Game Theory Answers

Question 1

This game has 3 Nash equilibria: two "pure strategy" equilibria in which one player veers and the other does not, and one "mixed strategy" equilibrium in which both players randomize between the actions with certain probabilities. The mixed strategy equilibrium should not concern you at this point.

Question 2

 This game is very much like the Prisioner's Dilemma game. It has only one Nash equilibrium, which is when the businessman doesn't bribe the politician, and the politician doesn't give the businessman the monopoly (don't bribe, don't give).

In reality, of course, this kind of political transaction occurs all the time. This is not necessarily because this game is a "wrong" representation of the interaction between businessmen and politicians: rather, it is probably because in real life the two players will interact repeatedly, so they can make future play depend on present actions, rewarding present cooperation with future cooperation, and punishing non-cooperation today with firther non-cooperation in the future (this would be a Nash equilibrium of the repeated game). The study of repeated games, however, is too messy for a class of this level.

Thus, this simple game is a good representation of the interaction between politician and businessman when they are only going to interact this once.

Question 3

1. Everything except nuking each other is a Nash equilibrium in this case.
2. The only Nash equilibrium is <don't nuke, don't nuke>.
3. There are now three Nash equilibria. One involves randomization, so we ignore it at this level. In both the others, one country nukes the other one, and the victim does not retaliate: <nuke, don't nuke> or <don't nuke, nuke>.

Notice that result (c) holds regardless of how small p may be: it is little surprise that the 60s, 70s and 80s were a jittery time. The problem was that some military commanders involved in decision making about nuclear issues might actually have a positive p, because participation in a nuclear holocaust would make them among the most famous and influential people in history (if there were any history left, of course.) In order to avoid this instability, both countries actually had mechanisms so that, in the event of a nuclear attack, retaliation was immediate and automatic. One way to interpret this is that they essentially eliminated the two boxes that are Nash equilibria in (c).

Incidentally, the US military used to fund rather a lot of research in game theory…

Question 4

1. The profit of the monopoly is revenue minus total cost, with the quantity sold at any particular price being determined by demand. Hence,

 

profit = p*q - cost
= (100-q)*q-2*q-10


If the monopoly is maximizing profits, it is choosing the level of q that maximizes this. So, we take the derivative and set it to zero:


0 = 98-2q^M
q^M = 49

Call this level of output q^M so that we can talk about it later. The profit at this level is

(100-49)*49-2*49-10 = $2391m

2. If Boeing is producing q^B, then Airbus faces a residual demand curve of the following form:


P = 100 - q^B - q^A

The profit of Airbus at any particular levels of q^B and q^A is:


Profit = (100-q^B-q^A)*q^A-2*q^A-10

 

Given q^B, Airbus should choose q^A* to maximize profits. Again, we derive this and set the detivative to zero:

0 = 98 -q^B -2*q^A*
q^A* = (98 -q^B)/2

Notice that, when I took the derivative, I treated q^B as a constant. This is what you do when confronting a multivariable calculus problem: If I have a function f(x,y) and I want its derivative with respect to x, I just pretend y is a constant and derive as usual.
The problem for Boeing is symmetric, and so is their best response function:

q^B* = (98 -q^A)/2

3. A Nash equilibrium of this game is any point where Airbus is best-responding to what Boeing is doing, and Boeing is best-responding to what Airbus is doing. In other words, we are looking for any two levels of q^A* and q^B* that are consistent with each other:

q^A* = (98 -q^B*)/2
q^B* = (98 -q^A*)/2

Solving these simultaneous equations, we find that there is only one answer:

q^A* = q^B* = 98/3 = 32 2/3

Ignore the fact that we usually think of planes as being indivisible: if our time frame is 3 years instead of one, then this poses no difficulty.
The equilibrium profits of each firm is:

Profit = (100-32 2/3-32 2/3)*32 2/3-2*32 2/3-10
= (34 2/3)*32 2/3 -65 1/3 -10

= $1057 1/9 million.

4. If they cooperate, each firm is producing half the monopoly output which is 24.5 planes, and they each get half the monopoly profits which is 1195.5. If one firm defects, it takes as given that the other firm is producing 24.5. If Airbus defects and produces q^A, it gets:

The defector's best strategy is to maximize this:

The defector's maximum profit is then:

The same applies to Boeing. If Boeing thinks that Airbus will cooperate, Boeing can cheat: if it cheats, its best deviation is to produce q^D = 36.75.

If one firm is cooperating and the other deviates, the sucker gets the following profits:

(e) Using the above information, we can fill in the table for the collusion game:

It should take you 2 minutes to show that there is only one equilibrium to this game, in which nobody cooperates and both firms produce the oligopoly output.

The point of this exercise is that, when setting up a simple game such as those you have seen in class, there is usually a more complete and sensible model behind it that yields the kinds of payoffs you find in the matrix. In class we are stuck with just the boxes, because the general games behind require more high-powered mathematical methods to solve…as you can see.