Continuity and robustness of Bayesian equilibria in Tullock contests
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Continuity and robustness of Bayesian equilibria in Tullock contests Ezra Einy1 · Diego Moreno2 · Aner Sela1 Received: 18 June 2020 / Accepted: 20 June 2020 © Society for the Advancement of Economic Theory 2020
Abstract We show that the Bayesian equilibrium correspondence of a Tullock contest with incomplete information is upper semicontinuous. Further, we show that when equilibrium is unique and players’ costs of effort are either state independent or uniformly bounded, then it is also lower semicontinuous, and it is robust to small perturbations of the players’ information, value for the prize, and cost of effort, as well as of the contest success function. Keywords Tullock contests · Incomplete information · Robustness of equilibria JEL Classification C72 · D44 · D82
1 Introduction In a contest, a group of individuals compete for a prize by exerting effort. In a Tullock contest the probability that an individual wins the prize is a non-decreasing function of the effort he exerts (see Tullock 1980). In many economic environments, Tullock contests arise either naturally or by design. Baye and Hoppe (2003), for example, have identified conditions under which a variety of rent-seeking contests, innovation tournaments, and patent-race games are strategically equivalent to Moreno acknowledges the financial support of the Ministerio de Ciencia e Innovación (Spain), Grants PGC2018-098510-B-I00 and MDM2014-0431. * Diego Moreno [email protected] Ezra Einy [email protected] Aner Sela [email protected] 1
Department of Economics, Ben-Gurion University of the Negev, Beersheba, Israel
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Departamento de Economía, Universidad Carlos III de Madrid, Getafe, Spain
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a Tullock contest. Most of the extensive literature studying the outcomes generated by Tullock contests focuses on the complete information case—see, for example, Nitzan (1994), Skaperdas (1996), Clark and Riis (1998), Konrad (2008), Fu and Lu (2012), and Fu et al. (2015). Recently, however, the literature has turned to study the equilibria of Tullock contests with incomplete information, as well as the impact of changes in the players’ information endowments on equilibrium outcomes—see, for example, Wasser (2013), Einy et al. (2017), and Aiche et al. (2018, 2019). A Tullock contest is identified by the players’ value for the prize, their cost of effort, and the impact of effort on the probability of winning the prize. When players have complete information about these attributes , Tullock contests define a complete information game. The Nash equilibria of this game are the equilibria of the contest. Szidarovszky and Okuguchi (1997), Cornes and Hartley (2005), Yamazaki (2008) and Chowdhury and Sheremeta (2011) have studied the existence and uniqueness of equilibria in Tullock contests with complete information. When players are uncertain about any of these attributes, a Tullock contest defines a Bayesian game. The Bayesian Nash equilibria of this game are the Bayesian equilibria of the contest. Einy et al. (2015, 2017
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