I heard this tale in India. A hat seller, on waking from a nap under a tree, found that a group of monkeys had taken all his hats to the top of the tree. In exasperation he took off his own hat and flung it to the ground. The monkeys, known for their imitative urge, hurled down the hats, which the hat seller promptly collected.
Half a century later his grandson, also a hat seller, set down his wares under the same tree for a nap. On waking, he was dismayed to discover that monkeys had taken all his hats to the treetop. Then he remembered his grandfather's story, so he threw his own hat to the ground. But, mysteriously, none of the monkeys threw any hats, and only one monkey came down. It took the hat on the ground firmly in hand, walked up to the hat seller, gave him a slap and said, "You think only you have a grandfather?"
This story illustrates an important distinction between ordinary decision theory and game theory. In the latter, what is rational for one player may depend on what is rational for the other player. For Lucy to get her decision right, she must put herself in Pete's shoes and think about what he must be thinking. But he will be thinking about what she is thinking, leading to an infinite regression. Game theorists describe this situation by saying that "rationality is common knowledge among the players." In other words, Lucy and Pete are rational, they each know that the other is rational, they each know that the other knows, and so on.
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The assumption that rationality is common knowledge is so pervasive in game theory that it is rarely stated explicitly. Yet it can run us into problems. In some games that are played over time, such as repeated rounds of Prisoner's Dilemma, players can make moves that are incompatible with this assumption.
I believe that the assumption that rationality is common knowledge is the source of the conflict between logic and intuition and that, in the case of Traveler's Dilemma, the intuition is right and awaiting validation by a better logic. The problem is akin to what happened in early set theory. At that time, mathematicians took for granted the existence of a universal set-a set that contained everything. The universal set seemed extremely natural and obvious, yet ultimately several paradoxes of set theory were traced to the assumption that it existed, which mathematicians now know is flawed. In my opinion, the common knowledge of rationality assumed by game theorists faces a similar demise. -K.B.
