Economics : Game-theory (Nash equilibrium)

Question

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way to solving following question.

Question: Six students are going on a foreign trip on which they will live close together. Where they are going, there is a disease which spreads easily among people who live close together. The value of the trip to a student who does not get the disease is 6. The value of the trip to a student who gets the disease is 0.

There is a vaccination against the disease. The vaccination costs different amounts for different students (perhaps they have different health plans). Let's call the students 1, 2, 3, 4, 5, and 6 respectively. The vaccination costs 1 for student 1; it costs 2 for students 2; etc...

If a student gets vaccinated, she will not get the disease. But, if she is not vaccinated then her probability of getting the disease depends on the total number in the group who are not vaccinated. If she is the only person not to get vaccinated then the probability that she gets the disease is 1/6. If there is one other person who is not vaccinated (i.e., two in all including her) then the probability that she gets disease is 2/6. If there are two other people who are not vaccinated (i.e., three including her)then the probability that she gets the disease is 3/6, etc...

For example, suppose only students 2 and 4 get vaccinated. Then 2's expected payoff is 6-[2] where the [2] is the cost of the vaccination. Student 4’s expected payoff in this case is 6-[4]. Student 5’s expected payoff in this case (recall she did not get vaccinated) is 6-[4]. Student 5’s expected payoff in this case (recall she did not get vaccinated) is (2/6)6+(4/6)0=2 where the fraction (4/6) is the probability that she gets the disease. To make this into a game , suppose that each student aims to maximize her expected payoff. The student decide, individually and simultaneously, whether or not to get a vaccination.

(a) Explain concisely whether or not it is a Nash equilibrium for students 1,2, 3 and 4 to get vaccinated and students 5 and 6 not to get vaccinated.

(b) Explain concisely whether or not it is a Nash equilibrium for students 1, 2 and 3 to get vaccinated and student 4, 5, and 6 not to get vaccinated.

(c) Which players in this game have weekly dominated strategies? Explain your answer.

Answer 1

For Part (a), examine each player's payoff for each of their two possible strategies with the other five players' strategies held fixed. So Player 1's payoff is $6-1=5$ if she chooses vaccination and $\frac{3}{6}6+\frac{3}{6}0=3$ if not. Player 2's payoff is $6-2=4$ if she chooses vaccination, and $\frac{3}{6}6+\frac{3}{6}0=3$ if not. The analysis is similar for Players 3 and 4. Player 5's payoff is $6-5=1$ if she chooses vaccination and $\frac{2}{6}6+\frac{4}{6}0=2$ if not. The analysis is similar for Player 6. We can use these numbers to start to make a table like the following, $$ \begin{array}{c|cc} \text{player} & V & U\\ \hline 1 & 5 & 3\\ 2 & 4 & 3\\ 3 & &\\ 4 & &\\ 5 & 1 & 2\\ 6 & & \end{array} $$ where strategy $V$ is vaccination and strategy $U$ is remaining unvaccinated.

You now ask whether the strategy played by each of the six players is a best response to what the other five players are doing. So for Player 1, $V$ is a best response since $5$ is a better payoff than $3$. For Player 5, $U$ is a best response since $2$ is a better payoff than $1$. The $6$-tuple of strategies $(V,V,V,V,U,U)$ is a Nash equilibrium if each of the six players' strategies is a best response with the five other players' strategies held fixed.

Part (b) can be answered by the same method.

For part (c), notice that the payoff for getting vaccinated does not depend on what the other players do. You can then determine whether $V$ is weakly dominated by examining the payoff for playing $U$ as the number of other players playing $U$ is varied. If the payoff for playing $U$ is always at least as high as the payoff for $V$, then $V$ is weakly dominated. Similarly, if the payoff for playing $V$ is always at least as high as the payoff for $U$, then $U$ is weakly dominated.

For example, Player 1 gets $5$ by playing $V$, and we already know of a situation where the payoff for playing $U$ is $3$. So $V$ is not weakly dominated for Player 1. On the other hand, $U$ is weakly dominated since the probability Player 1 remains healthy (and hence receives a payoff of $6$) varies from $\frac{5}{6}$ to $\frac{0}{6}$ depending on how many of the other players also go unvaccinated, and therefore the payoff for playing $U$ varies from $5$ to $0$.