# Explanation of Mixed Strategy Definition in Game Theory

Definition:

Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$.

• Where N is a finite set of n players, indexed by i
• $A=A_1\times...\times A_n$, where $A_i$ is a finite set of actions available to player i

The part that is unclear to me is this:

"...and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$"

Let's say that $X = \{0, 1\}$, what does "set of all probability distributions over $X$" mean? Is it a set $\{a, b\}$ of all possible values of $a$ and $b$, where both $a$ and $b$ are positive and add up to 1?

But what does it mean for $S_i$ that it's an infinite set that has all the positive numbers which add up to one? In mine example where $X = \{0, 1\}$, $S_i = \{(0.5,\ 0.5),\ (0.25,\ 0.75),\ (0.6,\ 0.4)\ ...\}$

As you can see, the "all" part confuses me.

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$S_i=\big\{(a,b): a\geq 0, b\geq 0, a+b=1\big\}$. What is the problem? – Michael Greinecker Dec 20 '12 at 10:35
The problem is that I would expect one particular player to have particular probabilities for move 0, and other for move 1, not all possible probabilities. I don't understand the purpose of defining $S_i$ as an infinite set. For example if you and I play a game, we have two cards, one is J and the other one is K and if its a king I get one dollar, you get nothing, and vice versa I would expect that $S_i = (0.5, 0.5)$, where first number is probability for payoff 1\$, and the second for payoff 0 dollar. Here I have a set of all possible probability distributions, not just particular one. – enedene Dec 20 '12 at 14:46
It is a description of what players can do, not what they actually do. While there are infintely many ways for a player to randomize over two startegies, the player will choose only one. – Michael Greinecker Dec 20 '12 at 15:09
OK, I think I understand now. Thanks. – enedene Dec 20 '12 at 15:50