¹ An extensive survey of that literature is given by O'Neill, Barry, “A Survey of Game Models of Peace and War,” in Aumann, Robert and Hart, Sergiu, eds., Handbook of Game Theory, vol. 2 (New York: Springer-Verlag, 1992).Google Scholar

² Strategies are in subgame-perfect equilibrium (SGPE) if each is best against the other(s) in any contingency. The concept of SGPE comes from Selten, Reinhart, “Re-examination of the Perfectness Concept for Equilibrium Points in Extensive Games,” International Journal of Game Theory 4 (1975).CrossRefGoogle Scholar

³ Jervis, Robert, “Realism, Game Theory, and Cooperation,” World Politics 40 (April 1988).CrossRefGoogle Scholar

⁴ Loosely speaking, this means that players share the same expectation of future interaction, thus determining the path of future play. See Morrow, James D., Game Theory for Political Science (Princeton: Princeton University Press, 1994).Google Scholar

⁵ If each side punishes its opponent for not meeting its expectation, those expectations must be identical or they will remain hopelessly out of equilibrium. This requires a coordination of expectations.

⁶ Our characterization of SGPEs in Appendix 2 also suggests how to test whether countervailing strategies have been adopted, by searching in the data for traces of coordination. Such tests, performed on data for U.S.-China relations for the period 1972 to 1988 and reported in Appendix 4, support the hypothesis that a pure countervailing strategy rather than any other type of strategy was adopted by the two countries during the period under consideration.

⁷ In 2 × 2 games, if C denotes cooperate, D defect, and XY the choices X by row and Y by column, a Prisoner's Dilemma structure for row means his preferences satisfy DC > CC > DD y CD. There are several Deadlock-type games. The relevant one here satisfies DD > CD > DC > CC.

⁸ We refer here to the noncooperative models that assume individualistic actors.

⁹ O'Neill (fn. 1); Brams, Steven J. and Kilgour, D. Marc, Game Theory and National Security (New York: Basil Blackwell, 1988).Google Scholar

¹⁰ See Powell, Robert, Nuclear Deterrence Theory: The Search for Credibility (New York: Cambridge University Press, 1990).CrossRefGoogle Scholar

¹¹ Axelrod, Robert, The Evolution of Cooperation (New York: Basic Books, 1984).Google Scholar

¹² The folk theorem asserts that any “individually rational” outcome (i.e., no worse than the minimax) can be supported by a trigger strategy equilibrium in a repeated game. See Fudenberg, Drew and Maskin, Eric, “The Folk Theorem in Repeated Games with Discounting or with Incomplete Information,” Econometrka 54 (1986).Google Scholar

¹³ For example, it is not enough to recognize that one's opponent has chosen to play a trigger strategy (which involves collaborating as long as the other players do and reverting to noncooperative play for a fixed number of turns to punish the opponent for deviating from cooperative play). Each player must understand which particular trigger strategy has been chosen (how many punishment turns and which precise outcome is considered as the cooperative point) and chose to adopt the same one, since, if not, the strategies do not form an equilibrium.

¹⁴ Downs, George W. and Rocke, David M., Tacit Bargaining, Arms Races, and Arms Control (Ann Arbor: University of Michigan Press, 1990).CrossRefGoogle Scholar

¹⁵ Jervis (fn.3).

¹⁶ While countervailing is inherently reciprocative, the additional punishment of unexpected play could lead to retaliations for cooperative initiatives! This paradox is inherent to all trigger strategy schemes in continuous games.

¹⁷ See, among others, Downs, George W.Rocke, David M. and Siverson, Randolph M., “Arms Races and Cooperation,” in Oye, Kenneth A., ed., Cooperation under Anarchy (Princeton: Princeton University Press, 1986)Google Scholar; and Kenneth A. Oye, “Explaining Cooperation under Anarchy: Hypotheses and Strategies,” in Oye, ed., Cooperation under Anarchy.

¹⁸ Beer, Francis A., “Games and Metaphors,” Journal of Conflict Resolution 30 (1986).CrossRefGoogle Scholar

¹⁹ Axelrod (fn. 11).

²⁰ See Ostrom, Charles W., “Evaluating Alternative Foreign Policy Decision-Making Models: An Empirical Test between Arms Race Models and an Organizational Politics Model,” Journal of Conflict Resolution 21 (1977)CrossRefGoogle Scholar; Majeski, Steven J. and Jones, David, “Arms Race Modeling: Causality Analysis and Model Specification,” Journal of Conflict Resolution 25 (1981)CrossRefGoogle Scholar; and Moll, Kendall D. and Luebbert, Gregory, “Arms Race and Military Expenditure Models: A Review,” Journal of Conflict Resolution 24 (1980).CrossRefGoogle Scholar

²¹ See Dixon, William J., “Reciprocity in United States-Soviet Relations: Multiple Symmetry or Issue Linkage?” American Journal of Political Science 30 (1986)CrossRefGoogle Scholar; and Ward, Michael D., “Cooperation and Conflict in Foreign Policy Behavior,” International Studies Quarterly 26 (1982).CrossRefGoogle Scholar See also Zinnes, Dina, “Three Puzzles in Search of a Researcher,” International Studies Quarterly 24 (1980)CrossRefGoogle Scholar; Russett, Bruce, The Prisoner's Insecurity: Nuclear Deterrence, the Arms Race and Arms Control (San Francisco: W. H. Freeman, 1983)Google Scholar; and Isard, Walter and Anderton, Charles, “Arms Race Models: A Survey and Synthesis,” Conflict Management and Peace Science 8, no. 2 (1985).CrossRefGoogle Scholar

²² Zinnes (fn. 21).

