Penalty Kicks—A Simultaneous Move Zero-Sum Game
and is replicated here as part of the SERC Pedagogic Service.
In this experiment, students take part in a simultaneous move, zero-sum game—the act of taking a penalty kick in the game of soccer. They each take several turns as a "player" and as a "scout" (i.e., an outside observer). The goal of a player is to accumulate as many points for their team as possible while the scout watches, trying to identify any strategy trends or patterns that may aid them (and their team) in future experiences. From their experiences students gain an understanding of the complex nature of simultaneous move games and the importance of randomized strategies.
Context for Use
The experiment can usually be completed in 15-20 minutes depending on how many rounds you choose to play and how many students are involved in the experiment. The experiment can be run in several different configurations:
- The classroom could be divided into groups of three students and all students could actively participate in the experiment. This is my recommended configuration as students learn by experience and their ability to identify predictable and repetitive behavior improves as the game is played.
- One group could take part in the experiment while other students observe from their seat. In this situation I recommend that all students not actively involved in the game act as scouts and see if they can identify a pattern in the kicker's and goal keeper's choices.
- Two or three groups could take part in the experiment while other students watch from their seat. The students not actively involved in the game should act as scouts and see if they can identify a pattern in the kicker's and goal keeper's choices.
- Two or three groups could participate in the experiment in succession. The groups who have already played should rejoin the students who have not played and should act as scouts. Having participated in the game already, they are often more aware of predictable behavior and are able to identify patterns faster than those who have not yet played.
Discussion can take anywhere from 5-10 minutes in a Principles of Microeconomics course, and as long as 15-30 minutes in an Intermediate Microeconomics or Game Theory course. The experiment can also be done over 2 classes, with the experiment completed in 1 class and the discussion being held in the next class meeting. This method will provide you the opportunity to look over the data gathered before discussing it with the students.
Description and Teaching Materials
The second is a record sheet for the scout. On this sheet the scout should note the strategy played by the kicker, the strategy played by the goalie, and the outcome of the interaction. They should use this sheet to observe any patterns that develop in either the goalie's or the kicker's actions.
The third handout is recommended and can help enhance the student's thinking about the experiment. This handout contains a few questions that ask students to predict the optimal strategies for each player before the experiment takes place. There is a follow up question that should be completed either half way through the experiment (i.e., after the first game) or at the end of the experiment which asks students whether their predictions were accurate and if not, what strategies they would recommend to future players.
Teaching Notes and Tips
This experiment provides a nice introduction to mixed strategies but requires the players and the "scout" in particular to focus on the game. It is easy for students to focus more on the fun aspects of the game than on the strategic interactions that are taking place. It is important to remind students why they are doing this experiment and what they should be looking for. Requiring students to complete and submit the recommended handout can help keep students focused and on task. The handout also helps cement the importance of randomized behavior by forcing students to formalize their observations from the experiment.
As an extension of the experiment environmental uncertainty can be added in a later round. There are several ways to do this, but the most straightforward is to add a fair coin to the game and use that to determine if the kicker misses the goal altogether and sends their kick out of bounds. If they miss the goal it is treated the same as a save by the goalie.
An alternative extension of the experiment is to change the number of strategies available to the players, which would also adjust the probability of a save occurring. The mechanics of the experiment should remain unchanged and students should observe that the likelihood of a save does not, necessarily, increase as the number of strategies available decreases.
There is normally at least one group where the "scout" has identified a pattern. If so, use that group to draw conclusions from the game and highlight what the implications are. If not, use this as motivation to explain why a pattern could not be identified (i.e., players were using mixed strategies).