Tuesday, 4 August 2009

'Forever' Means 'I'm Willing to Play this Iterative Prisoner's Dilemma Game Indefinitely'

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):

  1. 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?

  2. 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...


  1. I think a far more interesting topic is why you think this issue lends itself to analysis via "utility" and rationality.

    (Do I need to indicate the level of humour here?)

    More to the point, the analysis would be completely different if you introduce dynamics. There is no such thing as a uniform iteration in this context (it could vary for each party), and leads and lags might give you interesting push-pull dynamics and cyclic behavior. (Of the type in the Strogatz article on Reader.)

    The truly weird thing about commitment is that it often survives the defection of one or both parties.

  2. I think there is some element of rationality that goes into such decisions, or at least some sort of post facto rationalization that follows along such lines. And of course there's some sort of a pay-off, though I just crudely put it down as 'utility' so that I don't have to do too much math. The same holds true for introducing dynamics and so on - too much math, which I frankly am not very good at.
    As the standard explanation/excuse for all economic models goes, this model abstracts away from reality to be able to simplify some aspects and make a point. Hence the assumption of the iterations being uniform in pay-offs. Even with some non-uniformity, I think the point stands - if you get into a relationship with a clearly defined end-date there's very little incentive to commit fully. If you assume that the game is going t be played infinitely, you can allow for cases when either or both partners defect at any one iteration, but will choose to commit in subsequent iterations due to the incentive of future payoffs. If, on the other hand, you assume that the relationship has a defined end date, neither party has an incentive to commit in future iterations because they know the other party will continue to try to take advantage of them by defecting in each iteration.

    One point to ponder is whether both parties are aware of each other's intentions. As I stated in the first question, if one person believes that the game will be repeated infinitely, they might choose to continue with a commitment strategy, although the other person may plan to bail out at some point. To link it to the point I was trying to make about the MR post, if one party enters the relationship keeping in mind an end date (the woman who says that she wants a man who'll 'take the divorce well'), but doesn't make that end date clear to the other party (the future husband), what should the other party's strategy be?

    As for question 2, if you're the kind of person who needs other people badly, you might be willing to commit even if the other party opts to defect. Perhaps that could explain people remaining in abusive relationships, or becoming stalkers...

    Anyway, thanks for commenting. I'm getting a little shameless in soliciting feedback.

  3. Pretty brill post, actually :) .

    1) "if one party enters the relationship keeping in mind an end date (the woman who says that she wants a man who'll 'take the divorce well'), but doesn't make that end date clear to the other party (the future husband), what should the other party's strategy be?"
    Assume good faith? This works by your analysis. In real life, of course, there is the un-modeled issue of one's desirability going down with age, which might make it a worse call if you can't find anyone else when you're done. This would certainly call for exercising some caution. Which is what people do, anyway.

    2) Isn't the question a whole lot simpler, in this case? There are no losing or winning outcomes. (Well, depending on whether both Ux-single and Ux-relationship are both very high or quite low).

    Also, I believe most relationships DO have rather cyclic or push-pull behaviour. At least according to Men are from Mars etc.

  4. Hey thanks. It's not often that someone describes my writing as 'pretty brill'.
    I've almost forgotten my own analysis since I wrote it up, but I don't think it automatically implies assuming good faith. Frankly the model isn't good enough to help come up with an answer, it's just there to help frame the question better.

    As for relationships and stuff - please tell me you aren't actually referring to 'Men are from Mars...'