Mathematical Game Theory by Ulrich Faigle

Author:Ulrich Faigle [Faigle, Ulrich]
Language: eng
Format: epub, pdf
ISBN: 9789811246715
Publisher: World Scientific
Published: 2022-09-15T00:00:00+00:00

Convex and concave utilities. If the utility measure U under consideration is not implied by a potential function, not even the finiteness of may be a guarantee for the existence of an equilibrium (see Ex. 5.2).

EX. 5.2. Give the example of a utility measure U relative to a finite state set with no gain and no cost equilibrium.

We now derive sufficient conditions for utilities U on a system whose set of states is represented by a nonempty convex set ⊆ ℝm. We say:

•U is convex if every local function us : → ℝ is convex.

•U is concave if every local function us : → ℝ is concave.

THEOREM 5.1. Let U be a utility measure with continuous local utility functions us : → ℝ on the nonempty compact set ⊆ ℝm. Then

(1)If U is convex, a cost equilibrium exists.

(2)If U is concave, a gain equilibrium exists.

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