1. Obtain stable suitor optimal and reviewer optimal matchings for the matching games shown.

• Game 1:

• Game 2:

• Game 3:

• Game 4:

2. Consider a matching game where all reviewers have the same preference list. Prove that there is a single stable matching.

3. For the following cooperative games:

1. Verify if the game is monotonic.
2. Verify if the game is super additive.
3. Obtain the Shapley value.
4. Prove that the Shapley value has the following properties:

• Efficiency
• Null player
• Symmetry
For the game shown (a generalisation of “Pigou’s example”) obtain the PoA as a function of $$\alpha$$.