Penalty Kicks—A Simultaneous Move Zero-Sum Game
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Initial Publication Date: February 2, 2015
Summary
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.
CollapseLearning Goals
At the end of this experiment students should have expanded their understanding of the complex nature of simultaneous move games. They should be able to explain the extra layers of complexity that arise when neither they nor their opponent have a dominant strategy. They should also be able to explain the strategic weakness encountered by predictability and the importance of randomized strategies. Students should be able to apply this knowledge across a wide variety of game theory models in the classroom and in their own "real-world" experiences.
Context for Use
This experiment can be conducted in any class that includes a module on Game Theory. In a Principles of Microeconomics course the experiment can be used to illustrate the complexity of real world games and to pique their interest in the subject matter. In an Intermediate Microeconomics or Game Theory class the experiment can be used before or during a unit on simultaneous games of imperfect information. Prior to playing this game in a Game Theory course students should be familiar with the importance of reputation effects, the concepts of pure and mixed strategies, and the potential flaws of predictable behavior.
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.
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
There are two files needed for this experiment and one recommended file. The first is a handout that explains the rules of the game, the five strategies available to players, the potential payoffs received by the players, the role of the scout, and the mechanics of taking a penalty kick. The sheet specifies that there will be 15 rounds of play but this number can be altered. I often have students go through 30 rounds.
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.
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
In the classroom it can often be difficult to have students reveal their strategies simultaneously. One possible method is to make 2 sets of strategy cards, numbered 1-5 for each group (1 for the kicker and 1 for the goalie). In each round the student should select their strategy and lay it face down on their desk (away from their other cards). After both players have selected their strategy they turn their cards over, revealing their choices and the outcome of the game.
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).
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).
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Assessment
There are a number of assessment tools that can be used to determine how effectively students grasped the concept(s) from the experiment. Students can be asked to share and analyze their experience during the game and/or apply the concepts to recent experiences in their life. Alternatively, a short summary paper or test question asking students to explain how their classroom interactions aligned (or not) with the theory could be used.