Mixture of Public and Private Distributions in Imperfect Information Games

TL;DR

The paper addresses the challenge of inferring hidden information in imperfect-information games without leaking private data. It introduces a tunable mixture belief distribution $Δ_{\lambda}(s_i) = (1-\lambda) Δ_{i}(s_i) + \lambda Δ_{pub}(s_{pub})$ and shows how to adapt determinization-based algorithms, specifically PIMC and IS-MCTS, to operate with this mixture, including per-infostate adaptations and aggregation across possible infostates. Empirical results on Liar's Dice, a Card Game variant, and Leduc Poker demonstrate that intermediate $\lambda$ values can reduce information leakage and, in many cases, improve performance, with further gains when using multiple mixtures across game progress. The work highlights practical implications for building robust AI in trick-taking and bluffing domains and outlines avenues for dynamic λ strategies and scaling to larger games.

Abstract

In imperfect information games (e.g. Bridge, Skat, Poker), one of the fundamental considerations is to infer the missing information while at the same time avoiding the disclosure of private information. Disregarding the issue of protecting private information can lead to a highly exploitable performance. Yet, excessive attention to it leads to hesitations that are no longer consistent with our private information. In our work, we show that to improve performance, one must choose whether to use a player's private information. We extend our work by proposing a new belief distribution depending on the amount of private and public information desired. We empirically demonstrate an increase in performance and, with the aim of further improving performance, the new distribution should be used according to the position in the game. Our experiments have been done on multiple benchmarks and in multiple determinization-based algorithms (PIMC and IS-MCTS).

Figures (6)

• Figure 1: Multiple belief distributions for the game Liar's Dice with $1$ dice of $2$ sides each. Four world states possible $w_1$, $w_2$, $w_3$ and $w_4$. The Public-Private belief uses the mixture distribution with $\lambda=0.5$.
• Figure 2: Example of the tree constructed by IS-MCTS. The first player is acting in the red square, the second player is acting in the green diamond and the blue circle refers to the chance node.
• Figure 3: Average TSSR for IS-MCTS and PIMC on multiple benchmarks according to $\lambda$ of the mixture distribution.
• Figure 4: Heatmap of the expected utility against the best response when playing at the first position.
• Figure 5: Average TSSR according to $\lambda$ value of the mixture distribution.

Theorems & Definitions (3)

• Definition : Private Belief Distribution
• Definition : Public Belief Distribution brown_combining_2020
• Definition : Mixture Belief Distribution