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Games parties play: Nash equilibrium of the Nuclear Agreement

Prakarsh Singh was Visiting Scholar at the Institute for Defence Studies and Analyses, New Delhi.
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  • July 31, 2008

    Game theory is the analysis of strategic interaction among agents. These agents may be two or more people, firms or countries, with both taking into account how the other agent will “play”. Game theory has been employed to increase revenue at auctions, better equip challengers against incumbents in the market, and explain why bad dictators last so long.

    To make predictions, we commonly assume that both agents are rational. This means that both maximize their payoff or utility. Secondly, both know each other’s payoffs. In case of a prolonged conflict, this is a reasonable assumption since it would be unlikely that one party has its costs and benefits unknown to the other. Thirdly, all players know that others are rational.

    In Nash equilibrium, each player maximizes her utility given the behaviour of the other players. To illustrate Nash equilibrium, the following is a static game on the strategies of India and the United States in a simultaneous-move game of India conducting a nuclear test and US imposing sanctions.

    India   Sanctions No Sanctions
    Test 1,-1 2,0
    No Test -2,-1 -1,0

    Here, the first payoff in each box is India’s and the second is America’s. For example, if India were to test, then the United States should play “No sanctions” as it gets a higher payoff. Thus, testing for India and not placing sanctions for America are more profitable strategies for both countries. Hence, the Nash equilibrium is “Test, No Sanctions”. However, since this is a static game, it does not capture important real life scenarios. Limitations of this model include an American response after the test and not simultaneously. Here, we do not need to assume for instance that India believes that the US is rational or vice versa (the third assumption). Regardless of whether the other party is rational or not, it is always the best strategy for both countries to play the Nash equilibrium. This simple model also says nothing about the impact of the 123 agreement on India’s decision to test and the American decision to impose sanctions.

    To make it more realistic, we can construct a dynamic game. It can be shown that the India-US nuclear pact would increase the autonomy of India’s strategic nuclear programme in terms of testing a bomb. To illustrate the Nash equilibrium of the strategic consequences of the India-US nuclear agreement, a three-stage game is shown below. In the first stage, the deal passes through the remaining international hurdles (IAEA, NSG, US Congress) and either comes into place or there is “No Deal”. The second stage is India’s decision to test a nuclear device, which would be contingent upon prevalent circumstances and the utility it would provide to India. The third and final stage is American reaction to the test in the form of economic, military and diplomatic sanctions.

    In the event India conducts a test without the deal being passed, it is highly likely that the US will impose stringent sanctions. It is generally agreed that sanctions imposed in 1998 following the Pokhran-II tests did not have a significant long-run impact on the economy. For the purpose of modelling, let us say that these sanctions give -100 to India (short-term effects).

    Since the deal has been called “fair” by independent experts, let us assume that it gives a payoff of 100 to both countries. Now assume that there is a benefit of 50 for India if it goes ahead with a test. In the event of a test after the deal, the US has a choice: it either imposes sanctions at a cost of 50 for itself but 200 for India (resulting in a payoff of 50 for itself and -50 for India), or does nothing in which case it still gets 100. India, meanwhile, basks in the glory of the deal and the test with a payoff of 150. This argument would hold if other countries follow what the US does, as is usually the case.

    A relatively higher punishment for India has been assumed in case of American sanctions. India would stand to lose more in such a case because the United States is India’s major trading partner but India is not America’s major trading partner. It is also likely that many other countries would follow the American lead in imposing sanctions. Hence, India stands to lose not only because of differential trade relations between itself and the United States but also because of the close political, economic and diplomatic links among the countries of the West. However, the argument in favour of testing (after the deal) does not rest on how badly India would be hit by sanctions but on two other assumptions that could be considered realistic. First, the US would have no choice but to impose sanctions in the case of an Indian test without the deal. Second, if there is a deal, the US does have a choice of imposing or not imposing sanctions on India. With just these two assumptions, it can be argued that India has a greater incentive to test a nuclear device with the deal than without it.

    To find the Nash equilibrium path, we need to start at Stage 3 and use backward induction. In Stage 3 after the Deal, a rational America will not impose sanctions, thereby allowing India to go for a nuclear test and increase its payoff from 100 to 150. It will be in America’s interest to have no sanctions after the deal has been passed since a payoff of 100 is greater than 50. Even if America threatens sanctions, a rational India would know that the threat is not credible.

    On the other hand, if the deal does not pass, India may never be able to test if it wants to maximize its payoffs. This is so because US sanctions would be greater (as opportunity costs would be lower) and also because India would be energy deficient and economically less integrated, thus having lesser bargaining power.

    India can thus proceed with its strategic programme fully aware that a rational America will not impose sanctions. In very few historical instances has it been the case that a policy has been Pareto improving, that is, it has benefited some people without harming others. This deal seems to have accomplished a Pareto improvement over the existing energy security situation in India along with a military security enhancement. If the goal is enlightened national interest, the deal seems to be unambiguously good, both for India’s energy security as well as for its strategic autonomy.

    The shaded boxes show the Nash equilibrium path rational policies will follow.

    Prakarsh Singh is Researcher, London School of Economics and Visiting Scholar at the Institute for Defence Studies and Analyses, New Delhi.