T1 vs. T2: on the definition of mixed strategies in noncooperative games

This paper identifies a central role of the topological separation axiom T1 in the definition of mixed strategies in noncooperative games with arbitrary pure strategy spaces. Our main result says that the pure strategy space is T1 if and only if the canonical mapping that assigns to each point its Dirac measure is injective and takes values in the space of regular Borel prob- ability measures. In the absence of the T1 separation axiom, pure strategies may not be available as mixed strategies and different pure strategies may correspond to the same mixed strategy. Moreover, there may be no mixed strategies with finite topological support. The analysis therefore suggests that the T1 separation axiom is a natural minimum requirement on a pure strategy space when randomization is allowed for.