Game Theory

A reminder, because I always need to remind myself:

 B chooses a column
A chooses a rowA's payoff, B's payoffA's payoff, B's payoff
A's payoff, B's payoffA's payoff, B's payoff

Prisoner's Dilemmas

It won't be news to anyone here that altruism is useful for converting a prisoner's dilemma into a non-dilemma. A situation that looks to two selfish agents like this:

2, 20, 3
3, 01, 1

 could look to two perfectly selfless agents like this:

4, 43, 3
3, 32, 2

where we've just added their utility functions. Or maybe more realistically, if we add in the anger that even most selfless people would feel at being cheated:

4, 4-1, 3
3, -11, 1

In any case, after a dose of altruism, prisoner's dilemmas can disappear.

Battles of the Sexes

A battle of the sexes game is of the form

11, 100, 0
0, 010, 11

The row player would like to coordinate on the top left, and the column player would like to coordinate on the bottom right.

Let's add some labels. Suppose one member of a couple wants to eat vegetarian (which involves their joint meals being vegetarian), and the other member of the couple wants to regularly eat meat (which involves many of their joint meals containing meat). Either one of them will capitulate, or they'll have to break up. And if they're honest with you, both would rather stick together and make sacrifices on their diet if need be.

 MeatNo Meat
Meat11, 100, 0
No meat0, 010, 11

But there's another issue that arises because we want our partners to behave in certain ways in games with us. Suppose that the above is the payoff matrix "on its face". What I mean is: that's the payoff for each of the two people if some other entity (like a totalitarian state) picked both of their actions for them. Supposing that, the payoff matrix would look slightly different when they have to pick the actions themselves. Because if the row player picks "Meat", the column player would probably be thinking "I can't believe he wasn't willing to give up meat for me! I guess he likes meat more than he likes me. Do I really want to be with him?" And if the column player picks "No Meat", the row player would probably be thinking "I can't believe she's making my diet a condition of our relationship. I guess she cares more about food than me. Do I really want to be with her?" And both of them have a point. How could the other be putting a relationship at stake (worth 10 points) out of concern for a measly 1 point?

Here are a few more examples.

Do we sleep with other people?YesNo
Yes11, 100, 0
No0, 010, 11

"Why does she care more about having casual sex with other people than she cares about me?" "Why does he only want me if I'm not allowed to sleep with other people?"

And of course, people can talk through it and try to come to an understanding. "This is something I really care about." "Ok." But in some cases, there might be a really fundamental disagreement, and after all the talking through it, they just still disagree and feel really deeply about the issue.

Do the cats sleep in our bed?YesNo
Yes11, 100, 0
No0, 010, 11

"She wants to sleep with the cats so much she doesn't care if it drives me away?" "He'd make me give up sleeping with the cats if I want to stay with him?"

Where do we live?LANY
LA11, 100, 0
NY0, 010, 11

"So his family is supposed to take precedence over my career?" "So her career is supposed to take precedence over my family?"

Do we eat dinner at 7 or 10? Does an aging parent live with us? Do we hire a nanny? 


Battles of the Sexes are especially painful for couples. Capitulating on any of the things above could be really painful, in part because they get a smaller payout "on the face of it", and mostly their partner is giving them an unfair choice, and that hurts; the choice is to break up or do it their way, no matter how much tact might be employed to obscure that. A healthy relationship will involve the understanding "we both make sacrifices for each other", and couples can try to resolve pairs of these games in opposite ways for fairness. But they still put a strain, and attempts to keep score will usually end up biased and always require nursing hurts.

In the original example, I hope most readers think that obviously someone is allowed to inconvenience their partner with vegetarianism. But in many cultures, it would just be called "being a picky eater", and they'd side with the meat eater. Most of our grandparents probably grew up in such cultures.

I think convention is how we usually deal with most of these problems. One choice is widely held to be the "default" choice, and the person who opposes the default is the one who is to blame for staking the relationship on a smaller issue. In the circles I run in, it's considered more normal to take the side of the restricted diet (even if it's restricted by choice), monogamy, cats not in bed, career-based location, dinner around 7, aging parent not living with us, and hiring a nanny.

I've presented the payoff matrices as if others were deciding on the players' behalf, but remember the true payoffs are lower when the players decide themselves because of effects of the form "I can't believe she wouldn't...". That's one way of making sense of the value of deferring to conventional wisdom on these matters, above and beyond the wisdom-of-crowd-type reasons for deferring to convention: joint deference to a third party avoids a lot of pain.

At least this is how most people deal with these problems. They ask their friends "Am I crazy for wanting X?" Rationalist couples might be more inclined to debate these things on merits, but a) that could get exhausting, and b) it's hard to not frame things in one's own desired light, and c) when a game really does have the form of a Battle of the Sexes, no arguments can dissolve that.

I'll end with one more example to show that this is a bit broader than just for couples. I've invested in a couple companies, and the question arises: how are the contracts structured? There are plenty of terms that could go one of a few ways, favoring either existing investors or favoring the new investor. If the two parties can't agree, there's no deal, and the payoff is much worse. But it's costly to negotiate endlessly. The solution seems to be to just go with "the standard terms" whenever possible.

The Analogy

As we probably evolved altruism in part to avoid prisoner's dilemmas, we probably evolved convention in part to avoid battles of the sexes.

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Lol yep. And as usual, he writes much better.

Don't forget that we don't have access to each other's utility functions.  I've started to think of "X, Y" as an inferior way to express A's and B's utility - it's better to think of them as incomparable dimensions - "X + Yi".  Your description of altruistic players is therefore incorrect - each player has a function where they value the other's (perceived) reward.  Instead of a selfish 2,2 becoming an altruistic 4,4, it's a selfish 2+2i becoming an altruistic 2+Afn(2i)+2i+Bfn(2), where Afn and Bfn output i*some conversion).  One problem in this is that MANY (perhaps all) humans use non-great functions for this altruistic conversion.  It's easy to mix up "what rewards B" with "what I imagine I'd want if I were B" or "what B should want".  

I think further that you're mixing up utility with resources (or components of the world-state that affect utility, but aren't actually utility).  The 11+10i meat/no-meat result is utility ONLY if it includes all of the other crap that goes on in our minds to evaluate world-states, including "do they really love me, why don't I have power in this relationship, etc."  If it's just a component, then there's no reason to believe it's linear, nor that it's independent of all the other things.  

I do like some of the exploration of convention, and the recognition that it's not universal.  I don't think simplistic game theory adds much to that theory, though.  And I suspect you're massively underestimating the cost of creating, understanding, and negotiating within those conventions.  

Your description of altruistic players is therefore incorrect

Yeah, it's a very simplified version of altruism. I understand that real altruism in the wild does not look like adding another's utility function to one's own (which one somehow has read access to).

 The 11+10i meat/no-meat result is utility ONLY if it includes all of the other crap that goes on in our minds to evaluate world-states

No I'm saying: assume that the payoff matrix I've written down is correct for the situation in which some outside entity chooses each player's actions for them. This whole post is basically about how the "other crap" changes the payoff matrix when they players have to take responsibility for picking the actions themselves.

If it's just a component, then there's no reason to believe it's linear, nor that it's independent of all the other things.  

I know what linearity and statistical independence are, but I don't know what you mean by this.