Decision Making Assignment: Solutions to the exercises in chapter one of the book of Charles Holt "Markets, Games and Strategic Behavior"

Solved by Marta Suárez-Varela

Exercise 4:  Graph the best-response lines for the prisoner’s dilemma game in Table 3.1, and indicate why there is a unique Nash equilibrium.

We can find this game in Chapter 3 of the book. In this game, both players have to choose between a high effort (1) or a low effort (0). The cost of exerting the higher effort is $10, and the benefits to each person depend on the total effort for the two individual combined. Since each person can chose an effort of 0 or 1, the total effort must be 0, 1 or 2. The benefits per person are shown in this table:

Total Effort	0	1	2
Benefit per person	$3	$10	$18

With all this information, we must first obtain the payoff table of the game:

High            Low
High                 Low	8, 8	0, 10
	10, 0	3, 3

To obtain the Nash Equilibrium of the game, we calculate the best response function for both players.

Step 1: Summarize notation

                                p= Probability that Column Player chooses High

                                q= Probability that Row Player chooses High

Best response for every action of the other player:

•  Best response for Row Player given that Column Player chooses High

   àB_R(H)=Làp=1, q=0

•    Best response for Row Player given that Column Player chooses Low

   àB_R(L)=Làp=0, q=0

•    Best response for Column Player given that Row Player chooses High

   àB_c(H)=Là q=1, p=0

• Best response for Column Player given that Row Player chooses Low

   àB_c(L)=Là q=0, p=0

As we can see, there is only one Nash Equilibrium in this game, which is the point in which both best response functions intersect.

Nash Equilibrium (L,L)

This action profile is a Unique Nash Equilibrium, because it is the only action profile in the game for which none of the players have incentives to deviate in order to get a higher payoff.

Exercise 5: Find the Nash equilibrium in mixed strategies for the coordination game in Table 3.2, and illustrate your answer with a graph.

The game considered in this exercise is the same as in the previous one, but with some variations.

A lot of production processes have the property that one person’s effort increases the productivity of another’s effort. Recall that 1 unit of effort produced a benefit of $10 per person and 2 units produced a benefit of $18. In this case, we are going to suppose that the second unit of effort makes the first one more productive in the sense that the benefit is more than doubled when a second unit of effort is added. The new benefits are represented in the following table:

Total Effort	0	1	2
Benefit per person	$3	$10	$30

The cost of exerting the higher effort remains $10, as in the original example.

With this information, we obtain the payoff table:

High            Low
High                      Low	20, 20	0, 10
	10, 0	3, 3

In this game there are two Nash Equilibria in pure strategies. One in which both exert a high effort and one in which both exert a low effort.

We are asked to calculate the mixed strategy Nash Equilibrium of the game.

As before, the first step is to determine the notation:

                                p= Probability that Column Player chooses High

                                q= Probability that Row Player chooses High 

Secondly, we must calculate the expected payoffs for each decision for each player:

To obtain the probability distribution in the equilibrium, we equate the expected payoffs. This way, we get the probability at which the players are indifferent between choosing one action or the other.

The equilibrium in randomized strategies will be reached in the point in which the Row Player chooses High with a probability of 3/13 and Column Player chooses high with a probability of 3/13.

We can confirm in the graph that there are three Nash Equilibria in this game, two in pure strategies and one in mixed strategy, which are the points in with both best response functions intersect.