Game theory and whether to wear a tie

A bunch of us have to give short research presentations tomorrow to help convince our funders to keep giving us money. My officemate Costa and I had the following exchange:

“I don’t know if I should wear a tie. Are you going to wear a tie?”

“I was going to wear one because you said yesterday you were going to wear one.”

“Well I’m going to wear one if you wear one. Game theory, man.”

“Shit, what’s the Nash Equilibrium? I think it’s if we both wear ties.”

There was some general agreement around the room, and that’s where we left it. But because I’m an unreconstructed geek, I started thinking about this later. Is that the right answer? What kind of game is this? I reasoned that the best outcome is for everyone not to wear ties, but by far the worst outcome is to be the only one not wearing a tie (“better to be overdressed than underdressed”). I made a payoff table, simplifying it to two players. It looks something like this, where each box has the outcome for [Player 1, Player 2].

  Player 2 No Tie Player 2 Wears Tie
Player 1 No Tie good, good bad, okay
Player 1 Wears Tie okay, bad less good, less good

It turns out this is a “coordination game”: we’re both better off if we play the same strategy. Like any (2-player) coordination game, there are actually two Nash Equilibria, either of the boxes on the diagonal (top-left, bottom-right). Except I do feel I prefer to play “wear a tie” if I don’t know what Costa is going to do. That way, I avoid the risk of being under-dressed (generally with coordination games, you can rationally play either strategy if you don’t know anything about what the other player is doing. Interestingly, this game fits a special class of coordination games called “Stag Hunts”, where there is a conflict between safety and cooperation. We can cooperate for the best outcome (everybody agree not to wear ties) or we can play it safe and wear the ties, not trusting that everyone else will dress down. So there are generally two types of equilibrium strategies — the payoff-dominated one (lose the tie and take a shot at the best outcome), and the risk-dominant one (wear the tie just in case: forgo the best outcome but avoid the worst one). I guess in the setting of giving a talk, I’m feeling risk-averse.

Apparently the stag hunt can be used as a model for social cooperation and biological cooperation in a lot of settings, like it’s more-famous cousin, the “prisoner’s dilemma”.

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