Comparing the notions of optimality in CP-nets, strategic games and soft constraints

TL;DR

This paper explores and relates the concepts of optimality across CP-nets, strategic games, and soft constraints, establishing formal correspondences and enabling cross-application of solution techniques.

Contribution

It introduces a qualitative modification of strategic games, proves the equivalence of CP-net optimal outcomes and Nash equilibria, and relates soft constraints to graphical games.

Findings

Optimal outcomes of CP-nets are Nash equilibria of modified strategic games.

For certain soft constraints, optimal solutions are Nash equilibria and Pareto efficient.

Pareto efficient strategies correspond to optimal solutions of soft constraints.

Abstract

The notion of optimality naturally arises in many areas of applied mathematics and computer science concerned with decision making. Here we consider this notion in the context of three formalisms used for different purposes in reasoning about multi-agent systems: strategic games, CP-nets, and soft constraints. To relate the notions of optimality in these formalisms we introduce a natural qualitative modification of the notion of a strategic game. We show then that the optimal outcomes of a CP-net are exactly the Nash equilibria of such games. This allows us to use the techniques of game theory to search for optimal outcomes of CP-nets and vice-versa, to use techniques developed for CP-nets to search for Nash equilibria of the considered games. Then, we relate the notion of optimality used in the area of soft constraints to that used in a generalization of strategic games, called graphical games. In particular we prove that for a natural class of soft constraints that includes weighted constraints every optimal solution is both a Nash equilibrium and Pareto efficient joint strategy. For a natural mapping in the other direction we show that Pareto efficient joint strategies coincide with the optimal solutions of soft constraints.