Question

1 Two stores, Allfoods (A) and Barks&Sensor (B), are located on the two opposite ends of a straight road. The length of the road is 1mi. The road is populated by a continuum of consumers (a unit mass) that are distributed uniformly along this road (alternatively, you may assume that there is a single consumer living on the road, but the location of the consumer is uniform random and is not observed by A and B ). In order to buy from a store, each consumer has to travel to that store and back. The cost of a return trip is equal to the distance from consumer's home to the store in miles. The stores sell identical nondivisible products and each consumer wants to buy at most one unit of this product. The value of the product for the consumer is v. The cost of producing one unit of the product is c

i
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for store i∈A,B}. The stores maximize profits and the consumers maximize the value of the product net of expenses. (a) Assume that c
i
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=0 for store i∈A,B}, and that v=[infinity] (since the value is illdefined, you can assume that the consumers minimize expenses, rather than maximize value). Suppose the two stores set their prices simultaneously (the consumers can see the prices before they decide which store to attend). Formalize this as a game. Suggest an equilibrium notion. (b) Find an equilibrium. (c) Now assume that v is finite. Find an equilibrium.

Answer 1

The question asks to model a situation as a game where two stores compete on price for consumers distributed along a road. Nash Equilibrium is suggested for part (a) and finding an equilibrium in parts (b) and (c) involves considering both prices and travel costs to maximize store profits and consumer utility.

Step-by-step explanation:

The student's question involves constructing a game theory model for two stores competing for consumers along a straight road. In this scenario, the stores (Allfoods (A) and Barks&Sensor (B)) act as players in the game, competing by setting prices for an identical product while considering production costs and consumer expenses related to travel.

For part (a), since the production cost ci is zero and consumer value v is assumed to be infinite, the game focuses on minimizing expenses. An equilibrium notion suitable for this scenario is Nash Equilibrium, where no player has anything to gain by changing only their own strategy if the strategies of the others remain unchanged.

For part (b), finding the equilibrium would involve setting prices such that both stores are indifferent to changing them, often resulting in identical prices that maximize each store's market share without incentivizing consumers to switch due to price differentials.

For part (c), when v is finite, an equilibrium can be found by taking into account both the price of the product and the travel cost for consumers. The equilibrium price would be one where consumers maximize their net value (value of product minus expenses), and each firm maximizes profit considering the marginal utility and opportunity cost for the consumer.

In all cases, equilibrium could mean each store capturing nearly fifty percent of the customers when no further profitable moves exist.

Answer 2

(a) In the simultaneous pricing game where both stores set prices simultaneously, an equilibrium notion could be the Nash equilibrium. In this equilibrium, each store's pricing strategy is optimal given the pricing strategy of the other store, and no store has an incentive to unilaterally deviate.

(b) In the case where
`\(c_i = 0\)` both stores and
`\(v = \infty\)`, an equilibrium is for both stores to set the price at zero. Since the consumers aim to minimize expenses, they will choose the store with a price of zero, resulting in both stores having an equal number of customers.

Step-by-step explanation:

(a) The Nash equilibrium is a suitable equilibrium notion for this simultaneous pricing game. In a Nash equilibrium, each store's chosen pricing strategy is the best response to the pricing strategy chosen by the other store, and no store can unilaterally change its strategy to improve its profit.

(b) When the cost
`\(c_i\)` is zero for both stores and the value v is infinite, the profit-maximizing strategy for both stores is to set the price at zero. In this case, consumers will choose the store with the lower cost, resulting in an equal distribution of customers between the two stores.

(c) If v is finite, determining the equilibrium involves finding the pricing strategy that maximizes each store's profit given the other store's strategy. The solution may depend on specific values of v and the cost structure, requiring further analysis based on the specific parameters.