¹ For a treatment of the Nash bargain, see Nash, J. F., “The Bargaining Problem,” Econometrica, XVIII, NO. 2 (April 1950), pp. 155–62.CrossRefGoogle Scholar A Nash bargain represents a situation in which two players have commodities to trade. The bargaining space consists of all possible trades with “no trade”—“(0, 0)”—considered as the origin of the bargaining space. The Nash criteria given below in the text define a trade in such a way that a single point, and only a single point, of the negotiation set satisfies the criteria. The criteria seem in some sense to be the most reasonable. The situation described in this article is not quite analogous, but may be translated into the terms of a Nash bargain simply by treating complete destruction of both players as the origin of the bargaining space and any solution short of complete destruction of both as a trade.

² Invariance with respect to utility transformations means that two versions of the same bargaining space which differ only in the point of origin and utility function can be mapped into each other by one specific utility transformation. Pareto optimality in a trade means that the payoff to one player cannot be increased without decreasing the payoff to the other player. In other words, a Pareto optimal trade must be at least as good as the status quo, feasible, and not bettered by any other feasible point. Independence of irrelevant alternatives means that if some new feasible trades are added to a bargaining problem in such a manner that the status quo remains unchanged, either the arbitrated solution is unchanged or one of the new trades becomes the arbitrated solution. Symmetry in the solution means that if the players are in symmetric roles in an abstract version of the bargaining game, the arbitrated value will yield them equal utility payoffs where utility is measured in the units which made the game symmetric. See Duncan Luce, R. and Raiffa, Howard, Games and Decisions, New York, 1957, pp. 124–28.Google Scholar

³ Schelling, Thomas C., “Bargaining, Communication, and Limited War,” Journal of Conflict Resolution, 1, No. 1 (March 1957), pp. 19–36.CrossRefGoogle Scholar Schelling also discovered that if one player offers to accept $49, die other player tries to force him even lower. This is intuitively reasonable, since it seems a confession of weakness. Such seeming weakness is even more important where non-nuclear forces are unequal and will be discussed in Theorem 2.

⁴ Counterattack gives (80, 80) because each player loses 15 on defense and 5 on attack. For a discussion of the stochastic game, see Kaplan, Morton A., System and Process in International Politics, New York, 1957, ch. 1.Google Scholar

⁵ A threat is a communication that, under stated circumstances, an action will be taken that destroys resources of the other player.

⁶ Szilard, Leo, “Disarmament and the Problem of Peace,” Bulletin of the Atomic Scientists, XI, No. 8 (October 1955), pp. 297–307.CrossRefGoogle Scholar

⁷ I am indebted to Richard L. Meier of the Mental Health Institute of the University of Michigan for his exposition of the great value of continued testing of nuclear weapons.