Notation for mixed strategies in Game Theory

In summary, The conversation discusses the notation and definition of a strategic game, where players use mixed strategies that involve probability distributions over different actions. The Delta notation represents the set of all possible distributions, which is infinite when the set of actions is finite.
  • #1

Simfish

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Quote: "Let [tex]G = <N, (A_i), (u_i)> [/tex] be such a strategic game. We denote by [tex]\Delta (A_i )[/tex] the set of probability distributions over [tex]A_i[/tex] and refer to a member of [tex]\Delta (A_i )[/tex] 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, [tex]\Delta (A_i )[/tex] 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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  • #2
Simfish said:
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 [itex]A_i[/itex] 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.

Simfish said:
Where does the Delta notation come from?

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

Simfish said:
When A is a finite set, [tex]\Delta (A_i )[/tex] 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
[tex]\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\}[/tex]

Simfish said:
Is the set infinite?

Yes. Specifically, [itex]\Delta (A_i)[/itex] is an http://en.wikipedia.org/wiki/Simplex#The_standard_simplex".
 
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1. What is the purpose of notation for mixed strategies in Game Theory?

The purpose of notation for mixed strategies in Game Theory is to represent the strategies of players in a game with incomplete information, where players make decisions based on probabilities rather than known outcomes. This notation allows for the analysis and comparison of different strategies and their potential outcomes.

2. How is notation for mixed strategies different from notation for pure strategies?

Notation for pure strategies in Game Theory represents a player's choice of a single, specific action in a game. In contrast, notation for mixed strategies represents a player's choice of a probability distribution over all possible actions in a game. This allows for a more nuanced analysis of decision-making in games with incomplete information.

3. What symbols are commonly used in notation for mixed strategies?

The most commonly used symbols in notation for mixed strategies are lowercase letters, such as "p" and "q", to represent the probabilities of choosing specific actions. These probabilities are often represented as fractions or decimals between 0 and 1, with the sum of all probabilities equaling 1.

4. How do you interpret notation for mixed strategies?

Interpreting notation for mixed strategies involves understanding the probability distribution of actions chosen by a player in a game. For example, if a player has a mixed strategy of (p, 1-p), this means they will choose the first action with probability p and the second action with probability 1-p. This notation can also be extended to represent more than two actions in a game.

5. How is notation for mixed strategies used in game analysis?

Notation for mixed strategies is used to analyze different strategies and their potential outcomes in a game with incomplete information. By representing the probabilities of actions chosen by players, analysts can determine optimal strategies for players and the expected payoffs for each player in a game. This notation is also used to compare different strategies and make predictions about potential outcomes in a game.

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