这是一份nottingham诺丁汉大学MATH4064作业代写的成功案例

We extend the endowment function en to coalitions via the function $e n: 2^{N} \times R \rightarrow \mathbb{N}$.
$$
e n(C, r)=\sum_{i \in C} \operatorname{en}(i, r) .
$$
Similarly, we extend the req function to sets of goals via the function req : $2^{\Theta} \times R \rightarrow \mathbb{N}$.
$$
\operatorname{req}\left(\Theta^{\prime}, r\right)=\sum_{\theta \in \Theta^{\prime}} \operatorname{req}(\theta, r) .
$$
A set of goals $\Theta^{\prime}$ satisfies player $i$ if $\Theta^{\prime} \cap \Theta_{i} \neq \emptyset$; we say that $\Theta^{\prime}$ satisfies coalition $C \subseteq N$ if it satisfies every member of $C$. A set of goals $\Theta^{\prime}$ is feasible for coalition $C$ if that coalition is endowed with sufficient resources to achieve all the goals in $\Theta^{\prime}$ :
$$
\operatorname{feas}\left(\Theta^{\prime}, C\right) \text { iff } \forall r \in R: e n(C, r) \geq r e q\left(\Theta^{\prime}, r\right) \text {. }
$$
Notice that as we add players to a coalition, the set of goals that this coalition can achieve grows monotonically. That is,
$$
\text { feas }\left(\Theta^{\prime}, C\right) \text { implies feas }\left(\Theta^{\prime}, D\right) \text { for all } C, D \subseteq N \text { such that } C \subseteq D \text {. }
$$
In other words, CRGs are inherently monotone.

MATH4064 COURSE NOTES ：

1. Coalition-action pair $(C, a)$
   $$
   Q_{i}(C, a)=\sum_{\mathbf{t}{C} \in T{C}} B_{i}\left(\mathbf{t}{C}\right) \sum{s} \operatorname{Pr}\left(s \mid a, \mathbf{t}{C}\right) R(s) . $$ This is $i$ ‘s expectation regarding the value of coalition $C$ taking action $a$, given $i$ ‘s beliefs about the profile $\mathbf{t}{C}$ that describes the types of $C$ ‘s members.
2. Coalition
   $$
   V_{i}(C)=\max {a \in A{C}} Q_{i}(C, a) .
   $$
   This is the immediate value that $i$ assigns to coalition $C$, i.e., the value of that coalition executing its optimal action, given $i$ ‘s beliefs.
3. Coalitional agreement $\left\langle C, a, \mathbf{d}{C}\right\rangle$ $$ p{i}^{i}\left(C, a, \mathbf{d}{C}\right)=r{i} V_{i}(C)
   $$
   This is what $i$ expects to get out of an agreement specifying that: (a) $i$ will be a member of $C$,
   (b) $C$ will execute action $a$, and (c) the payoffs will be divided using the demand vector $\mathbf{d}_{C}$.