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.


1 Answer 1


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 $\{ \text{Defect} \}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.