Nash Equilibria with Incomplete Preferences: A Strategic Completion Approach

We introduce a strategic completion approach to deal with incomplete preferences relations in noncooperative games through appropriate value functions. We ensure that each Nash equilibrium of the original game with incomplete preferences is characterized as an equilibrium of a corresponding scalar game, obtaining both necessary and sufficient Nash equilibrium conditions. We compare our representation results with other approaches grounded on preference completion processes, including the special case where value functions are pseudo-utilities or utilities [in the sense of Peleg (Econom J Econom Soc 38:93-96, 1970)]. We identify minimal requirements for value functions to provide full charaterization of Nash equilibria of a game with incomplete preferences through scalarization, with no convexity assumptions. As a special case, we characterize Nash equilibria in vector games by scalarization, extending to the non convex case the results originally developed by Shapley (Naval Res Logist Q 6:57-61, 1959), pointing out some weaknesses of the use of mixed extension in vector games. Finally, we illustrate our approach in vector games by means of special families of scalarizing functions. To conclude, we apply our method to study a bicriteria Cournot duopoly in the context of the managerial theory of firms.