While killing time on Google Reader, I came across two posts, one on Marginal Revolution and the other on Overcoming Bias, on relationships. They got me thinking about this idea I had a while ago about how it might be fun to apply the game theoretic framework of the Prisoners' Dilemma to how a relationship might form and survive*. I finally got around to typing it up into a post (sub-scripts and super-scripts are killer), so here goes:
(Important Disclaimer: I'm not taking myself seriously in this post, and neither should you).
Consider an individual x with a utility function broadly as follows:
U(x) = Ux if x is single, and
U(x) = U'x + UxR if x is in a relationship
U(x) is x's total utility (Yes, ok that's crappy notation, see disclaimer above)
Ux is the utility that x gets from generally getting on with the day-to-day aspects of his/her life,
U'x is the utility that x gets from generally getting on with the day-to-day aspects of his/her life when in a relationship, and UxR is the added utility that x gets from being in a relationship because the other person commits.
Assumption 1: Consider that committing to a relationship usually involves some sort of change in one's daily routine and possibly even more sacrifice, so we can assume that usually,
Ux > U'x
Although, one would presume that
U'x + UxR > Ux
(call this presumption 1, if you will; without this presumption, of course, further analysis would be meaningless)
Now consider another individual, y, with a similarly formed utility function V(y) with components Vy, V'y and VyR.
Assuming that x and y are of the right gender to suit their respective orientations and are open to getting into a relationship, a one-off encounter between them could be considered within the simple Prisoner's Dilemma framework as follows:
U'x + UxR , V'y + VyR
U'x , Vy + VyR
Ux + UxR , V'y
Ux , Vy
Here, since the bonus utility ( UxR or VyR) comes from having the other person commit to the relationship, if either party defects but the other commits, the defector gets the bonus but not the one who commits. Think of this in terms of the committing partner having to make sacrifices but not getting much of the rewards from being in the relationship. Obviously, then, as long as assumption 1 above holds, both x and y would defect in a one-off encounter, as in the standard single-iteration PD game, resulting in the two getting utilities of Ux and Vy, respectively. That's one way of explaining why it's very rarely that something like 'love at first sight' might happen (perhaps a relaxation in assumption 1 is required?).
The single-iteration PD game can be extended by considering iterative game play. Firstly, let us consider iterative game play with a defined number of iterations, say t. If I remember my introductory game theory classes, this does not arrive at a 'satisfactory' solution. While both players may consider committing since it means that they can get higher gains, since the number of iterations is fixed, it becomes rational to defect in the t-th iteration and aim for the highest possible gain. But if you know that your partner is going to defect at t, you could opt to defect at the (t-1)-th iteration itself, so you can try for the maximum gain in that iteration and avoid being duped in t. Since both parties would think this way, they will end up defecting from the first iteration itself. Not very romantic, but then again there are very few cases where you'll find people getting into a relationship with a clearly defined end-date. (There are examples, of course, but I'll leave you to find them and post them in the comments. I would guess, though, that in most of those cases assumption 1 would not hold).
On then to the next case: the infinitely-repeated PD game. Here, if both the players profess undying love and commit to each other, they get pay-offs of U'x + UxR and V'y + VyR in each iteration. They can also set credible threats for the other player, so that any defection by the other player could be met with some sort of punishment- a consequent defection in the next n iterations, say, or a 'grim' trigger strategy where any defection by one player will be met with the other player also defecting for all further iterations. These punishments ensure that the players are better off committing instead of defecting in any one iteration. And how do both players knows that the game is infinitely repeated? By repeatedly asserting the same, and/or locking in the commitment through a contract aka marriage. In such a framework, then, as long as neither player defects, both will maximize utility for all future iterations – or, as they say in the literature, they go on to live happily ever after.
Homework questions (Answer in the comments, if you please):
What happens if one person thinks that the game is infinitely repeated, and the other knows it's going to be finite? Read the post on MR again. How would this analysis apply there?
- How does the analysis change if we relax assumption 1? What inferences would you make of a person for whom Ux <>x ?
*Incidentally, I was considering naming this post 'Prem Qaidi', but I wasn't sure how many people would get the joke...