# Notation for mixed strategies in Game Theory

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Quote: "Let $$G = <N, (A_i), (u_i)>$$ be such a strategic game. We denote by $$\Delta (A_i )$$ the set of probability distributions over $$A_i$$ and refer to a member of $$\Delta (A_i )$$ as a mixed strategy of player i; we assume that the players' mixed strategies are independent randomizations. "

So notationally there is a difference between pure and mixed strategies; however, how does this translate to a physical difference between a pure and a mixed strategy? Where does the Delta notation come from? (I'm reading a really abstract book on game theory)

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EDIT:

From http://www.virtualperfection.com/gametheory/1.StrategicFormGames.1.0.pdf ,
When A is a finite set, $$\Delta (A_i )$$ is the set of all probability distributions over A. So how then, would we list the set of all probability distributions over A? Is the set infinite? (because you can assign any probability from [0,1] to the elements of A, and the range of the set has an infinite number of elements)

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So notationally there is a difference between pure and mixed strategies; however, how does this translate to a physical difference between a pure and a mixed strategy?

Well, there are variety of ways that a mixed strategy can arise, but in this example, the phrase "we assume that the players' mixed strategies are independent randomizations" makes things more specific. What they mean is, at each turn, each player picks a strategy $A_i$ at random, according to some (fixed) distribution. I.e., if there were 2 possible strategies, and the distribution was 50/50, that player would flip a coin at each turn to decide which strategy to pursue. In more complicated situations, the mechanism for picking a strategy at each turn could be a function of the previous picked strategies, as well as the state of the game. But in this example they are simply picked from some fixed distribution at each turn.

Where does the Delta notation come from?

Presumably because it's Greek for D, as in distributions.

When A is a finite set, $$\Delta (A_i )$$ is the set of all probability distributions over A. So how then, would we list the set of all probability distributions over A?

Well, there are any number of ways, but the most straightforward thing to write is
$$\Delta (A_i) = \left\{ (p_1,p_2,\ldots,p_n) \in \mathbb{R}^n \left| \, \sum_{i=1}^n p_n=1\mbox{ and } p_i \geq 0, \, \forall i \in 1,\ldots,n \right. \right\} \right\}$$

Is the set infinite?

Yes. Specifically, $\Delta (A_i)$ is an http://en.wikipedia.org/wiki/Simplex#The_standard_simplex".

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