= Unique outcome under strict outcome preferences
Suppose a finite <perfect-information game> has no chance moves, and every player strictly ranks the possible outcome labels. Different terminal histories may have the same outcome label and thus the same <payoff>. Every subgame then has a unique <subgame perfect equilibrium> outcome, even though its equilibrium strategy need not be unique.
The proof is <backward induction>. Each child subgame has a unique continuation outcome. The moving player's strict ranking selects exactly one best label among those outcomes. Several actions can tie only by leading to that same label, which proves uniqueness at the current node. For a sequential election, the labels may be winning candidates: a voter can have several optimal ballots, all producing its same favourite achievable winner. If preferences have ties between different labels, the conclusion can fail.
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