Was Cristiano Ronaldo Right?

After their 120-minute scoreless draw, Portugal and Spain needed a penalty shootout to decide which team would go through to the Euro 2012 final. Cristiano Ronaldo, the Portuguese captain and superstar, was supposed to be the last penalty-taker in the first round of five kicks, against the Spanish captain and his Real Madrid teammate, Iker Casillas.  But the football gods intervened.  With Spain going first and missing the first attempt through Xabi Alonso but converting the next three including the fourth one through Sergio Ramos a la Panenka, Portugal managed to miss both the first one through Joao Moutinho and the fourth through Bruno Alves.  When Cesc Fabregas converted for Spain, Portugal lost the shootout with their arguably best penalty-taker unused.  When asked later by reporters, Ronaldo explained that he was told by his coach to take the last penalty, and volunteered that it was not reason why Portugal went out to Spain in the shootout.  Was Ronaldo right?  Is the order of penalty-taking irrelevant for winning a penalty shootout?  

To answer the above question, imagine that the eleven Portuguese players are ranked by their fixed chance of scoring against Casillas, and vice versa the Spaniards against the Portuguese keeper.  The two respective coaches independently choose their first five players, the order of them taking penalties, and the order of remaining players for the sudden death round if the two teams tie after the first round of five kicks.  It is not hard to see that given any choices of players for the first round of five kicks, the coach will choose the remaining players for the sudden death round in the decreasing order of their chance against the opposite keeper.  If a player with a greater scoring chance is chosen immediately after another with a smaller chance, then switching them around would increase the total chance of winning in the two rounds while decrease the chance of losing, because the shootout continues to the second round only if the two teams both score or both miss.  The total chance of tying after the two rounds may either increase or decrease, and tying could be either good or bad depending on whether the team has better or worse remaining penalty-takers than the opposing team, but the overall impact on the total chance of tying is small relative to the increased chance of winning and decreased chance of losing.  By a similar logic, the coach should choose his best penalty-takers for the first round of five kicks.  However, the order of these five penalty-takers is irrelevant. With fixed chances of scoring against the opposing goalkeeper, switching around scoring and missing in a given sequence of outcomes of the first five kicks changes nothing of the score difference at the end of the round.  Except in the case like what happened in the Spain-Portugal Euro 2012 semi-final shootout, Ronaldo would probably have scored against Casillas if he were to go earlier and would have prolonged the Portuguese misery by one kick.  So, Ronaldo was right.  

But the above analysis also means that the order between two teams in a shootout is irrelevant.  This, however, contradicts the fact that teams win the coin toss before a shootout always choose to take their penalties first.  Most coaches explain this preference by saying that going first and succeeding would put pressure on the opposing side.  Since it is easier for a penalty-taker to put the ball in in the net than for the opposing keeper to save the day, the pressure that coaches talk about is on penalty-takers.  So, instead of having fixed scoring chances against the opposing keeper, imagine that penalty-takers have slightly lower chances when their side is trailing. Then it is not hard to see that in the sudden death round, teams prefer to go first than second.  Going first increases the chance of winning by putting the opposing penalty-taker under pressure when scoring first, and reduces the chance of losing by avoiding the pressure that arises when the opposing side goes first and scores.  The chance of tying and going into the next round can either go up or go down, and tying could be either good or bad, but the overall impact on tying is small so teams always want to go first.  By a similar logic, they also prefer to go first in the first round of five penalty kicks.  

More to the point of our analysis, the order of the first five penalty-takers is no longer irrelevant, because switching around scoring and missing in a given sequence of outcomes of the first five kicks now can change the overall score difference after five kicks for each team.  Imagine that the score is tied after the first three rounds, and the fourth and fifth penalty-takers are Ramos and Fabregas for Spain, and Alves and Ronaldo for Portugal.  In the event that the Spaniards win the shootout, the order between Ramos and Fabregas, who is arguably a better penalty-taker, is irrelevant if both end up scoring.   But if the Spaniards win the shootout because both Alves and Ronaldo miss their attempts, then Ramos missing the fourth kick and Fabregas scoring the fifth is no longer the same as, and in fact less advantageous than, Fabregas scoring the fourth and Ramos missing the fifth.  This is because by scoring the fourth kick Fabregas would put Alves under pressure, which would increase the chance of winning the shootout after the last two rounds.  Of course, for the same reason, if the Spaniards win because both Alves and Ronaldo miss, then Ramos scoring the fourth kick and Fabregas missing the fifth would be more advantageous than Fabregas missing the fourth and Ramos scoring the fifth.  However, since Fabregas is a better penalty-taker than Ramos, switching the order between them would increase Spain's overall chance of winning the shootout.  A little bit more calculation is needed to verify that switching Ramos and Fabregas will also reduce the chance of losing, and as before, would have only a relatively small impact on tying.  The same should be true for Portugal as well; that is, since Ronaldo has a greater scoring chance than Alves, switching the order between the two would increase the winning chance for Portugal.  So, Ronaldo was wrong after all, unless of course the true reason he went AWOL is that he was really a big-game choker.