Tournament of Numbers, part 1...

In my previous posts about the game of Traveler’s Dilemma I argued that the Nash equilibrium value of 2 is not a very rational choice.  It seems that way intuitively, and when people play the game for real they pick much higher numbers and win higher returns.  But as interesting as it is to see how humans actually play, is there any way we can determine which number or numbers are objectively the best?  We’d like to bypass formal analysis and get a purely empirical result that we can compare to the high values typically chosen by human decision-making and the value of 2 preferred by game theory.  It turns out we can do this by holding a series of virtual tournaments.

In Traveler’s Dilemma two players each pick a number from 2 to 100 and they both win the minimum of the two, plus a two point penalty for the one picking the higher number and a two point bonus for the one picking the lower number.  The best strategy is to pick a number that yields the maximum outcome against any other strategies, themselves chosen for yielding maximum outcomes.  Using a virtual tournament and allowing different strategies to compete will determine which one is best directly by measuring how well they do.  By running the actual experiment, albeit in a computer, we don’t have to rely on analysis – the best strategy will simply be the winner.

Each round of the tournament starts with a fixed pool of strategies, some of which may be the same.  A strategy in this case is just a single number from 2 to 100 which will be played in each game, but in principle it could be something more complex.  The round consists of a fixed number of games, and two strategies are chosen at random to play against each other each game.  The total points each strategy wins in the games it plays determines its overall ranking for that round.  The strategies scoring in the bottom half are then replaced by those in the top half.  The total number of strategies stays the same, but the most successful strategies move on to compete in the next round.  This tournament continues until a winner emerges.

To assure that we’re fully exploring possible alternative strategies, each round contains a few ringers.  These are variants of the successful strategies which are created infrequently but often enough so that there are several in each round.  One type of variation is to increase or decrease a previously winning number by one, and the other type is to just pick a completely new number.  The first type assures that if there’s a better strategy near a number that came out ahead in the last round we’ll find it next round.  The second type of ringer assures that our winners don’t work themselves into a cul-de-sac where they do well against each other but can be easily beaten by a radically different strategy.

This virtual tournament is an example of a genetic algorithm, a type of global optimization that is highly applicable to game theory problems.  The randomness inherent in the process means that it cannot give a single closed form solution, but it will tell us a great deal about the types of strategies that will tend to be successful and how they fare against rival strategies.  I have programmed and run this experiment, and the results are unequivocal: humans are rational, the Nash equilibrium is irrational, and cooperation is a quantifiable good.

- jack*

UPDATE: Part 2 here.

Comments

[Marilyn]

Keep up the good work.