Expectimax

A game tree evaluation rule using maxima at the controlled player’s decision nodes and expected values at chance nodes:respectively. Terminal utilities fix the boundary values. These rules exactly optimize a finite-horizon expected utility objective by backward induction. For cyclic infinite-horizon reachability problems, a finite Markov decision process formulation and value iteration replace naive expansion of an infinite game tree. Start with one at successful terminal states and zero elsewhere, keep terminal values fixed, and apply the maximizing/expectation backups repeatedly. The $t$th iterate is the greatest success probability achievable within $t$ transitions, by backward induction. These values are nondecreasing and bounded by one. For every policy, eventual success is the increasing set union of success within finite horizons, so its probability is the limit of those finite-horizon probabilities. Taking the supremum over policies and horizons in either order proves that the iterates converge to the optimal eventual-success probability. The limiting fixed-point equations can also admit larger solutions, so the initialization from terminal-success indicators matters.