Homework 2  Nash equilibrium in normal form games

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

For what values of \(\alpha\) does a Nash equilibrium exist in pure strategies for the following game:

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

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

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:

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?

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.


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.