The Frequency of Convergent Games under BestResponse Dynamics
Abstract
Generating payoff matrices of normalform games at random, we calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of $n$player, $m$strategy games. These are perfectly predictable as they must converge to the Nash equilibrium. We then consider a wider class of games that converge under a bestresponse dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games goes to zero as the number of players or the number of strategies goes to infinity. In the $2$player case, we show that for large games with at least $10$ strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an $n$partite graph to describe games.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 arXiv:
 arXiv:2011.01052
 Bibcode:
 2020arXiv201101052W
 Keywords:

 Economics  Theoretical Economics;
 91A10
 EPrint:
 16 pages