Published online by Cambridge University Press: 05 June 2012
ABSTRACT
In this chapter we discuss the sensitivity of the minimax theorem to the cardinality of the set of pure strategies. In this light, we examine an infinite game due to Wald and its solutions in the space of finitely additive (f.a.) strategies.
Finitely additive joint distributions depend, in general, upon the order in which expectations are composed out of the players' separate strategies. This is connected to the phenomenon of “non-conglomerability” (so-called by deFinetti), which we illustrate and explain. It is shown that the player with the “inside integral” in a joint f.a. distribution has the advantage.
In reaction to this asymmetry, we propose a family of (weighted) symmetrized joint distributions and show that this approach permits “fair” solutions to fully symmetric games, e.g., Wald's game. We develop a minimax theorem for this family of symmetrized joint distributions using a condition formulated in terms of a pseudo-metric on the space of f.a. strategies. Moreover, the resulting game can be solved in the metric completion of this space. The metrical approach to a minimax theorem is contrasted with the more familiar appeal to compactifications, and we explain why the latter appears not to work for our purposes of making symmetric games “fair.” We conclude with a brief discussion of three open questions relating to our proposal for f.a. game theory.
INTRODUCTION
In this essay we derive results for finitely additive (mixed) strategies in two-person, zero-sum games with bounded payoffs.
To save this book to your Kindle, first ensure no-reply@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.
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.
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.