# Game theory pointer: Tit-for-tat may be India's best strategy with Pakistan

Do unto others what they do unto you is a workable strategy under game theory where neither party can trust the other. This holds good for India and Pakistan.

Rajeev Srinivasan June 06, 2015 15:52:56 IST

The recent death of John Nash, the mathematician who came up with the idea of the Nash equilibrium, has brought game theory into focus as a mechanism for dealing with interactions in antagonistic situations. A good example of this is the simmering conflict between India and Pakistan: and let's see how the theory may predict how the two behave.

John Nash. AFP

Game Theory, shorn of the mathematics, is a probabilistic prediction of how two or more parties act in a 'game', that is, an interaction between them. It is assumed – and that seems reasonable – that the parties are all rational and wish to maximise the benefit to themselves. There are zero-sum games, where party A winning is guaranteed to make party B lose by an equal amount, so that there is no net gain or loss. A lawsuit may well be a zero-sum game: the winner wins by more or less the same amount that the loser loses (ignoring transaction costs).

Non-zero-sum games are more interesting. Your winning or losing may not have an equal and opposite impact on someone else. In some cases, there may be win-win; in others lose-lose. Sometimes it may yet be win-lose, although not necessarily with zero net impact.

A widely used game is the 'Prisoners' Dilemma', wherein two people have been caught committing a serious crime, but there are no witnesses except each other. The two, A and B, are arrested and held in separate cells, with no way to communicate with each other. The police tell each (separately) the following: there are no other witnesses, and the way to go forward is through confessions.

What A and B both are told is the following, all from the point of the penalties they may be subject to:

• If neither accuses the other, they both get one year as the case against them is not strong
• If A betrays B and B does not betray A, then B gets 10 years and A gets to walk away, free. The reverse is also true.
• If A and B both betray each other, they get five years each.

How would you play this game? The police may add the masala that the other guy has already betrayed you, and you don't trust the police, so you don't know if that is true.

On the face of it, it appears that the best technique would be to cooperate with each other and not betray the other. Unfortunately, since you can't communicate with each other, you cannot 'conspire'; you are left with the non-trivial possibility that the other party will betray you. Since you don't trust the other party to cooperate, the most rational outcome is mutual betrayal, surprisingly enough.

That, of course, is not an optimal outcome: the best outcome for both is they both get away with the light one-year sentence. Unfortunately, the most likely outcome is that they both betray each other. That is a Nash Equilibrium: a stable situation which is non-optimal, but in which neither party will dare to unilaterally change its strategy because of the possibility of a worse outcome.

Now consider two parties that deal with each other repeatedly in the same game: a repeated Prisoner's Dilemma game. A long-running buyer-seller relationship can be considered an example. Diplomatic moves between two rival countries can be another example. In these repeated games, each party has information about how the other player acted in previous games, and that can be used to predict their behaviour this time around.

What would be the best strategy in a repeated Prisoner's Dilemma game? Theorists have run computer simulations with millions of interactions, with all sorts of complicated algorithms as strategy, and looked at the outcomes from them. I'd like you to consider this question carefully before reading on: what, in your opinion, would be the best strategy to pursue to maximise your benefit?

Now consider what should be done if the two parties were India and Pakistan. There have been innumerable interactions between the two, and there are instances in which the two have cooperated, some in which they betrayed each other, and some in which one cooperated, the other betrayed. Is there a simple strategy to ensure that they would have maximised the benefit to each other?

Well, it turns out that the best, the very best, strategy in a repeated Prisoner's Dilemma game is the simplest: Tit-for-tat. That is, you start off by cooperating; thereafter you simply follow what the other player did in the last round. It turns out this is the most effective solution, and it can possibly lead to a situation where the two are not condemned to a Nash equilibrium of mutual distrust and betrayal; in fact, in an ideal world, it will lead to the two of them cooperating, with mutual benefit.

In fact, India has not followed tit-for-tat. For years, every Pakistani betrayal (Kargil, 26/11, Sharm -el-Sheikh etc.) has been followed by an Indian cooperation (resumption of talks, rail links, cricket ties), either under pressure from the motivated or clueless self-proclaimed intelligentsia or because the powers-that-be actually believed unilateral concessions would somehow result in better relations. Thus the Indians cooperated, and the Pakistanis betrayed merrily, with negative outcomes for the former. A simple application of game theory would have cured this behaviour.

It appears that Prime Minister Narendra Modi has attempted tit-for-tat with better results. He started by inviting the Pakistani prime minister for his inauguration – a signal that he wished to cooperate. But Pakistan responded with a betrayal: their envoy openly meeting with separatist Kashmiris. This caused India to also betray: it cancelled talks. Pakistan then tried another betrayal: heavy firing across the Line of Control. India's tit-for-tat was withering return mortar fire.

Now Pakistan has gone back to its old standby: killing Indian soldiers in Jammu and Kashmir. If the pattern holds, India will now retaliate with something that hurts them in return. Thus we have tit-for-tat, and a betrayal-betrayal, point-counterpoint. Not optimal, but a lot better than giving unilateral concessions and being betrayed continuously. And this may be all that can be expected with an irrational player like Pakistan.

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