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:


    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