[Paper Review] Computing best-response strategies in infinite games of incomplete information

Daniel M. Reeves, Michael P. Wellman|arXiv (Cornell University)|Jul 7, 2004

Auction Theory and Applications33 references53 citations

TL;DR

This paper presents an algorithm to compute best-response strategies in two-player infinite games of incomplete information, where payoffs are piecewise linear in types and actions, and depend on linear comparisons of actions. The method enables equilibrium computation in games like auctions and novel allocation mechanisms, with demonstrated efficacy on both existing and new game types.

ABSTRACT

We describe an algorithm for computing best-response strategies in a class of two-player infinite games of incomplete information, defined by payoffs piecewise linear in agents' types and actions, conditional on linear comparisons of agents' actions. We show that this class includes many well-known games including a variety of auctions and a novel allocation game. In some cases, the best-response algorithm can be iterated to compute Bayes-Nash equilibria. We demonstrate the efficacy of our approach on existing and new games.

Motivation & Objective

• Address the challenge of computing best-response strategies in infinite games where players have incomplete information about each other's types.
• Develop a general-purpose algorithm applicable to a broad class of games defined by piecewise linear payoffs and linear action comparisons.
• Enable iterative computation of Bayes-Nash equilibria in selected games through the best-response algorithm.
• Demonstrate the method's applicability to both well-known games (e.g., auctions) and novel game types, such as a new allocation game.

Proposed method

• Model games using payoff functions that are piecewise linear in agents' types and actions, with dependencies based on linear comparisons of actions.
• Formulate best-response computation as a mathematical optimization problem over continuous type and action spaces.
• Leverage the piecewise linear structure to decompose the strategy computation into tractable subproblems over convex regions.
• Use linear programming or equivalent techniques to solve for optimal actions in each region defined by the linear comparisons.
• Iterate the best-response computation across players to converge toward Bayes-Nash equilibria in suitable game classes.
• Validate the algorithm’s correctness and efficiency through implementation and testing on benchmark and novel game instances.

Experimental results

Research questions

• RQ1 Can an efficient algorithm compute best-response strategies in infinite games with incomplete information and piecewise linear payoffs?
• RQ2 To what extent can this best-response algorithm be extended to compute Bayes-Nash equilibria through iteration?
• RQ3 Which well-known game classes, such as auctions, fall within the scope of this algorithmic framework?
• RQ4 How does the algorithm perform on novel game types, such as the proposed allocation game, compared to known solution methods?

Key findings

• The algorithm successfully computes best-response strategies in games with piecewise linear payoffs and linear action comparisons.
• The method is applicable to a wide range of games, including various auction formats and a newly proposed allocation game.
• Iterative application of the best-response algorithm yields Bayes-Nash equilibria in certain classes of games.
• The approach demonstrates practical efficacy through successful computation on both existing and novel game instances.