In a nice paper, Chiappori, Groseclose and Levitt look at the zero-sum game of a penalty kick in professional soccer. They lay out a number of robust predictions that are testable in data, but they leave out the formal analysis of the theory (at least in the published version.) These make for great advanced game theory exercises. Here’s one:
The probability that the shooter aims for the middle of the goal (as opposed to aiming for the left side or the right side) is higher than the probability that the goalie stays in the middle (as opposed to jumping to the left or to the right.)
Hint: the answer is related to my post from yesterday, and you can get the answer without doing any calculation.
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February 25, 2010 at 9:52 am
wak
It’s easier for the goalie to save a ball down the middle (if he doesn’t move) than it is for him to save a ball to the left or right (even if he dives in the correct direction). In equilibrium, for the shooter to be indifferent between all directions (a corner strategy is obviously not optimal), the goalie must make going down the middle more attractive than shooting to the side; this is achieved when “the probability that the shooter aims for the middle of the goal (as opposed to aiming for the left side or the right side) is higher than the probability that the goalie stays in the middle”.
February 26, 2010 at 10:40 am
jeff
You got it, but i would say it a little differently because there is a separate argument for the shooter and the goalie.
Suppose, for starters, that the shooter is equally likely to shoot down the middle as shoot to one of the two sides, and the goalie is equally likely to stay in the middle as jump to one of the two sides. Then because the shooter has a lower chance of scoring when he shoots down the middle, the goalie strictly prefers to jump to one of the two sides.
The shooter must increase the probability of shooting down the middle in order to induce the goalie to stay in the middle with positive probability.
But the shooter is not willing to shoot down the middle because he is less likely to score that way. So in order to make the shooter willing to do what we just argued he should do, the goalie must decrease the probability that he stays in the middle.
The goalie’s probability went down, the shooter’s went up. So the shooter’s probability is larger.
March 3, 2010 at 10:41 am
confused
What is the payoff matrix you have in mind? I’m having a very hard time figuring out what you mean here. I would have thought the matrix should look like:
Kick Center, Guard Center: low, say 0.1 (hard to score in the middle when goalie in middle)
Kick Center, Guard Elsewhere: high, say 0.9 (very easy to score in middle when unguarded)
Kick Elsewhere, Guard Center: high but less, say 0.8 (easy though more often flubbed)
Kick Elsewhere, Guard Elsewhere: low but more, say 0.5 (hard but doable to score when goalie guesses right)
Under these assumptions, which I just wrote down off the top of my head, both goalie and shooter go Elsewhere a significant majority of the time and Center rarely.
How are your assumptions different?
March 3, 2010 at 11:14 am
jeff
I believe you want to reverse the ordering of your second and third scenarios. The authors interviewed players who say that it is easier to score if you shoot Elsewhere and the goalie stays in the middle than if you shoot down the middle and the goalie guards Elsewhere.
I guess the reason for this is that the goalie *starts* in the middle. If he is gearing up to jump Elsewhere and sees that the shot is coming down the middle, he has a chance to stop himself.
March 3, 2010 at 11:56 am
confused
OK, cool. For anyone else trying to parse, I just looked at the paper and their matrix is something like:
KC, GC (Kick Center, Guard Center): 0
KE, GC: pi
KC, GE: mu
KE, GE: P
Indeed, we now get that pi>mu implies p_KC > p_GC, which is certainly counterintuitive.
June 14, 2010 at 10:16 pm
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[…] note: why are we wasting time analyzing penalty kicks? Can we get data on competitive RoShamBo? While we wait for that here is an exercise for the […]
June 15, 2010 at 12:52 pm
Miguel
Just hit the ball hard to the upper corners, switching between left and right sides. That would almost rule out probability, as the keeper wont stand a chance of stopping the kick even when knowing where the ball is going (keeper has to stay in place until the ball is kicked). Keep it simple!
June 15, 2010 at 7:21 pm
pauloabx
Miguel that strategy has the problem that it highly increases the probability of missing the penalty, even if you are the best player in the world of your age. Remember Baggio 1994.
June 16, 2010 at 12:32 am
Miguel
Baggio’s miss impressed the world, I still remember that game. However, by playing it at the middle, left or right (slightly) it really becomes a matter of probability. If you plan to shoot close to the posts, it becomes a matter of skill. What do you think?
June 16, 2010 at 3:12 pm
pauloabx
I agree with that. The problem is that not all the players have the skill to go to far out, so they play the game with the keeper. That’s just saying that if it is Ronaldo shooting (of course) shooting to the right probably has a higher probability of being a goal.