## Sunday, November 29, 2015

### Maximum entropy better than game theory (again)

Today Nick Rowe mentions the dictator/ultimatum game (I choose to divide a pot and if you refuse the division, we both get nothing ... or the dictator version where you get no input). It's another case where the maximum entropy guess is better than game theory. Game theory says the solution is 99.9% (or more) for the dictator and x = 0.1% (or less) for the other person if people were truly rational. Maximum entropy guesses x ≈ 50%, but allows x ≤ 50% if information transfer is non-ideal. It also would guess x = 33% for three players, x = 25% for four, etc.

More on the rest of Nick's post later, but it brings up this again. And this.

1. Interesting post Jason. I like these maximum entropy vs game theory ones.

O/T: please let us know if Allan Gregory takes you up on a forecasting challenge!

1. Thanks, Tom.

2. Game theory only finds solutions for two person zero sum games. Economists act as though n person non-zero sum games have solutions, though.

1. Hi Bill,

This is not correct -- it's true that zero sum games can sometimes be solved easily using minimax. However, minimax can solve n-player zero sum games:

https://en.wikipedia.org/wiki/Nash_equilibrium#Formal_definition

And n-player non-zero sum games do have solutions because they are equivalent to zero sum games with n+1 players:

"In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalized form of a zero-sum game for two players, and that any non-zero-sum game for n players can be reduced to a zero-sum game for n + 1 players; the (n + 1)th player representing the global profit or loss."

https://en.wikipedia.org/wiki/Zero-sum_game#Extensions

Basically all games can be represented as zero sum games for two players, even non-zero sum games for n-players.

2. If the Nash equilibrium were a solution, then college freshmen would not do better in the Prisoner's Dilemma than senior economics majors. A soldily reproduceable result.

Also, I am not referring to the Iterated Prisoner's Dilemma, either, which has its own properties.

3. My remark about economists had to do with their treatment of Nash equilibria as solutions. Once they have found an equilibrium, they assume that they have discovered how people will behave. In fact, people often do not choose their Nash equilibrium options, and often are better off because of that. That could not happen if the Nash equilibrium were a solution.