<p>A mixed strategy given by a continuous random variable is represented by the random variable's probability density function or by its associated cumulative distribution function, here <span class="math-container">$F$</span>. While every single action <span class="math-container">$a_i$</span> is played with probability zero, it can nevertheless have a positive probability <em>density</em>. In a mixed Nash equilibrium, a player's expected payoffs must be equal for all actions with positive probability density.</p>

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2024-10-17T08:11:24.763Z
2024-10-17T08:11:24.763Z
VARulle
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<p>The following excerpt is taken from Martin J. Osborne's book on game theory. When considering a mixed strategy Nash equilibrium under a continuous random variable, why do we consider actions with zero probability? Here, <span class="math-container">$F(0)$</span> represents the probability of playing an action less than or equal to zero but since <span class="math-container">$a_i&gt;=0$</span>, this is the probability of playing an action equal to zero.</p>
<p>I thought that the definition of a mixed strategy specifies that expected payoffs must be equal only under actions played with positive probability so why does this work?</p>

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Question about mixed strategies in all-pay auctions
secretrevaler
# Question about mixed strategies in all-pay auctions

(Asked by: username_0 on 2024-10-16)

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The following excerpt is taken from Martin J. Osborne's book on game theory. When considering a mixed strategy Nash equilibrium under a continuous random variable, why do we consider actions with zero probability? Here, $F(0)$ represents the probability of playing an action less than or equal to zero but since $a\_i>=0$, this is the probability of playing an action equal to zero.

I thought that the definition of a mixed strategy specifies that expected payoffs must be equal only under actions played with positive probability so why does this work?

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2024-10-17. Score: 1 | Accepted answer

A mixed strategy given by a continuous random variable is represented by the random variable's probability density function or by its associated cumulative distribution function, here $F$. While every single action $a\_i$ is played with probability zero, it can nevertheless have a positive probability *density*. In a mixed Nash equilibrium, a player's expected payoffs must be equal for all actions with positive probability density.


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