Pure Strategy Nash Equilibria Refinement in Bimatrix Games by Using Domination Efficiency along with Maximin and the Superoptimality Rule

Abstract

Background. Multiple Nash equilibria bring a new problem of selecting amongst them but this problem is solved by refining the equilibria. However, none of the existing refinements can guarantee a single refined Nash equilibrium. In some games, Nash equilibria are nonrefinable. Objective. For solving the nonrefinability problem of pure strategy Nash equilibria in bimatrix games, the goal is to develop an algorithm which could facilitate in refining the equilibria as much further as possible. Methods. A Nash equilibria refinement is suggested, which is based on the classical refinement by selecting only efficient equilibria that dominate by their payoffs. The suggested refinement exploits maximin subsequently. The superoptimality rule is involved if maximin fails to produce just a single refined equilibrium. Results. An algorithm of using domination efficiency along with maximin and the superoptimality rule has been developed for refining Nash equilibria in bimatrix games. The algorithm has 10 definite steps at which the refinement is progressively accomplished. The developed concept of the equilibria refinement does not concern games with payoff symmetry and mirror-like symmetry. Conclusions. The suggested pure strategy Nash equilibria refinement is a contribution to the equilibria refinement game theory field. The developed algorithm allows selecting amongst nonrefinable Nash equilibria in bimatrix games. It partially removes the uncertainty of equilibria, without going into mixed strategies. There are only two negative cases when the refinement fails. For a case when more than a single refined equilibrium is produced, the superoptimality rule may be used for a player having multiple refined equilibrium strategies but the other player has just a single refined equilibrium strategy.

Acyclic-and-Asymmetric Payoff Triplet Refinement of Pure Strategy Efficient Nash Equilibria in Trimatrix Games by Maximinimin and Superoptimality

“…To draw a generalized scheme of an algorithm of using the developed approach refinement in trimatrix games. 7. To show illustratively how the algorithm works on real-valued examples of trimatrix games having efficient Nash equilibria, which are nonrefinable by the known concepts [1,2,5,7,14,15].…”

“…8. To discuss the developed approach and its algorithm, and emphasize some unsolved issues, if any, in them [7].…”

“…The existing approaches to refining do not guarantee that the refined efficient Nash equilibrium will be single [1,2,6,7]. And sometimes efficient Nash equilibria are nonrefinable because of lack of an additional information that could have helped in distinguishing among efficient Nash equilibria [8,9].…”

“…And sometimes efficient Nash equilibria are nonrefinable because of lack of an additional information that could have helped in distinguishing among efficient Nash equilibria [8,9]. Simple examples of the nonrefinability are easily shown with bimatrix games, wherein efficient Nash equilibria produce identical/symmetric payoffs [7].…”

“…A novel approach to refining pure strategy efficient Nash equilibria in bimatrix games was suggested in article [7]. It exploits the maximin rule aiming at guaranteeing payoffs.…”