A Battle of the Sexes, in game theory terms, is a game where two players have no perfect solutions. They only have acceptable and unacceptable solutions.
The common example, from which the scenario derives its name, is this:
A husband and wife have agreed to go out for an evening. Neither can remember where they have agreed to go. They would both rather go anywhere together than be apart. The husband recalls they agreed to go to a boxing match, which he would prefer; the wife remembers agreeing to go to an opera, which she prefers.
In this scenario, the husband achieves maximum happiness in only one condition, going to the boxing match with his wife; and the wife only achieves maximum happiness at the opera with her husband. Neither achieves any happiness if they go their separate ways. In the end, there are two ‘best’ solutions. They both go to either the opera or the boxing match. Mathematically, it doesn’t really matter which, though of course to people it matters whose preference is observed.
In a narrative, we want to see protagonists aligned this way. We want all of the protagonists rowing in the same direction, but we don’t want them all perfectly happy about it. We want some to think they’re going the wrong way or at the wrong speed. Dissatisfaction is human, and to see it in our protagonists reaffirms their humanity. Sticking together despite their disagreements is part of what makes them protagonists. They are willing to sacrifice for the greater good. They get more satisfaction from going with the team than striking out on their own.
We can think about the Fellowship of the Ring. Merry, Pippin, Sam, Frodo, Gandolf, Aragorn, Boromir, Gimli, and Legolas often disagreed about how the should get to Mordor, but once a decision was made, they, while the Company lasted anyway, stuck together despite their individual reservations. Different characters determined the direction the Fellowship took, too. At different points Gandolf, Aragorn, and Frodo all pick a course and the rest follow.
For our heroes, being together is strictly better than any outcome where they are separated.
Jean Jacques Rousseau proposed a scenario wherein two men went hunting a stag. They could either lay in wait for the stag together, or either one could hunt a hare. The stag will feed both men and provide a surplus; a hare will feed only the hunter who kills it with no surplus. Hunting a hare ends any possibility of catching another hare or the stag. In order for both hunters to eat, each must trust the other to continue to wait for the stag. It is clearly the best solution to the situation, but as time continues each hunter must continue to gamble that the other hunter will not grow too impatient. There are variants of the game where the hunters may, or may not, talk to one another, which can have an important effect on their coordination.
For the narrative, we should understand that readers want protagonists to work together for the greater good- for them to make the best choice in this scenario and wait for the stag. Dramatic tension can be inserted by making it look as if the protagonists wont wait for the stag, or, better yet, making one of the hunters an antagonist, which would lead the reader to believe the protagonist will lose the game based on his overabundant trust. The antagonist might, believably, choose to wait for the best solution, or he might, also believably, choose to be selfish after waiting ‘long enough’.
In effect, the final book of Brandon Sanderson’s Mistborn trilogy has two, or more, protagonists engaging in a high-stakes, non-coordinated stag hunt. The effect is thrilling, because the reader is never sure all protagonists will all continue to seek the best overall solution (the ‘stag’). Sanderson does a great job of giving different protagonists options that seem appealing but less full-filling (‘hares’).
I’ve decided to spend some time exploring how game theory can inform an author’s perspective on the narrative. I started this in my discussion of dragon-slaying, and I enjoyed writing that. My intent with these is to learn a bit about human nature and how that can and should impact my writing.
Today, I’m going to focus on one of the best known game theory problems: the prisoner’s dilemma. As Wikipedia tells us a classic example of the prisoner’s dilemma is this:
Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated the prisoners, visit each of them to offer the same deal. If one testifies for the prosecution against the other (defects) and the other remains silent (cooperates), the defector goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?
Rational players, in repeated experiments, consistently choose to defect. While being silent and cooperating could have a nice payoff for both, too much trust is required to take that gamble.
What does this mean for the writer? Generally speaking, people will choose some bad over the possibility of a good if they can also defer a greater bad. In the realm of fantasy, we might rephrase the dilemma to this:
Two protagonists are captured by the antagonists and are separated. Each is offered a deal: each can choose to join the forces of darkness. If one defects and join the forces of darkness and the other stays silent, then the silent one will be killed and the defector will remain in place, perhaps to affect the final fate of the world. If both remain silent, the forces of darkness will keep them prisoner, but won’t kill them- the knowledge they have is too valuable. If both defect, the forces of darkness will have all of the knowledge it needs to destroy the world, but they are both still alive and in place with the small hope of affecting the final fate of the world.
As a translation, that’s a bit rough around the edges, perhaps, but it tells us rational protagonists may do things we, emotionally, don’t want them to do on the hope that they can achieve a pyrrhic victory.
Another thing this tells me is this: antagonists are often rational actors, protagonists rarely are. Emotionally, we want to boo the person who makes the rational choice here. Conversely, a fantasy protagonist will often hope and trust beyond reason. This is something we enjoy in them. We respect them for it. Trust and hope in others in the face of betrayal is Christ-like. Christian and non-Christian alike want such in their fantasy protagonists. Which means, in this case, a protagonist will likely short-circuit the prisoner’s dilemma. The best narrative would be to establish such a dilemma and have it seem as if the protagonist will do the rational thing. Then have him do something unexpected and heroic.