Consider three major department stores—Big Giant, Titan, and Frieda’s— contemplating opening a branch in one of two new Boston-area shopping malls. Urban Mall is located close to the large and rich population center of the area; it is relatively small and can accommodate at most two department stores as “anchors” for the mall. Rural Mall is farther out in a rural and relatively poorer area; it can accommodate as many as three anchor stores. None of the three stores wants to have branches in both malls because there is sufficient overlap of customers between the malls that locating in both would just mean competing with itself. Each store prefers to be in a mall with one or more other department stores than to be alone in the same mall, because a mall with multiple department stores will attract sufficiently many more total customers that each store’s profit will be higher. Further, each store prefers Urban Mall to Rural Mall because of the richer customer base. Each store must choose between trying to get a space in Urban Mall (knowing that if the attempt fails, it will try for a space in Rural Mall) and trying to get a space in Rural Mall right away (without even attempting to get into Urban Mall). In this case, the stores rank the five possible outcomes as follows: 5 (best), in Urban Mall with one other department store; 4, in Rural Mall with one or two other department stores; 3, alone in Urban Mall; 2, alone in Rural Mall; and 1 (worst), alone in Rural Mall after having attempted to get into Urban Mall and failed, by which time other nondepartment stores have signed up the best anchor locations in Rural Mall. The three stores are sufficiently different in their managerial structures that they experience different lags in doing the paperwork required to request an expansion space in a new mall. Frieda’s moves quickly, followed by Big Giant, and finally by Titan, which is the least efficient in readying a location plan. When all three have made their requests, the malls decide which stores to let in. Because of the name recognition that both Big Giant and Titan have with the potential customers, a mall would take either (or both) of those stores before it took Frieda’s. Thus, Frieda’s does not get one of the two spaces in Urban Mall if all three stores request those spaces; this is true even though Frieda’s moves first.
(a) Draw the game tree for this mall location game.
(b) Illustrate the rollback pruning process on your game tree and use the pruned tree to find the rollback equilibrium. Describe the equilibrium by using the (complete) strategies employed by each department store. What are the payoffs to each store at the rollback equilibrium outcome?