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

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  • Christian Ewerhart

Abstract

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 a pure strategy space is topologically T1 if and only if (i) all singleton strategy sets are Borel, (ii) all Dirac measures are regular, and (iii) the canonical mapping from pure strategies to Dirac measures is one-to-one. The analysis therefore suggests that the T1 separation axiom is a minimum requirement on the topology of a pure strategy space when randomization is allowed for. Using an example, we show that the T1 assumption is indeed missing from the minimax theorem of Mertens (1986).

Suggested Citation

  • Christian Ewerhart, 2025. "T1 vs. T2: on the definition of mixed strategies in noncooperative games," ECON - Working Papers 468, Department of Economics - University of Zurich.
    • Handle: RePEc:zur:econwp:468
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    File URL: https://www.zora.uzh.ch/id/eprint/276896/1/econwp468.pdf
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    Keywords

    Mixed strategies; Hausdorff spaces; T1 separation axiom; minimax theorem;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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