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Probability of Obtaining a Pure Strategy Equilibrium in Matrix Games with Random Payoffs

  • Srijit Mishra
  • T. Krishna Kumar
Part of the Theory and Decision Library book series (TDLC, volume 18)

Abstract

If the payoffs in an mXn zero-sum matrix game are drawn randomly from a finite set of numbers, N, then the probability of obtaining a pure strategy equilibrium, p, will be a weighted sum of the probabilities of obtaining a pure strategy equilibrium, ps, with s distinct payoffs, the weights, qs being the probabilities of obtaining s distinct payoffs from N. However, as N → ∞ the probability qmn → 1. In this limiting case p = Pmn Although pmn has been derived by Goldman (1957) and Papavassilopoulos (1995), our method is more general. We show that pmn= Σtp mn t where p mn t denotes the probability of obtaining a pure strategy equilibrium for the tth (t = 1,...,s(= mn)) ordinal payoff, the ordinality being the rank when the payoffs are put in an ascending order.

Further, we introduce the notion of separation of arrays, S(rk ,cl), which is a necessary and sufficient condition for the equilibrium of an mXn zero-sum matrix game to be associated with a mixed strategy solution. This generalizes the notion of sepatation of diagonals for 2X2 zero-sum matrix games derived by Von Neumann and Morgenstern (1953).

It can be easily verified that as m or n increases pmn decreases. Then given the importance of strong equilibrium, which is always a pure strategy equilibrium, a possible behaviourial interpretation is that players may prefer to play games with less number of strategies.

Keywords

Pure Strategy Matrix Game Strong Equilibrium Pure Strategy Equilibrium Pure Strategy Nash Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Srijit Mishra
    • 1
  • T. Krishna Kumar
    • 2
  1. 1.Centre for Development StudiesThiruvananthapuramIndia
  2. 2.Indian Statistical InstituteBangaloreIndia

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