The 2014 FIFA World Cup has been no short of excitement! Only 5 games (out of 48) ended in goal-less draws in the 1st round. As a result, the better sides moved to the Round of 16. Here though, 5 out of 8 games extended to extra time and 2 of them were settled by Penalties. Remarkable head to head clashes, with the tiniest of differences between teams on the pitch!

The penalty shoot-out is perhaps the most critical stage of a game since it brings a definitive end to it. Goalkeepers and spot-kick takers are trained specifically for the shoot-out. Different players follow different strategies while taking a penalty or saving one. In this post, I’m going to explore various strategies adopted by goalkeepers and strikers during a penalty shoot-out.

Penalty shootouts were introduced in the FIFA World Cup in 1978. Before 78, all ties were decided in 120 minutes (73 matches) or by replays (4 matches). Germany has been on the winning side the most number of times (4 times) whereas England and Italy have been on the losing side the most (3 times). You can check out a brief history of penalty shoot-outs in the final version (as it is played today) of the FIFA World Cup in the above info-graphic.

Last time FIFA World Cup was played in Brazil (in 1950), there were no knock-out matches.

**Strategies for a Penalty shoot-out- Which side to shoot and which side to defend?**

When a penalty kick is taken there are 2 possible outcomes that can happen:

- It results in a goal
- It is blocked by the goalkeeper (a third outcome, but of equivalent result, is that the kick is off target; very embarrassing for the penalty taker)

Each of these outcomes is favorable to one of the teams only. For a kicker, to convert a shot into a goal or a goalkeeper to block a shot, they can either:

- Outplay the other with their speed and agility OR
- Develop a strategy which can potentially work in favor of each of them

Here’s what Rhett Allan had to say about ‘Outplaying the Opponent’ – How Do You Block a Penalty Kick? This is of course a valid way to win, but a lot of times the right strategy might just do enough for one to win the game. Let us now explore this problem from a Game Theory standpoint.

Assumptions:

- The Goalkeeper can definitely save if he jumps in the right direction. That is, if a goalkeeper has jumped to the ‘Left’ and the shot was also in the same direction, then he will save it.
- The striker never shoots outside the post. The only way to stop the striker from scoring is by blocking his shot.

In this game the following cases may arrive:

- The kicker shoots in one direction and the goalkeeper jumps in the same direction
- The kicker shoots in one direction and the goalkeeper jump in the opposite direction.

Based on the above cases, lets try to plot the pay-off matrix for this game.

From above, when both the goal-keeper and the striker are on the same direction, then it is a favorable output for the Goalkeeper which is indicated in the boxes ‘a’ and ‘d’ as (-1,1, i.e -1 for the Striker and +1 for the Goalkeeper). Similarly, when they are in opposite directions, then it is a favorable output for the striker which is indicated in the boxes ‘b’ and ‘c’ as (1,-1). We can see that there exists no Nash’s equilibrium in this game. If this game is played for an infinite number of times, the system will not adjust to settle at equilibrium. So we need to arrive at a mixed strategy for each of the players – the goalkeeper and the striker. Solving for it we can see that the goalkeeper and the striker should randomly jump or shoot in either directions in this game. See the solution below:

This is not fun! The strategies derived from the above makes the game very random. Can there be some other way of looking at this game? Lets try to revisit our assumptions again.

- The Goalkeeper can definitely save if he jumps in the right direction. That is, if a goalkeeper has jumped to the ‘Left’ and the shot was also in the same direction, then he will save it –
*This is possible if the Goalkeeper always pre-decides his direction for jump.* - The striker never shoots outside the post. The only way to stop the striker from scoring is by blocking his shot –
*This is may not be true always! Often a player misses a penalty, not because it is saved by the Goalkeeper, but because he shoots off the target.*

Based on the above condition, lets assume that the striker is strong on the left side and is never off target. But on the right side, he has a 50% chance of hitting the target. Lets now recreate our payoff matrix with the new conditions.

Solving for the pay-offs like above, we will see that it is preferable for the striker to shoot left on his weaker side more often than on his dominant side. Think about it?

Apparently you might think, that the striker must shoot left in the direction of his stronger side to get the best out of this game, but if you think about the goalkeeper, he knows that the striker is naturally weaker on his right. So there is no need to save that side. God will take care of it! Whereas, the left needs to be protected as the striker never misses a chance in that direction. So he will always jump on that side. Hence the striker should take his chances and shoot on the right, God has lesser chances of saving his shot than the Goalkeeper if he kicks towards the left side!

Goalkeepers study individuals from the opponents and have a good knowledge about their preferred side. They apply this technique during penalty shoot-outs. That is why you will always see that the Goalkeeper jumps before a shot is taken. In a penalty shoot-out it is harder to beat the opponent by outplaying them. The right strategy plays the most important role during such a situation.

*Do you have an alternate solution to this problem? Share this post and express your views!*

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