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经济代写|博弈论代考Game theory代写|Terminology and Notation for Strategies

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经济代写|博弈论代考Game theory代写|Terminology and Notation for Strategies

It is now time to get acquainted with some standard notation. If you need help understanding the basic mathematical symbols and concepts used here, please consult Appendix A. We must be formal and precise going forward.

Given a game, we let $S_i$ denote the strategy space (also called the strategy set) of player $i$. That is, $S_i$ is a set comprising each of the possible strategies of player $i$ in the game. For the game shown in Figure 2.7(a), the strategy space of player 1 is $S_1={\mathrm{H}, \mathrm{L}}$, and the strategy space of player 2 is $S_2=\left{\mathrm{HH}^{\prime}, \mathrm{HL}^{\prime}, \mathrm{LH}^{\prime}, \mathrm{LL}^{\prime}\right}$. We use lowercase letters to denote single strategies (generic members of these sets). Thus, $s_i \in S_i$ is a strategy for player $i$ in the game. We could thus have $s_1=\mathrm{L}$ and $s_2=\mathrm{LH}^{\prime}$, for instance.

A strategy profile is a vector of strategies, one for each player. In other words, a strategy profile describes strategies for all of the players in the game. For example, suppose we are studying a game with $n$ players. A typical strategy profile then is a vector $s=\left(s_1, s_2, \ldots, s_n\right)$, where $s_i$ is the strategy of player $i$, for $i=1,2, \ldots, n$. Let $S$ denote the set of strategy profiles. Mathematically, we write $S=S_1 \times S_2 \times \ldots \times S_n$. Note that the symbol ” $\times$ ” denotes the Cartesian product. ${ }^1$

Given a single player $i$, we often need to speak of the strategies chosen by all of the other players in the game. As a matter of notation, it will be convenient to use the term $-i$ to refer to these players. Thus, $s_{-i}$ is a strategy profile for everyone except player $i$ :
$$s_{-i}=\left(s_1, s_2, \ldots, s_{i-1}, s_{i+1}, \ldots, s_n\right) .$$

knows only whether firm 1 is in or out of the market; firm 2 does not observe firm l’s competitive stance before taking its action. In this game, there is one information set for firm 1 (the initial node) and one for firm 2 . The strategy sets are $S_1={\mathrm{A}, \mathrm{P}, \mathrm{O}}$ and $S_2={\mathrm{A}, \mathrm{P}}$

经济代写|博弈论代考Game theory代写|The Normal Form

The extensive form is one straightforward way of representing a game. Another way of formally describing games is based on the idea of strategies. It is called the normal form (or strategic form) representation of a game. This alternative representation is more compact than the extensive form in some settings. As we develop concepts of rationality for games, you will notice the subtle differences between the two representations.

For any game in extensive form, we can describe the strategy spaces of the players. Furthermore, notice that each strategy profile fully describes how the game is played. That is, a strategy profile tells us exactly what path through the tree is followed and, equivalently, which terminal node is reached to end the game. Associated with each terminal node (which we may call an outcome) is a payoff vector for the players. Therefore, each strategy profile implies a specific payoff vector.

For each player $i$, we can define a function $u_i: S \rightarrow \mathbf{R}$ (a function whose domain is the set of strategy profiles and whose range is the real numbers) so that, for each strategy profile $s \in S$ that the players could choose, $u_i(s)$ is player i’s payoff in the game. This function $u_i$ is called player $i$ ‘s payoff function. As an example, take the game pictured in Figure 3.1(b). The set of strategy profiles in this game is
$$S={(\mathrm{OA}, \mathrm{O}),(\mathrm{OA}, \mathrm{I}),(\mathrm{OB}, \mathrm{O}),(\mathrm{OB}, \mathrm{I}),(\mathrm{IA}, \mathrm{O}),(\mathrm{IA}, \mathrm{I}),(\mathrm{IB}, \mathrm{O}),(\mathrm{IB}, \mathrm{I})}$$

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经济代写|博弈论代考Game theory代写|Terminology and Notation for Strategies

$$s_{-i}=\left(s_1, s_2, \ldots, s_{i-1}, s_{i+1}, \ldots, s_n\right) .$$

经济代写|博弈论代考Game theory代写|The Normal Form

$$S={(\mathrm{OA}, \mathrm{O}),(\mathrm{OA}, \mathrm{I}),(\mathrm{OB}, \mathrm{O}),(\mathrm{OB}, \mathrm{I}),(\mathrm{IA}, \mathrm{O}),(\mathrm{IA}, \mathrm{I}),(\mathrm{IB}, \mathrm{O}),(\mathrm{IB}, \mathrm{I})}$$

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