In soccer, players sometimes take penalty kicks, where a player shoots at the go

Question

In soccer, players sometimes take penalty kicks, where a player shoots at the goal and only the goalie is allowed to try to stop the shot. Because penalty kicks are taken from very close to the goal, the goalie usually must jump to where the ball is headed at almost exactly the same time that the ball is kicked to have any chance to prevent a goal. Therefore you will often see the goalie jump to his left while the ball is being kicked to his right, for example. 2 We can think of the interaction between the goalie and the player taking the penalty kick as a game where the two players simultaneously decide whether to go right or left. If they go in opposite directions, a goal is scored; if they play the same direction, the shot is blocked. Is there a Nash Equilibrium in pure strategies?

Explanation / Answer

In the above game there is no nash equilibrium in pure strategies because there is a tendency to move for both players in another strategy. We are taking two players one is Goalie and another is Penalty Kicker. It has been said that there is two option for both players, for Goalie to jump either in right side or left side. Similarly for penalty kicker there is two option i.e to take the shot in right side or left side to the goal post. It has been also said that when there is a same move for both players the shot is blocked. It means Goalie save the kick. It also means in terms of outcome we can say full outcome for Goalie and 0 outcome for penalty kicker. It will happen just opposite when they move opposite direction i.e Goalie will get zero outcome and penalty kicker will get full outcome . We are denoting here full outcome as 1 and zero outcome as 0. So we can write down the as following form.

In this above game lets take strategy Move left and Kick left , here the penalty kicker can increase its payoff from 0 to 1 by taking kick right rather than left. So penalty kicker will have tendency to move. So it will be not nash equilibrium. Now we take strategy Move left and Kck right , here also the Goalie can increase its payoff by moving right rather than left. So here also the Goalie will have tendency to move. Similarly it will not be nash equilibrium. Now we take the strategy move right and kick left, here the Goalie can increase its payoff by moving to left. So its not his best option. Now we take the option Move Right and Kick Right , here also we get player 2 i.e the penalty kicker has beeter option or to get more payoff by moving or taking the shot left. So it will not be nash equilibrium.

So we can say in this game we will not get no pure strategy nash equilibrium.

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