The following situation is played out between two students at a school. They are both working together on a school project. They must decide how much effort they both put into a group project. If the student does low effort and the other student does high effort, then they will get a good grade for little work. The table below shows what the persons happiness would be if they give a specific action by themselves and the other player.
For example If john gives low effort and jill gives high effort, john will get 8 happiness since he does no work but gets a good grade and jess will get low happiness since she needs to do more work for johns lack of work.
What kind of game is this? How does this differ from normal markets?
What is the players jesses dominant strategy? What is the games Nash equilibrium?
Suppose the teacher changes how grades are given out, now if one person gives low effort, they will be penalized. So, the game becomes this:
How does this change jesses dominate strategy? How does this change the Nash equilibrium?
Full Answer Section
Type of Game:
- Non-cooperative Game: This term indicates that the students are not forming a binding agreement to cooperate. Each focuses on maximizing their own individual happiness, leading to potential conflict of interest.
- Simultaneous Move Game: The students choose their effort levels simultaneously, unaware of each other's decisions until the grades are revealed. This adds an element of uncertainty and surprise.
- Two-Player, Finite Matrix Game: The game involves only two players (John and Jill) and has a finite set of actions and outcomes represented by the payoff matrix.
Normal Markets vs. Grade Game:
While sharing some similarities with economic markets, the Grade Game has distinct characteristics:
- Externalities: John's low effort negatively impacts Jill's happiness (externality), unlike normal markets where transactions typically affect only the involved parties.
- No Bargaining: Unlike markets where negotiation and compromise can occur, the Grade Game involves only individual choices with no direct interaction.
- Incomplete Information: Students are unaware of each other's chosen effort level until the grades are revealed, unlike market participants who usually have access to relevant information.
Dominant Strategies and Nash Equilibrium:
- Dominant Strategy for Jill:
- In the original payoff matrix, Jill's dominant strategy is High Effort. Regardless of John's choice, High Effort always yields her a higher happiness (7 vs. 5 or 4 vs. 1).
- However, in the revised matrix with teacher penalty, Low Effort becomes potentially dominant for Jill if the penalty for John exceeds the difference in John's high and low effort grades (e.g., penalty > 6).
- Nash Equilibrium:
- The Nash equilibrium represents a combination of choices where neither player has an incentive to deviate from their chosen strategy given the other player's choice.
- In the original matrix, the Nash equilibrium is (High Effort, High Effort). Both players choosing High Effort maximizes their individual happiness.
- In the revised matrix, depending on the penalty size, two equilibria can emerge:
- (High Effort, High Effort): If the penalty is insufficient to outweigh the happiness gained from minimal effort, the original equilibrium remains.
- (Low Effort, Low Effort): If the penalty makes John's Low Effort less rewarding than Jill's High Effort, both choosing Low Effort becomes the new equilibrium.
Further Analysis:
The Grade Game offers a simplified model of real-world cooperation dilemmas. Exploring this example can be expanded upon by:
- Introducing more players: Analyze the game with additional students, exploring coalition formation and potential for exploitation.
- Changing the payoff structure: Modify the happiness values to represent different preferences and risk tolerance.
- Incorporating communication: Model a scenario where students can briefly discuss their intentions before choosing their effort levels.
- Real-world application: Discuss how principles from the Grade Game can be applied to other situations involving cooperation and competition, such as teamwork in organizations or international treaties.
By analyzing the dynamics of the Grade Game, we gain valuable insights into human behavior in strategic situations. Understanding these dynamics can help us promote cooperation, manage conflict, and make informed decisions in collaborative environments.
Sample Answer
The situation you described, where two students decide their effort levels on a school project, can be modeled as a strategic game. These games analyze decision-making scenarios where the outcome depends not only on your own choice but also on the choices of others. Let's delve into this particular game, exploring its dynamics and comparing it to "normal markets".