Figure 1: Game tree with mimimax highlighted in red, and prospective better path in green |

What does it mean to include "risk" in a turn based two player zero sum game? Consider Figure 1, which highlights the typical minimax approach to game trees in red. Each player has competing goals, Max is looking to choose a path that maximizes his score, thus picking the maximum nodes in the next level, while Min is trying to minimize the score on his turn. Scores are bubbled up from the bottom. We see from this perspective, Max's optimal choice would be to pick the left branch, because given optimal play from Min, the score of 1.2 produces a slightly superior position. If the scores were perfect, indeed this analysis would be correct, and risk would be completely mitigated.

The reality in games where the search depth rarely allows one to reach terminal nodes is that we use a heuristic as a proxy for the true score. When the frontier node is finally reached, we repeat the process of creating a tree to search, bubbling up a new score that serves as a better estimation of the actual value of the position at that node. This means there is a variance in the heuristic that must be considered when choosing our path through the tree. When we eventually reach a pre-frontier node, we will have reached the last responsible moment from which we can change our decision. The score of pre-frontier nodes as viewed from the root must then be some combination that accounts for the likelihood that the proxy score for any frontier node has improved by the time we reach the pre-frontier level.

Consider node A vs node B. If we were to reach node A and then discover that our heuristic evaluation overestimated the score of the child node valued 1.2, we have left ourselves without a viable alternative, as the variance in our heuristic score is unlikely to overcome the 6.2 point gap we predicted in the alternative branch. If the true score of the best choice at node A was -1.0, we would have no recourse but to continue down that path. Node B, on the other hand, provides two choices of similar value. If we were to reach Node B and discover that our 1.1 option actually produced a score of -1.0, we would still have a viable alternative in the 1.0 node. Minimax fails to capture this characteristic.

Now consider nodes C and D. From the prospective of Max, we again recognize that Min's options are restricted at node C due to the large difference between the first and second best nodes. It is not likely that the variance in the 4.0 node will overcome the gap in scores, which means that a miss-evaluation of the frontier node in Max's favor cannot be mitigated by a second choice from Min. Node D however does provide that alternative, in the similar score of the 1.3 node.

Figure 2: Normal Distribution |

Figure 3: Cumulative Distribution Function |

Figure 4: Three choices represented by three cumulative distribution functions |

Equation 1: Recursively defined CDF accumulating all move alternatives |

Equation 1 expresses a combined cumulative probability distribution recursively, where Fi represents CDF for the ith alternative move choice. Here, we can say at any point that the probability that a random draw from a distribution is at least as great as X is equal to the probability that Fi is greater than X, plus the probability that Fi is less than X times the probability of the combination of the remaining alternatives is greater than X. A concrete example can be again taken from Figure 3. Consider the blue, red, and orange distributions, each with its mean at 0 (ie: F1(0) = F2(0) = F3(0) = 0.5), and assume those represent the probability distributions for Figure 4. Each by itself has a 50% chance of scoring higher than 0. Combining those into Equation 1 and expanding the probability around 0, we find that we have an 87.5 percent chance of scoring better than zero. This completely upends our understanding of the score at node A in Figure 4.

At every terminal node, the Quiesce function instead returns the cumulative probability function mapping to the score produced by our heuristic evaluation. Every non-terminal node produces a combined CDF using the methodology described by Equation 1. Bounds are set by evaluating a CDF at the prescribed confidence level, giving us a fixed expected value to compare against as Beta. Then, when a child is iterating through its move choices, it updates the probability that one of its subnodes will be greater than Beta. If the probability is greater than our prescribed confidence, we can do a beta cutoff. One of the nice consequences of this strategy is that as the variance of our heuristic evaluation drops to 0, the search converges back to a normal alpha beta search.

B* Search

ProbCut Pruning

Conspiracy Search

Similar approaches in the literature:

An optimal game tree propagation for imperfect players