1. Compute the Nash equilibrium (if they exist) in pure strategies for the following games:

2. For what values of $$\alpha$$ does a Nash equilibrium exist in pure strategies for the following game:

3. Consider the following game:

Suppose two vendors (of an identical product) must choose their location along a busy street. It is anticipated that their profit is directly related to their position on the street.

If we allow their positions to be represented by a points $$x_1, x_2$$ on the $$[0,1]_{\mathbb{R}}$$ line segment then we have:

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By considering best responses of each player, identify the Nash equilibrium for the game.

4. Consider the following game:

Plot the expected utilities for each player against mixed strategies and use this to obtain the Nash Equilibria.

5. Assume a soccer player (player 1) is taking a penalty kick and has the option of shooting left or right: $$S_1=\{\text{SL},\text{SR}\}$$. A goalie (player 2) can either dive left or right: $$S_2=\{\text{DL}, \text{DR}\}$$. The chances of a goal being scored are given below:

1. Assume the utility to player 1 if the probability of scoring and the utility to player 2 the probability of a goal not being scored. What is the Nash equilibrium for this game?

2. Assume that player 1 now has a further strategy available: to shoot in the middle: $$S_1=\{\text{SL},\text{SM}, \text{SR}\}$$ the probabilities of a goal being scored are now given:

Obtain the new Nash equilibrium for the game.

6. In the notes the following theorem is given:

Every normal form game with a ﬁnite number of pure strategies for each player, has at least one Nash equilibrium.

Prove the theorem for 2 player games with $$|S_1|=|S_2|=2$$. I.e. prove the above result in the special case of $$2\times 2$$ games.

Solution available