Tuesday, December 1, 2009

Spouses Dilemma - 2

Read part 1 here

In a finitely repeated Spouses Dilemma there can be no other equilibrium.
Consider this: In the course of their dispute, if John and the Lady(L) learn that they aren’t soul mates and decide to be together just for a finite period of time, say 10 days, they’ll loaf till the 10th day.
We can determine this by reasoning in reverse: On the last they, both of them will know that they have no chances to reward or punish each other further, so both of them will loaf. (If either one of them cleans, then he/she is not sticking to the plan and cares enough about the future so as to influence the behaviour of the other. Here we assume both of them behave like androids)
On the second-last day, L knows that John will loaf on that day as well as the next day so L has no incentive to clean. So she loafs. John reasons similarly and does the same. Reasoning in a similar fashion backwards, we find that both of them loaf on all days, regardless of whether they part after 10 or 100 days. Thus any definite stopping point causes cooperation to perish.

It is not surprising any more why pre-nuptial agreements invariably lead to divorces. In fact the very instrument of divorce snapped out the effect of the ‘infinitely repeatable’ from the institution of marriage.

Anyway, would things be different if John and the lady knew for certain that they were soul mates? The answer is yes, for they would care enough about the future. If the lady is smart enough, she would use strategies that threaten permanent punishment for selfish behaviour. These are known as grim strategies.
One example could be this:
“Clean on the first day. From the next day onwards, clean as long as John and I have an unbroken history of cleaning on every previous day. If this is broken, loaf for all days thereafter.”

That is, the punishment for either of them loafing is punishment. Better cooperate or else the house turns into a garbage dump. Obviously both of them clean of the first day. On the start of the second day, they have an unbroken record of cleaning, and so they clean again and so forth (cooperation*).

Assuming that both of them use these strategies, will they be tempted to loaf? After all the fact that they can cooperate doesn’t mean that they will cooperate. If you remember the simple pay-off matrix, both of them receive a pay-off of 2 on all days because they clean the house (upper left square). Suppose that one day John decides to watch a play instead at Rangshankara with his friends. He increases his pay-off to 3 (lower left square). If John cares about his future interactions with L he would decide otherwise. Because according to the strategy, L would begin to loaf on all subsequent days for the rest of their lives. (Remember they are soul mates?:p) This would reduce John’s best available pay-off from 2 to 1 (lower right square) Thus we have a Nash equilibrium, in which the threat of a punishment (and not love) – mutual loafing – causes them to cooperate.

Of course it is always better to rely on temporary punishments. Both of them may use strategies that tell them to punish selfish choices by loafing for a few days and then start cleaning again (aha!) But that is a different story. The question is whether John still sees his dream muse as an Economics Major.

*Nobel Laureate Robert Aumann pioneered the study of cooperation and conflict in repeated games.

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