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A Dutch book for group decision-making?

Author

Editor

  • Benedikt Löwe
  • Eric Pacuit
  • Jan-Willem Romeijn

Summary, in English

Distribute white and black hats in a dark room to a group of

three rational players with each player having a fifty-fifty chance of

receiving a hat of one colour or the other. Clearly, the chance that,

as a result of this distribution,

(A) "Not all hats are of the same colour"

is 3/4. The light is switched on and all players can see the hats of

the other persons, but not the colour of their own hats. Then no matter

what combination of hats was assigned, at least one player will see two

hats of the same colour. For her the chance that not all hats are of

the same colour strictly depends on the colour of her own hat and hence

equals 1/2.





On Lewis's principal principle, a rational player will let her degrees

of belief be determined by these chances. So before the light is

switched on, all players will assign degree of belief of 3/4 to (A) and

after the light is turned on, at least one player will assign degree of

belief of 1/2 to (A). Suppose a bookie offers to sell a single bet on

(A) with stakes $4 at a price of $3 before the light is turned on and

subsequently offers to buy a single bet on (A) with stakes $4 at a price

of $2 after the light is turned on. If, following Ramsey, the degree of

belief equals the betting rate at which the player is willing to buy and

to sell a bet on a given proposition, then any of the players would be

willing to buy the first bet and at least one player would be willing to

sell the second bet. Whether all hats are of the same colour or not,

the bookie can make a Dutch book - she has a guaranteed profit of $1.





However, it can be shown that a rational player whose degree of belief

in (A) equals 1/2 would not volunteer to sell the second bet on (A),

neither when her aim is to maximise her own payoffs, nor when she wants

to maximise the payoffs of the group. The argument to this effect shares

a common structure with models (i) for the tragedy of the commons and

(ii) for strategic voting in juries.

Publishing year

2009

Language

English

Pages

91-101

Publication/Series

Foundations of the formal sciences VI : probabilistic reasoning and reasoning with probabilities

Volume

Studies in logic, 16

Document type

Book chapter

Publisher

College Publications

Topic

  • Philosophy

Status

Published

ISBN/ISSN/Other

  • ISBN: 9781904987154