The value of a draw
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The value of a draw Casilda Lasso de la Vega1 · Oscar Volij2 Received: 2 May 2017 / Accepted: 31 July 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract We model a match as a recursive zero-sum game with three possible outcomes: Player 1 wins, player 2 wins, or there is a draw. We focus on matches whose point games also have three possible outcomes: Player 1 scores the point, player 2 scores the point, or the point is drawn in which case the point game is repeated. We show that a value of a draw can be attached to each state so that an easily computed stationary equilibrium exists in which players’ strategies can be described as minimax behavior in the point games induced by these values. Keywords Matches · Stochastic games · Recursive games · Draws JEL Classification C72 · C73
1 Introduction A match is a recursive zero-sum game with three possible outcomes: player 1 wins, player 2 wins, or the game never ends. Play proceeds by steps from state to state. In each state, players play a “point” and move to the next state according to transition probabilities jointly determined by their actions. Examples of matches include tennis, penalty shootouts and, you will forgive the repetition, chess matches.1 In a chess match, two players play a sequence of chess games until some prespecified score is 1 The game of chess itself is also a match. In fact, in the first article to appear on game theory, Zermelo (1913) models chess as a zero-sum recursive game.
The authors thank the Spanish Ministerio de Economía y Competitividad (Project ECO2015-67519-P) and the Gobierno Vasco (Project IT568-13) for research support.
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Oscar Volij [email protected] Casilda Lasso de la Vega [email protected]
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University of the Basque Country, Bilbao, Spain
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Ben-Gurion University of the Negev, Beer sheva, Israel
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C. L. de la Vega, O. Volij
reached. For instance, the Alekhine–Capablanca match played in 1927 took the format known as first-to-6 wins, according to which the winner is the first player to win six games. Some matches are finite horizon games. As an example, we have a best-ofseven playoff series. Indeed, this match will necessarily end in at most seven stages. A penalty shootout, on the other hand, is an infinite horizon game. It will never end if, for instance, every penalty kick is scored. Similarly, a first-to-6-wins chess match is also an infinite horizon game.2 Matches can further be classified into binary and non-binary games. A penalty shootout is an example of the former and a chess match of the latter. The reason is that while each penalty kick has only two outcomes, either the goal is scored or it is not scored, a chess game may also end in a draw. Matches have been the object of several empirical studies. For instance, Walker and Wooders (2001) and Gauriot et al. (2016), using data on tennis, and Palacios-Huerta (2003), using data on penalty kicks, show that players’ behavior is broadly consistent with the minimax hypothesis. Specifically, they show that professional players reg
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