Your and your partner are working together on a project. Where your effort towards the project is y and your partner's effort is , the project's value will equal (+)+. You will share this value equally (each of you getting half of it) once the project is complete.

For both you and your partner, effort is costly: units of effort bears a cost of ^2 for you, and similarly units of effort comes with a cost of ^2 for your partner. Therefore your payoff is 0.5[(+)+]^2 and your partner's is0.5[(+)+]^2.

Wherever rounding is needed, please round to 3dp.

1. Suppose =4 and the only effort options available are 0, 1 and 2. Suppose decisions are made simultaneously. Find the set of pure strategy Nash equilibria of this game. Add up both players' payoffs in all of these equilibria. This sum is _____.

2. How many strictly dominated strategies does your partner have in the game described in the previous question?

3. Now suppose =10 and that instead of being limited to three effort levels, both you and your partner can choose any nonnegative effort level. Find the effort level you exert in the unique Nash equilibrium of the game.