A soccer penalty kick is a real-world version of rock paper scissors: the kicker and goalkeeper choose sides at essentially the same moment, and any predictable pattern can be exploited, so the best strategy for both is to randomize. Economists have tested this with thousands of professional penalties and found that players behave remarkably close to what mixed-strategy game theory predicts.
Why the two games have the same structure
In rock paper scissors, neither player can react to the other's throw, and every fixed choice has a counter. The equilibrium is to mix your throws unpredictably, one third each. You can read the full derivation in the math of rock paper scissors.
A penalty is similar. The ball reaches the goal too fast for a keeper to wait and react, so the keeper must commit to a side roughly as the kicker strikes. If a kicker always went left, keepers would always dive left. If a keeper always dove right, kickers would always go left. Neither pure strategy survives, so both sides mix.
The main difference is that penalties are not symmetric. Most kickers are more accurate to their "natural" side (for a right-footer, usually the keeper's right), and the centre is a third option that is risky if the keeper stays put. So the equilibrium is not an even split. Theory predicts something subtler: each player should mix so that the other is indifferent, meaning the kicker's scoring rate should be the same whichever way they kick, and the keeper's save rate the same whichever way they dive.
What the research found
Chiappori, Levitt and Groseclose (2002). Pierre-André Chiappori, Steven Levitt and Timothy Groseclose published "Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer" in the American Economic Review (vol. 92, no. 4). Using penalties from top-flight European league matches, they built a model that allows kickers and keepers to differ in skill and tested its predictions. The data were broadly consistent with players randomizing as theory predicts, including some counter-intuitive predictions such as how often kickers should aim for the centre.
Palacios-Huerta (2003). Ignacio Palacios-Huerta's "Professionals Play Minimax", in the Review of Economic Studies (vol. 70), analysed 1,417 penalty kicks from professional European league matches. He tested two implications of the minimax theorem: that scoring rates should be equal across the choices each player makes, and that each player's choices should be serially independent, with no detectable pattern from one kick to the next. The professionals passed both tests, which is striking, because laboratory subjects asked to randomize usually fail.
What this means for your rock paper scissors game
The lesson from penalties is not "be random". It is more specific:
- Randomize with respect to what your opponent can observe. Professional kickers did not alternate or streak in detectable ways. Casual RPS players almost always do. Our own data on throw patterns shows players reacting strongly to their last result.
- Unequal options call for unequal mixes. In variants where throws are not symmetric, as in some variations, the right mix is the one that leaves your opponent indifferent, not necessarily one third each.
- Exploit, but carefully. If you spot a pattern, lean against it. The moment you become predictable yourself, you have handed the edge back.
Our game theory page covers mixed strategies in more depth, and you can test your own randomness by playing free online.
FAQ
Do penalty takers really randomize?
The evidence from Chiappori, Levitt and Groseclose (2002) and Palacios-Huerta (2003) says professionals come close to equilibrium mixing, with no exploitable sequence patterns in aggregate.
Why is a penalty like rock paper scissors?
Both are simultaneous-move games in which every predictable choice has a counter, so the stable strategy is to mix.
Is the best penalty strategy to kick to each side half the time?
No. Because kickers are stronger to one side, the optimal mix is uneven. What should be equal is the success rate across choices.