Adversarial search

Exploration of a game tree containing choices by competing decision makers. A maximizing node chooses the largest successor value; a minimizing opponent chooses the smallest. Known random outcomes instead require chance nodes and expected values, as in expectimax. A game tree representation can combine controlled choices and chance without assuming that chance maliciously chooses its worst outcome.

Backward induction on a finite game tree containing maximizing choices, minimizing choices and chance nodes. Terminal utilities are backed up by $\max$, $\min$ and ${\sum }_j{p}_j{V}_j$, respectively. Exact chance node evaluation generally needs every positive-probability outcome; ordinary alpha-beta pruning does not transfer unchanged because one child's value does not bound the full weighted sum.

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.