Rational vs. Irrational Behavior - Behavioral Mathematics for Game AI - page 88

In the Prisoner's Dilemma, this manifested itself rather well. The strictly dom-
inant strategy was to betray the other player because that promised the best results
no matter what the other person did. However, as we explored, the optimal strategy
was for both players to keep quiet. The only way a player would select this option,
however, was if he knew that the other person was going to act in the completely ra-
tional fashion and keep quiet as well. If the other person's rationality is not known
(or he is known to be a loose cannon anyway), then the fall back is the strictly dom-
inant strategy.
In the Guess Two-Thirds Game, we are faced with a similar dilemma—but on
a larger scale. We need to decide if the other players of the game are going to be
acting rationally or not before we can decide if we should play the rational strategy
or not. What's more, now that we have strayed away from the simplicity of von
Neumann's game theory examples, we have to account for far more variables. With
the Prisoner's Dilemma, we had to ascertain the rationality of one person to deter-
mine which of the two choices he would make. In this case, how many of the other
players are going to be rational? And to what extent?
Iteratively Eliminating Irrational Answers
The problem here is that there is no strictly dominating strategy from which to start.
That is, we can't say, “This is the best way to play no matter what other people do. �
Unlike in the Prisoner's Dilemma, we can't say, “Betraying gives us the best chance.�
Interestingly, there is a unique pure strategy. In the Prisoner's Dilemma, this strat-
egy was to keep silent with the knowledge that our partner was going to do so as
well—because he was rational. Similarly, in the Guess Two-Thirds Game, this
approach leads us to the best possible result if everyone in the game is acting purely
rationally. However, we get a far different answer than we would expect.
To arrive at this answer, we need to walk through the exercise from the begin-
ning, just like we did with our miserly brigands. In this case, we do this by iteratively
eliminating strictly dominating strategies. To do that, we must find the strictly
dominated strategies. By using that information to our advantage, we can narrow
our solution set down significantly.
Just as a strictly dominating strategy was one that was the best for the situation
no matter what , a strictly dominated strategy is one that is the worst for the situation
no matter what. In the Guess Two-Thirds Game, there is no strictly dominating
strategy, but there are strictly dominated strategies. That is, there are ways to play
that will always lose— no matter what.
The reason for this is that there are mathematical impossibilities in the game.
Because the rules of the game state that people select numbers between 0 and 100,
we know that it is impossible for the average guess to be above 100. Certainly, it