Summary of Beautiful Game Theory by Ignacio Palacios-Huerta

BookSummaryClub Blog Summary of Beautiful Game Theory by Ignacio Palacios-Huerta

Economics and soccer is not something that you would not usually think about in the same train of thought. Soccer seems to be more of a matter of skill, maybe a bit of statistics when you try to predict the outcome of a match. Surprisingly though, the same questions about the chances of someone scoring or the odds of the goalkeeper stopping a goal can be related to economics. How? Well, all you have to do is take a closer look at soccer data.

Economic thinking comes into play more often than you would expect with even abstract economic theories being proven by the sport. 

In this book summary readers will discover:

  • The minimax theory
  • What penalty kicks have to do with economics and psychology
  • Decision making
  • How hooliganism affects statistics

Key lesson one: The minimax theory

John von Neumann’s minimax theorem is a good place to start. This game theorem is based on two-player games in which a positive payoff for one player always means a negative payoff for the other player. These two-player games are called zero-sum games and a good example of the game is Rock Paper Scissors. The theorem assumes that whilst playing, players will aim to minimize their opponent’s maximum payoff whilst also trying to minimize their own maximum loss. This basically means that whilst playing a game of Rock Paper Scissors each player is trying to minimize the wins of the other. 

In order to achieve this, there are two strategies that players can use. Pure or mixed. Pure strategies refer to those strategies where players play the same move and mixed strategies see them vary their moves. According to the minimax theorem, if it is a disadvantage for your opponent to know what you choose in advance, then you will benefit from choosing random strategies. To explain this as an example using Rock Paper Scissors, if you are playing an opponent who you know usually plays Rock often, then by using the pure strategy you could always play Paper. If, however, one player is playing pure and the other mixed, the mixed strategy would win two out of three times. And if both players use a mixed strategy there’s a fifty per cent chance of winning. Hence, a mixed strategy is better with a 50 per cent chance of winning as compared to a pure strategy with a 33 per cent chance. 

Penalty kicks in soccer verify the minimax theorem. Just like a game of Rock Paper Scissors, a penalty has a limited number of strategies. These strategies are also chosen without knowing the other person’s choice. Just think about it. A penalty kick involves a goalkeeper against a striker. They each have a strategy with three options – left, right or centre. The positive payoff for either player is a negative payoff for the other. The goalkeeper cannot react to the striker’s strategy, therefore his strategy decision has to be made before the kick is made. This makes their strategy decision independent of one another. Given the limited number of strategies they have, their decisions can therefore be evaluated and used to verify the minimax theorem. 

The whole penalty kick situation verifies the theorem by proving that mixed strategies by both players are the best option. This is further seen in soccer because no striker employs a pure strategy when playing. Striker’s actually create serially independent sequences. This basically means that they will neither kick in the same direction nor always change direction – their choices are random. This is also in line with the minimax theorem as the choice of strategy is not influenced by previous decisions. After analyzing 9000 penalty kicks, it was found that the average probability of scoring is approximately 80 per cent. 

Key lesson two: What penalty kicks have to do with economics and psychology

When it comes to penalty kicks, the decisions made are something that economists and psychologists can study. To better understand decisions, both experts tend to evaluate the environment in which the decision is made. Penalty kicks actually fall into the category of tournament settings as it involves competition. This category is also used by social scientists to analyze other competitive situations like recruitment or job promotions. 

In terms of sports tournaments though, economists and psychologists have found that they cannot be compared to real-life situations because it’s nearly impossible to document the strategies behind real-life situations. You won’t be able to distinguish if a candidate got the job because they had a great interview or if they had a higher incentive to achieve. Whereas something like a penalty kick has clear strategies that can be analyzed. This makes it easy for psychologists to study the psychological influence in tournament settings. There are just the psychological factors and the result. The randomness of the kicks, due to the coin toss to determine who kicks first, also makes the data more reliable. Interestingly enough, 97 per cent of the coin toss wins ends with the decision to kick first. 

This decision to kick first has been shown to affect an athlete’s performance. Looking at over four decades of penalty shoot outs, the team to kick first won 60.6 per cent as compared to the 39.4 per cent for the second. This is because if you choose to go first, you put a huge amount of pressure on the person who goes next. In fact, all soccer players know this. There have only been two exceptions in all documented coin tosses where the winner of the toss chose to go second. The simple choice to go first puts the second team at a psychological disadvantage. This has been further proven by studies that have randomized the way in which kicks are taken during a penalty shootout.  Considering team A and team B, instead of an ABABABAB order, they asked La Liga professionals to take kicks in an ABBABAAB order. This new order almost completely eliminated the disadvantage by changing the win percentage to 51 per cent and 49 per cent split for team A and team B respectively.

The ABBABAAB sequence is actually referred to as the Prouhet-Thue-Morse or PTM sequence makes competition between two players as fair as possible. The simple ABABABAB sequence is referred to as strict alternation. Strict Alternation clearly gives a psychological advantage to team A which has a negative effect on team B.

Key lesson three: Decision making

As stated before, the way in which decisions are made is important for economists to analyze. A soccer game provides unique insight into the effects of social pressure on decision making. The best way to explain this is to consider a referee during a game. In soccer, a game can have up to 100 000 live spectators depending on which country you are in. The supporters of the home team can apply so much social pressure to the referee that biases may arise. 

Looking at Spanish soccer games, the data has shown that the greater the reward for the match, an equal increase in the referee’s bias towards the home team also exists. Another example is injury time. In the second half of a game, the statistical mean of injury time is 2.93 minutes and it tends not to change by much if a team is in the lead or if the game is tied. However, if the home team is ahead by a goal, then extra time at the end of the game is 29 per cent below average. If the home team is behind by a goal, then extra time is about 35 per cent above average. This just goes to show how social pressures can unconsciously influence a referee’s decisions. 

Key lesson four: How hooliganism affects statistics

Becker and Rubinstein put together an economic model in 2013 to describe the effect fear has on one’s behaviour. The theorem states that with the right incentives, people will control their fears. This means that their fear is dependent on the economic cost or benefit that would occur from the control of the fear. This can be proven when considering violence and hooliganism in soccer.

Only 51 per cent of people who were married chose to renew their tickets to a match after an act of hooliganism. Their attachment to their marriages outweighed their interest in soccer. In contrast, the results of singles were the opposite because they were not invested in a marriage. The attendance of people who had purchased a single ticket also dropped by approximately 40 per cent. People who only buy one ticket tend not to go to soccer matches regularly and will not benefit from overcoming their fear. On the other hand, people who had single ticket season tickets were 94 per cent more likely to renew their season ticket. This data confirms the Becker Rubinstein model that controlling our fears is dependent on the incentive. 

The Becker Rubinstein model, however, is made up of two hypotheses. The second hypothesis states that people who are better educated can assess objective risk more accurately. The season ticket holders were split into groups of high and low education to see what their attendance would be like after an act of hooliganism. The attendance of the low education group actually dropped more than that of the high education group showing that the theory that higher educated groups are better at assessing objective risk. 

The key takeaway from Beautiful Game Theory is:

Surprisingly, soccer can give us great insight into economic behaviour. Soccer provides important empirical data which can be used to prove economic theorems. From penalty kicks, shoot-outs and simple game attendance, the evidence of association with economic theory cannot be denied. 

How can I implement the lessons learned in Beautiful Game Theory:

This newfound knowledge will definitely change your perspective about the beautiful game. Try implementing von Neumann’s theorem when watching a soccer game and see if you can accurately predict the outcomes. Might be great for your fantasy football team!

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