Game Theory and Divorce

by W.F. Price on January 16, 2013

William Press and Freeman Dyson (still going strong at 89, apparently) have come out with a new explanation of the Prisoner’s Dilemma that explains a lot about what happens in divorce. The Prisoner’s Dilemma game deals with betrayal and cooperation, and attempts to explain why cooperation might have evolved.

In a simple, one-shot game it pays to betray the other party, but when a sequence of games is played the results get more complicated, because the other player’s strategy becomes recognizable.

The new analysis shows how a sentient player (one who knows the other player’s intent) can consistently maximize payoff by “extorting” the other player.

Press and Dyson derive a simple formula for the long-term scores of two IPD players, in terms of their one-memory strategies. Their formula naturally suggests a special class of strategies, called zero determinant (ZD) strategies, that enforce a linear relationship between the two players’ scores. The existence of such strategies has far-reaching consequences. For example, if a player X is aware of ZD strategies, then she can choose a strategy that determines her opponent Y’s long term score, regardless of how Y plays. There is nothing Y can do to improve his score, although his choices may affect X’s score. This is similar to the “equalizer” strategies, discussed in the context of folk theorems (11). Such a strategy is essentially just mischievous, because it does not guarantee that X will receive a high payoff, or even that she will outperform Y.

However, ZD strategies offer more than just mischief. Suppose once again that X is aware of ZD strategies, but that Y is an “evolutionary player,” who possesses no theory of mind and instead simply seeks to adjust his strategy to maximize his own score in response to whatever X is doing, without trying to alter X’s behavior. X can now choose to extort Y. Extortion strategies, whose existence Press and Dyson report, grant a disproportionate number of high payoffs to X at Y’s expense (example in Fig. 1). It is in Y’s best interest to cooperate with X, because Y is able to increase his score by doing so. However, in so doing, he ends up increasing X’s score even more than his own. He will never catch up to her, and he will accede to her extortion because it pays him to do so.

This is applicable to divorce, because in most cases the female spouse is the “sentient” partner; she is more subjectively aware than the other “player,” who might not even realize that he is involved in a game. It is very helpful that Press and Dyson have set X as a sentient female and Y as an evolutionary male, because it allows us to get a more accurate idea of how the game is played in real life.

No fault divorce, in removing punishments for betrayal, has given X a permanent advantage whereby she can extort Y, forcing him to accede over and over again to her great advantage. Y’s only defensive response to being taken advantage of is to sabotage the game so that X will receive a lower payoff, and perhaps consider cooperation with Y.

Another possibility is that X alone is aware of ZD strategies, but Y does at least have a theory of mind. X can once again decide to extort Y. However, Y will eventually notice that something is amiss: whenever he adjusts his strategy to improve his own score, he improves X’s score even more. With a theory of mind he may then decide to sabotage both his own score and X’s score, by defection, in the hopes of altering X’s behavior. The IPD has thus reduced to the ultimatum game (12), with X proposing an unfair ultimatum and Y responding either by acceding or by sabotaging the payoffs for both players.

Finally, if both players are sentient and witting of ZD strategies, then each will initially try to extort the other, resulting in a low payoff for both. The rational thing to do, in this situation, is to negotiate a fair cooperation strategy. Players ignorant of ZD strategies might eventually adopt something like the Tit-For-Tat strategy, which offers each player a high score for cooperation but punishes defection. However, knowledge of ZD strategies offers sentient players an even better option: both can agree to unilaterally set the other’s score to an agreed value (presumably the maximum possible). Neither player can then improve his or her score by violating this treaty, and each is punished for any purely malicious violation.

Marriage 1.0 used the “Tit-For-Tat” strategy to keep spouses in line, but that has been legally eliminated in favor of a zero determinant strategy that, when unrestrained, encourages extortion and sabotage.

However, if both players are aware of ZD strategies, they can agree on a preset value (e.g. prenuptial agreements). Unfortunately, for those who are aware of ZD strategies without agreements, the best tactic when confronted with a problem is to immediately engage in sabotage, because that will force cooperation on the part of the other player.

For the mindless Y types (typical husbands), X can consistently extort to her advantage in perpetuity.

Extortion strategies work best when other players do not realize they are being extorted. Press and Dyson discuss how extortion strategies allow a sentient player to dominate an evolutionary player, who mindlessly updates his strategy to increase his payoff. However, this is not the only use of ZD strategies. Had Press and Dyson
kept their results to themselves, they may have enjoyed an advantage in tournaments like those set up by Axelrod (6, 7), in which a variety of fixed strategies compete. To test whether this is true, we reran Axelrod’s original tournament, but with the addition of some ZD strategies (Fig. 1). We found ZD strategies that foster cooperation and receive the highest total payoff in the tournament—higher even than Tit-For-Tat’s payoff. In addition, we found extortion strategies that win the largest number of the head-to-head competitions in the tournament.

This suggests that for men, avoiding victimization begins with foreknowledge of the nature of the game and the rival, and understanding of the fact that ZD strategies have begun to dominate marital relationships.

Once informed, the options, therefore, are:

1. Do not play the game

2. agree on preset scores

and, as a last resort,

3. Sabotage

Number 3 is the least favorable, but it also happens to be the tactic that most married men would need to use, since they don’t have prenups. What this means is that without some sort of construct that enforces cooperation on both players, it is always in the player’s interests to not cooperate.

Example: she initiates the game by saying “I’m unhaaaapy, so you need to do this and this and this or I’ll leave.” This is pure extortion. The man may stave off disaster by acceding to her demands, but she will come out on top, and will be emboldened to repeatedly use the same tactic.

Rather than cooperate, the man should immediately say something like this (or actually do it): “OK, then I’m going to quit my job, sell everything and disappear, or you could come to terms with me and we could avoid that kind of mess.” To prevent a pattern of extortion from occurring, the man must be prepared to sabotage the relationship every single time an extortion attempt arises.

The implications of the game in regards to marriage are pretty clear, and suggest that mutual interests are best served by controlling scores with prenups. However, it is also evident that keeping men ignorant of the nature of the game is possibly the best strategy for wives, because it allows endless extortion from mindless males. Sabotage is an effective counter to extortion, but because it incurs high cost on both parties it should be used only when extortion is initiated.

Leave a Comment

Previous post: