SITUATION: PRICE COMPETITION WITH IDENTICAL PRODUCTS
Suppose you’re vacationing in Niagara Falls and figure that it’s time to pick up a few souvenirs for family and friends. You wander over to the string of sou- venir shops at Clifton Hill, all selling basically the same stuff: Niagara Falls pens (tilt it and the person in the barrel goes over the falls), Niagara Falls re- frigerator magnets, Niagara Falls snow globes—you name it. As you browse from store to store, you notice that they charge pretty much the same price. Coincidence? Collusion? Competition? What’s up?
To analyze this situation, suppose there are two souvenir shops offering the same products. To keep things simple, we’ll focus on just one of those prod- ucts. Each shop orders the product from the same manufacturer in China at a per-unit cost of $10. For example, if Wacky Tacky Souvenir Shop (or shop 1, for short) sells 13 units of the item, then its total cost is $130. Tasteless Trinkets (aka shop 2) faces the same cost.
Shoppers survey both stores and, since the goods are identical, buy from the one with the lowest price (although a shopper may not buy at all if prices from both stores are too high). Assume that the lower the price, the greater is the number of shoppers who will buy. The rela- tionship between price and the number of units sold is summarized in the concept of a market demand curve, which is plotted in FIGURE 6.1. If the lowest price among the two shops is then the number of units sold is
The next step is to define the demand for an individual shop. Since consumers base their decision only on price, if shop 1 has a lower price than shop 2 then all con- sumers buy only from shop 1, so shop 1 sells
units. If, however, shop 1’s price is higher than shop 2’s price then no shoppers buy from shop 1, so it doesn’t sell any units. Finally, if both shops set the same price, then total demand is assumed to be split equally between them. For example, if both charge $40, then shoppers want to buy 60 units and each shop sells 30 units. Summarizing this description, we note that shop 1’s demand curve, denoted takes the form
Shop 2’s demand curve is analogously defined.
D1(p1, p2) � •100 � p1 if p1 6 p2(12)(100 � p1) if p1 = p2 0 if p2 6 p1
.
D1(p1, p2),
(� 100 � 40)
(p2 6 p1), 100 � p1
(p1 6 p2),
100 � p. p,
FIGURE 6.1 The Market Demand Curve for Souvenirs
Pr ic
e ($
)
100
40
0 60 100
100 � p
Number of units
150 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
Let’s look more closely at shop 1’s demand curve by plotting it when shop 2 charges a price of 40. (See FIGURE 6.2.) Notice the jump in the curve at a price of 40. For a price slightly above 40, shop 1 sells nothing; for a price slightly below 40, it sells all 60 units being demanded. At the same price of 40, it gets half of market demand, or 30 units, rep- resented by the colored dot (as opposed to the open circles).
In specifying a shop’s strategy set, a strat- egy is a price, and prices are allowed to take any value in the interval [0,100]. Finally, sup- pose that each shop cares only about how much money it makes. Shop 1’s profit equals the revenue shop 1 collects—which is the price it charges times the number of units it sells, or —minus its cost, which equals 10 times the number of units it
sells, or Shop 1’s payoff (or profit) function is then
which is plotted in FIGURE 6.3 when Shop 2’s payoff function is analo- gously defined:
This game is known as the Bertrand price game, because it was developed by Joseph Bertrand in 1883.
A candidate for Nash equilibrium is any pair of prices from the interval [0,100]. Let’s see if we can start eliminating some possibili- ties. If a firm prices the product below its cost of 10 and sells any units, then its payoff is neg- ative. Such a price cannot be optimal, be- cause, by pricing at 10, a shop can always en- sure a zero payoff, regardless of the other shop’s price. Thus, if the lowest price being charged is less than 10, the shop with the low- est price is not pricing optimally. From this ar- gument, we can conclude that any strategy pair in which either of the shops price below 10 is not a Nash equilibrium.
We just managed to eliminate an infinite number of strategy pairs as candidates for a
•0 if p1 6 p2(12)(p2 � 10)(100 � p2) if p1 = p2 (p2 � 10)(100 � p2) if p2 6 p1
.
p2 � 40.
