# game theory problem

Suppose there is a circle with numbered locations from 0-60 (with 0/60 at the top; like a clock in minutes). Customers are evenly distributed at locations around the circle. Three firms will locate at locations in order, so that firm F knows where previous firms located (i.e. 1 goes first, then 2 knowing what 1 did, then 3 knowing what 1 and 2 did). All the customers which are closest on the circle to a particular firm shop there (e.g. if it a customer is at 18 and firms are at 12 and 20, the customer shops at firm location 20). The firms desire to maximize the * expected * percentage of customers who shop at their store. If firms locate between two other firms, they choose a spot that maximizes their market share of customers, and equalizes the market share to the other two firms.

Without loss of generality, suppose firm 1 locates at 0 (=60).

(a) Where should firm 2 locate?

(b) Where should firm 3 locate?

(c) Is there an advantage or disadvantage—in theory—to going first or last?

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#### Solution Preview

Firm 1 locates at position 0, now, we consider where firm 2 should locate.

Note that it does not matter where firm 2 locates, it (firm 2) will have the same amount of market share and it cannot try to get a bigger share by choosing one location over others. For example, if firm 2 locates at position 4, then 0-2 will go to firm 1, 2-4 will go to firm 2. 4-32 will go to firm 2, and 32-0 will go to firm 1. Each firm gets 30 (which is a half). If firm 2 tries to locate to another place, say, position 30, it still gets 50% of the market share, because firm 2 gets 15-30 and 30-45.

If there are only two firms, then firm 2 will be indifferent between locating in any of ...

#### Solution Summary

This problem investigates the optimal locations of firms on a circle in order to maximize their market share.