Solution
Loyal Customers and Shoppers
Show the problem again
A market has 100 consumers, each willing to pay up to 10 for one unit. Of them, 20 are loyal to firm A and 20 are loyal to firm B: a loyal consumer buys from their firm if its price is at most 10. The other 60 are shoppers who buy from the cheaper firm (splitting equally on a tie). Production is costless and the two firms choose prices in [0, 10] simultaneously. There is no pure-strategy equilibrium; in the symmetric mixed equilibrium, what is the probability that a firm charges a price below 5? Give a reduced fraction.
Worked solution
A firm charging 10 sells only to its 20 loyal customers (a shopper would find the rival cheaper almost surely), earning 200. In a mixed equilibrium every price in the support yields exactly that, so for a price p in the support p[20 + 60(1 − F(p))] = 200, where 1 − F(p) is the chance the rival charges more. At the bottom of the support F = 0, which gives p = 200/80 = 2.5. Solving, F(p) = 1 − (200/p − 20)/60. Then F(5) = 1 − (40 − 20)/60 = 2/3. Equilibrium prices are random, so sales occur: shoppers benefit, and a firm never wins them at the highest price.
Source: Inspired by Hal Varian's model of sales (1980). Statement written for AxiomIQ.