Refer to the payoff matrix above. In each cell of the table, the first number is the Firm 1’s payoff, and the second number is the Firm 2’s payoff. For example, if both firm chose to enter, we are in the top left cell of the payoff matrix, and Firm 1’s payoff is 20, and Firm 2’s payoff is 40. Further assume that payoffs represent economic profits of the respective firms, i.e. in the example above Firm 1 earns profit of 20, and Firm 2 earns profit of 40.

How do we determine the dominant strategy?

First, fix the strategy of one of the firms, say Firm 2. Assume Firm 2 is going to enter. Then what would be the best for the Firm 1?

If we assume that Firm 2 is going to enter, we can eliminate the bottom row of the table and obtain the reduced payoff matrix:

Now, if the Firm 1 enters, it will receive a payoff of 20 (first number in the pair in the cell corresponding to enter, enter). If it does not enter, it will receive a payoff of 0. Since 20 > 0, Firm 1 will choose to enter if Firm 2 enters.

What happens if we assume instead that Firm 2 does not enter? Similarly to the above, this assumption would effectively eliminate the row of the original payoff table corresponding to the "enter" choice of the Firm 2:

If Firm 2 does not enter, Firm 1 is facing the payoff of 100 if it enters, and a payoff of 0 if it does not enter. Since 100 > 0 , Firm 1 chooses to enter if Firm 2 does not enter.

The strategy of Firm 1 can be summarized as follows:

If Firm 2 enters, enter

If Firm 2 does not enter, enter

Let us consider the choices Firm 2 faces. We will proceed in the same fashion as above, and consider optimal choices of Firm 2 in response to fixed strategies of Firm 1. Suppose Firm 1 enters. What would be the best choice for the Firm 2 given the "ent er" choice of Firm 1? The reduced payoff matrix would be as follows:

If Firm 2 enters, it gets a payoff of 40. If Firm 2 does not enter, it receives a payoff of 0. Since a firm would prefer a larger payoff to a smaller one, Firm 2 will choose to enter and receive a payoff of 40.

What would happen if Firm 1 instead decided not to enter? Then the relevant column would be one that contains payoffs corresponding to "not enter" choice of the Firm 1

If Firm 1 does not enter, Firm 2 will receive a payoff of 60 if it enters, and a payoff of 0 if it does not enter. Since 60 > 0, Firm 2 chooses to enter.

Firm 2 strategy can be summarize as follows:

If Firm 1 enters, enter

If Firm 1 does not enter, enter

What happens when both firms make their choices simultaneously and independently? Each of the firms will play its dominant strategy, which, in this case, is "enter" for each of the firms. The actual outcome would be the following: Both firms enter the market, Firm 1 receives a payoff of 20, and Firm 2 receives a payoff of 40.

Suppose now that only one of the firms had a dominant strategy. How would we arrive at the equilibrium? Assume Firm 1 has a dominant strategy "enter" as before, but (due to change in its payoffs) Firm 2 does not have a dominant strategy. Both firms kno w the (new) payoff matrix, and each of the firms can figure out if the other has any dominant strategies.

When Firm 2 sees that Firm 1 has a dominant strategy "enter", it understands that since Firm 1 will choose enter, the "not enter" selection on the part of Firm 2 will not happen, and considers the reduced payoff matrix consisting of the column correspo nding to the "enter" choice of Firm 1:

Firm 2 now chooses between "enter" and "not enter", and selects "enter" since 40 > 0. Now, it might have been that if Firm 1 chose "not enter", Firm 2 would have also chosen "not enter". But since "enter" is a dominant strategy for Firm 1, it will n ever choose "not enter", and Firm 2 problem is simplified to selecting the larger of two payoffs and not worrying about the fact that Firm 1 may change its strategy because of Firm 2’s actions.

If the firms would be able to form a binding agreement providing transfer payments from one firm to the other, what pair of strategies would they choose?

Let us find the size of the "profit pie" (recipe, anyone? :P ).

If both firms choose to enter, total profit is 20 + 40 = 60

If Firm 1 enters and Firm 2 does not, total profit is 100 + 0 = 100

If Firm 1 does not enter and Firm 2 enters, total profit is 0 + 60 = 60

If neither firm enters, total profit is 0 + 0 = 0

It appears that the largest pie can be had when Firm 1 enters and Firm 2 stays out. Since this pair of strategies provides us with more total profit, each firm can be made better off. For example, if the firms wanted to split the "extra" 40 in the midd le to be fair, Firm 2 may agree to stay out of the market in exchange for cash payment from Firm 1 in the amount of 60. After the transfer payment, the payoffs to this strategy in the collusive situation are: Firm 1 gets 40 and Firm 2 gets 60, which is be tter than 20 and 40 respectively they were making in the dominant strategy equilibrium.

Who might want to cheat on this agreement? Perhaps Firm 1 would decide not to pay 60 to Firm 2 after the moves have been made, and keep the entire 100 for itself. Above, we assumed a possibility of a binding contract, so the Firm 2 can sue Firm 1 in su ch a case. However, since collusion is illegal most of the times, it may well be a situation under which you can’t write a legally binding contract for such a collusive purpose (hmm, although you might want to "spin off" part of your company that would en ter into this market and sell it to your competitor, and your competitor may pay you well above what it’s worth, accounted for by "Good Will" term accountants use. Then you effectively achieve a collusive arrangement). If such binding contract was not pos sible, the firms would resort to their dominant strategies, since it is in the best interest of the other firm to cheat on the unenforceable agreement and obtain higher profits.

Now how else would the firms try to circumvent the restriction on collusion? They may try to lobby the government to allow only one of the firms to enter, to obtain a patent of some sort perhaps. In this case, how much would Firm 1 be willing to pay fo r the right to be the sole firm in the market?

We know that when both firms are allowed to be in the market and collusion is not possible, both of the firms will play their dominant strategies and enter, resulting in payoffs of 20, 40 to Firm 1 and 2, respectively. If Firm 1 is successful in legall y preventing Firm 2 from entering the market, Firm 1 will reap the payoff of 100. It is better off by 80. Therefore, it will be willing to spend up to 80 if it can pass the appropriate law (Suppose Firm 1 spends 70 and the law passes. Then it gets a payof f of 100, defrays the cost of 70 with it, and left with a residual payoff of 30, which is still larger than 20 it would have gotten had the Firm 2 been able to enter the market).

Similarly, if Firm 2 is successful in keeping Firm 1 out of the market, it will earn a payoff of 60 compared to its payoff of 40 in the dominant strategy equilibrium. 60 is bigger than 40 by 20, and therefore, Firm 2 will be willing to spend up to 20 t o keep Firm 1 out of the market.

If the government auctions off the concession (the right to be the sole operator in the market), Firm 1 will outbid Firm 2. Since the government decided that this market will be a monopoly, only 2 cells in the payoff matrix matter. These cells correspo nd to the situations when only one of the firms enters. If the dominant strategy equilibrium (enter, enter) is no longer an option, each firm would compare the payoffs it will receive if it wins the auction and the payoff of 0 if it doesn’t. So the Firm 1 will be willing to bid up to 100, and Firm 2 will be willing to bid up to 60. Firm 1 will win this auction with a bid just above 60.

Questions, comments? Email me at economist@asu.edu

Web Posted 5:45 p.m. Arizona time 04/16/99