Example 2

EXAMPLE 2 The Restricted Location/Schedule Game

Assume that Mark and Lisa revisit the location/schedule game once Mark and Lisa have their first child. To limit the number of hours their child must spend in daycare, Mark and Lisa agree that Mark cannot work the 8 A.M. to 4 P.M. shift, because this conflicts with Lisa’s work schedule. A new game is formed by eliminating the 8 A.M. to 4 P.M. shift column in Table 15.1. As before, Mark and Lisa can each do a worst-case analysis. Mark is worried about the minimum number in each row, and Lisa is concerned with the maximum number in each column. The result, along with minimum row and maximum column information, appears in Table 15.3.

Table 15.6: TABLE 15.3 Hourly Wage (in dollars) from Table 15.1, with the Row Minima (Maximum Circled) and Column Maxima (Minimum Circled)

		Lisa Shifts
	Hospitals	12 A.M.–8 A.M.	4 P.M.–12 A.M.	Row Minima
Mark	Rural	23	22	22
	Suburban	27	29	27
	Downtown	30	25	25
	Column Maxima	30	29

628

Mark sees that his maximin is now 27, so he can guarantee an hourly wage of $27 per hour, instead of the $26 per hour from before. Lisa notices that her minimax is now 29. When the maximin and the minimax are not the same, then the game does not have a saddlepoint, but it does have a value (described in the next section).

If Mark plays his maximin strategy, the suburban hospital, and Lisa plays her minimax strategy, the 4 P.M. to 12 A.M. shift, then the resulting payoff is 29. However, Lisa may be motivated to gamble in this case by playing her other strategy, the 12 A.M. to 8 A.M. shift. If she switches while Mark still selects the suburban hospital, then Mark’s hourly wage will decrease to $27 per hour.

If Mark thinks that Lisa will select the 12 A.M. to 8 A.M. shift, then he has an incentive to select the downtown hospital instead. The highest hourly wage that Mark can receive is by working the 12 A.M. to 8 A.M. shift at the downtown hospital.

However, if Lisa looks far enough ahead, she can respond to Mark’s reasoning by playing her minimax strategy. The result is $25 per hour if Mark selects the downtown hospital and Lisa selects the 4 P.M. to 12 A.M. shift. The $25 per hour is less than the $29 per hour when Lisa plays her minimax strategy and Mark plays his maximin strategy. This ? means that Mark has an incentive to return to his maximin strategy.