For several years, a professor at Johns Hopkins University used the following grading scheme for his final exam: He would give an \(A\) to the student with the highest score. The grades of the remaining students were then based on what percentage their scores were of the top student's score. But at the end of one semester, the students in his class decided to boycott the final exam. They stood in the hallway outside the classroom but did not enter the room to take the exam. After waiting for a time, the professor cancelled the exam and, applying his grading scale, gave everyone in the class an \(\mathrm{A}\) on the exam. An article in the New York Times about this incident observes: "This is an amazing game theory outcome, and not one that economists would likely predict." Do you agree with this observation that game theory indicates the students' strategy was unlikely to work? Briefly explain.

Imagine an intriguing scenario: two criminals are arrested and held in separate cells, unable to communicate with one another. The prosecutor offers each a deal – if one testifies against the other (defects), they'll go free while the other receives a heavy sentence. However, if both stay silent (cooperate), they'll get lighter sentences due to lack of evidence. If both defect, both get a moderate sentence. This setup is the classic 'prisoner's dilemma', a staple in game theory used to analyze strategic interactions where the outcome of one participant's choice is contingent on the concurrent choice of the other.

In the Johns Hopkins final exam story, the students faced a similar predicament. If they all boycotted the exam in solidarity (cooperated), they all received an A. But, like the prisoners, if one student took the exam (defected), they could snag the top grade, while the others could end up with a lower grade based on the percentage system. The dilemma rests on trust and the risk of betrayal, reflecting real-life strategic decisions and cooperation challenges.

In the realm of game theory, 'Nash Equilibrium', named after mathematician John Nash, is a pivotal concept. It's reached within a game when players select strategies where no one can benefit from changing their own strategy while others keep theirs constant. It's like reaching a standstill where everyone's choices lead to a situation where no one wins or loses by unilaterally switching tactics. The equilibrium is a state of optimal decision-making given the strategies of other players.

Applying this notion to the university exam case, if every student expects others to boycott, each has no incentive to diverge from the collective strategy and take the exam. If any student considered breaking the boycott, they'd risk getting a lower grade should others stick to the plan. Hence, the boycott itself represents a Nash Equilibrium, provided that all students anticipate the others will maintain their commitment to the boycott.

Decisions in game theory often involve weighing risk versus reward. In 'risk analysis in game theory', strategic thinkers evaluate the potential gains or losses from various actions under uncertainty. Players consider not only their own possible actions but also those of their opponents, developing a forecast of outcomes and analyzing the risks involved in each choice. They must predict others' behaviors, and strategies can be quite complex, blending psychology, probability, and logic.

In the context of the exam boycott, risk analysis played a key role. Each student had to consider the risk of someone breaking the pact against the potential reward of everyone getting an A. They also had to trust their peers - a miscalculation or betrayal by a single participant could undermine the collective strategy. The students’ successful maneuver demonstrated acute risk assessment and a deep understanding of the mutual benefits of sticking together.

When individuals or groups work together to achieve a common goal, they employ 'collaborative strategies in game theory'. These strategies involve a mutual agreement to cooperate in order to improve the outcomes for all involved parties. Such collaboration often requires communication, a shared understanding of the common goal, and trust among participants. It advances the notion that the collective benefit can be greater than the sum of individual gains when acting alone.

The students at Johns Hopkins harnessed a collaborative strategy by forming a pact to boycott the exam. Their unity was essential: if even one student had decided to take the exam, it could have disrupted the agreement and led to lower grades for others. The success of their collective action demonstrates the power of cooperation and collaborative strategy in situations where individual incentives might otherwise lead to less optimal outcomes for the group as a whole.