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Five political parties, A, B, C, D, E take part in the following cooperative game. The number of votes controlled by individual parties are a = 9, b = 9, c = 10, d = 60, e = 60, Any coalition that controls a majority of the votes will be able to form a government, receiving a payoff of 1. Other coalitions’ payoffs are zero.

  1. Characteristic function

v(a)=v(b)=v(c)=v(d)=v(e)=v(a,b)=v(a,c)=v(a,d)=v(a,e)=v(b,c)=v(b,d)=v(b,e)=v(c,d)=v(c,e)=v(0)=0

else =1

is it correct? and how should I find the core and Shapley value?

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  • $\begingroup$ Is the core empty ? $\endgroup$ – Keti Dzebniauri May 12 at 22:20
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First of all, notice that your characteristic function is almost correct, you forgot $v(a,b,c)=0$.

Moreover, you are right with your remark that the core is empty. To see that, we have to notice that your specified type of weighted majority game belongs to the class of essential constant sum games.

A game is called essential whenever $v(N)>\sum_{i \in N}\,v(\{i\})$ is satisfied.

A game is called constant sum whenever $v(S) + v(N\backslash S) = v(N)$ is given for all $S \subseteq N$.

Having introduced these definitions, we can go along to prove that the core of a weighted majority games is empty.

For doing so, assume by the way of contradiction that $x \in C(v)$ is satisfied, where $C(v)$ denotes the core of the game $v$. This implies that $x(N\backslash \{i\})\ge v(N\backslash \{i\})$ holds for all $i \in N$. However, in connection with the above constant sum property of the game and efficiency, we get

$$x(N \backslash \{i\}) \ge v(N \backslash \{i\}) = v(N) - v(\{i\}) = x(N) - v(\{i\}). $$

But then we have $x_{i} \le v(\{i\})$ for all $i \in N$. The game is essential, this implies that

$$ x(N) \le \sum_{i \in N}\,v(\{i\}) < v(N),$$

must hold. Contradicting the fact that $x$ is an imputation. Thus, we conclude that $C(v)=\emptyset$ holds true.

To get the Shapley value of the game, you need just to apply its formula. For a five person game, this is a little bit laborious, so I will just give you the result.

The Shapley value is quantified through the following power index

$$sh(v) = (4,4,4,9,9)/30 $$

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