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Competition for a Majority

  • Paulo Barelli
  • Srihari Govindan
  • Robert Wilson

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 such games, we establish sufficient conditions for existence of a value of the game, maximin and minimax strategies for the players, and a Nash equilibrium. If all discontinuities favor one player, then a value exists and that player has a maximin strategy. A property called payoff approachability implies existence of an equilibrium, and that the resulting value is invariant: games with the same payoffs at points of continuity have the same value and ɛ‐equilibria. For voting games in which two candidates propose policies and a candidate wins election if a weighted majority of voters prefer his proposed policy, we provide tie‐breaking rules and assumptions about voters' preferences sufficient to imply payoff approachability. These assumptions are satisfied by generic preferences if the dimension of the space of policies exceeds the number of voters; or with no dimensional restriction, if the electorate is sufficiently large. Each Colonel Blotto game is a special case in which each candidate allocates a resource among several constituencies and a candidate gets votes from those allocated more than his opponent offers; in this case, for simple‐majority rule we prove existence of an equilibrium with zero probability of ties.

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File URL: http://hdl.handle.net/10.3982/ECTA11008
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Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 82 (2014)
Issue (Month): 1 (01)
Pages: 271-314

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Handle: RePEc:wly:emetrp:v:82:y:2014:i:1:p:271-314
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  1. 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.
  2. Kvasov, Dmitriy, 2007. "Contests with limited resources," Journal of Economic Theory, Elsevier, vol. 136(1), pages 738-748, September.
  3. Sergiu Hart, 2006. "Discrete Colonel Blotto and General Lotto Games," Levine's Bibliography 321307000000000532, UCLA Department of Economics.
  4. 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.
  5. 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.
  6. 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.
  7. Brian Roberson, 2006. "The Colonel Blotto game," Economic Theory, Springer, vol. 29(1), pages 1-24, September.
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