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² Garwin, , “Is There a Way Out?” (symposium), Harpers (June 1985), 35–47Google Scholar. Specifically, Garwin suggested the following:

The United States ought to build 400 Midgetmen, the small, not very accurate, singlewarhead missile proposed by the Scowcroft Commission, and put them in Minuteman II silos. Meanwhile, we should reduce our forces temporarily by 50 percent—send half our submarines to cruise in the Antarctic, pile 20 meters of earth over half our minuteman silos, and put half our strategic bombers in mothballs. Then we should invite the Soviet Union to follow suit within six weeks, and if it does, we should make the arrangement permanent (p. 44).

³ Schelling, , The Strategy of Conflict (New York: Oxford University Press, 1963)Google Scholar. See also Schelling, Thomas and Halperin, Morton, Strategy and Arms Control (New York: Twentieth Century Fund, 1961)Google Scholar.

⁴ Osgood, , An Alternative to War or Surrender (Urbana: University of Illinois Press, 1962)Google Scholar.

⁵ Axelrod, , The Evolution of Cooperation (New York: Basic Books, 1984)Google Scholar; Axelrod, Robert, “The Emergence of Cooperation Among Egoists,” American Political Science Review 75 (June 1981), 306–18CrossRefGoogle Scholar; Axelrod, Robert and Hamilton, William D., “The Evolution of Cooperation,” Science, No. 211 (March 1981), 1390–96CrossRefGoogle Scholar.

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⁸ Formally, the only requirement for an arms race is that each side prefer mutual escalation to unilateral reduction. This leaves open the possibility that a given arms race could be motivated by a wide variety of games. Of course, each participant in an arms race may have a different set of preferences and may thus be playing a different game.

⁹ Cobden, Richard, The Political Writings of Richard Cobden (London: William Ridgway, 1868)Google Scholar; Hirst, F. W., The Six Panics (London: Methuen, 1913)Google Scholar; Gooch, G. P., Franco-German Relations, 1817–1914 (London: Longmans, 1923)Google Scholar.

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¹⁶ For simplicity, in the simulation the rival states were assumed already to be proceeding at their individually optimal rate. This means that substantial deterioration could not occur. Had we started the simulation at a lower rate, it would clearly have reinforced our conclusions at the cost of some additional complexity of presentation.

¹⁷ Box, George E. P., Hunter, William G., and Hunter, J. Stuart, Statistics for Experimenters (New York: John Wiley, 1975)Google Scholar. In the simulation, we examine the effects of three parameters. In order to simplify the presentation, we use a base case and three sensitivity analysis cases that vary one parameter at a time. A full factorial design was not used because of the difficulty of grasping eight charts simultaneously.

¹⁸ Formally, we suppose that a state achieves a net benefit of

when arms are increased, and

when arms are decreased, where T_A and T_B are the new levels of arms for the two sides, and T°_A is the last turn's arms level for side A. The cost function is quadratic for arms increases, to represent the difficulties that the industrial infrastructure and the political system have with arbitrarily large increases. Values used in the simulation are b = 1.0, c = 1.0, and d = 0.5 in all cases, with a = 6 in the Base Case and a = 3 in the Low-Benefit Case. It is easily seen that the “optimal” increase of arms for side A is given by

regardless of the level chosen by side B. Thus, the simulation is begun by assuming that the arms race has been proceeding at this “optimal” level for some time. Note that it is possible for A to choose an arms increase greater that Although it might seem irrational, some punishment-based strategies could have this result.

¹⁹ Downs et al. (fn. 7), 123.

²⁰ In the first case, when B employs a reciprocal strategy, the increase will be

T°_A — T*_A

matching A's increase (or decrease) from last time, where the o and * superscripts refer to the last move and the one before that. In the second case, B will continue its past build-up no matter what, so the increase will be

T°_A — T*_B

Thus, the current increase is exactly the same as the last turn's increase. In modeling cautious reciprocity, we imagine that B examines A's change last time compared to B's own change two turns ago. If A were playing a reciprocal strategy, these two quantities would be identical. If they are, or if A's move is larger, then B plays reciprocally, subject to staying below the optimal increase; otherwise, B makes an increase midway between A's last move and B's own last move. Although this strategy seems complex, it only amounts to following up a cooperative gesture with a smaller cooperative gesture.

²¹ A larger bias makes tacit bargaining almost impossible; anything less would be historically naive.

²² The numbers were calculated from simulations run separately for the case in which the opponent is playing Deadlock and the case in which the opponent is employing the reciprocal strategy.

²³ If the payoff for making no cooperative gesture is P_o and the payoff for making a cooperative gesture of size G is P_G, then the quantity graphed against G is (P_G — P_o)/P_o. The required numbers were estimated by simulation.

²⁴ Payoffs were calculated by simulation for each of the seven strategies against each of the opponent's strategies for each environment. A typical point (p₁, p₂, p₃) in the triangle represents an estimated probability of p, that State B is playing its first strategy, etc. The expected payoff to a given strategy is a weighted average, by p₁, p₂, and p₃, of the outcomes against the three possible strategies of State B. From the payoff formula for each strategy at each point, the areas of optimality for each strategy within the triangle were easily computed.

²⁵ For a list and discussion of these arms races, see Huntington, Samuel, “Arms Races: Prerequisites and Results,” Public Policy 8 (1958), 41–86Google Scholar; Kennedy, Paul, Strategy and Diplomacy (Aylesbury, England: Fontana, 1984), 163–78Google Scholar; Downs et al. (fn. 7), 119–21. For one viewpoint on the role of cooperation-based tacit bargaining in the U.S.-Soviet arms race, see Adelman (fn. 1).