<p>From <a href="https://en.wikipedia.org/wiki/Support_(mathematics)#Formulation" rel="nofollow noreferrer">Wikipedia</a>:</p>
<blockquote>
<p>Suppose that <span class="math-container">$f : X \to \mathbb{R}$</span> is a real-valued function whose domain is an arbitrary set <span class="math-container">$X.$</span> The set-theoretic support of <span class="math-container">$f$</span>, written <span class="math-container">$\operatorname{supp}(f),$</span> is the set of points in <span class="math-container">$X$</span> where <span class="math-container">$f$</span> is non-zero:
<span class="math-container">$$
\operatorname{supp}(f) = \{ x \in X \,:\, f(x) \neq 0\}.
$$</span></p>
</blockquote>
<p>As you write, in game theory a mixed strategy <span class="math-container">$\sigma$</span> 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 <span class="math-container">$\operatorname{supp}(\sigma)$</span> is the set of pure strategies played with non-zero probability.</p>
<p>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.</p>
<p>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 <span class="math-container">$\{ \text{Defect} \}$</span>.</p>

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2021-09-04T07:54:41.500Z
2021-09-04T07:54:41.500Z
Giskard
<p>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 <span class="math-container">$\sigma^*$</span>, which denotes the probability distribution of the mixed strategies. In other words <span class="math-container">$\sigma^*$</span> serves also as a probability measure. Can we find the support of the set of the pure strategies as well?</p>
<p>I would appreciate it if someone could give an example apart from an intuitive answer.</p>

<mathematical-economics><game-theory><probability><real-analysis>
What is the meaning of the support set in game theory?
Nav89
# What is the meaning of the support set in game theory?

(Asked by: username_0 on 2021-09-04)

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.

**Tags:** mathematical-economics, game-theory, probability, real-analysis


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## 1 answers

### Answer by username_1 on 2021-09-04. Score: 3 | Accepted answer

From [Wikipedia](https://en.wikipedia.org/wiki/Support_(mathematics)#Formulation):

> 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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