Skip to main content Accessibility help
×
Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-10T23:37:24.548Z Has data issue: false hasContentIssue false

Chapter 5 - The Dutch Book Argument

from Part 1 - The Basics

Published online by Cambridge University Press:  05 May 2015

Jeffrey Paris
Affiliation:
University of Manchester
Alena Vencovská
Affiliation:
University of Manchester
Get access

Summary

Having derived some of the basic properties of probability functions we will now take a short diversion to give what we consider to be the most compelling argument in this context, namely the Dutch Book argument originating with Ramsey [122] and de Finetti [25], in favour of an agent's ‘degrees of belief’ satisfying (P1–3), and hence being identified with a probability function, albeit subjective probability since it is ostensibly the property of the agent in question. Of course this could really be said to be an aside to the purely mathematical study of PIL and hence dispensable. The advantage of considering this argument however is that by linking belief and subjective probability it better enables us to appreciate and translate into mathematical formalism the many rational principles we shall later encounter.

The idea of the Dutch Book argument is that it identifies ‘belief’ with willingness to bet. So suppose, as in the context of PIL explained above, we have an agent inhabiting some unknown structure M ∈ T L (which one imagines will eventually be revealed to decide the wager) and that θ ∈ SL, 0 ≤ p ≤ 1 and for a stake s > 0 the agent is offered a choice of one of two wagers:

(Bet1p) Win s(1 − p) if Mθ, lose sp if Mθ.

(Bet2p) Win sp if M ⊭ θ, lose s (1 − p) if M ⊧ θ.

If the agent would not be happy to accept Bet1p we assume that it is because the agent thinks that the bet is to his/her disadvantage and hence to the advantage of the bookmaker. But in that case Bet2p allows the agent to swap roles with the bookmaker so s/he should now see that bet as being to his/her advantage, and hence acceptable. In summary then, we may suppose that for any 0 ≤ p ≤ 1 at least one of Bet1p and Bet2p is acceptable to the agent. In particular we may assume that Bet10 and Bet21 are acceptable since in both cases the agent has nothing to lose and everything to gain.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×