Oligopoly prices and Solutions to Pricing Games

We are concerned in this chapter about oligopoly pricing and market strategy. Here is an example. It is a very simplified model of price competition. Like Augustin Cournot (writing in the 1840's) we will think of two companies that sell mineral water. Each company has a fixed cost of $5000 per period, regardless whether they sell anything or not. We will call the companies Perrier and Apollinaris, just to take two names at random.

The two companies are competing for the same market and each firm must choose a high price ($2 per bottle) or a low price ($1 per bottle). Here are the rules of the game:

1) At a price of $2, 5000 bottles can be sold for a total revenue of $10000.

2) At a price of $1, 10000 bottles can be sold for a total revenue of $10000.

3) If both companies charge the same price, they split the sales evenly between them.

4) If one company charges a higher price, the company with the lower price sells the whole amount and the company with the higher price sells nothing.

5) Payoffs are profits -- revenue minus the $5000 fixed cost.

Here is the payoff table for these two companies

Table 2

price = $1 price = $2
Apollinaris price = $1 0,0 5000,-5000
price = $2 -5000,5000 0,0

Once again, as in the Prisoners' Dilemma, each company has a strong rationale to choose one strategy -- and in this case it is a price cut. For example, Appolinaris might reason "Either Perrier will cut to $1 or it will not. If it does, then I had better cut, too -- otherwise I'll lose all my customers and lose $5000. On the other hand, if Perrier doesn't cut, I'm still better off to cut, since I'll take their customers away and get a profit of $5000." Thus, the price cut is a dominant strategy.

But this is, of course, a very simplified -- even unrealistic -- conception of price competition. Let's look at a more complicated, perhaps more realistic pricing example:

Another Price Competition Example

Following a long tradition in economics, we will think of two companies selling "widgets" at a price of one, two, or three dollars per widget. the payoffs are profits -- after allowing for costs of all kinds -- and are shown in Table 5-1. The general idea behind the example is that the company that charges a lower price will get more customers and thus, within limits, more profits than the high-price competitor. (This example follows one by Warren Nutter).

Table 3

Acme Widgets
price = $1 price = $2 price = $3
Wiley Widgets price = $1 0,0 50,-10 40,-20
price = $2 -10,50 20,20 90,10
price = $3 -20,40 10,90 50,50

Unlike the mineral-water example (and more realistically), industry profits in this example depend on the price and thus on the strategies chosen by the rivals. Profits may add up to 100, 20, 40, or zero, depending on the strategies that the two competitors choose. We can also see fairly easily that there is no dominant strategy equilibrium. Widgeon company can reason as follows: if Acme were to choose a price of 3, then Widgeon's best price is 2, but otherwise Widgeon's best price is 1 -- neither is dominant.

Nash Equilibrium

We will need another, broader concept of equilibrium if we are to do anything with this game. The concept we need is called the Nash Equilibrium, after Nobel Laureate (in economics) and mathematician John Nash. Nash, a student of Tucker's, contributed several key concepts to game theory around 1950. The Nash Equilibrium conception was one of these, and is probably the most widely used "solution concept" in game theory.

DEFINITION: Nash Equilibrium If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.

Let's apply that definition to the widget-selling game. First, for example, we can see that the strategy pair p=3 for each player (bottom right) is not a Nash-equilibrium. From that pair, each competitor can benefit by cutting price, if the other player keeps her strategy unchanged. Or consider the bottom middle -- Widgeon charges $3 but Acme charges $2. From that pair, Widgeon benefits by cutting to $1. In this way, we can eliminate any strategy pair except the upper left, at which both competitors charge $1.

We see that the Nash equilibrium in the widget-selling game is a low-price, zero-profit equilibrium. Many economists believe that result is descriptive of real, highly competitive markets -- although there is, of course, a great deal about this example that is still "unrealistic."

Let's go back and take a look at that dominant-strategy equilibrium in Table 4-2. We will see that it, too, is a Nash-Equilibrium. (Check it out). Also, look again at the dominant-strategy equilibrium in the Prisoners' Dilemma. It, too, is a Nash-Equilibrium. In fact, any dominant strategy equilibrium is also a Nash Equilibrium. The Nash equilibrium is an extension of the dominant strategy equilibrium.