Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

It’s 2020 and our mythical El Clasico game between Real Madrid and Barcelona is in the 23rd minute at the Santiago Bernabeu when Lionel Messi is brought down in the penalty box. He is rewarded with a spot kick against the custodian of the Los Blancos net, Keylor Navas.

Messi knows from the team statistician that if he aims straight and the goalkeeper stands still, his chance of scoring is just 30%. But if he aims straight and Navas dives to one corner, his chance of converting the penalty rises to 90%.

On the other hand, if Messi aims at a corner and the goalkeeper stands still, his chance of scoring is a solid 80%, while it falls to 50% if the goalkeeper dives to a corner.

We are here simplifying the choices to two distinct options, for the sake of simplicity and clarity.

Navas also knows from his team statistician that if he dives to one corner and Messi aims straight, his chance of saving is just 10%. But if he stands still and Messi aims at one corner, his chance of saving the penalty rises to 50%.

On the other hand, if Navas stands still and Messi aims at a corner, his chance of making the save is just 20%, while it rises to 70% if Messi aims straight.

So this is the payoff matrix, so to speak, facing Messi as he weighs up his decision.

 Goalkeeper – Stands Still Goalkeeper – dive to one corner Lionel Messi – Aims straight 30% 90% Lionel Messi – Aims at corner 80% 50%

So what should he do? Aim straight or to a corner. And what should Navas do? Stand still or dive?

Here is the payoff matrix facing Navas.

 Messi – Aims straight Messi – Aims at a corner Navas – Stands still 70% 20% Navas – Dives to one corner 10% 50%

Game theory can help here.

Neither player has what is called a dominant strategy in game-theoretic terms, i.e. a strategy that is better than the other, no matter what the opponent does. The optimal strategy will depend on what the opponent’s strategy is.

In such a situation, game theory indicates that both players should mix their strategies, in Messi’s case aiming for the corner with a two-thirds chance, while the goalkeeper should dive with a 5/9 chance.

These figures are derived by finding the ratio where the chance of scoring (or saving) is the same, whichever of the two tactics the other player uses.

The Proof

Suppose the goalkeeper opts to stand still, then Messi’s chance (if he aims for the corner 2/3 of the time) = 1/3 x 30% + 2/3 x 80% = 10% + 53.3% = 63.3%

If the goalkeeper opts to dive, Messi’s chance = 1/3 x 90% + 2/3 x 50% = 30% + 33.3% = 63.3%

Adopting this mixed strategy (aim for the corner 2/3 of the time and shoot straight 1/3 of the time), the chance of scoring is therefore the same. This is the ideal mixed strategy, according to standard game theory.

From the point of view of Navas, on the other hand, if Messi aims straight, his  chance of saving the penalty kick (if he dives 5/9 of the time) = 5/9 x 10% + 4/9 x 70% = 5.6% + 31.1% = 36.7%

If Messi opts to aim for the corner, Navas’ chance = 5/9 x 50% + 4/9 x 20% = 27.8% + 8.9% = 36.7%

Adopting this mixed strategy (dive for the corner 5/9 of the time and stand still 4/9 of the time), the chance of scoring is therefore the same. This is the ideal mixed strategy, according to standard game theory.

The chances of Messi scoring and Navas making the save in each case add up to 100%, which cross-checks the calculations.

Of course, if the striker or the goalkeeper gives away real new information about what he will do, then each of them can adjust tactics and increase their chance of scoring or saving.

To properly operationalise a mixed strategy requires one extra element, and that is the ability to truly randomise the choices, so that Messi actually does have exactly a 2/3 chance of aiming for the corner, and Navas actually does have a 5/9 chance of diving for the corner. There are different ways of achieving this. One method of achieving a 2/3 ratio is  to roll a die and go for the corner if it comes up 1, 2, 3 or 4, and aim straight if it comes up 5 or 6. Or perhaps not! But you get the idea.

For the record, Messi aimed at the left corner, Navas guessed correctly and got an outstretched hand to it, pushing it back into play. Leo stepped forward deftly to score the rebound. Cristiano Ronaldo equalised from the spot eight minutes later. And that’s how it ended at the Bernabeu. Real Madrid 1 Barcelona 1. Honours even in El Clasico.

Appendix

Messi’s strategy

x = chance that Messi should aim at corner

y = chance that Messi should aim straight

So,

80x + 30y (if Navas stands still) = 50x + 90y (if Navas dives)

x + y = 1

So,

30x = 60y

30x = 60 (1-x)

90x = 60

x = 2/3

y=1/3

Navas’ strategy

x = chance that Navas should dive to corner

y  = chance that Navas should stand still

So,

10x + 70y (if Messi aims straight) = 50x + 20y (if Messi aims at corner)

x+y = 1

So,

10x + 70y = 50x + 20y

40x = 50y

40x = 50(1-x)

90x = 50

x = 5/9

y = 4/9