²³ See Freeman, John R.Williams, John T., and Lin, Tse-Min, “Vector Autoregression and the Study of Politics,” American Journal of Political Science 33 (November 1989).CrossRefGoogle Scholar

²⁴ See Goldstein, Joshua S. and Freeman, John, “U.S.-Soviet-Chinese Relations: Routine Reciprocity, or Rational Expectations?” American Political Science Review 85 (March 1991)CrossRefGoogle Scholar; and idem, Three-Way Street: Strategic Reciprocity in World Politics (Chicago: University of Chicago Press, 1990).Google Scholar

²⁵ See Williams, John and McGinnis, Michael, “Sophisticated Reaction in the U.S.-Soviet Arms Race: Evidence of Rational Expectations,” American Journal of Political Science 32 (November 1988).CrossRefGoogle Scholar

²⁶ See McGinnis, Michael and Williams, John, “Change and Stability in Superpower Rivalry,” American Political Science Review 83 (December 1989).CrossRefGoogle Scholar

²⁷ Goldstein and Freeman (fn, 24, 1990 and 1991); and Goldstein, Joshua S., “Great Power Cooperation under Conditions of Limited Reciprocity: From Empirical to Formal Analysis,” International Studies Quarterly 39 (December 1995).CrossRefGoogle Scholar

²⁸ See Shapiro, James E. et al., Direct Investment and Joint Ventures in China: A Handbook for Corporate Negotiators (New York: Quorum Books, 1991).Google Scholar

²⁹ Ibid.

³⁰ Milholin, Gary and White, Gerard, Bombs from Beijing: A Report on China's Nuclear and Missile Exports, report prepared for the House Foreign Affairs Committee, U.S. Congress, May 1991.Google Scholar

³¹ See O'Leary, Greg, “China's Foreign Relations: The Reintegration of China into the World Economy,” in Brugger, Bill, ed., China since the Gang of Four (London: Crown Helm, 1980)Google Scholar.

³² Well-known mixed-motive 2×2 games include the Prisoner's Dilemma, Chicken, and the Stag Hunt.

³³ Miller, William G., The People's Republic of China's United Front Tactics in the United States, 1972–1988 (Bakersfield, Calif.: Charles Schlacks, 1988)Google Scholar.

³⁴ Choudhury, Golam W., China in World Affairs: The Foreign Policy of the PRC since 1970 (Boulder, Colo.: Westview Press, 1982)Google Scholar.

³⁵ Miller (fn. 33).

³⁶ Harrison Wagner, “The Theory of Games and the Problem of International Cooperation, American Political Science Review 77 (1983), argues that games such as Harmony and Deadlock should be considered before any mixed-motive game. And in a game such as Deadlock, the very meaning of the term cooperate becomes questionable: if both sides prefer the defect-defect outcome, is this not mutual accommodation?

³⁷ Goldstein and Freeman (fn. 24,1990 and 1991).

³⁸ The data used by Goldstein and Freeman (fn. 24,1990 and 1991) run on a scale from −6 to +6, where +6 represents full cooperation. In our notation, and if we did not aggregate events, this translates into x_us = 1/2 − UC/12, UC being the coded WEIS series representing U.S. behavior, and x_ch= 1/2 − UC/12, CU being the coded WEIS series representing Chinese behavior.

³⁹ Countervailing does not require a Prisoner's Dilemma structure in order to succeed. Indeed, our empirical estimates for the pre-open-door period (1972–78) point to a different game that, although listed in exhaustive studies, has received little attention; see Steven J. Brams, Theory of Moves (Cambridge: Cambridge University Press, 1994). That game has a Nash equilibrium in which both sides defect, leaving one side (China) with its best outcome and the other (U.S.) with its next to worst. But countervailing succeeds in promoting a relatively cooperative outcome while providing an SGPE in that game.

⁴⁰ When trying to define an initial date, the question is how far back one should consider past events and political regimes to be relevant to the day-to-day management of a relationship. By focusing on stationary strategies, we avoid this problem, although it is possible to reconstruct strategies that account for a starting date (see fn. 41).

⁴¹ If an initial date t = 0 can be meaningfully defined, we need to describe X_ch⁰ separately. But as it turns out, an arbitrary x_ch⁰ will be optimal against the U.S. countervailing strategy we describe below.

