July 7, 2008
Field centipedes and the perils of rationality
Game theorists know all about the centipede game:
One instance of the centipede game is as follows. A pile of $4 and a pile of $1 are lying on a table. Player I has two options, either to “stop” or to “continue.” If he stops, the game ends and he gets $4 while Player II gets the remaining dollar. If he continues, the two piles are doubled, to $8 and $2, and Player II is faced with a similar decision: either to take the larger pile ($8), thus ending the game and leaving the smaller pile ($2) for Player I, or to let the piles double again and let Player I decide. The game continues for at most six periods. If by then neither of the players have stopped, Player I gets $256 and Player II gets $64. Figure 1 depicts this situation. Although this game offers both players a very profitable opportunity, all standard game theoretic solution concepts predict that Player I will stop at the first opportunity, getting just $4.
Except, nobody really thinks this is the way players would behave in reality. The optimal strategy seems sociopathic; isn’t it worth playing cooperatively in the hope that the other player will do the same thing? (Unlike much real human interaction, standard game theory does not accomodate the “hope” that someone else will play suboptimally: optimal play is to be expected at all times. )
But Ignacio Palacios-Huerta (best known to Undercover Economist readers as discovering that strikers and goalkeepers play optimal strategies in penalty-taking) and Oscar Volij gave the centipede game to skilled chess players. They found that the chess players were far more likely to play optimally; grandmasters always played optimally and took the $4. Hyper-rationality can be a disadvantage. (Or did the experiment discover something else: that chess grandmasters are sociopaths?) Palacios-Huerta and Volij don’t speculate. My guess is that they have discovered something about the rationality rather than morality or empathy of chess players, but I may be wrong.
I was also reminded of my own investigations into another hyper-rational professional games player, Poker champion Chris Ferguson, whose exploits are described here and in The Logic of Life.
So rationality is akin to sociopathy and hyper-rationality even more so?
Does that mean irrational behaviour is more socially responsible and morally acceptable?
Posted by: Andy Warren | July 7th, 2008 at 12:33 pm | Report this commentThat has always been true, Andy Warren.
“Or did the experiment discover something else: that chess grandmasters are sociopaths?”
Why is this presented as an either/or?
Posted by: Ken Houghton | July 7th, 2008 at 2:52 pm | Report this commentThis is only optimal if the game is only played once. If played repeatedly, the optimal solution is to alternate winners between the last two stages. This does take cooperation and would still leave the second player less well off. If the first player alternates between games though, they would end up equally better off by playing to the end. It is this enlargement of the game that most people anticipate, that the limited rationality of only thinking of one game is surpassed.
Posted by: Lord | July 7th, 2008 at 11:45 pm | Report this commentActually, it is only optimal to the individual step. Even two steps makes more stages the rational choice. This can be viewed as a change in opponent from that of the other player to that of the designer. The winner then becomes the one that ends up with the most money they can rather than only more than the other player. If the players can colude, it even makes sense for them to outwit the designer’s intent.
Posted by: Lord | July 8th, 2008 at 2:53 am | Report this commentI would suspect that chess players are transferring their intuitions about the other player from their experience in chess. Their experience in chess is that the other player will _always_ take advantage of _any_ flaw in strategy, so no flaws are permitted.
Also, the object of chess is always to win, regardless of how much is at stake.
So it would make sense that chess players would play this game (1) assuming the counterparty cannot be trusted at all, and (2) to win, i.e., get more than the opponent, rather than maximize the amount won.
I dislike the premise of the article, trying to associate rationality as a presumably undesirable trait. The problem with chess players is not that they’re “too rational”. It’s that they aren’t being rational enough to forget the lessons learned in chess and adapt their behavior to a different environment.
Then again, if you _know_ that you are playing this game against a chess grandmaster, then the best outcome _is_ to stop as soon as you have opportunity. You know that he is going to.
Posted by: denis bider | July 8th, 2008 at 7:40 am | Report this commentI wouldn’t equate rationality with sociopathic behaviour. Very clumsy use of the term.
Sociopath’s are defined by a lack of empathy for other human beings, therefore are incapable of understanding how their actions affect others. Often serial killers are sociopathic, therefore murdering, raping or stealing become viable options to them as they just dont feel empathy with others.
Nothing to do with when should you stick or twist in a zero-sum game. poor journalism.
Posted by: Veritas Jissumley | July 9th, 2008 at 2:34 pm | Report this commentWhat about a chess grandmaster who plays second to a player who has just opted to continue? Does logic dictate that he too should then continue: if player 1 opted to continue at his first opportunity, then why would he opt to stop on his second go? Player 1 has “signalled” co-operation and friendship, and there would be little benefit in doing so falsely. Does such a first move not engender trust and alter the “optimal” strategy for the second player through information given about intent? And on the basis that the only logical move for player 2 is to “continue” if player 1 first does so, is not in fact the optimal first move also to opt to “continue” in the happy knowledge that co-operative intent will be acknowledged by player 2 resulting in ultimate mutual gain.
PS I wonder if women play differently from men? Speaking personally, I would always opt not to stop even though I know that by so doing the other player may gain more than I do - I would be just as concerned about our mutual profits. Oh, unless I hated the other player for some reason, or thought that the setter of the game couldn’t afford the prize money. Is that all very girlie reasoning?
Posted by: Amy L. | July 9th, 2008 at 3:38 pm | Report this commentrelated to rationality and sociopathology:
“He may be mad, but there’s method in his madness. There nearly always is method in madness. It’s what drives men mad, being methodical.”
Posted by: Dave McDougall | July 9th, 2008 at 6:56 pm | Report this comment- GK Chesterton
Well I think this shows that rationality can be, well, irrational. What’s rational is to have faith, in the goodness of yourself and others, even if it appears more sensible and profitable to be cynical. It’s true both for individual success and success with others. I feel like a lot of people who are rich and famous had faith in themselves from an early age, never believing they could fail to become famous basketball players or actors or whatever, fields with enormous competition. and here, it would pay for both players to love their neighbor. if they had faith in the other, they could get a prize that completely outweighed the initial gains. so the chess players got 4 bucks. nice. they could have so much more if they had a little faith in their partner. of course these games , by not having repeats and not allowing the players to talk first, encourage ruthless/’sociopathic’, rational behavior..
Posted by: Jon | July 9th, 2008 at 9:41 pm | Report this commentmaybe the stakes were not high enough
chess grandmasters are ntoriously poor. what if you were player 1 and the initial stake was 30000 dollars. would you still be so trusting?
Posted by: labglaas | July 12th, 2008 at 9:55 pm | Report this comment[…] 1. Hyper-rationality can be a disadvantage as shown in Field centipedes and the perils of rationality. […]
Posted by: Assorted Links | Monevnomics | October 1st, 2008 at 2:51 pm | Report this comment