• (p1 � 10)(100 � p1) if p1 6 p2(12)(p1 � 10)(100 � p1) if p1 = p2, 0 if p2 6 p1
10 � D1(p1, p2).
p1 � D1(p1, p2)
FIGURE 6.2 Shop 1’s Demand Curve when Shop 2 Charges a Price of 40
Pr ic
e ($
)
100
40
0 60 70 80 9030 40 5010 20 100 Number of units
D1(p1, p2 � 40)
FIGURE 6.3 Shop 1’s Payoff Function when Shop 2 Charges a Price of 40
Pa yo
ff
0
0 40 50 60 70 80 9010 20 30 100 p
1
Nash equilibrium! The only problem is that we’re still left with an infinite number: all price pairs for which price is at least as great as 10. To simplify matters, from here on we’ll limit our attention to symmetric strategy pairs—strategy pairs in which both shops charge the same price.
Consider any symmetric strategy pair ex- ceeding 10. Suppose, for example, both shops price the product at Then the payoff function faced by shop i is depicted in FIGURE 6.4 when the other shop charges a price of In that case, shop i earns a positive payoff of since it sells units at a per-unit profit of
Let’s compare this with a price of where When is really small, the profit earned from each unit
sold is about the same between pricing at and at (cf., e.g., and However, with a price of the number of units sold is al-
most twice as large (cf. and As a result, the pay- off from pricing at exceeds that from pricing at when is really small:
The preceding argument identifies a powerful incentive for a shop to under- cut the price of its rival: doing so allows it to double its demand compared with matching its rival’s price, with almost no reduction in its profit per unit. But as long as one shop undercuts, the other shop is not going to be content, since it’ll then have zero sales. Thus, any symmetric price pair in which the price exceeds a cost of 10 is not a Nash equilibrium, since both firms have an incentive to charge a slightly lower price instead.
To summarize, we first argued that a symmetric price pair below 10 is not a Nash equilibrium and then argued that a symmetric price pair above 10 is not a Nash equilibrium. This leaves only one possibility: both shops price at 10. Is that a Nash equilibrium? By symmetry, we need only consider one of the shops, so let it be shop 1. With its rival pricing at 10, shop 1 sells 45 units by also pricing at 10, but its payoff is zero because it is selling at cost. It can in- stead price above 10, but then it doesn’t sell any units—since all shoppers buy from shop 2—so shop 1’s payoff is again zero. It can price below 10 and cap- ture the entire market, but now it is losing money on each unit it sells, which means that its payoff is negative. Thus, shop 1 cannot do any better than to price at cost, given that its rival does the same. We have a Nash equilibrium!
What is striking about this result is that competition is incredibly intense even though there are only two shops. A price equal to cost of 10 is the lowest level consistent with them operating. If the price were below 10, then a shop would prefer to shut down than to incur losses. This sce- nario is a consumer’s dream world! However, life is not al- ways so grand for consumers: in Chapter 14, we’ll see how,
3 (p¿ � e) � 10 4 3100 � (p¿ � e) 4 7 a1 2 b(p¿ � 10)(100 � p¿).ep¿p¿ � e
100 � (p¿ � e)).(12)(100 � p¿) p¿ � e,p¿ � e � 10).
p¿ � 10p¿ � ep¿ ee 7 0.p¿ � e,
p¿ � 10. (12)(100 � p¿)
(12)(p¿ � 10)(100 � p¿), p¿.
p¿ 7 10.
6.2 Solving for Nash Equilibria Without Calculus 151
FIGURE 6.4 Shop i ’s Payoff Function when the Other Shop Charges a Price of p�
Pa yo
ff
0
0 40 50 60 70 80 9010 20 30 100 p
i
p��� p�
Now suppose there are three shops in Clifton Hill, all selling the same product. The game is exactly as just specified, with the addition that if all three shops set the same price, each receives one-third of market demand. Find both symmetric and asymmetric Nash equilibria.
6.1 CHECK YOUR UNDERSTANDING
152 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
in a more realistic description of the environment, stores can get around com- petition and sustain a much higher price.
If you go to the online sporting goods sites of Dick’s Sporting Goods, <www.dickssportinggoods.com>, Fogdog.com <www.fogdog.com>, and Sports Authority <www.sportsauthority.com>, you’ll notice that the products have not only identical prices (competition at work!), but also identical store product numbers (say what?). What is going on here? (This is one conundrum for which there is an answer.2)