Competition for a Majority
We define the class of two-player zero-sum games with payoffs having mild discontinuities, which in applications typically stem from how ties are resolved. For games in this class we establish sufficient conditions for existence of a value of the game and minimax or Nash equilibrium strategies for the players. We prove first that if all discontinuities favor one player then a value exists and that player has a minimax strategy. Then we establish that a general property called payoff approachability implies that the value results from equilibrium. We prove further that this property implies that every modification of the discontinuities yields the same value; in particular, for every modification, epsilon-equilibria exist. We apply these results to models of elections in which two candidates propose policies and a candidate wins election if a weighted majority of voters prefer his policy. We provide tie-breaking rules and assumptions on voters' preferences sufficient to imply payoff approachability, hence existence of equilibria, and each other tie-breaking rule yields the same value and has epsilon-equilibria. These conclusions are also derived for the special case of Colonel Blotto games in which each candidate allocates his available resources among several constituencies and the assumption on voters' preferences is that a candidate gets votes from those constituencies allocated more resources than his opponent offers. Moreover, for the case of simple-majority rule we prove existence of an equilibrium that has zero probability of ties.
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- Brian Roberson & Dmitriy Kvasov, 2010.
"The Non-Constant-Sum Colonel Blotto Game,"
Purdue University Economics Working Papers
1252, Purdue University, Department of Economics.
- Brian Roberson & Dmitriy Kvasov, 2008. "The Non-Constant-Sum Colonel Blotto Game," CESifo Working Paper Series 2378, CESifo Group Munich.
- Brian Roberson & Dmitriy Kvasov, 2010. "The Non-Constant-Sum Colonel Blotto Game," School of Economics Working Papers 2010-31, University of Adelaide, School of Economics.
- Duggan, John, 2007. "Equilibrium existence for zero-sum games and spatial models of elections," Games and Economic Behavior, Elsevier, vol. 60(1), pages 52-74, July.
- Sergiu Hart, 2006.
"Discrete Colonel Blotto and General Lotto Games,"
Discussion Paper Series
dp434, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
- Kvasov, Dmitriy, 2007. "Contests with limited resources," Journal of Economic Theory, Elsevier, vol. 136(1), pages 738-748, September.
- Brian Roberson, 2006. "The Colonel Blotto game," Economic Theory, Springer, vol. 29(1), pages 1-24, September.
- Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
- Roger B. Myerson & Daniel Diermeier, 1999. "Bicameralism and Its Consequences for the Internal Organization of Legislatures," American Economic Review, American Economic Association, vol. 89(5), pages 1182-1196, December.
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