Game theory and empirical observations (this might be interesting for you, Parcival)

Encouraged by Parcival's comment on my post about modal logic, I'll pass on some wisdom of Joe Halpern's recent course in Lausanne (see also this post).

The short version is this: In game theory, the concept of Nash equilibrium (now famous by the movie A Beautiful Mind) is considered as the "best" solution (in some sense), but empirical observations showed that players may use completely different solutions to a particular game than the Nash equilibrium supposes.

Consider for example the following game: Two people are asked a (natural) number between 2 and 100. If both answer the same number, both will get as many dollars, as the number they said. If they answer different numbers, whoever said the lower number will get as many dollars as the number he said and additionally 2$ bonus, while the other one also gets the lower number, but minus 2$ penalty. Say for example, the two players both say 100, then they will get 100$. If the first player says 98 and the second says 99, then the first one will get 100$ and the second one will get 96$. Are we still together? (to quote Joe Halpern.) By the way, this is known as Traveler's dilemma.

Now game theory tells us, both players should say 2, as this is the only Nash equilibrium (for details, see below). But even without empirical studies, you clearly see that almost no player will say 2. Even members of the Game Theory Society didn't stick to this strategy, as a survey shows, the most popular strategies were to play 90 or higher. What's the problem about playing 2? It's quite simple: you will regret it, because you could have made a lot more money (even though you might make less than you opponent). So, people will usually choose strategies, that minimize their regret: If you play 2 and your opponent plays 95, you best response would have been 94. So you could have won 96$, but as you played 2, you will win 4$. So the amount of money you regret is 92$. People seem to be very sensitive to regret, it's not only important, what they won, but also what they could have won. So, you're trying to minimize your regret. And some simple calculations later you will see, that the optimal strategy to minimize your regret is to play 97. And this also agrees with the empirical studies.

More details can be found in this paper by Halpern and Pass.