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Playing for RealGame Theory$
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Ken Binmore

Print publication date: 2007

Print ISBN-13: 9780195300574

Published to Oxford Scholarship Online: May 2007

DOI: 10.1093/acprof:oso/9780195300574.001.0001

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 Mixing Things Up

 Mixing Things Up

(p.177) 6 Mixing Things Up
Playing for Real

Ken Binmore (Contributor Webpage)

Oxford University Press

This chapter develops the idea of a mixed strategy using the entry into a sealed-bid auction as a non-trivial example. Reaction curves are first illustrated for the case of pure strategies and then applied to computing mixed Nash equilibria. The Hawk-Dove Game is equivalent either to the Prisoner's Dilemma or Chicken, depending on parameter values. The mixed-strategy reaction curves are plotted in each case. The interpretation of mixed Nash equilibria as polymorphic equilibria in a game played by a large population is considered. The matrix algebra necessary for handling mixed strategies is reviewed and illustrated with O'Neill's Card Game. Convexity ideas are reviewed and applied to the geometric representation of mixed strategies. Cooperative and noncooperative payoff regions are introduced and illustrated using Chicken and the Battle of the Sexes. Correlated equilibria are introduced after a discussion of self-policing agreements, cheap talk, and preplay randomization. The possibility of correlation without a referee using techniques from cryptography is discussed.

Keywords:   mixed strategy, sealed-bid auction, reaction curve, Hawk-Dove Game, polymorphic equilibrium, O'Neil's Card Game, payoff region, preplay randomization, Battle of the Sexes, Robert Aumann

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