Fig 1.
Examples of board configurations.
Left, board I (6x6 squares) full condition, ‘X’ wins within 4 moves (correct solutions are ‘f3’ or ‘f5’). Right, board IV (10x10 squares) full condition, ‘X’ wins within 5 moves (correct solution is ‘e8’).
Fig 2.
Participants search size does not depend on the board’s complexity.
A) Participants’ search size (blue) is small and fixed compared with the search size of alpha-beta pruning (gray). B) The percent of participants who solved correctly each of the board configurations, by board complexity. As board complexity increases, participants’ success rate decreases, yet at a slower rate compared to the board’s complexity increase. Error bars depict 95% confidence intervals.
Fig 3.
Participants’ moves indicate the use of scoring strategies.
A) The distributions of first moves predicted by the “Interaction” scoring strategy and the distributions of participants’ actual first moves on this board. B) Log-likelihoods of scoring strategies’ predictions of participants’ moves. C) Percent of participants fitted to each scoring strategy. D) Log-likelihoods of scoring strategies’ predictions of participants’ moves. There are two configurations for each scoring strategy: using the scoring strategy as is (“base”) and adding the best fitted shutter value per each participant (see Methods). For all scoring strategies, adding the shutter significantly improved the log-likelihood and AIC scores (in all cases p < 10−5 using a likelihood ratio test). All error bars are 95% confidence intervals.
Fig 4.
Participants’ search is influenced by their previous choices and is not explained by an internal search model or by reduced attention to the opponent.
A) Distribution of participants’ moves on the full version of board configuration II. B) Distribution of participants’ moves on the truncated version of board configuration II. C) Mean entropy of the distribution of participants’ first moves in the truncated boards and the distribution of moves in the equivalent board state in the full boards. D) Mean entropy of the distribution of participants’ first moves in the truncated boards and the distribution of moves in the equivalent board state in the full boards as predicted by the “Interaction” scoring strategies with an added shutter heuristic (shutter = 0). **, p < 0.01. E) Mean entropy of the distribution of participants’ first moves in the truncated boards and the distribution of moves in the equivalent board state in the full boards as predicted by the “Interaction” scoring strategy when ignoring opponent’s winning paths, thus focusing only on one’s own pieces. F) Mean entropy of the distribution of participants’ first moves in the truncated boards and the distribution of moves in the equivalent board state in the full boards as predicted by an internal search model, using alpha-beta pruning search with branching factor k = 7 and limited depth d = 1. All error bars are 95% confidence intervals.
Fig 5.
Participants who exhibited a narrower shutter were more likely to find the winning move, but were also more likely to miss winning moves for the ‘O’ player.
A) Illustration of the shutter heuristic: assuming the player’s last move was f5 (shown in a black circle), there are three potential paths to win induced by this move, f6–f3, f5–f2 and c5–f5 (their squares marked in blue). Squares on these paths are considered at distance 0 from the last move. Squares adjacent to these squares (Manhattan distance = 1) are considered at distance 1 from the last move (marked in orange). B) The probability to find the winning move of participants with different shutter sizes. All differences were statistically significant: **, p < 0.005 ***, p < 0.001. C) The proportion between the probability for missed ‘X’ winning moves and the probability of missed ‘O’ winning moves in the computational simulations (left) and by participants (right). Narrow shutter shown in blue, medium in orange and wide in green. Differences between narrow and wide shutter size and medium and wide shutter size were statistically significant, ***, p < 0.001. All error bars are 95% confidence intervals.
Fig 6.
For most of possible search parameters, search with a narrow path shutter is the dominant search strategy.
Computation reduction is computed as A) Example of a board configuration where a narrow shutter is the dominant search strategy since it shows similar probabilities for finding the winning move at substantially lower number of computations. Simulation parameters were: complexity [830], noise level [0.5], branching [5], limit number of moves [30]. B) Example of a board configuration with a trade-off between computation amount and the probability to find a winning move. For the given simulation parameters, an increase from shutter size 0 to shutter size 1 incurs a significantly higher probability to find a winning move but the number of computations increases as well. Simulation parameters were: complexity [49], noise level [0.5], branching [5], limit number of moves [30]. C) The phase space of board complexity vs. noise levels (aggregated over branching factor and search size limitations, see Methods). Each square shows the proportion of configurations in which there was a trade-off between using narrow vs. wider shutter size values: dark blue indicates configurations where the narrow shutter dominates the Pareto front (no trade-off), dark red indicates a trade-off between narrow and wider shutter values (trade-off between computation resources and accuracy). All error bars are 95% confidence intervals.
Fig 7.
Aggregated results for AlphaZero with different sizes of additional search using MCTS against pure MCTS with 1000 simulations.
A) Limiting the trained deep learning models with a shutter of size zero (blue bars) significantly improved their performance against an MCTS algorithm on all experimental board configurations (all p < 10−5). B) Adding a shutter heuristic did not improve performance on empty boards (i.e., from the initial state of a game), except when no additional MCTS search was done. All error bars are 95% confidence intervals.