⁴² As the Chinese position x_ch^t evolves with time t, so does the U.S. discounted payoff in (4). Therefore, this Chinese countervailing strategy does not make the U.S. values constant in time. In fact, if the Chinese are provoked into higher and higher x_ch^t 's by a tough U.S. stand, the discounted values in (4) decrease as a result.

⁴³ Recall that this means that play prescribed by such strategies maximizes discounted payoff regardless of the prior developments of the game.

⁴⁴ Since we wish to discuss the Prisoner's Dilemma structure in this section, we must assume b_ch > 0 and b_us > 0. But it is possible to relax this restriction and to consider Deadlock-type games.

⁴⁵ In this approach, where the strategy is derived from the countervailing assumptions (9) and (12), we express x_ch^t+1 as a function of X^t instead of X_ch^t as a function of X^t−1 as in (3a). Of course, the two approaches are equivalent.

⁴⁶ Again we emphasize that payoff structures other than the Prisoner's Dilemma can be successfully dealt with using our countervailing approach. In other game structures the fear and temptation concepts do not necessarily apply and other relationships emerge.

⁴⁷ All other remarks made about the robustness of the countervailing strategy (3a) extend to (14b). In particular, no U.S. strategy could promote a better steady state for the U.S.

⁴⁸ g_us could also be written as a sum. The particular form chosen is for notational convenience when writing out China's move at time (t + 1) as a function of the past history of play.

⁴⁹ We could have generalized the payoff function and the g_us and g_ch functions to include terms in X_usX_ch. Such cross terms allow for estimation of a wider variety of payoff functions. However, the inferred coefficient on the cross term for the payoff functions were not found to be significantly different from zero.

⁵⁰ The denominator 6 in the k-term accounts for the range. The use of monthly data leads to aggregation of events and a widening of the range of the data. This would magnify the denominator 6 and would not affect the rest of our analysis. Of course, we still need to assume some bounded interval to guarantee the boundedness of payoffs required by the theorem in Appendix 2.

⁵¹ An autoregression following (18) will identify the parameters of a countervailing strategy. As the main theorem in Appendix 2 demonstrates, strategies that form SGPEs are either countervailing or coordinated. If the players do not adopt a countervailing strategy but instead coordinate play, then the data should reveal punishing responses by each player. As mentioned above, we provide, in Appendix 3, a test of the hypothesis that China and the U.S. could be coordinating their moves rather than implementing countervailing strategies. The data do not support the hypothesis of coordination.

⁵² In that case, countervailing would not succeed in providing an SGPE. A finding of negative b-coefficients for both sides would therefore refute the countervailing hypothesis.

⁵³ Vincent, Jack E., Project Theory (Landam: University Press of America, 1979)Google Scholar; Goldstein and Freeman (fn. 24,1990).

⁵⁴ Goldstein, Joshua S., “A Conflict-Cooperation Scale for WEIS Events Data,” Journal of Conflict Resolution 36, no. 2 (1992)CrossRefGoogle Scholar.

⁵⁵ Such a restriction does not imply that the effects of an aggressive move will necessarily evaporate after six months. First, if the defector persists in an aggressive position, countervailing will continue and even deepen with time. And if the defector moves back to cooperation, the countervailing responses will only slowly reestablish cooperation.

⁵⁶ See Greene, William, Econometric Analysis (New York: Macmillan, 1993)Google Scholar.

⁵⁷ See Granger, Clive and Newbold, Paul, “Spurious Regressions in Econometrics”, Journal of Econometrics 2 (1974)CrossRefGoogle Scholar.

⁵⁸ It turns out that when the equations estimated for the period 1972 to 1978 are used to explain subsequent years, a break in the data is apparent about 1979 and is confirmed by a formal Chow test.

⁵⁹ Even under the standard assumption of normal distribution for the coefficients estimated in (20), the resulting distribution of the b-coefficients is not normal since their derivation involves nonlinear transformations.

⁶⁰ Miller (fn. 33).

⁶¹ Brams (fn. 39),

⁶² Goldstein and Freeman (fn. 24,1990 and 1991).

⁶³ Goldstein and Freeman (fn. 21,1991); and Goldstein (fn. 27,1994).

⁶⁴ This is the case when payoffs are continuous and defined over a compact decision space. Here, for instance, the linear payoffs (13) are continuous and the decision space [0,1] × [0,1] is compact (as closed and bounded in ℜ²).

⁶⁵ At time t = 0, H⁰ = θ. Also note that the history H^t is of length t and that no stationarity assumption is necessary for either the strategies or the g_i and π_i functions.

⁶⁶ Since H^t is as arbitrary as H^t+1, we may use either one in the equality of the second line of (31).

⁶⁷ In other words, we assume that π_us = μ_us|X_us^t — ζ_us(X_ch^t, …, X_ch^t-n, X_us^t-1, …, X_us^t-n)|, with μ_us ≥ 0.