## Tuesday, September 15, 2015

### Maximum entropy better than game theory

 From NPR's Planet Money.

Richard Thaler has an article up where he discusses the Keynesian beauty contest. He does the test that he did with the Financial Times in 1997 again in 2015. That test is as follows: You win if you correctly guess 2/3 of the average of all entrants' choice of a number between 0 and 100.
• The Nash equilibrium of this game is zero (essentially an infinite regress of guessing and second-guessing 33, 22, 15, ...).
• The maximum entropy (information equilibrium) solution is 2/3*(50) = 33 (all states are equally likely, therefore 2/3 of the ensemble average of 50 is 33).

The final results of the contest were 18.9 (in 1997 with 1382 contestants) and 17.3 (in 2015 with 583 contestants) which means the error is:

• 19 (17 in 2015) for Nash equilibrium
• 14 (16 in 2015) for maximum entropy

So the two models are about the same.

However! This also illustrates the negative impact of expectations via non-ideal information transfer as the "price" (i.e. guess) in this case "should" be 33 -- the average guess should be 50 if we weren't second guessing each other and driving the price to zero.

The information transfer model in its full generality would say x ≤ 33 if you don't know that the market is ideal. An ideal market would have x = 33.

Since the information transfer framework understands its own limitations, it is a better model than the game theory result of x = 0. Basically x ≤ 33 beats x = 0 as an answer.

This also explains the the Keynesian idea that Paul Krugman put on his blog today:
Economies sometimes produce much less than they could, and employ many fewer workers than they should, because there just isn’t enough spending. Such episodes can happen for a variety of reasons; the question is how to respond.

Sometimes markets aren't ideal and you have non-ideal information transfer. That results in lower prices and less output (measured in money).

...

Update:

I am thinking about how MaxEnt would be applied to the cutest animal contest in the picture at the top (a picture I also reference here). My best guess is 1) something like the Monty Hall problem leads us to 2/3 = 67% and 1/6 = 16% for the other two or 2) that the logic that 1/3 will choose a given animal and 2/3 of those people will choose a different animal, leading to an approximate floor of 1/9 = 11% for the low performers (and a ceiling of 7/9 = 78% for the best).