Optimal Threats

Many conflicts, ranging from biblical strife to modern superpower confrontations, are significantly affected by the threat of retaliation. We analyze the policy of tit-for-tat in a Threat Game based on Chicken, in which each player can initially choose any level of preemption or non-cooperation. If one player is more preemptive than the other, the less preemptive player can retaliate; if the players preempt at the same level, neither can retaliate. After these choices, the game terminates, with the players' payoffs based on both the payoffs of Chicken and their levels of preemption and retaliation. An analysis of Nash equilibria shows the Deterrence Equilibrium (neither player preempts, but both threaten to retaliate if preempted) to be the most compelling solution to this game; a player's choice of his strategy associated with this equilibrium renders any level of preemption by his opponent costly. Deterrence Equilibrium strategies entail, in some cases, more-than-proportionate threats of retaliation against low levels of preemption and less-than-proportionate threats against high levels. We consider the optimality of such threats in light of existing prescriptions and evidence on different conflicts, especially those that occur in the international arena and may involve nuclear threats.