# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### A Dutch book for group decision-making?

and Bovens, Luc (2009) In Foundations of the formal sciences VI : probabilistic reasoning and reasoning with probabilities Studies in logic, 16. p.91-101
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)
author
organization
publishing date
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
; ; and
volume
Studies in logic, 16
pages
91 - 101
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
2008-01-28 08:51:54
date last changed
2016-04-16 10:34:01
```@inbook{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        = {91--101},
publisher    = {College Publications},
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},
}

```