Iterated Elimination of Weakly Dominated Strategies in Well-Founded Games

TL;DR

This paper extends key results on iterated elimination of weakly dominated strategies from finite to well-founded games, including infinite and multi-equilibrium cases, using transfinite methods.

Contribution

It generalizes finite game results to well-founded games, allowing transfinite eliminations and addressing infinite and multi-equilibrium scenarios.

Findings

Finite zero-sum games can be solved in 'n-1' steps.

Transfinite elimination applies to infinite well-founded games.

Results extend to classes with multiple equilibria.

Abstract

Recently, in [K.R. Apt and S. Simon: Well-founded extensive games with perfect information, TARK21], we studied well-founded games, a natural extension of finite extensive games with perfect information in which all plays are finite. We extend here, to this class of games, two results concerned with iterated elimination of weakly dominated strategies, originally established for finite extensive games. The first one states that every finite extensive game with perfect information and injective payoff functions can be reduced by a specific iterated elimination of weakly dominated strategies to a trivial game containing the unique subgame perfect equilibrium. Our extension of this result to well-founded games admits transfinite iterated elimination of strategies. It applies to an infinite version of the centipede game. It also generalizes the original result to a class of finite games that may have several subgame perfect equilibria. The second one states that finite zero-sum games with 'n' outcomes can be solved by the maximal iterated elimination of weakly dominated strategies in 'n-1' steps. We generalize this result to a natural class of well-founded strictly competitive games.