Hardware Store Pricing Game and a Price-Match Guarantee
Two hardware stores, Ace and BuildCo, dominate a small town. Each independently chooses an everyday price: High or Low. Daily profit in thousands of dollars is as follows. If both choose High, each earns $8,000. If Ace chooses High and BuildCo chooses Low, Ace earns $3,000 and BuildCo earns $11,000. If Ace chooses Low and BuildCo chooses High, Ace earns $11,000 and BuildCo earns $3,000. If both choose Low, each earns $5,000. The stores set prices simultaneously each morning and cannot observe the other's choice until sales are rung up. Both owners have read that 'competition is good for consumers' and that 'collusion is illegal,' but they are not sure what the game predicts. A consultant proposes that they each sign a public 'price match guarantee': if a customer finds a lower price at the other store, the store will refund the difference. The town council is considering a minimum markup law that would prohibit prices below a specified floor.
1. Does either store have a dominant strategy? What is the Nash equilibrium? Explain in terms of incentives.
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For Ace: if BuildCo chooses High, Ace earns $8,000 from High and $11,000 from Low, so Low is better. If BuildCo chooses Low, Ace earns $3,000 from High and $5,000 from Low, so Low is better. Low is a dominant strategy for Ace. BuildCo's payoffs are symmetric: if Ace chooses High, BuildCo earns $11,000 from Low versus $8,000 from High; if Ace chooses Low, BuildCo earns $5,000 from Low versus $3,000 from High. Low is also dominant for BuildCo. A Nash equilibrium is a pair of strategies where neither can gain by changing unilaterally. (Low, Low) gives $5,000 each; if either switches to High while the other stays Low, it gets $3,000, so no one gains. (High, High) is not a Nash equilibrium because either store can raise profit from $8,000 to $11,000 by switching to Low.
2. Explain why the (High, High) outcome is not a Nash equilibrium and why it may still be observed in a repeated setting. What conditions make tacit collusion sustainable?
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In a one-shot game, (High, High) cannot be an equilibrium because each has a unilateral incentive to cheat. In a repeated game, the promise of future cooperation can make High sustainable. Suppose both use a trigger strategy: keep High unless someone cheats, then play Low forever. Let delta be the discount factor, the value today of next period's dollar. Cooperating yields 8/(1 - delta). Cheating yields 11 today plus 5 delta/(1 - delta) thereafter. Cooperation is better when 8/(1 - delta) > 11 + 5 delta/(1 - delta), which simplifies to delta > 0.5. So patient owners, frequent interaction, easy detection, and a credible punishment can sustain High. If the game is played once, or owners discount the future heavily, Low is likely.
3. Evaluate the price-match guarantee and the proposed minimum markup law as ways to keep prices high. Are they likely to be legal or stable?
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A price-match guarantee changes the payoff from undercutting. If BuildCo lowers its price but Ace refunds the difference, customers do not switch to BuildCo, so BuildCo gains little or nothing and may lose margin. The Low payoff is reduced, which can make High more stable without an explicit agreement. But the guarantee must be public and credible, and competition authorities may scrutinize it if its purpose or effect is to soften competition. A minimum markup law is a government-imposed price floor; it legally restricts Low and can sustain higher prices, but it creates deadweight loss and may attract entry or noncompliance. From the game's perspective, both devices alter payoffs to make High less costly to maintain. Consumers lose from higher prices in either case. The minimum markup law is more clearly a regulatory intervention, while the price-match guarantee is a business practice whose legality depends on jurisdiction and evidence of intent.