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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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    OK, I think I understand now. Thanks.2012-12-20

0 Answers 0