# Game Theory—Fourth Week

During class, the professor reviewed the notion of Nash equilibrium and noted the possible existence of multiple Nash equilibria in a game. In some games, for example, there are two Nash equilibria rather than one.

A typical example is as follows:

From the payoff matrix, we can see that when two players choose ‘up, left’ strategy or payoff ‘1, 1’ strategy, they are at a Nash equilibrium, because neither player has any incentive to deviate from his/her strategy played. If player 1 deviates from the strategy he plays (if he/she chooses down), he/she will earn a payoff of 0, which is lower than the payoff when he/she chooses up. Interestingly, when two players choose ‘down, right’ or payoff ‘0, 0’ strategy, they are also at a Nash equilibrium, for they are both playing their best responses in accordance to the other’s strategy. Even if anyone changes his/her strategy in this game, he/she will still earn a payoff of 0. As a result, there is no incentive for any player to choose another strategy.

Why are there two Nash equilibria? Let’s bring this example to a larger social context.

If I organize a party and invite my friends to come to my party, what are Nash equilibria in this situation?

The answer is either all my friends come or none of them comes. When all my friends come to my party, the party will be fun, because people can engage with each other and enjoy themselves. None of them will choose to leave the party, because no one dislikes having fun. When all my friends choose not to come to my party, however, this situation is also at a Nash equilibrium, because if one of my friends deviates from his/her strategy (if he/she chooses to come to my party) while there is no one else, he/she will be worse off because he/she will have one to play with.

This is a typical situation in our real life, especially in a school setting. If most of the students in Westtown School decide to support their basketball team, everyone will have an incentive to go because more students means more enthusiasm and fun. On the other hand, if a student realizes that no one will go to the game, no one will want to go, because he/she may be the only one to go to the game if he/she decides to go.

Sometimes, this kind of situation can get really tricky. What about if 40% of the student body wants to go while the other 60% does not? As a result, communication becomes extremely important. A good way to overcome this awkward situation is to persuade everyone to go to the game. If 80% of the student body wants to go, the remaining 20% will probably go, because the remaining 20% is worse off by not being able to experience enthusiasm and the opportunity to enjoy themselves.

Therefore, whenever a teacher is creating a weekend event, he/she probably has to consider the notion of Nash equilibrium. If there are virtually no people signing up for an event, the teacher should either cancel the event or persuade more people to sign up, because when students see no one signs up for a certain event, they tend not to sign up. However, if they see many students sign up for the event, the rest of the student body might all sign up.