An interesting few pages in the aforementioned Soccernomics book I am reading concerns penalties, and the best strategies to adopt when taking them. A penalty taker who always aims for the same side will of course soon be found out, so all penalty takers have to mix it up. But how often should they stray from their natural side given that their chances of scoring are somewhat reduced by shooting to the unnatural side?
This is called a “mixed-strategy” by game theorists, and are “peculiar because they require the actor to incorporate randomness into decision making. Should I go to the pub or the cinema? A mixed strategy requires me to toss a coin, which sounds odd, since one might expect that I prefer one to the other.” *
The research showed that the optimal mixed-strategy (after taking into account the probability of the goalkeeper guessing right) is for a penalty taker to hit 61.5 percent of his kicks to his natural side, and 38.5 percent to the other. In reality, the penalty takers in the study (of 1,417 penalties) got remarkably close to this, hitting 60 percent to the natural side, versus 40% to the other. (No mention of how penalties straight down the middle were counted).
* I’ve tried tossing a coin myself to help make decisions. It doesn’t work, or rather, it does, but not directly.
If I am ever torn between two choices, and elect to decide the issue on a coin toss, as soon as the coin lands, there is always a slight feeling of elation or of disappointment. It’s the revelation of that emotion that makes my decision for me, not the direct result of the coin toss. I hasten to add that I’m not usually indecisive. No, wait, actually that’s not always true. Well, sometimes it is…
This is called a “mixed-strategy” by game theorists, and are “peculiar because they require the actor to incorporate randomness into decision making. Should I go to the pub or the cinema? A mixed strategy requires me to toss a coin, which sounds odd, since one might expect that I prefer one to the other.” *
The research showed that the optimal mixed-strategy (after taking into account the probability of the goalkeeper guessing right) is for a penalty taker to hit 61.5 percent of his kicks to his natural side, and 38.5 percent to the other. In reality, the penalty takers in the study (of 1,417 penalties) got remarkably close to this, hitting 60 percent to the natural side, versus 40% to the other. (No mention of how penalties straight down the middle were counted).
All this shows the extraordinary amount of subconscious thought that goes into playing top-level soccer. Previous studies in game theory have shown that people could construct random sequences if the problem was first explained to them in some detail. Nobody is suggesting that footballers have sat at home reading up on mixed-strategy equilibria. Rather, the best players intuitively grasp the truth of the theory and are able to execute it.Apparently Franck Ribery would not know which way he was going to shoot even after he had started on his run-up!
* I’ve tried tossing a coin myself to help make decisions. It doesn’t work, or rather, it does, but not directly.
If I am ever torn between two choices, and elect to decide the issue on a coin toss, as soon as the coin lands, there is always a slight feeling of elation or of disappointment. It’s the revelation of that emotion that makes my decision for me, not the direct result of the coin toss. I hasten to add that I’m not usually indecisive. No, wait, actually that’s not always true. Well, sometimes it is…
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