Monopoly with No Barriers to Entry

We will now describe a game on so-called extensive form, where the question is whether a monopolist can uphold her monopoly if there are no barriers to entry. In a game on extensive form there is, in contrast to games on normal form, an order to the choices. One could say that we have added a time dimension. There are two firms, The Incumbent (J) and the Entrant (E). J has, at the beginning, a monopoly in the market and E has to choose whether to enter the market or not. If she decides to enter it, J can choose to start a price war, i.e. lower the price to punish E, or to accept the competitor. The problem with a price war is that it also hurts J herself.

• The players; The Incumbent (J) and the Entrant (E).
• Actions; For E: choose "enter" or "not enter"; for J: choose "pricewar" or "accept.”
• Information; E knows what the game structure looks like, but not howJ will decide later on. J, on the contrary, knows how E has chosen when it is her time to choose. J consequently has more information than E.
• Strategies. For E there are two strategies:

1. Choose "enter.”
2. Choose "not enter.”
For J, there are also two strategies:
1. Choose "price war.”
2. Choose "accept.”

• Payoffs. Here we need to know the players’ preferences. Assume these are as in Figure 13.2.

This type of game is usually represented with a so-called game tree. The present game will look like in Figure 13.2. In the game tree, we have indicated where E and J decide, and what they can decide between at that point. At the far bottom, there are two rows of numbers. The number in the first row indicates the first player’s (E’s) payoff and the number in the second row the second player’s (J’s). The game tree is read from top to bottom. It begins with E choosing between “not enter” and “enter.” If the she chooses “not enter,” the game ends and E gets 50 while J gets 100. If E, instead, chooses “enter,” J gets to choose between “price war” and “accept.” If she chooses “price war,” the game ends and E gets 25 and J gets 50. Compared to the case when E chooses not to enter, both E and J get a lower payoff. If J, instead, chooses “accept,” the game ends with E and J sharing the market and both getting a payoff of 75.
It is clear that J prefers that E does not enter the market (which gives J 100) to accepting (75), and both of these to a price war (50). E prefers to be accepted (75) to not entering (50), and both of these to a price war (25).

Finding the Nash Equilibrium for a Game Tree

To find the Nash Equilibrium for a game tree, we compare all different combinations of strategies. E has two strategies and J has two. It is then possible to “translate” the game tree to a game on matrix form, as in Figure 13.3. Each strategy is translated into a row or a column. It now looks similar to the game from Section 13.2, but with other payoffs and strategies.

Note that in the case when E chooses “not enter”, it does not matter what J chooses. Looking at the game tree in Figure 13.2, this is obvious since the game ends after such a choice and J never gets to choose. In the matrix, this translates into identical payoffs in all columns of the corresponding row, i.e. (50, 100). To find the Nash equilibrium, we use the same method as in the last section and check each square separately. E chooses in the vertical direction and J in the horizontal. In Figure 13.3, we have inserted arrows from squares that have a better alternative to that alternative. Squares that have no arrows going out will then be Nash equilibria. In this case, there are two Nash equilibria:

• E chooses "enter" and J chooses "accept.” If E unilaterally changes her strategy, she will diminish her payoff from 75 to 50, and if J does so, she will diminish her payoff from 75 to 50.
• E chooses “not enter” and J chooses “price war.” If E changes her strategy, she will diminish her payoff from 50 to 25, and if J does so, she will get the same as before, i.e. 100. The latter is due to the fact that it does not matter what J chooses when E has chosen “not enter.” There is something odd with the latter Nash equilibrium. E chooses not to enter since J implicitly threatens with a price war. However, if E had established herself, J would have lost utility by actually starting a price war. According to the definition, this is a Nash equilibrium, but this objection leads us to introduce an alternative method of solving games on extensive form.
  

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