Equilibrium Design for Concurrent Games

TL;DR

This paper explores how to design incentives in concurrent games to ensure desired equilibria, especially those satisfying temporal logic properties, with complexity results and applications to rational synthesis and game repair.

Contribution

It introduces a framework for equilibrium design in concurrent games with temporal logic specifications and analyzes the computational complexity of related problems.

Findings

Equilibrium design can be achieved in PSPACE for LTL properties.

For GR(1) specifications, the problem is in NP/$\\Sigma^{P}_{2}$.

Complexity of related decision problems lies within the polynomial hierarchy.

Abstract

In game theory, mechanism design is concerned with the design of incentives so that a desired outcome of the game can be achieved. In this paper, we study the design of incentives so that a desirable equilibrium is obtained, for instance, an equilibrium satisfying a given temporal logic property -- a problem that we call equilibrium design. We base our study on a framework where system specifications are represented as temporal logic formulae, games as quantitative concurrent game structures, and players' goals as mean-payoff objectives. In particular, we consider system specifications given by LTL and GR(1) formulae, and show that implementing a mechanism to ensure that a given temporal logic property is satisfied on some/every Nash equilibrium of the game, whenever such a mechanism exists, can be done in PSPACE for LTL properties and in NP/$\Sigma^{P}_{2}$ for GR(1) specifications. We also study the complexity of various related decision and optimisation problems, such as optimality and uniqueness of solutions, and show that the complexities of all such problems lie within the polynomial hierarchy. As an application, equilibrium design can be used as an alternative solution to the rational synthesis and verification problems for concurrent games with mean-payoff objectives whenever no solution exists, or as a technique to repair, whenever possible, concurrent games with undesirable rational outcomes (Nash equilibria) in an optimal way.