SITUATION: NEUTRALIZING PRICE COMPETITION WITH PRICE-MATCHING GUARANTEES
Best Buy: “Store Price Guarantee—If you’re about to buy at a Best Buy store and discover a lower price than ours, let us know and we’ll match that price on the spot.”3
Circuit City: “Price Guarantee—If you’ve seen a lower advertised price from an- other local store with the same item in stock, we want to know about it. Bring it to our attention, and we’ll gladly beat their price by 10% of the difference.”4
Some retail stores have the policy that if you can find one of their products cheaper elsewhere, they’ll match that lower price. This practice—known as a price-matching guarantee—has been used by electronics stores, office super- stores, supermarkets, tire dealers, and many more retailers. On the surface, the practice seems like highly competitive behavior. Won’t it drive prices down? In fact, it can drive prices up! A bit of game theory ought to convince you.
Consider the same two shops that were competing to sell identical goods in the previous example. Suppose these stores have each instituted a price- matching guarantee. This means that shop 1 will sell at a price of when
(so that its sticker price is not higher than its rival’s sticker price), but will charge only when (so that the low-price guarantee kicks in). Although shop 1 may choose to price the good at the good may actually sell for the lower of the two shops’ prices, which is denoted for the minimum of and
Shop 1’s payoff function is then
and similarly for shop 2:
With the low-price guarantee, note that both shops end up selling at the same price—lowest one posted—and each gets 50% of sales. This certainly sounds like it ought to result in low prices. But before jumping to conclu- sions, we need to carefully consider a store’s incentives when it comes to set- ting prices.
It’ll prove useful to first derive how a shop would price if it were the only shop on the street. In that case, its payoff function would be
,(p � 10)(100 � p)
3min5p1, p26 � 10 4a12b3100 � min5p1, p26 4 . 3min5p1, p26 � 10 4a12b3100 � min5p1, p26 4 ,
p2.p1
min5p1, p26,p1, p1 7 p2p2
p1 � p2
p1
which is plotted in FIGURE 6.5. Notice that the payoff function is hill shaped and reaches its highest level at a price of 55. A shop that is a monopolist would then price at 55, sell 45 units, and make a profit of or 2,025.
Let’s focus on deriving symmetric Nash equilibria and consider a strategy pair in which both stores price at where
so that the price lies between the cost and the monopoly price. FIGURE 6.6 depicts the payoff function faced by shop 1 when The blue curve is shop 1’s profit when it is a monopolist, and the red curve is shop 1’s profit given that shop 2 prices at When (i.e., shop 1’s price is not higher than shop 2’s price), shop 1 sells at and gets half of the total profit of
the red curve is half the distance between the blue curve and zero. (Recall that shop 2 will be selling at as well because of the low-price guarantee.) When
shop 1 sells at (matching shop 2’s price because of its guarantee), and its payoff is The payoff func- tion is flat in that case, because, regardless of the sticker price shop 1 ends up selling the good for shop 2’s lower price of
Notice in Figure 6.6 that any price at or above is a best reply for shop 1, as its payoff is then maximized. In particular, is a best reply. Since the strat- egy pair is symmetric and the game is symmetric, it is also the case that is a best reply for shop 2. Thus, both shops charging a price of is a Nash equi- librium, regardless of whether is 10, 55, or any price in between.
There are, then, many symmetric Nash equilibria when shops are commit- ted to price-matching guarantees. However, if we focus on payoff-dominant Nash equilibria, then the solution is that both shops charge 55, because that
p¿ p¿
p¿ p¿
p¿ p¿.p1,
(12)(p¿ � 10)(100 � p¿).
p¿p1 7 p¿,
p1
(p1 � 10)(100 � p1);
p1
p1 � p¿p¿.
10 6 p¿ 6 55.
10 � p¿ � 55, p¿,
(55 � 10)(100 � 55),
6.2 Solving for Nash Equilibria Without Calculus 153
FIGURE 6.5 A Shop’s Payoff Function If It Were the Only Shop on Clifton Hill
Pa yo
ff
0
2,025
0 40 50
55
60 70 80 9010 20 30 100 p
FIGURE 6.6 The Blue Curve Is the Total Profit If Both Shops Priced at p1 (or If Shop 1 Were a Monopolist), and the Red Curve Is Shop 1’s Profit when Shop 2 Prices at p� and Both Shops Have Price-Matching Guarantees
Pa yo
ff
0
0 40 50
55
60 70 80 9010 20 30 100
(p1 � 10) (100 � p1)
p1
p�
154 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
is the Nash equilibrium that yields the highest payoff to both players. But no- tice that 55 is the monopoly price! With price-matching guarantees, competi- tion between shops evaporates, and they end up charging the same price as when there is only one shop.
To understand this surprising result, go back to the previous model without price-matching guarantees and consider stores’ incentives. Charging an iden- tical price above cost wasn’t an equilibrium for the two shops, because each had an incentive to undercut its rival’s price. Undercutting would double a store’s sales with only a trivial fall in profit per unit sold. But with price- matching guarantees, undercutting doesn’t work. Even if shop 1 sets a lower price than shop 2, customers who would have bought from shop 2 still do so, since shop 2 will match shop 1’s lower price. All that shop 1 has done by un- dercutting is to cause it to sell at a lower price to the same set of shoppers. Thus, price-matching guarantees destroy the incentive to undercut a rival’s price and allow shops to sustain higher prices. What appears to enhance com- petition actually destroys it!
One study of the adoption of price-matching guarantees by supermarkets found that the practice did indeed raise prices.5 In 1983, Big Star, a North Carolina grocer, introduced a price-matching policy and published a weekly circular (known as the Price Finder) that listed the prices of over 9,000 prod- ucts. For these products, Big Star guaranteed to match the prices of Food Lion, its primary rival. In 1985, another competitor, Winn-Dixie, introduced a similar policy, once again promising to match the prices of Food Lion. The theory predicts that the prices for the products listed in the Price Finder should go up more than the prices for those not listed in the Price Finder, because the former, but not the latter, were subject to price-matching guarantees. Well, that is what occurred. Although the average effect was not large—it was about 2%—it was there; and some items were affected considerably. For example, prior to the adoption of price matching, Maxwell House Coffee sold for $2.19 at Food Lion, $2.29 at Winn-Dixie, and $2.33 at Big Star. After its adoption, all three were selling Maxwell House for $2.89.