# 经济代写|博弈论代写Game Theory代考|ECON6025

## 经济代写|博弈论代写Game Theory代考|Boltzmann values

The probabilistic analysis of the previous section shows that the value assessment concepts of the BANZHAF power index and the SHAPLEY value, for example, implicitly assume that players just join – but never leave – an existing coalition in a cooperative game $(N, v)$.
In contrast, the model of the present section assumes an underlying probability distribution $\pi$ on the set $2^N$ of all coalitions of $N$ and assigns to player $i \in N$ its expected marginal value
$$E_i(v, \pi)=\sum_{S \subseteq N} \partial_i v(S) \pi_S .$$
Ex. 8.24. Let $\pi$ be the uniform distribution on $\mathcal{N}$ :
$$\pi_S=\frac{1}{|\mathcal{N}|} \quad \text { for all } S \in \mathcal{N} .$$

In view of
\begin{aligned} \sum_{S \subseteq N} \partial_i v(S) &\left.=\sum_{i \in S} v(S)-v(S \backslash i)\right)+\sum_{i \notin S}(v(S)-v(S \cup i)) \ &=\sum_{T \subseteq N \backslash i}(v(T \cup i)-v(T))+\sum_{T \subseteq N \backslash i}(v(T)-v(T \cup i)) \ &=0, \end{aligned}
one has
$$E_i(v, \pi)=\sum_{S \in \mathcal{N}} \partial_i v(S) \pi_S=\frac{1}{|\mathcal{N}|} \sum_{S \in \mathcal{N}} \partial_i v(S)=0 .$$
So the expected marginal value of any particular player is zero, if all coalitions are equally likely.

## 经济代写|博弈论代写Game Theory代考|Coalition formation

In the cooperative game $(N, v)$ it may not be clear in advance which coalition $S \subseteq N$ will form to play the game and produce the value $v(S)$. The idea behind the theory coalition formation is the viewpoint of a dynamic process in which players join and part in discrete time steps until a final coalition is likely to have emerged.

If this coalition formation process has BolTZMANN temperature $T>0$, the MEtropolis process suggests the model where, at the moment of a current coalition $S \subseteq N$, a randomly determined player $i \in N$ considers to enact a possible transition
$$S \rightarrow S \Delta{i}= \begin{cases}S \backslash{i} & \text { if } i \in S \ S \cup{i} & \text { if } i \notin S\end{cases}$$
by either becoming inactive or active. The transition is made with probability
$$\alpha_i(S)=\min \left{1, e^{\partial_i v(S) / T}\right} .$$
In this case, the coalition formation process converges to the corresponding BoltZmanN distribution on the family $\mathcal{N}$ of coalitions in the limit.

Cost games. If $(N, c)$ is a cost game, a player’s gain from the transition $S \rightarrow S \Delta{i}$ is the negative marginal cost
$$-\partial_i c(S)=c(S)-c(S \Delta{i}) .$$
So the transition $S \rightarrow S \Delta{i}$ in the Metropolis process would be enacted with probability
$$\alpha_i^{\prime}(S)=\min \left{1, e^{-\partial_i c(S) / T}\right} .$$

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Boltzmann values

$$E_i(v, \pi)=\sum_{S \subseteq N} \partial_i v(S) \pi_S .$$

$$\pi_S=\frac{1}{|\mathcal{N}|} \quad \text { for all } S \in \mathcal{N} .$$

$$\left.\sum_{S \subseteq N} \partial_i v(S)=\sum_{i \in S} v(S)-v(S \backslash i)\right)+\sum_{i \notin S}(v(S)-v(S \cup i)) \quad=\sum_{T \subseteq N \backslash i}(v(T \cup i)-v(T))+\sum_{T \subseteq N \backslash i}(v(T)$$

$$E_i(v, \pi)=\sum_{S \in \mathcal{N}} \partial_i v(S) \pi_S=\frac{1}{|\mathcal{N}|} \sum_{S \in \mathcal{N}} \partial_i v(S)=0 .$$

## 经济代写|博弈论代写Game Theory代考|Coalition formation

$$S \rightarrow S \Delta i={S \backslash i \quad \text { if } i \in S S \cup i \quad \text { if } i \notin S$$

$$\backslash \text { \alpha_i }(S)=\backslash \min \backslash \operatorname{left}\left{1, e^{\wedge}{\text { partial_i } v(S) / T \backslash \backslash \text { right }}\right. \text { 。 }$$

$$-\partial_i c(S)=c(S)-c(S \Delta i) .$$

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