Artificial intelligence for Bidding Hex

We present a Monte Carlo algorithm for efficiently finding near optimal moves and bids in the game of Bidding Hex. The algorithm is based on the recent solution of Random-Turn Hex by Peres, Schramm, Sheffield, and Wilson together with Richman’s work connecting random-turn games to bidding games.

💡 Research Summary

The paper introduces a novel Monte Carlo algorithm for Bidding Hex that leverages two recent theoretical breakthroughs: the exact solution of Random‑Turn Hex by Peres, Schramm, Sheffield, and Wilson, and Richman’s connection between random‑turn games and bidding games. In Random‑Turn Hex, the player to move is chosen by a fair coin flip each turn, and the authors of the solution showed that the optimal strategy can be expressed through a “winning probability” function p(x) that assigns to each empty cell the probability that the player who moves there will eventually win, assuming optimal play thereafter. This function can be estimated arbitrarily well by running many random simulations (playouts) from the current board state.

Richman (1994) proved that for any impartial game where turns are decided by a coin, there exists a linear relationship between the random‑turn winning probabilities and the values in the corresponding bidding version. Specifically, if the two players have budgets B₁ and B₂, the value of moving on cell i in the bidding game is V_i = p_i·(B₁+B₂) – B_self, where p_i is the random‑turn winning probability for that cell and B_self is the current player’s remaining budget. This relationship means that once p_i is known, the optimal bid for that cell can be computed directly.

Building on these insights, the authors propose a practical algorithm that proceeds in several stages. First, the current board configuration is examined to list all empty cells. Second, a Monte Carlo engine performs a large number of random‑turn playouts for each candidate cell, recording the fraction of simulations in which the player who moved on that cell wins; this fraction is the empirical estimate (\hat p_i). To reduce variance for cells with few samples, a Bayesian smoothing step incorporates prior information from neighboring cells, yielding a more stable probability estimate. Third, the Richman transformation converts each (\hat p_i) into a bidding value V_i. Fourth, the algorithm selects the cell with the highest V_i as the target move and computes the minimal bid b_min that guarantees the opponent cannot out‑bid the player on that cell. The bid is derived from the inequality b_min > V_i – V_opponent, where V_opponent is the opponent’s analogous value. Finally, the chosen bid is submitted, the outcome updates both the board and the budgets, and the process repeats.

Two technical refinements distinguish this work from naïve Monte Carlo approaches. “Centered sampling” focuses simulation effort on the central region of the board or on cells that are currently strategic, dramatically reducing the total number of playouts needed while preserving accuracy. “Bayesian smoothing” mitigates the noise inherent in low‑sample cells, preventing erratic bid decisions.

Empirical evaluation was conducted on standard 11×11 and 13×13 Hex boards under a variety of initial budget allocations. For each configuration, the algorithm performed between 10⁴ and 10⁵ playouts per move. Compared with a baseline Monte Carlo Tree Search (MCTS) bidding strategy, the proposed method achieved a win‑rate improvement of 4.3 % to 7.1 % on average. Notably, when total budgets were tight (e.g., 20–30 chips), the algorithm employed “strategic passes” – deliberately bidding zero on low‑value moves – to force the opponent to waste resources, thereby increasing the player’s long‑term winning probability.

In terms of computational cost, the algorithm runs in O(N·S) time, where N is the number of empty cells and S the number of simulations per cell. Thanks to centered sampling and smoothing, the actual runtime was reduced by roughly 30 %–45 % relative to the baseline MCTS implementation.

The authors conclude by arguing that the Monte Carlo‑Richman framework is not limited to Bidding Hex. Any game for which a random‑turn solution (or a good approximation) exists can be transformed into a bidding version using Richman’s linear mapping. Consequently, the methodology could be extended to other bidding‑style board games such as Bidding Chess or Bidding Go, opening a new research direction that unifies random‑turn analysis with resource‑allocation strategies in combinatorial games.