Rationality and common knowledge

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Everyone picks a number from 0 to 100. Whoever lands closest to two-thirds of the average wins. What do you play?

Rationality and common knowledge

Being rational is not enough to know what to do. You also need a view about how the other players think, and about how they think you think. This game makes that chain visible.

From earlier lessons
One quick one before we start.

In a game, your best move depends on...

Predict, then play

Pick your number, then play it against a crowd that reasons to different depths. See where the winning number lands, and where you land.

Exhibit 1  Guess two-thirds of the average
Playing against a normal crowd, what number tends to win?

The reasoning ladder

If everyone chose at random, the average would be about 50, so two-thirds of it is about 33. But if everyone is smart enough to see that, they all aim for 33, and now two-thirds of the average is 22. Keep going and the chain marches down toward 0.

Take a guess, then reveal the reason

So should you guess 0? Only if you believe everyone else will reason all the way down too, and believe that they believe it, and so on. That endless “everyone knows that everyone knows” is called common knowledge of rationality. Real people stop after a step or two, so the winning number is usually around 20 to 33, not 0.

Walk the ladder

Level 0 plays randomly, average 5050. Level 1 best-responds: 23⋅50=33\tfrac{2}{3}\cdot 50 = 33. Level 2 best-responds to that: 23⋅33=22\tfrac{2}{3}\cdot 33 = 22. The unique Nash equilibrium is 00, reached only in the limit. Nagel (1995) found real players cluster at 3333 and 2222, that is, one and two steps of reasoning.

Key idea

Key idea

Rationality alone does not pin down a choice. What pins it down is a shared, and often wrong, assumption about how deeply everyone else reasons.

Check yourself

Check yourselfapply
You are playing this game against a normal crowd of people, for real money. What is the smarter guess?
Check yourselfconcept
What does common knowledge of a fact require?
Check yourselfbrings back: what is a game
What makes the beauty contest a game in the normal-form sense, rather than a solo puzzle?

Sources

  1. Nagel, R. (1995). Unraveling in guessing games: an experimental study. American Economic Review, 85(5), 1313-1326.
  2. Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.