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What is the meaning of the support set in game theory? I have seen it, in many papers, however none there explains how did they find it or why did the define it in a specific way. I understand that the support of the mixed strategies contains all the pure strategies that are chosen with positive probability under $\sigma^*$, which denotes the probability distribution of the mixed strategies. In other words $\sigma^*$ serves also as a probability measure. Can we find the support of the set of the pure strategies as well?

I would appreciate it if someone could give an example apart from an intuitive answer.

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From Wikipedia:

Suppose that $f : X \to \mathbb{R}$ is a real-valued function whose domain is an arbitrary set $X.$ The set-theoretic support of $f$, written $\operatorname{supp}(f),$ is the set of points in $X$ where $f$ is non-zero: $$ \operatorname{supp}(f) = \{ x \in X \,:\, f(x) \neq 0\}. $$

As you write, in game theory a mixed strategy $\sigma$ can be considered a probability measure over pure strategies. That is, it is a function that assigns a number (probability) to all pure strategies, hence $\operatorname{supp}(\sigma)$ is the set of pure strategies played with non-zero probability.

All pure strategies can be considered mixed strategies where a pure strategy is played with probability one. In this case the support set is a singleton consisting of the pure strategy.

An example: in the equilibrium of Prisoner's Dilemma both players play Defect with probability 1, hence the support set of either player's strategy is $\{ \text{Defect} \}$.

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