## 经济代写|博弈论代写Game Theory代考|The value of Banzhaf

As an example, let us assume that a player $i$ joins any of the $2^{n-1}$ coalitions $S \subseteq N \backslash{i}$ with equal likelihood, i.e., with probability
$$\pi_S^B=\frac{1}{2^{n-1}} .$$
Consider the unanimity game $v_T=\widehat{\delta}T$ and observe that $\partial_i v_T(S)=0$ holds if $i \notin T$. On the other hand, if $i \in T$, then one has $$\partial_i v_T(S)=1 \Longleftrightarrow T \backslash{i} \subseteq S .$$ So the number of coalitions $S$ with $\partial_i v_T(S)=1$ equals $$|{S \subseteq N \backslash{i} \mid T \subseteq S \cup{i}}|=2^{n-|T|-1} .$$ Hence we conclude $$E_i^{\pi^B}\left(v_T\right)=\sum{S \subseteq N \backslash{i}} \partial_i v_T(S) \pi_S^B=\frac{2^{n-|T|-1}}{2^{n-1}}=\frac{1}{2^{|T|}},$$
which means that the random value $E^{\pi^B}$ is identical with the BANZHAF power index. The probabilistic approach yields the explicit formula
$$\Phi_i^B(v)=F_i^{\pi^B}(v)=\frac{1}{2^{n-1}} \sum_{S \subseteq N \backslash{i}}(v(S \sqcup i)-v(S)) \quad(i \in N)$$

## 经济代写|博弈论代写Game Theory代考|Marginal vectors and the Shapley value

Let us imagine that the members of $N$ build up the “grand coalition” $N$ in a certain order
$$\sigma=i_1 i_2 \ldots i_n$$
and hence join in the sequence of coalitions
$$\emptyset=S_0^\sigma \subset S_1^\sigma \cdots \subset S_k^\sigma \subset \cdots \subset S_n^\sigma=N$$
where $S_k^\sigma=S_{k-1}^\sigma \cup\left{i_k\right}$ for $k=1, \ldots, n$. Given the game $(N, v), \sigma$ gives rise to the marginal vector ${ }^{21} \partial^\sigma(v) \in \mathbb{R}^N$ with components
$$\partial_{i_k}^\sigma(v)=v\left(S_k^\sigma\right)-v\left(S_{k-1}^\sigma\right) \quad(k=1, \ldots, n) .$$
Notice that $v \mapsto \partial^\sigma(v)$ is a linear value by itself. We can randomize this value by picking the order $\sigma$ from the set $\Sigma_N$ of all orders of $N$ according to a probability distribution $\pi$. Then the expected marginal vector
$$\partial^\pi(v)=\sum_{\sigma \in \Sigma_N} \partial^\sigma(v) \pi_\sigma$$
represents, of course, also a linear value on $\mathbb{R}^{\mathcal{N}}$.
Ex. 8.22. Show that the value $v \mapsto \partial^\pi(v)$ is linear and efficient. (Hint: Recall the discussion of the greedy algorithm for network connection games.)

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|The value of Banzhaf

$$\pi_S^B=\frac{1}{2^{n-1}} .$$

$$\partial_i v_T(S)=1 \Longleftrightarrow T \backslash i \subseteq S .$$

$$|S \subseteq N \backslash i| T \subseteq S \cup i \mid=2^{n-|T|-1} .$$

$$E_i^{\pi^B}\left(v_T\right)=\sum S \subseteq N \backslash i \partial_i v_T(S) \pi_S^B=\frac{2^{n-|T|-1}}{2^{n-1}}=\frac{1}{2^{|T|}},$$

$$\Phi_i^B(v)=F_i^{\pi^B}(v)=\frac{1}{2^{n-1}} \sum_{S \subseteq N \backslash i}(v(S \sqcup i)-v(S)) \quad(i \in N)$$

## 经济代写|博弈论代写Game Theory代考|Marginal vectors and the Shapley value

$$\sigma=i_1 i_2 \ldots i_n$$

$$\emptyset=S_0^\sigma \subset S_1^\sigma \cdots \subset S_k^\sigma \subset \cdots \subset S_n^\sigma=N$$
${ }^{21} \partial^\sigma(v) \in \mathbb{R}^N$ 带组件
$$\partial_{i_k}^\sigma(v)=v\left(S_k^\sigma\right)-v\left(S_{k-1}^\sigma\right) \quad(k=1, \ldots, n) .$$

$$\partial^\pi(v)=\sum_{\sigma \in \Sigma_N} \partial^\sigma(v) \pi_\sigma$$

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