In soccer, a penalty kick can be described as follows: A striker lines up with the ball in front of him

In soccer, a penalty kick can be described as follows: A striker lines up with the ball in front of him. He runs forwards and kicks the ball towards the net. The goalie tries to stop it. So, both players pick a side, and the goalie wants to match sides (stop the ball), while the striker wants to mismatch sides (score a goal). In other words, the goalie wins if the striker kicks to the left side of the goal and the goalie jumps to the left or if the ball is kicked to the right side of the goal and the goalie jumps to the right. Vice versa, if the goalie jumps in the wrong corner, the striker wins. Let’s say the payoff is 1 for the player who wins and the payoff is 0 for the player who loses.

a. So , the pay-off matrix for the game is :

The Goalie is the column player and the Striker is the row player. The first term of each cell represents the pay-off of the Striker , while the second term represents the pay-off of the Goalie.

We indicate the optimal choices by underlinning them. The Nash Equilibirum will be given by the cell that contains two underlines.

• When the Striker chooses Left , the Goalie would also choose Left since his pay-off will be higher in that case (1>0)
• When the Striker chooses Right , the Goalie would choose Right. (1>0)
• When the Goalie chooses Left , the Striker chooses Right. (1>0)
• When the Goalie chooses Right , the Striker chooses Left. (1>0)

Thus , there exists no pure strategy Nash Equilibrium in this game.
Suppose , the Striker chooses Left with a probability of 'p' and the Goalie chooses Left with a probability 'q'.

At the optimal ,
Expected Utility of the Striker from choosing Left = Expected Utility of the Striker from choosing Right

EU_s(L) = EU_s(R)

Therefore , 0.q + 1.(1-q) = 1.q + 0.(1-q)

Which implies , 1- q = q

Therefore , 2q = 1

So , q = 1/2

Similarly , Expected Utility of the Goalie from choosing Right = Expected Utility of the Goalie from choosing Left

EU_G(L) = EU_G(R)

Therefore , 0.p + 1.(1-p) = 1.p + 0.(1-p)

Which implies , 1 - p = p

Therefore , 2p = 1

So , p = 1/2

So , the mixed strategy Nash Equilibrium for this game is (1/2 , 1/2)

b. This game is an example of a Matching Pennies game.

Matching Pennies is a zero-sum game because each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the participants' total gains are added up and their total losses subtracted, the sum will be zero.

This game has no pure strategy Nash equilibrium since there is no pure strategy (heads or tails) that is a best response to a best response. In other words, there is no pair of pure strategies such that neither player would want to switch if told what the other would do. Instead, the unique Nash equilibrium of this game is in mixed strategies: each player chooses heads or tails with equal probability. In this way, each player makes the other indifferent between choosing heads or tails, so neither player has an incentive to try another strategy.