Exponential Lower Bounds on the Double Oracle Algorithm in Zero-Sum Games

TL;DR

This paper establishes exponential lower bounds on the double oracle algorithm's performance in zero-sum games, highlighting limitations depending on tiebreaking schemes in different game types.

Contribution

It provides the first exponential lower bounds for the double oracle algorithm in both POSGs and EFGs, considering various tiebreaking assumptions.

Findings

Exponential lower bounds are proven for EFGs with adversarial tiebreaking.

Lower bounds for POSGs hold regardless of tiebreaking.

Results reveal fundamental limitations of the double oracle algorithm in certain game settings.

Abstract

The double oracle algorithm is a popular method of solving games, because it is able to reduce computing equilibria to computing a series of best responses. However, its theoretical properties are not well understood. In this paper, we provide exponential lower bounds on the performance of the double oracle algorithm in both partially-observable stochastic games (POSGs) and extensive-form games (EFGs). Our results depend on what is assumed about the tiebreaking scheme -- that is, which meta-Nash equilibrium or best response is chosen, in the event that there are multiple to pick from. In particular, for EFGs, our lower bounds require adversarial tiebreaking, whereas for POSGs, our lower bounds apply regardless of how ties are broken.