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The puzzle of the hats


The Puzzle of the Hats is a betting arrangement which seems to show that a Dutch book can be made against a group of rational players with common priors who act in the common interest and have full trust in the other players’ rationality. But we show that appearances are misleading—no such Dutch book can be made. There are four morals. First, what can be learned from the puzzle is that there is a class of situations in which credences and betting rates diverge. Second, there is an analogy between ways of dealing with situations of this kind and different policies for sequential choice. Third, there is an analogy with strategic voting, showing that the common interest is not always served by expressing how things seem to you in social decision-making. And fourth, our analysis of the Puzzle of the Hats casts light on a recent controversy about the Dutch book argument for the Sleeping Beauty.

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Author information

Correspondence to Luc Bovens.

Additional information

Our Puzzle of the Hats is inspired by a mathematical puzzle introduced in 1998 by Todd Ebert. See Robinson (2001).

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Bovens, L., Rabinowicz, W. The puzzle of the hats. Synthese 172, 57 (2010). https://doi.org/10.1007/s11229-009-9476-1

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  • Puzzle of the Hats
  • Credences
  • Betting rates
  • Sequential choice
  • Strategic voting
  • Dutch Book
  • Sleeping Beauty
  • Rational choice
  • Rationality