Nash Equilibrium Pricing Games


The concept of a Nash equilibrium can also be applied to simultaneous games where players have many choices. Consider a game of price competition where two firms sell widgets.

The parameters for such a game are these:

  • Two firms can produce widgets of identical quality for $1 each.

  • Both firms choose what price to sell their widget for.

  • The customers will buy widgets from whomever charges less.

If the firms charge the same price, one-half of the customers will go to each firm.

If Firm One charges $100, and Firm Two charges $200, then all the customers would go to Firm One. Consequently, it’s not a Nash equilibrium for one firm to charge more than its rival does. The firm charging the higher amount would get no customers and so wouldn’t be happy with its strategy. If Firm One charges $100, what is Firm Two’s optimal response?

  • If Firm Two charges more than $100, it gets zero customers.

  • If Firm Two charges exactly $100, it splits the customers with Firm One.

  • If Firm Two charges less than $100, it gets all the customers.

Obviously, if Firm One sets a price of $100, Firm Two should charge $99.99. By undercutting its rival, Firm Two gets all the customers for itself. To have a Nash equilibrium, however, both players have to be happy with their choice given their opponent’s strategy. If Firm One charges $100, and Firm Two, $99.99, then Firm One won’t be happy, so we still don’t have a Nash equilibrium. Once Firm Two undercuts Firm One, Firm One would then want to undercut Firm Two. This process will continue until both firms charge $1 a widget and make zero profit.

If Firm One charges $1, then Firm Two can’t possibly do better than charge $1 itself. If it charges more than $1, then it will get no customers. If it charges less than $1, it will lose money on every sale. Given that Firm Two sets a price of $1, Firm One is destined always to make zero profit and will be as happy as it can be charging $1. Similarly, if Firm Two charges $1, Firm One can’t do any better than charge $1 itself. Thus, both firms setting their price equal to their cost is the only Nash equilibrium.

The consumers were willing to pay any amount, yet the two producers still sold their product at cost. If both firms charged $1,000, then the consumers in this example would pay it, and the firms would make a large profit. The logic of Nash equilibrium, however, dooms both of these firms to make zero profit.

In our pricing game both firms greatly benefit from undercutting each other. If I sell my product for a penny less than you do, then I get all the customers. Because the benefit to undercutting an opponent is so large, both firms continue to do it until they are each selling the good at cost. This game provides another example of the damage that price competition can do to firms; damage that’s magnified when the firms have high sunk costs.

Let’s add to our example by assuming that to start making widgets a firm needs to spend $50,000 to build a factory. After building the factory, it will still cost $1 extra to manufacture each widget. What price will the firms charge? If both firms still end up charging $1, they will each lose $50,000. At a price of $1 the firms would break even on every widget sold, but would still have to pay the cost of the factory. Each firm charging only $1, however, is still the only Nash equilibrium. To see this, consider, is each firm charging $2 a Nash equilibrium? If they both charge the same amount, they must split the customers. Thus, if each firm is charging $2, one firm could acquire all of the customers by cutting its price to $1.99. Would a firm want all the customers if it could get only $1.99 per widget? Yes. Every additional customer you serve costs you $1. If you can sell a widget for $1.99 to a new customer you are better off by $.99. What about the $50,000 you spent building the factory? This is a strategically irrelevant sunk cost. Once you have built the factory you can’t get back the $50,000 regardless of what you do. You should ignore this sunk investment and instead worry about your future gains and losses. Hence, if the other firm charges $2.00, you would want to charge $1.99 because this way you could acquire all the customers. Because of the $50,000 spent building the factory you may lose money if you charge only $1.99. You would lose less money, however, getting all the customers and charging $1.99 than you would by charging $2 and serving only one-half of the customers.

Industries with high sunk costs are extremely vulnerable to price competition because it is rational for companies to ignore their sunk costs when setting prices. As in the previous example, competition in the presence of high sunk costs can easily drive prices to a point where everyone loses money.

The airline industry suffers from high sunk costs because of the high cost of planes. Once you have an airplane and have decided to fly, it costs relatively little to add extra passengers.

Imagine that your airline always has one flight daily from New York to Paris. Assume that if this plane were always full, you would need to charge $400 a passenger to break even. What if, however, your flights were only one-half full, but you could sell additional seats for $300 each? Should you fill the extra spaces? Yes. Since you are already going to have the flight, you’re better off getting $300 for a seat than leaving it empty. Of course, since everyone in the industry will feel this same way, the market price could easily be driven below $400. Nothing in game theory, however, guarantees that firms will make a profit or even survive.

The airline companies might go beyond the logic of Nash equilibrium, for in a Nash equilibrium the players never cooperate. The airlines might grasp how destructive competition could be. Should these companies formally agree to restrict competition, then they would be in violation of antitrust laws. Each firm, however, could decide not to reduce its price in hopes that other firms will follow course. The next chapter extensively considers the stability of firms’ charging high prices.




Game Theory at Work(c) How to Use Game Theory to Outthink and Outmaneuver Your Competition
Game Theory at Work(c) How to Use Game Theory to Outthink and Outmaneuver Your Competition
ISBN: N/A
EAN: N/A
Year: 2005
Pages: 260

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