EXAMPLE 1 Determining Work Schedule and Location
After moving to a new town so that Lisa can take a job as a newspaper editor, newlyweds Mark and Lisa must decide which position Mark should accept as an emergency room nurse. There are three area hospitals, each with openings for three 8-hour shifts. Concerned about saving for a down payment on a house, Mark wants to choose the shift and hospital location that pays the most money. Lisa is less concerned about finances and more concerned about their quality of life. A tougher schedule and a more active emergency room translate into a higher hourly wage, but a lower quality of life. In this way, their preferences are diametrically opposed. What is better for Mark is worse for Lisa, and vice versa.
Lisa and Mark make a table with the shifts listed along the top row and the hospitals listed along the left column (see Table 15.1). The entries in the table give the hourly wage Mark will receive for working at the hospital, given by the row, and for the shift, given by the column. The hourly wages are the payoffs to Mark for the different outcomes.
| Shifts | |||
|---|---|---|---|
| Hospitals | 12 A.M.–8 A.M. | 8 A.M.–4 P.M. | 4 P.M.–12 A.M. |
| Rural | 23 | 24 | 22 |
| Suburban | 27 | 26 | 29 |
| Downtown | 30 | 23 | 25 |
Payoff Matrix DEFINITION
A payoff matrix (illustrated by Table 15.1) is a table whose rows and columns correspond to the strategies of the two players. For a total-conflict game, the numerical entries of the matrix give the payoffs to the row player when these strategies are chosen.
Lisa and Mark turn the table into a competitive game to determine which job Mark should accept—such a game represented by a payoff matrix is called a game in strategic form. Mark will select one of the three hospitals—Rural, Suburban, or Downtown—and Lisa will simultaneously choose one of the three shifts—beginning at 12 A.M., 8 A.M., or 4 P.M. Because their choices will be made simultaneously, neither knows beforehand what the other will do.
624
Because Lisa’s preferences are diametrically opposed to Mark’s preferences, Lisa’s payoffs are represented by −1 times the hourly wage. For example, if the outcome is for Mark to work at the rural hospital from 12 A.M. to 8 A.M., then Mark’s payoff is 23 I while Lisa’s payoff is −23. This is the origin of the term zero-sum game.