Adv. Math. Econ. 6, 39-53 (2004)

The compactness of P r(K)

Dionysius Glycopantis^{1} and Allan Muir^{2}

1 Department of Economics, City University, Northampton Square, London EC1V 0HB, UK (email: d.glycopantis@city.ac.uk)

2 Department of Mathematics, City University, Northampton Square, London EC1V 0HB, UK (email: muirlynn@llanfynydd.u-net.com)

Received: April 4, 2003 Revised: May 20, 2003

JEL classification: C72

Mathematical Subject Classification (2000): 91A10

Summary. We prove the compactness of P r(K), the set of Borel probability measures on a compactum K endowed with the weak* topology, without embedding this set in rca(K), the space of regular, countably-additive, signed measures with their finite total variation as norm. P r(K) can be extended to a convex, Hausdorff, linear topological space. Then Glicksberg’s fixed point theorem is applied to prove the existence of Nash equilibria.

Key words: Borel probability measures, weak* topology, compactness, payoff functions, reaction correspondences, Nash equilibrium, fixed point theorem.