Dominated Strategy in Random Game

Abstract

This study explores the prevalence of strictly dominated strategies in random games, highlighting two primary contributions: computational and economic. As the game size increases, if the number of strategies for Player 1 and Player 2 are similar, the probability of having no strictly dominated strategies approaches 1. This is particularly evident in large square games where the row size is proportionate to the column size within a specific range. Consequently, the probability of a game being dominance solvable diminishes to zero. Furthermore, we show that this result is very nearly tight: small deviations lead to a non-zero chance of strictly dominated strategies, while larger deviations make their existence nearly certain. Our findings initially emphasize the significance of the parameters in the underlying probability distribution.

We also examine the behavior of the fixed proportion q of dominated strategies as M, N → ∞, where M and N are row and column sizes. Specifically, for the row player, we show that the probability of existence of q-portion dominated strategy approaches 0 as N ≫ M/ln(M) , and it approaches 1 when M ≫ (N/(1– δ– q))^N, for fixed δ > 0.

Finally, we propose an efficient algorithm that reduces computational complexity, a topic of interest in economics and computer science since Yu and Zeleny (1975). This algorithm improves upon the minimum/maximum point and conventional approach, reducing the time complexity for the row player from O(M^2N) to O(M^2N/2).

Keywords: Dominated Stratagy, Random Game, Time Complexicity

JEL Classification: C70