Cristiano Ronaldo was vilified for choosing to take the 5^{th} penalty in Portugal’s Confederations cup shoot-out against Chile, in the wake of defeat after missing their first 3 spot-kicks. Ronaldo didn’t get a chance to take one.

The media based its derision on a view that Ronaldo was seeking glory by choosing to take the 5^{th} (and potentially winning) penalty, implying that the team would have benefited by scheduling its best penalty taker earlier. For example, Alan Shearer was quoted on Twitter as saying, ‘What’s the point in having your best penalty taker last?”

Was this vilification justified? Probably not.

In a Penalty Shoot-out, as long as a player’s ability isn’t affected by when they take their spot-kick, **the order of the first 5 penalty takers makes no difference to the expected outcome** – even if the players are of varying ability. There is no benefit in playing your best penalty taker first.

This can be shown by treating each of the first 10 penalties as independent, and building a probability tree. As each penalty only has 2 possible outcomes (score or miss) the total possible combinations of 10 penalties is 2^10 = 1024. Although, in practice, only 504 of the 1024 would be fully played out, because the shoot-out ceases as soon as one team achieves an unassailable lead.

If we know the expected conversion rate of each team’s first 5 penalty takers, we can work out the likelihood of each team winning the shoot-out. To illustrate this, suppose each Chile player had a penalty conversion rate of 72.5% (near the average tournament shoot-out success rate), Ronaldo had a conversion rate of 80%, but the other 4 Portugal players only had a conversion rate of 60%

If Ronaldo takes the 5^{th} penalty the likelihood of Portugal winning within the first 10 penalties is 26%, Chile’s chances are 48% (the remaining 26% is a level outcome after 10 kicks). If Ronaldo, instead takes the first (or second, third or fourth) spot-kick, these probabilities would be exactly the same.

The only difference in outcome between Ronaldo shooting first or fifth, is that the shoot-out is likely to last longer if Ronaldo goes first – but the overall win likelihood is the same, because the weaker penalty takers go later. This might create the illusory view that going first is better. The chart below illustrates this effect.

To take an even more extreme example, suppose Claudio Bravo has the number of all 5 Portugal shooters except Ronaldo. He knows where they will shoot, so will save each one. Conversely Ronaldo, being the supreme penalty taker, will beat Bravo every time. That is, we know with certainty that Portugal will score one penalty out of five. If this is the case, the only way that Portugal can win is if Chile miss all 5 spot-kicks, So with a conversion rate of 72.5%, the chances of this happening are (1-72.5%)^5= 0.16%. The chances of Portugal winning are 0.16% irrespective of whether Ronaldo goes first or last.

Now, in the analysis above, I’ve made a couple of assumptions that won’t really be borne out in practice. It’s unlikely that each of the 10 penalties will be independent – what’s happened previously surely impacts how the next penalty is taken. And, players are likely to be affected by the situation of a spot-kick (e.g. whether it’s a sudden death scenario).

To illustrate the psychological effect of Penalty shoot-outs, theoretically it should make no difference whether a team shoots first or second. In reality, it does – with an average success rate of around 60% for the team kicking first.

So back to Ronaldo’s choice of shoot-out position. Would going earlier reduce the pressure on the later kickers in some way? – possibly, but they are still likely to face the pressure of a sudden death situation.

But placing Ronaldo as the fifth taker makes perfect strategic sense if Portugal believe that he will be less affected by a pressure sudden-death kick, than other players – because the strongest likelihood is that the shootout will be won on a team’s 5^{th} spot-kick.

So, Ronaldo’s shoot-out position is logical, despite a pervading view that the best shooters should go first.

I expect to dust off this article at next year’s World Cup.