Modeling Precomputation In Games Played Under Computational Constraints

TL;DR

This paper introduces a new model for precomputation in computationally constrained games, revealing a trade-off between randomness and precomputation susceptibility, with algorithms and experiments on chess algorithms like Stockfish.

Contribution

It presents a novel model for precomputation in constrained games and algorithms to measure strategy susceptibility and compute approximate equilibria.

Findings

Randomization is necessary in constrained games.

Algorithms effectively measure precomputation susceptibility.

Trade-offs between randomness and precomputation are demonstrated.

Abstract

Understanding the properties of games played under computational constraints remains challenging. For example, how do we expect rational (but computationally bounded) players to play games with a prohibitively large number of states, such as chess? This paper presents a novel model for the precomputation (preparing moves in advance) aspect of computationally constrained games. A fundamental trade-off is shown between randomness of play, and susceptibility to precomputation, suggesting that randomization is necessary in games with computational constraints. We present efficient algorithms for computing how susceptible a strategy is to precomputation, and computing an $\epsilon$-Nash equilibrium of our model. Numerical experiments measuring the trade-off between randomness and precomputation are provided for Stockfish (a well-known chess playing algorithm).