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The prisoners dilemma with a more hopeful infinite horizon

This week in class, we have been covering the prisoners dilemma with an infinite horizon.  In this post I will discuss how the extended prisoners dilemma problem represents the global applies to countries and how the equilibrium strategies do not express the realities of real world cooperation.

First for anyone who is unfamiliar with the prisoners dilemma, basically the extended prisoners dilemma models what happens when 2 players have high, high-medium, low-medium, and low payoffs, such that if they cooperate they each get high-medium, if one cooperates and the other does not, the cooperating guy gets low payoffs and the jerk gets high payoffs, and if they both don’t cooperate they both get low-medium payoffs. Now in the extended form of the game both players decide whether to cooperate or not at each time step and the time goes on to infinity.

It can pretty easily be seen how the prisoners dilemma applies to country politics. If two countries, such as china and the philippines, cooperate they can both benefit from the strengths of the other and both be better off than going it alone. If one country helps the other but the other does not help, the unhelpful country will be much better off as it will get something without giving anything, and the helpful country will be strictly worse off. If both countries do not help each other, lets say the institute a policy of isolationism, than they might be okay independently but not as well off as if theyd cooperated. And as countries constantly make the choice to cooperate or not over time, they are a perfect example of the prisoners dilemma with an infinite horizon.

Now we have learned that there are two equilibrium strategies in the extended prisoners dilemma problem that allow cooperation, the “grim-trigger” strategy and tit-for-tat. The “grim-trigger” strategy basically says I will cooperate until the other guy doesn’t cooperate, than I will never cooperate again. Tit for Tat on the other hand says that you do the thing the other player did to you in the last round, with you initially being nice. I will not attempt to ptove these are equilibrium strategies, but i will say that it is important to note that the reason these strategies work is that the threat of total non-cooperation deters the other player from not cooperating at any time, so you end up always cooperating.

But these strategies do not seem to hold with the practices of reality. In real life there are countries which have done terrible things to each other, like war, such that if we followed the Grim trigger strategy, it would mean these countries should never again cooperate. Heck, the united states dropped two nuclear bombs on Japan and we are still business partners today. And if we lived by the strategy of tit-for-tat the minute one country did something to the other, they would constantly be exchanging blows from that point on. So if reality doesn’t follow the two equilibrium strategies that call for cooperation, what strategy does it follow? I propose one possible strategy that aims to model real world observations.

So the strategy I propose is called the hopefully-trusting strategy, the idea behind the hopefully-trusting strategy is that each player has a decreasing function f=c_i – x where x is how many times the other player has not cooperated since you last didn’t cooperate, and an increasing function g=y where y is how many times you’ve not cooperated since you last cooperated. basically the strategy is to cooperate until f=0 than stop cooperating until g=d_i, than cooperate again and repeat. Basically this strategy attempts model the concept of tolerance and trust. The function f is basically a buffer measuring how much you are willing to tolerate from the other player before you’ve had enough, where there are different tolerances t_i over time depending on the severity of relations. The function g on the other hand is a timer that basically tries to model the act of mending wounds and building trust, where there are again different times t_i over time depending on the severity of relations.

The hopefully-trusting strategy is not an equilibrium strategy as the other player is better off by  just occasionally not cooperating while still being in your tolerant zone, but i believe this strategy more closely models the actuality of relationships between countries in the real world.



One Response to “ The prisoners dilemma with a more hopeful infinite horizon ”


    I really like your proposal of a function for modeling a strategy in the extended prisoners’ dilemma. It takes more of the aspects of the relationship between two countries into account. I also think that this could be applicable elsewhere, like in intimate relationships or even friendships.
    The only concern I had was that world affairs is not a two player game. There are many players with many different agendas, and international pressure from the United Nations and other countries forces countries to cooperate in a lot of cases. That’s a lot of factors to go into strategy and would be extremely difficult, if not impossible, to model into a function. For a two country relationship, I really like the ideas behind your function

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