## Monday, 23 November 2015

### Lecture 19

I had worried that my proof of Theorem 18.2 was not sufficient. (I have not taken this from any textbook, but made it up.) On reflection, I think that it is. Do you agree? It is not the shortest proof of this result since it relies on first understanding the algorithm for finding a nucleolus on page 85. However, since we have that algorithm, we may as well use it. Notice that this algorithm does two things: it proves the nucleolus exists, and it shows that the nucleolus is unique. Another way to show the nucleolus exists is by appealing to compactness of the set in which we are looking for the lexicographic minimum.

I should have mentioned that the set of imputations might be empty. However, it is non-empty if the game is superadditive. There is a simple $x$ that will work as an imputation. Can you guess it? I will make this a question for Examples sheet 3. Notice that in (P1) on page 85 there is no requirement that $x$ be an imputation, only that it be efficient, in the sense that $\sum_{i\in N} x_i = v(N)$.

There are some very nice notes and slides for 12 lectures on cooperative games, by Stéphane Airiau. His notes for lectures 4 and 5 are particularly relevant to what I have presented about bargaining games, core and nucleolus. You might enjoy browsing the slides for a second view and to see more examples.