Ivars Peterson's MathTrek

May 13, 1996

## Toward a Fairer Expansion Draft

When new teams are brought into an established league, the team owners usually take turns picking players from a pool of available talent. The conventional method for such drafts, however, often produces unfair results.

For example, if one of two owners of new teams wins the coin toss, that owner gets to pick first, and then the picks alternate between the two owners. It's quite possible that the owner making the first selection will consistently get the better player of each pair available, building a team that may be considerably stronger than that chosen by the owner going second.

Such imbalances can occur even in a modified basic draft, in which the winner of the coin toss selects one player, the second owner picks two players, and so on, with the first owner taking one player in the final turn. Though generally perceived as fairer than the basic draft, this procedure can still put one owner at a disadvantage. Mathematician C. Bryan Dawson of Emporia State University in Kansas has worked out what he thinks is a fairer way of dividing up a talent pool to create two new teams. Dawson had become interested in the issue of fair division back in his high school days, when he and his friends would draft teams to play baseball in a virtual league. With as many as six "owners," the person choosing first would have a great advantage over the person choosing sixth.

"We tried various methods to make it more fair," he says. "We would go 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, and then repeat the sequence. We tried 1, 2, 3, 4, 5, 6, 6, 1, 2, 3, 4, 5, 5, 6, 1, 2, 3, 4, 4, and so on. We tried other patterns, but we were never really satisfied with any of the methods."

Last year, a math paper by Steven J. Brams and Alan D. Taylor revived Dawson's interest in the issue of fair division. Their report concerned a novel, mathematical procedure for dividing up a cake among three or more people so that all the participants are satisfied with the result.

There's a familiar strategy for two persons: "I cut, you choose." The first person divides the cake into two pieces that appear equally desirable to him. The pieces may not seem equally desirable to the second person, so she picks the one she prefers. Both parties automatically end up with a piece that they think is at least as good as the piece they didn't get.

Such a strategy can't be applied directly to an expansion draft. Obviously, one can't slice up a player to ensure that two teams appear equally strong. Moreover, the number of players available in the so-called waiver pool is generally much larger than the number of players needed to stock the new teams. Teams have the option at any time of waiving a player by sending him back to the pool and choosing a replacement from the pool.

Also, the value of an individual player may depend on who else is on the team. In basketball, for example, it doesn't make sense to draft a team in which all the players are centers. Moreover, the right combination of players may create a team stronger than the sum of its individual talents.

Dawson has worked out a scheme that he calls the "cut-and-choose draft protocol" for selecting two teams:

Step 0: The two owners flip a coin with the winner choosing whether to be Owner 1 or Owner 2.

Step 1: Owner 1 chooses two teams of n players.

Step 2. Owner 2 chooses one of the two teams and gives the other to Owner 1.

Step 3. The owners take turns at the waiver pool, beginning with Owner 2, exchanging as many players as they wish (possibly none) at each turn. The process ends when neither owner wishes to make a change or the owners return to a pair of teams they held simultaneously before.

"I was able to prove that my method is fairer than the usual drafting methods for two teams," Dawson says. But he can't guarantee complete satisfaction. For example, if both owners value any team with a particularly talented, superior player (say, Michael Jordan) as worth more than any team without that player, someone will end up disatisfied.

Dawson has had difficulty extending his scheme to drafts that involve more than two teams without giving up some degree of fairness. But he's ready to help out in major league baseball's impending two-team expansion draft. "I'm willing to offer my services to them," Dawson says.

While he's awaiting the phone call from the big leagues, Dawson has other applications of his protocol in mind.

Suppose two brothers go together to a flower shop on Valentine's Day to buy their sweethearts the six-flower special. To their dismay, they discover that there aren't enough red roses to fill both orders. But there are other types of flowers available to complete the bouquets. In this case, the cut-and-choose draft protocol, with the brothers as "owners" and the flowers as "players," rescues the men from fighting over their choices.

### References:

Brams, Steven J., and Alan D. Taylor. 1996. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge, Eng.: Cambridge University Press.

______. 1995. An Envy-Free Cake Division Protocol. American Mathematical Monthly 102 (January 1995): 9-18.

Dawson, C. Bryan. 1996. A Better Draft: Fair Division of the Talent Pool. Abstracts of Papers Presented to the American Mathematical Society 17 (Winter 1996): 148.

Peterson, Ivars. Formulas for Fairness. Science News 149 (May 4, 1996): 284-285.

C. Bryan Dawson can be reached at dawsonbr@esumail.emporia.edu.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.