A Dutch book for group decision-making?
(2009) In Foundations of the formal sciences VI : probabilistic reasoning and reasoning with probabilities Studies in logic, 16. p.101-91- Abstract
- 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.
... (More) - 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. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/757354
- author
- Rabinowicz, Wlodek ^{LU} and Bovens, Luc
- organization
- publishing date
- 2009
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- in
- Foundations of the formal sciences VI : probabilistic reasoning and reasoning with probabilities
- editor
- Löwe, Benedikt; Pacuit, Eric and Romeijn, Jan-Willem
- volume
- Studies in logic, 16
- pages
- 101 - 91
- publisher
- College Publications
- ISBN
- 9781904987154
- language
- English
- LU publication?
- yes
- id
- 5c7390dc-6e66-4d1b-971a-f3081a424c01 (old id 757354)
- alternative location
- http://eprints.lse.ac.uk/31622/1/Bovens_A_Dutch_book_for_group_2009.pdf
- date added to LUP
- 2008-01-28 08:51:54
- date last changed
- 2016-04-16 10:34:01
@misc{5c7390dc-6e66-4d1b-971a-f3081a424c01, abstract = {Distribute white and black hats in a dark room to a group of<br/><br> three rational players with each player having a fifty-fifty chance of<br/><br> receiving a hat of one colour or the other. Clearly, the chance that,<br/><br> as a result of this distribution,<br/><br> (A) "Not all hats are of the same colour"<br/><br> is 3/4. The light is switched on and all players can see the hats of<br/><br> the other persons, but not the colour of their own hats. Then no matter<br/><br> what combination of hats was assigned, at least one player will see two<br/><br> hats of the same colour. For her the chance that not all hats are of<br/><br> the same colour strictly depends on the colour of her own hat and hence<br/><br> equals 1/2.<br/><br> <br/><br> <br/><br> On Lewis's principal principle, a rational player will let her degrees<br/><br> of belief be determined by these chances. So before the light is<br/><br> switched on, all players will assign degree of belief of 3/4 to (A) and<br/><br> after the light is turned on, at least one player will assign degree of<br/><br> belief of 1/2 to (A). Suppose a bookie offers to sell a single bet on<br/><br> (A) with stakes $4 at a price of $3 before the light is turned on and<br/><br> subsequently offers to buy a single bet on (A) with stakes $4 at a price<br/><br> of $2 after the light is turned on. If, following Ramsey, the degree of<br/><br> belief equals the betting rate at which the player is willing to buy and<br/><br> to sell a bet on a given proposition, then any of the players would be<br/><br> willing to buy the first bet and at least one player would be willing to<br/><br> sell the second bet. Whether all hats are of the same colour or not,<br/><br> the bookie can make a Dutch book - she has a guaranteed profit of $1.<br/><br> <br/><br> <br/><br> However, it can be shown that a rational player whose degree of belief<br/><br> in (A) equals 1/2 would not volunteer to sell the second bet on (A),<br/><br> neither when her aim is to maximise her own payoffs, nor when she wants<br/><br> to maximise the payoffs of the group. The argument to this effect shares<br/><br> a common structure with models (i) for the tragedy of the commons and<br/><br> (ii) for strategic voting in juries.}, author = {Rabinowicz, Wlodek and Bovens, Luc}, editor = {Löwe, Benedikt and Pacuit, Eric and Romeijn, Jan-Willem}, isbn = {9781904987154}, language = {eng}, pages = {101--91}, publisher = {ARRAY(0x950ba98)}, series = {Foundations of the formal sciences VI : probabilistic reasoning and reasoning with probabilities}, title = {A Dutch book for group decision-making?}, volume = {Studies in logic, 16}, year = {2009}, }