Playing with game theory

1 October 2008

In a recent article Varoufakis (2008, p.1259—1260) says:

To give an example, consider the following simple N-person game known as the Race-to-Zero. N players are asked to write on a piece of paper (in isolation from one another) a real number between 0 and 100 (inclusive). The player whose chosen number is nearest the maximum choice among all players divided by two wins $1m times her choice of number. (Joint winners divide the spoils.) Is there a “solution’ to this game? Is there an equilibrium towards which the players’ choices will tend the more rationally they think? What number should one write down? Nash suggests that rational players would immediately decide that it makes no sense to choose a number in excess of 50, thinking that: “Since the largest number that can be chosen is 100, and I win if my choice is nearest to that maximum choice divided by 2, I should never choose a number above 50.” However, this thought immediately begets another, infinitely longer, thought:

‘If I am clever enough to work this out, then the rest will also work this out too. Therefore none will select a number greater than 50, in which case I must not choose any number above 25. But if this is so, will the others not know this to be so too? And if they do, will they not restrict their choices to a maximum of 25? Then I must not go beyond 12.5.”

And so on. Asymptotically, one’s optimal choice of number tends to zero just as surely as the proverbial rock rolls down a hill until, asymptotically, it hits rock-bottom. “Choose zero’ is, therefore, the game’s equilibrium.

I often have problems following such ad infinitum reasoning. What Varoufakis is saying is that rational thinkers will select zero as the best option. And yet it clearly isn’t the best solution.

Firstly, there is no payout at zero, so it makes no sense to choose zero as ones best choice. Why would anyone make a choice where the payout is guaranteed to be zero (and this isn’t even a zero sum game). Any number other than zero is a better number (in fact one could argue that one is a where things will come to rest).

Secondly, If I truly believe that everyone else will chose zero then I should choose 100. If everyone else chooses zero, and I choose 100, we are all the same distance from the maximum choice divided by two (0—50—100), so I am a winner (as is everyone else). But my payout is 50 x $1m, and not 50 x 0.

Now, if everyone thinks as I do, and they all pick 100, then we are all winners, albeit that we have to share the money. Of course, if one some “smart Alec’ goes it alone and picks 50 then I am “stuffed’… and the circle starts again, and I should then pick 50. And if I do, then everyone else. At which point we’re back to what Varoufakis says should happen, everyone gravitates to zero.

Except of course the zero choice makes no sense (as there is no pay-off). Consequently, I don’t think there is a “rational’ solution. There is no point of equilibrium (Nash or otherwise—but since I’m not a mathematician, nor a game theorist perhaps I shouldn’t be so bold in my assertion).

As an aside, if one is actually the number that the rest of the players choose, then it still makes sense for me to go for 99 . I’ll leave it to you, the reader, to figure out where (if at all) is the point of equilibrium exists in this version of the game.

My argument is that there is no logical (aka rational) solution except not to pick zero. More generally I would say that there is no point of equilibrium for all players in this game. Either that, or if the “rational’ thing to do is to go with zero—and that we expect rational behaviour from everyone else—then best thing to do is be “irrational’ and go with 100.

Someone, please point the error in my-logic.

References

Varoufakis, Y. (2008). Game Theory: Can it Unify the Social Sciences? Organization Studies, 29(8—9), 1255—1277. doi: 10.11770170840608094779.