Skip to content

Penalty Game Solver

Any habit at a penalty gets punished, so both taker and keeper should mix it up. Set how often the taker scores in each case, and see exactly how often each should go which way.

The taker can also go down the middle, and the keeper can stay put.

Weaker and natural are the taker's sides: his natural side is the one he's stronger at. The starting values are from 1,417 professional penalties (Palacios-Huerta, 2003); those for the middle from 459 (Chiappori, Levitt and Groseclose, 2002).

Try an example

Taker to weaker side
Keeper to that side
Scored

Always the weaker side, keeper knows
Always the natural side, keeper knows

How it works

If the keeper dives to the taker's weaker side a share \(g\) of the time, the taker's chance of scoring when he aims there is \(g \times \text{ww} + (1 - g) \times \text{wn}\), and when he aims at his natural side \(g \times \text{nw} + (1 - g) \times \text{nn}\). The keeper's best \(g\) makes the two equal, so the taker gains nothing either way:

$$g = \frac{\text{wn} - \text{nn}}{(\text{wn} - \text{nn}) + (\text{nw} - \text{ww})}$$

The taker's share \(k\) comes from the same reasoning the other way round, leaving the keeper nothing to gain from either dive:

$$k = \frac{\text{nw} - \text{nn}}{(\text{nw} - \text{nn}) + (\text{wn} - \text{ww})}$$

In plain football

  • ww, wn, nw, nn are the four scoring chances you set: the first letter is where the taker aims, the second where the keeper dives (weaker or natural side).
  • That pair of mixes is the Nash equilibrium: neither player can do better by changing his own plan alone.
  • If either player has a choice that's best whatever the other does, there's nothing to mix: the solver checks for that first and says so.

With the middle switched on, each player has three choices and there's no short formula. The keeper's best mix makes every place the taker actually uses equally good, and some places may get none at all; the solver finds it by checking every point where two of the taker's options pay the same. Why keepers rarely stay put explains it with real numbers.

Where it goes wrong

  • Even with the middle it's a simplification. Real takers also choose height and power, and some wait for the keeper to move.
  • The middle's starting values rest on few kicks: keepers stayed put for only 11 of the 459.
  • It says how often, not which kick. The choice each time has to be unpredictable, or the keeper will learn it.
  • The four chances are averages. A particular taker against a particular keeper will have his own.

The maths behind it: Where should he put it? Penalties and the Nash equilibrium

These figures come from a game-theory model and are for analysis and education. Real penalties depend on far more than four numbers.