Strategic Exit and Unilateral Control

Repeated-game strategies can impose global payoff relations by balancing controlled rewards across time. The question is what remains when the opponent decides, after observing each round, whether the relationship continues. Against every history-dependent opponent strategy with finite expected duration, a cumulative linear identity is enforceable if and only if its expected stage increment vanishes at every reachable history. Strategic exit therefore converts global control into local control. In a two-action game with a unique local equalizer, the controller must use the same mixed action at every reachable history, without any memory restriction on either player. For the Prisoner's Dilemma with $(T,R,P,S)=(5,3,1,0)$, the surviving identities are $(1+p)U_X+(4-p)U_Y+5(p^2-3p-1)E[\tau]=0$ The duration coefficient never vanishes on $[0,1]$. Hence strategic bilateral exit leaves a one-dimensional payoff-duration relation, while ruling out every nontrivial payoff-only cumulative identity.