of equilibrium selection in games. Cambridge: MIT Press.

Konrad, K. A. (2009). Strategy and dynamics in contests. Oxford, UK: Oxford University Press.

Maynard Smith, J. (1974). Theory of games and the evolution of animal contests. Journal of Theoretical Biology, 47, 209–221.CrossRef

Maskin, E., & Tirole, J. (2001). Markov perfect equilibrium I. Observable actions. Journal of Economic Theory, 100, 191–219.CrossRef

Park, A., & Smith, L. (2008). Caller number five and related timing games. Theoretical Economics, 3, 231–256.

Rinott, Y., Scarsini, M., & Yu, Y. (2012). A Colonel Blotto gladiator game. Mathematics of Operations Research, 37, 574–590.CrossRef

Rosenthal, R. (1981). Games of perfect information, predatory pricing, and the chain store paradox. Journal of Economic Theory, 25, 92–100.CrossRef

Sela, A., & Erez, E. (2013). Dynamic contests with resource constraints. Social Choice and Welfare, 41, 863–882.CrossRef

von Stackelberg, H. (1934). Marktform und Gleichgewicht. Vienna and Berlin: Springer Verlag.

Titel: Optimal stealing time
verfasst von: Andrea Gallice
Publikationsdatum: 02.06.2015
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 3/2016
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-015-9507-y

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