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 is considerably stronger than the second owner's team.

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 worked out what he thought would be 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," Dawson said. "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."

A 1995

mathematics 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.

The

Brams-Taylor method is an extension of the familiar "I cut, you choose" strategy for two persons. 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. For one thing, you can't slice up a player to ensure that two teams appear equally strong. Moreover, owners usually choose from a much larger number of players than the number they need to stock the new teams.

An added complication is that 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, a harmonious combination of players may create a team stronger than the sum of its individual talents.

The so-called waiver pool is a collection of players who have been released by their original teams and are then available to other teams. Owners have the option at any time of waiving a player by sending him to the pool and then choosing a replacement from the pool.

Dawson worked out a scheme that he called the "cut-and-choose draft protocol" for selecting two teams from a waiver pool:

**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 on a previous turn.

"I was able to prove that my method is fairer than the usual drafting methods for two teams," Dawson said. He couldn't guarantee complete satisfaction, however. One problem is that if the waiver pool includes just one outstanding player whom both owners value more than any team without that player, then someone will end up dissatisfied.

Dawson also had difficulty extending his scheme to drafts that involve more than two teams without giving up some degree of fairness. But he was ready to help out in any major-league expansion draft involving two new teams.

In 1997, after his

article about a fairer expansion draft appeared in the

*College Mathematics Journal*, Dawson received an invitation from the

Western Professional Hockey League (WPHL) to help with its upcoming expansion draft. Unfortunately, the league was expanding by four teams,and Dawson's work had concerned only two teams.

"I then came up with a method for an arbitrary number of teams, but the method wasn't optimal in the same sense as the method for two teams," Dawson said.

Dawson did some computer simulations to find out what would happen under various scenarios. In one scenario, for example, there were a few star players and many ordinary players. In another scenario, a lot of players had average ability.

"Under the first scenario, my method performed better than the traditional one-player-at-a-time draft," Dawson said. "Under the second scenario, however, the differences in rating of talent by the various teams had enough of an influence that the traditional drafting methods were superior to my new method."

Because the WPHL quickly loses its star players to the

National Hockey League and the remaining players constitute a fairly homogeneous talent pool, Dawson recommended that the WPHL use a traditional method. Dawson, however, did rearrange the draft order from round to round.

"In my discussions with the WPHL director of operations after the draft, he seemed pleased with my recommendations and stated that the draft went much quicker and more smoothly than expected," Dawson said. "He said that the teams seemed to know who they wanted and were able to draft [accordingly]."

This remark appears to validate Dawson's assumption that the main differences affecting choices involved not so much the actual talent of the players as the owners' evaluations of that talent.

*Original version posted May 13, 1996*