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# 经济代考|博弈论代考GAME THEORY代考|ECON0200 n-Person Games

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## 经济代考|博弈论代考GAME THEORY代考|Dynamics of n-person games

If the game $\Gamma=\left(u_{i} \mid i \in N\right)$ is played, a game instance yields a sequence of state transitions. The transitions are thought to result from changes in the strategy choices of the players.

Suppose $i \in N$ replaces its current strategy $x_{i}$ by the strategy $y \in X_{i}$ while all other players $j \neq i$ retain their choices $x_{j} \in X_{j}$. Then a state transition $\mathbf{x} \rightarrow \mathbf{y}=\mathbf{x}{-i}(y)$ results, where the new state has the components $$y{j}=\left{\begin{array}{cc} y & \text { if } j=i \ x_{j} & \text { if } j \neq i \end{array}\right.$$
Two neighboring states $\mathbf{x}$ and $\mathbf{y}$ differ in at most one component. In particular, $\mathbf{x}{-i}\left(x{i}\right)=\mathbf{x}$ holds under this definition and exhibits $\mathbf{x}$ as a neighbor of itself. Let us take the set
$$\mathcal{F}{i}(\mathbf{x})=\left{\mathbf{x}{-i}(y) \mid y \in X_{i}\right}$$
as the neighborhood of the state $\mathbf{x} \in \mathfrak{X}$ for the player $i \in N$. So the neighbors of $\mathbf{x}$ from $i$ ‘s perspective are those states that could be achieved by $i$ with a change of its current strategy $x_{i}$, provided all other players $j \neq i$ retain their current strategies $x_{j}$.

The utility functions $u_{i}$ thus provide the natural utility measure $U$ for $\Gamma$ with the values
$$U(\mathbf{x}, \mathbf{y})=u_{i}(\mathbf{y})-u_{i}(\mathbf{x}) \quad \text { for all } i \in N \text { and } \mathbf{x}, \mathbf{y} \in \mathcal{F}_{i}(\mathbf{x})$$

## 经济代考|博弈论代考GAME THEORY代考|Randomization of matrix games

An equilibrium of $\Gamma=\Gamma\left(u_{i} \mid i \in N\right)$ is an equilibrium of the utility measure $U$ as in (41). The joint strategic choice $\mathbf{x} \in \mathfrak{X}$ is thus a gain equilibrium if no player has an utility incentive to switch to another strategy, i.e.,
$$u_{i}(\mathbf{x}) \geq u_{i}(\mathbf{y}) \text { holds for all } i \in N \text { and } \mathbf{y} \in \mathcal{F}{i}(\mathbf{x}) .$$ Completely analogously, a cost equilibrium is defined via the reverse condition: $$u{i}(\mathbf{x}) \leq u_{i}(\mathbf{y}) \text { holds for all } i \in N \text { and } \mathbf{y} \in \mathcal{F}{i}(\mathbf{x}) .$$ Aggregated utilities. There is another important view on equilibria. Given the state $\mathbf{x} \in \mathfrak{X}$, imagine that each player $i \in N$ considers an alternative $y{i}$ to its current strategy $x_{i}$. The aggregated sum of the resulting utility values is
$$G(\mathbf{x}, \mathbf{y})=\sum_{i \in N} u\left(\mathbf{x}{-i}\left(y{i}\right)\right) \quad\left(\mathbf{y}=\left(y_{i} \mid y_{i} \in X_{i}\right)\right) .$$

## 经济代考|博弈论代考GAME THEORY代考|Equilibria

An $n$-person game $\Gamma=\left(u_{i} \mid i \in N\right)$ is said to be a (generalized) matrix game if all individual strategy sets $X_{i}$ are finite.

For a motivation of the terminology, assume $N={1, \ldots, n}$ and think of the sets $X_{i}$ as index sets for the coordinates of a multidimensional matrix $U$. A particular index vector
$$\mathbf{x}=\left(x_{1}, \ldots, x_{i}, \ldots, x_{n}\right) \in X_{1} \times \cdots \times X_{i} \times \cdots \times X_{n}(=\mathfrak{X})$$
thus specifies a position in $U$ with the $n$-dimensional coordinate entry
$$U_{\mathbf{x}}=\left(u_{1}(\mathbf{x}), \ldots, u_{i}(\mathbf{x}), \ldots, u_{n}(\mathbf{x})\right) \in \mathbb{R}^{n}$$

## 经济代考|博恋论代考GAME THEORY 代考|Dynamics of n-person games

$y$ if $j=i x_{j}$ if $j \neq i$

Twoneighboringstates\$x\$and $\$ \mathbf{y} \$$dif ferinatmostonecomponent. Inparticular, \ \mathbf{x}-i(x i)=\mathbf{x} \mathrm{~ I m a t h c a l { F } { i } ( | m a t h b f { x } ) =} \ \$$

$$U(\mathbf{x}, \mathbf{y})=u_{i}(\mathbf{y})-u_{i}(\mathbf{x}) \quad \text { for all } i \in N \text { and } \mathbf{x}, \mathbf{y} \in \mathcal{F}{i}(\mathbf{x})$$

## 经济代考|博帟论代考GAME THEORY 代考|Randomization of matrix games

$$u_{i}(\mathbf{x}) \geq u_{i}(\mathbf{y}) \text { holds for all } i \in N \text { and } \mathbf{y} \in \mathcal{F} i(\mathbf{x}) .$$

$$u i(\mathbf{x}) \leq u_{i}(\mathbf{y}) \text { holds for all } i \in N \text { and } \mathbf{y} \in \mathcal{F} i(\mathbf{x}) .$$

$$G(\mathbf{x}, \mathbf{y})=\sum_{i \in N} u(\mathbf{x}-i(y i)) \quad\left(\mathbf{y}=\left(y_{i} \mid y_{i} \in X_{i}\right)\right) .$$

## 经济代考|博恋论代考GAME THEORY 代考|Equilibria

$$\mathbf{x}=\left(x_{1}, \ldots, x_{i}, \ldots, x_{n}\right) \in X_{1} \times \cdots \times X_{i} \times \cdots \times X_{n}(=\mathfrak{X})$$

$$U_{\mathbf{x}}=\left(u_{1}(\mathbf{x}), \ldots, u_{i}(\mathbf{x}), \ldots, u_{n}(\mathbf{x})\right) \in \mathbb{R}^{n}$$

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