Economist Bill Samuelson suggests a problem centering around three air carriers competing for passengers on a given city-pair route. Namely, the fare that can be charged on the route is fixed at $$\$ 225$$, while the size of the market is fixed at 2,000 passengers per day. There are three competing airlines: A, B, and C. Each airline gets passengers in proportion to its share of total flights. For example, if all three airlines offered the same number of flights, then they would each get one-third of the passengers. If Airline A offered six flights and B and C each offered three, then A would get 50 percent of the market, while $\mathrm{B}$ and $\mathrm{C}$ would get 25 percent each. Each plane holds a maximum of 300 passengers. Each plane trip costs $$\$ 20,000$$, whether the plane is full or not.
a. Confirm firm A's profit equals $$\$ 450,000[a /(a+b+c)]-$$ $$\$ 20,000 a$$, where $a, b$, and $c$ represent the number of flights by firms A, B, and C, respectively.
b. Confirm to yourself that the table below gives the profits to A as a function of its flights and its competitors' flights per day.
c. Consider a strategy for any one of the firms to be a policy of flying a certain number of flights per day. Is there a dominant strategy for A-that is, a number of flights that gives higher profits no matter what the competitors do?
d. Is there a Nash equilibrium in this game? That is, is there a set of strategies (numbers of flights $a, b$, and $c$ ) such that each airline's strategy is optimal given what the others are doing? Or, said another way, is there a set of strategies in which "unilateral defection" does not pay?
e. Are there any strategies of A's that are dominated by other strategies? That is, can you rule out one or more of A's strategies because they are always worth less than something else?
f. Follow the foregoing logic to its end: if you rule out some of A's strategies, can you also rule out some for B and C? And if you can do that, can you then go back and rule out more of A's strategies? Can you continue this process of "iterated dominance" to convince yourself of how many flights A should fly?