## 经济代写|博弈论代写Game Theory代考|Individual greediness and public welfare

Let us assume that $N$ is a society whose common welfare is expressed by the potential $v$ on the family $\mathcal{N}$ of all possible coalitions: If the members of $N$ decide to join in a coalition $S \subseteq N$, then the value $v(S)$ will be produced.

If all members of $N$ act purely greedily, an $i \in N$ has an incentive to change its decision with respect to the current coalition $S$ depending on its marginal value $\partial_i v(S)$ being positive or negative. This behavior, however, will not guarantee a high public welfare.

The METropolis process suggests that the public welfare can be steered if an incentive is provided such that $i$ enacts a move $S \rightarrow$ $S \Delta{i}$ (i.e., changes its decision) with a non-zero probability
$$\alpha_i^T(S)=e^{\partial_i v(S) / T} \quad \text { (even) if } \partial_i v(S)<0 .$$ If the control parameter $T>0$ is sufficiently small, the behavior of an $i \in N$ is “almost purely greedy” in the sense
$$T \rightarrow 0 \quad \Longrightarrow \quad \alpha_i^T(S) \rightarrow 0 \quad \text { if } \partial_i v(S)<0 .$$ Moreover, a small temperature $T>0$ in the coalition formation process allows us to expect a high public welfare.

## 经济代写|博弈论代写Game Theory代考|Equilibria in cooperative games

In many cooperative games, the grand coalition offers an obvious equilibrium if the players’ utilities are assessed by their marginal values:
LemMA 8.5. Let $(N, v)$ be a cooperative game. Then the two statements are equivalent:
(1) $N$ is a gain equilibrium with respect to the individual utility functions $u_i(S)=\partial_i v(S)$.
(2) $v(N) \geq v(N \backslash i)$ for all $i \in N$.
In general, we may view $(N, v)$ as an $n$-person matrix game with individual utilities
$$u_i(S)=\partial_i v(S)=v(S \Delta i)-v(S) .$$
Hence we know from NASH’s Theorem $6.1$ that the randomization of $(N, v)$ admits an equilibrium.

Remark 8.15. The randomization of $(N, v)$ means that each $i \in N$ selects a probability $0 \leq w_i \leq 1$ for the probability to become active.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Individual greediness and public welfare

MEtropolis 流程表明，如果提供激励措施，可以引导公共福利：采取行动 $S \rightarrow S \Delta i$ (即，改变其决 定) 具有非零概率
$$\alpha_i^T(S)=e^{\partial_i v(S) / T} \quad \text { (even) if } \partial_i v(S)<0 .$$ 如果控制参数 $T>0$ 足够小，一个行为 $i \in N$ 在某种意义上是“几乎纯粹的贪婪”
$$T \rightarrow 0 \quad \Longrightarrow \quad \alpha_i^T(S) \rightarrow 0 \quad \text { if } \partial_i v(S)<0$$ 此外，温度低 $T>0$ 在联盟的形成过程中，我们可以期待很高的公益性。

## 经济代写|博弈论代写Game Theory代考|Equilibria in cooperative games

(1) $N$ 是关于个体效用函数的增益均衡 $u_i(S)=\partial_i v(S)$.
(2) $v(N) \geq v(N \backslash i)$ 对所有人 $i \in N$.

$$u_i(S)=\partial_i v(S)=v(S \Delta i)-v(S) .$$

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