Math Trek

The tell-tale anecdote

An Edgar Allan Poe story reveals a flaw in game theory.

9:25am, June 20, 2008
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During the whole of a dull, cramped and wearisome flight from Israel to New York, as the night pressed heavily against the airplane windows, Ariel Rubinstein had been toiling through a singularly dreary article on game theory; and at length the economist found himself, as the sharpness of his focus waned, seeking respite from the tedium in Edgar Allan Poe’s short story “The Purloined Letter.

But the economist’s work, it seemed, wouldn’t let him rest. For in the middle of the detective story, Poe launched into an analysis of game theory! Rubinstein read:

“I knew one about eight years of age, whose success at guessing in the game of ‘even and odd’ attracted universal admiration. This game is simple, and is played with marbles. One player holds in his hand a number of these toys, and demands of another whether that number is even or odd. If the guess is right, the guesser wins one; if wrong, he loses one.
“The boy to whom I allude won all the marbles of the school. Of course he had some principle of guessing; and this lay in mere observation and admeasurement of the astuteness of his opponents. For example, an arrant simpleton is his opponent, and, holding up his closed hand, asks, ‘are they even or odd?’ Our schoolboy replies, ‘odd,’ and loses; but upon the second trial he wins, for he then says to himself, ‘the simpleton had them even upon the first trial, and his amount of cunning is just sufficient to make him have them odd upon the second; I will therefore guess odd;’ — he guesses odd, and wins.
“Now, with a simpleton a degree above the first, he would have reasoned thus: ‘This fellow finds that in the first instance I guessed odd, and, in the second, he will propose to himself, upon the first impulse, a simple variation from even to odd, as did the first simpleton; but then a second thought will suggest that this is too simple a variation, and finally he will decide upon putting it even as before. I will therefore guess even;’ — he guesses even, and wins.
“Now this mode of reasoning in the schoolboy, whom his fellows termed ‘lucky,’ — what, in its last analysis, is it?’
‘It is merely,’ I said, ‘an identification of the reasoner’s intellect with that of his opponent.’” [From “The Purloined Letter.” Paragraph breaks added]

A sense of insufferable gloom pervaded Rubinstein’s spirit. For now, his personality and interests being as they were, he was helpless to resist the urge to do his own analysis of the game, using game theory. Upon finishing it, he gasped — for according to game theory, Poe got it wrong.

Rubinstein imagined himself playing the game with Poe. Being grown men, they would be more interested in honor than marbles. So instead of picking a number of marbles, Rubinstein would choose “even” or “odd” and Poe would guess which one he chose. To prevent cheating, they'd call out their guesses at the same moment. If they matched, Poe would win, having guessed Rubinstein’s pick. Otherwise, Rubinstein would win.

But wait! Now who’s the chooser and who’s the guesser? The game is perfectly symmetric: Poe tries to predict what Rubinstein will pick and then match it. But in the meantime, Rubinstein — wily guy that he is — can try to predict what Poe will pick and then choose the opposite!

So while Poe seemed to think that the guesser was in a unique position to make clever guesses, game theory says there’s no difference.

Rubinstein told his colleague Kfir Eliaz of BrownUniversity about the story, and they decided to put the game to the test. “If game theory has any chance of predicting the outcome of a game, this is such a game,” Eliaz and Rubinstein eventually wrote in a research article. After all, it might be the simplest game there is, so players should be able to find the best strategies quickly. But the pair also knew that Poe was no fool.

Time for a play-off! The researchers rounded up students for an experiment. Since neither marbles nor honor usually enflame students’ desires, Eliaz and Rubinstein had them play for good coin.

They made the game similar to Poe’s description. The first player was labeled the “misleader,” who chose numbers he thought the second player wouldn’t guess, and the second player was the “guesser.” The misleader entered 0 for even or 1 for odd into a computer, and then the guesser tried to infer the choice at his own computer. Fifty cents went to the guesser if the entries matched and to the misleader if they didn’t.

The result? Guessers won 53 percent of the time, enough to be statistically significant. Down with game theory, up with Edgar Allen Poe!

“We figured, maybe human beings are somehow better guessers than misleaders,” Rubinstein says. “It’s an optimistic view of human nature.” In this theory, the person described as a misleader simply tries to behave randomly, not realizing the strategic potential of his role.

To test the conjecture, they varied the game. This time, they didn’t label the students “misleader” or “guesser,” instead simply telling them that if the sum of their choices was odd, the first player would win, and if even, the second player would. Game theory would say this game is identical with the first, but psychologically, the researchers reasoned, it might seem different.

The result this time? Player 2, the one who was formerly the guesser, won even more often — 54 percent of the time. Game theory goes down again!

Well, the researchers thought, maybe the problem is that mentally, it’s a bit easier to try to match the other player’s move than oppose it. So they tried another variation. This time, the two players together create a word. The first player goes first and chooses the first letter, a or i. The second player then chooses the second letter, s or t. If they create the word at or is, player 1 scores. If it’s as or it, the point goes to player 2.

Finally, game theory prevailed. There was no statistically significant difference between how often each player won.

Eliaz says the games point out that although the way a problem is presented is irrelevant for game theory’s predictions, it can be quite important for human beings.

Rubinstein, for his part, has altogether revised his opinion of Poe, realizing that there is nearly half as much of the wise as of the entertaining about the man — in short, that “he’s brilliant.”

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