There was a nervous minute this past Monday for the 20 million Italians who watched their national football (soccer) team compete in the European Championships. The Italian team already won their match against Ireland, 2-0, a clinical if uninspiring victory. However, the final match in their group between Spain and Croatia had still one minute to go. Spain was ahead 1-0. If the results held, then Spain and Italy would finish top two in their group, and advance to the quarterfinals. Why were the Italians nervous?
Fate was no longer in their hands (They never came close to scoring the third goal, which would ensure passage.) If Croatia were to tie the match at 1-1, the math would work out in a way that would send the Italians out of the tournament! Spain would advance whether it's a Spanish victory or a tie. Croatia would advance with a draw but crash out with a loss. Many fans of Italy, including me, felt that Spain should have gifted the draw to Croatia in order to send Italy, a 4-time World Cup winner, home. The Italians call this arrangement a "biscotto". (This article describes the atmosphere.)
In reality, Croatia didn't score, they failed to advance, and Spain and Italy are still in the tournament. I happen to think Spain might eventually regret not feeding the biscotto. We'll find out in the next two weeks.
What should Spain have done if it acts like a "rational" economist? The scenario facing Spain and Croatia can be analyzed using game theory invented by John Nash. The fear in Italy is that these two teams would coordinate in order to draw the match because if they manage a high-scoring draw (2-2 or higher), they ensure that they both advance at the expense of Italy. If there is a winner, the losing team is sent packing (assuming Italy won against Ireland, which was a safe bet.) The key question is whether any side has an incentive to cheat and back away from the coordinated outcome.
I'll present a simplified version of this problem here. Those who know all the possibilities for the Monday match will notice that I left out some details (which I will come back to at the end). It is always a good idea to start with a basic model, and then add complexity to it.
The biggest assumption I'll make is to ignore the difference between a high-scoring tie and a low-scoring tie. I'll just assume that a tie will suit both Spain and Croatia while a win by one team will dissatisfy the other team. I'll also assume Italy will win the match against Ireland so we can focus on Spain and Croatia tactics.
Here are the possible outcomes faced by Spain and Croatia:
Now, Spain and Croatia will pick one of two approaches to the match: play to win (Attack) or play to draw (Hold). In the following "pay-off matrix", we noted the relative level of satisfaction in each of four scenarios and for each team:
Start at the bottom right corner (SH+CH), in which both sides play to draw. This is the feared coordinated outcome. A draw sends both squads through to the quarterfinals and they both achieve their goals, and so each team rates this outcome a perfect 10. In the matrix, the red numbers refer to Spain and the blue numbers to Croatia.
The key question is whether one team is tempted to walk away from the Hold strategy. Look at the right column (CH). If Croatia were to play to draw, would Spain be tempted to switch from a Hold strategy to an Attack strategy? I think not. I think Spain would rate the SA+CH scenario as marginally worse than the mutual Hold scenario. Whether Spain wins or draws, Spain would advance. If Spain goes on attack, the match may still end in a draw, and there is now a small chance that Spain might lose. So at best, Spain would consider this even with the mutual Hold scenario but probably slightly worse.
Croatia would not be happy to see Spain switching from Hold to Attack. Because if that happens, and Croatia continues with its Hold strategy, then its chance of losing- and thus crashing out of the Euros - increases, and therefore, they will rate the SA+CH scenario lower, say 7 out of 10.
A symmetric situation exists if Spain plays Hold and Croatia switches from Hold to Attack. Croatia is not better off while Spain is definitely worse off. So, we see that the mutual Hold scenario is "stable".
If one side cheats, then most likely both sides will end up playing Attack but a mutual Attack scenario has a higher chance of delivering a loser (who crashes out) so both teams would rate this scenario as worse than mutual Hold. I give this scenario a score of 8 all around.
Mutual Attack is also "stable" in the sense that neither side would unilaterally switch from Attack to Hold. If a side did switch to Hold (without coordinating with the other side), then this side would have a larger chance of losing the match.
In summary, it is rational for Spain and Croatia to coordinate and try to draw the match.
The Spain-Croatia match was a 0-0 draw until two minutes before the end of regulation time. Soon after, Italy won their match when Balotelli scored a late spectacular second goal. If Spain wanted Italy out of the tournament, they had a few minutes to concede a goal to make it 1-1. It was the perfect time to feed them the biscotto. Why Spain went ahead and won the match can be due to sportsmanship, ego, or perhaps their view that they prefer to meet Italy rather than Croatia in a future round. (Here is the Guardian's account of the match.)
A few refinements can be made to this analysis. First, there should be three strategies because a low-scoring tie has different outcomes than a high-scoring tie. Second, the outcomes can be described as probabilities of a win, draw or loss, which each results in a level of satisfaction. Third, instead of a single-move game, one can think of this dynamically and assume that the teams can change strategy over the course of 90 minutes.