A Dutch book for group decisionmaking?
Research output: Chapter in Book/Report/Conference proceeding › Book chapter
Abstract
Distribute white and black hats in a dark room to a group of
three rational players with each player having a fiftyfifty 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.
three rational players with each player having a fiftyfifty 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.
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Research areas and keywords  Subject classification (UKÄ) – MANDATORY

Original language  English 

Title of host publication  Foundations of the formal sciences VI : probabilistic reasoning and reasoning with probabilities 
Editors  Benedikt Löwe, Eric Pacuit, JanWillem Romeijn 
Publisher  College Publications 
Pages  91101 
Volume  Studies in logic, 16 
ISBN (Print)  9781904987154 
Publication status  Published  2009 
Publication category  Research 
Peerreviewed  No 
Publication series
Name  

Volume  Studies in logic, 16 
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