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

Vadim V. Romanuke


Background. A problem of selecting amongst efficient Nash equilibria is solved by refining them. The existing approaches to refining do not guarantee that the refined efficient Nash equilibrium will be single. Nevertheless, a novel approach to refining pure strategy efficient Nash equilibria in bimatrix games suggested before exploits the maximin and superoptimality rule that, at least partially, remove the uncertainty of the equilibria.

Objective. The goal of the article is to develop the bimatrix game approach expanding it over trimatrix games for refining efficient Nash equilibria as much further as possible.

Methods. An efficient Nash equilibria refinement is suggested for trimatrix games, which is based on expanding the refinement approach for bimatrix games, exploiting the maximinimin and superoptimality. Games with acyclic-and-asymmetric payoff triplets are only considered.

Results. Series of trimatrix game simulations allow concluding on that whereas the refinement is needed in about between 33 % and 65 % of trimatrix games where players possess between 4 to 12 pure strategies (this rate increases as the game size increases), it is perfectly accomplished to a single metaequilibrium in roughly between 46 % and 52 % of those cases (this rate decreases as the game size increases), using maximinimin only, without the superoptimality rule. Exploiting the maximin, expanded to the maximinimin principle, and superoptimality using now double-summing, the main work for the refinement is off the maximinimin principle.

Conclusions. An algorithm for the developed approach refinement in trimatrix games is very simple. It consists of four generalized items. Although a total fail of the refinement is not excluded, the aggregate efficiency of removing the uncertainty of equilibria by the maximinimin principle and superoptimality rule seems satisfactory.


Trimatrix game; Efficient Nash equilibria; Refinement; Maximinimin; Superoptimality rule

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