# India-Pakistan Game-Theoretic Interplay – Analysis

By Manohar Parrikar Institute for Defence Studies and Analyses (MP-IDSA)

By Sarita Azad and Arvind Gupta

India and Pakistan have once again embarked upon the path of dialogue. But will the process last? In this context, game theory—the mathematical study of strategies and decision-making—may offer some insights. In negotiations, game theory models the strategic interactions by predicting how people should ‘rationally’ behave in situations of conflict.

Cooperation is usually analysed in game theory by means of a non-zero-sum game, which involves two players with two choices. It addresses the class of different situations in which there is a fundamental conflict between what is a rational choice for an individual member of a group and for the group as a whole. The individual is relatively better off adopting the dominating strategy, regardless of what the opponent chooses. This leads to the so-called Nash Equilibrium, named after John Nash, a mathematician who received the Nobel Prize in economics in 1994 for his work on game theory. There is also, however, a Pareto-optimal solution, which is relatively better for both players. The outcome of a game is Pareto-optimal if there is no other outcome that makes every player at least as well off and at least one player strictly better off. Often, Nash Equilibrium is not Pareto-optimal, implying that the players’ payoffs can be increased.

To illustrate, a ‘game’ is formulated between India and Pakistan as follows: the two countries have strained bilateral relations because both mutually do not consent to negotiate on critical issues like terrorism, water, trade, etc. The two possible strategies for both countries can be to ‘Negotiate’ or ‘Not Negotiate’ on a certain issue. These strategies are taken as alternative courses of action, which they (the ‘players’) can choose. For the India-Pakistan ‘game’, the possible options for both countries are expressed in terms of a two-dimensional Table (see Table 1), where a two-person, nonzero-sum game is represented in the so-called payoff (profit) matrix. The objective is to determine a fair and reasonable outcome for the ‘game’.

Table 1 assumes that if both countries compromise on a given issue and choose to ‘Negotiate’, each will have a payoff of 2. On the other hand, if both choose the strategy ‘Not Negotiate’, the standoff continues and each would get a lesser payoff of 1. However, if only India were to compromise and adopt the strategy ‘Negotiate’ while Pakistan adopts the ‘Not Negotiate’ strategy, India loses by getting a payoff of 0, whereas Pakistan will get some incentive (payoff 3). Similarly, if only Pakistan were to compromise its stance and India were to defect, India gets a payoff of 3. Needless to say, the strategy choices and associated payoffs shown in Table 1 are only assumptions; a clearer picture of a given situation can be drawn by estimating the more or less real payoffs. Also, there can be more than two alternatives listed, as well as several variations on each. In such a situation, Nash Equilibrium seeks to find a solution where each player’s strategy is the best response to the other player’s strategy. Simply put, in Nash Equilibrium, no player has an incentive to unilaterally deviate from its current strategy. That is, if each player plays a best response to the strategies of all other players, we have Nash Equilibrium.

Now, Nash Equilibrium is identified by going through each cell of Table 1. Suppose that Pakistan thinks that India will choose to ‘Negotiate’. Then Pakistan has two options to ‘Negotiate’ or ‘Not Negotiate’ and will get payoffs 2 and 3, respectively. Obviously, Pakistan will opt the ‘Not Negotiate’ strategy because of the higher payoff 3. On the other hand, if Pakistan thinks that India will choose the strategy ‘Not Negotiate’, it has the option to ‘Negotiate’ or ‘Not Negotiate’ with payoffs 0 and 1, respectively. In this case too, Pakistan will choose the strategy ‘Not Negotiate’ because of higher payoff 1. Therefore, in either case, Pakistan will always choose the strategy ‘Not Negotiate’. Similarly, the best response for India is also found to be ‘Not Negotiate’. Thus, when both countries choose the dominant strategy ‘Not Negotiate’ as the best option, the negotiations collapse. The dominant strategy ‘Not Negotiate, Not Negotiate’ is then a Nash Equilibrium. However, this doesn’t lead to the conclusion that people necessarily play Nash Equilibrium. As can be seen from Table 1, if both countries were to choose ‘Negotiate’, both would be better off, with payoffs of 2 each rather than 1. In such a situation the strategy pair ‘Negotiate; Negotiate’, the Parteo-optimal solution, would be achievable.

India and Pakistan are for the moment stuck in a low pay-off Nash Equilibrium, whereas the obvious choice should be to move to Pareto-optimal solution. Neither side is able to make the transition because of the hardened position each has acquired over time, which prevents appropriate compromises. The hardline domestic constituencies, which see any concession as a defeat, prevent the two sides from making a compromise. Therefore, in India-Pakistan situations, the low paying ‘Nash Equilibrium’ is preferred by the players over the more rational ‘Pareto-optimal’ solution. In order to make the transition to Pareto-optimal, India and Pakistan will need to break away from traditional thinking of making no concessions.

*Originally published by Institute for Defence Studies and Analyses (www.idsa.in) at *http://www.idsa.in/idsacomments/IndiaPakistanGameTheoreticInterplay_sazad_200411