# Game Theory—Third Week

This week, I continued making progress. The professor introduced mathematical language into the course, which slightly confused me. I am used to crunching numbers in my calculus class; however, using mathematical language to describe an abstract definition seemed unintuitive to me. Later on, I overcame this problem by simply copying down the definition while actively reflecting upon it. This way allowed me to grasp the mathematical idea behind the definition and get used to writing definitions in a mathematical way.

In the lecture, the professor introduced me to the most important concept in game theory—Nash equilibrium. Instead of simply explaining the term (everyone who has a economics degree can explain it), the professor used a game and analyzed a Nash equilibrium using a graphical approach. In the first few lessons, I learned about how people tend to make decisions based on the notion of the dominant strategy. A Nash equilibrium is basically an extension of the dominant strategy.

When two players play strategies in a Nash equilibrium, neither of them has any incentive to deviate from their played strategies. In other words, each player is playing the best response in accordance with the other’s strategy at a Nash equilibrium.

The Nash equilibrium is an interesting idea because it can be seen everywhere. We can think of traffic lights and drivers as in a Nash equilibrium. If a car encounters a red light and another car encounters a green light at an intersection, drivers of the cars are best off by following traffic rules. If any of them deviate from their “regular strategies,” for example, if a car kept going during the red light, the two cars would crash. This situation leaves both drivers worse off. If a car decided to stop at green light, neither car would go. This situation also leaves those drivers worse off. As a result, neither driver has an incentive to deviate from obeying traffic rules. In other words, they are at a Nash equilibrium.

In reality, if we give this notion enough thought, most rules nowadays exemplified the concept of Nash equilibrium. Therefore, when we are making a rule, we have to think about the concept of Nash equilibrium in order to determine whether people have any incentive to disobey that rule.

The picture above shows a list of dorm rules at Westtown School. We can think of students in the school as being in a Nash equilibrium, since students are best off by not breaking rules. If a student, however, breaks a rule and receives three late night violations, he/she has to go to a small discipline group while others who did not break rules will not. Therefore, he/she is worse off than other students since they have to go to a small discipline group, which will definitely take up a student’s free time and put him/her in a bad mood. By enacting punishments, Westtown School gives all students an incentive to not “deviate from” the rules. In other words, they are in a Nash equilibrium.

In case you are interested in knowing the derivation of Nash equilibrium from a mathematical perspective, here is an example:

Partnership Game:

-Two people jointly own a firm, and they share 50% of the firm’s profit.

-Each person chooses the effort level they put into the firm between 1-4. Or using mathematical symbol, S=[1,4] .

-The firm’s profit is given by 4*(S1+S2+b*S1*S2), 0<b<0.25. S1 and S2 describe the profit earned through their individual efforts. b*S1*S2 shows the profit they gained as a result of their synergy. The b in the equation indicates the degree of synergy.

-When the idea above is written in a function notation, the payoff equation of the first person is given by U1=(1/2)*[4*(S1+S2+b*S1*S2)]-S1^2. This equation shows that each person reaps 1/2 of the profit. In the same time, the work that each person does costs his/her individual efforts S^2. By the same reasoning, the payoff for the second person is given by U2=(1/2)*[4*(S1+S2+b*S1*S2)]-S2^2.

Then, I can derive their payoff maximizing equations by taking the derivative and setting them to zero. I will get two equations that are the functions of their profit maximizing strategies:

S1=1+b*S2

S2=1+b*S1

In order to see this game in a more intuitive way, the professor drew these two equations on a graph when b=1/4:

In picture 1, we delete the dominated strategies for the first person, because he/she will never play those strategies.

In picture 2, we delete the dominated strategies for the second person, because he/she will never play those strategies.

In picture 3, we combine the information in picture 1 and picture 2.

In picture 4, we zoom in at the strategies that are possible to play.

This process can be repeated infinitely, and the result will eventually converge at an intersection point. That point is a Nash Equilibrium, as both of the players will be playing their best responses.

Work Cited:

“Traffic Light Eating.” myNutratek Blog. The myNutratek Team, n.d. Web. 22 Sept.
2014. <http://mynutratek.com/blog/tag/traffic-light-eating&gt;.

## One thought on “Game Theory—Third Week”

1. margaretjhaviland

What happens to the partnership, if one partner is a workaholic determined and the other partner is a smoke and mirrors slacker (all sizzle no pop). What happens to the profits if one partner decides to pursue the dominant strategy?