Quantitative Methods for Negotiating Trades in Pro Sports

I recently had some thoughts about negotiating trades in the NBA. Specifically, I heard that the Lakers and the Bulls were having daily discussions about a trade involving Kobe Bryant, for at least a week; that seemed like a long time to me. Was this week-long series of conversations productive and/or necessary? Are there no quantitative methods for structuring trade negotiations that could have been used to save these teams some time and energy? I’ve outlined a potential solution, which can probably be improved using methods from the literature on (1) statistical models for rankings and (2) bargaining and negotiating.

My idea is this: First, construct a list of all possible trades to be made between two teams, which would involve closely examining the entire roster of each team, accounting for salary cap restrictions that preclude certain trades, and including or excluding specific key players. Then, instruct each team to rank these possible trades from the most desirable trade to the least desirable trade. These two sets of rankings will naturally be each other’s approximate inverse, because a very good trade for one team will most likely be a very bad trade for the other team (Kobe Bryant for Chris Duhon, anyone?). Lastly, the negotiations consist of each team taking turns eliminating the lowest-ranked trade from their respecitve lists, until the two lists have only one trade in common. If this trade is rejected by either team, then – and here comes the part that I think could be powerful – no trade can be made between these two teams until at least one of them changes their rankings. It is a framework that could be used to, at the very least, save two teams some time when negotiating a trade.

The only similar setting that I can think of is when opposing lawyers eliminate potential jurors from a jury pool (they call these “peremptory challenges”). Does anyone know of another situation in which opposing agents rank items and eventually must agree on a compromise? Maybe there is something in the bargaining literature.

The statistical question of interest is this: What is the percentile of the rank (for each team) of the jointly optimal trade? (That is, the last trade that remains on both lists after eliminations are made). It would be nice if, in the pro sports example, both teams could improve significantly. This would probably only happen in an “apples for oranges” type of trade. Some preliminary work in the Lakers-Bulls example shows that the jointly optimal trade is in the 47th percentile for the Lakers and the 48th percentile for the Bulls – not too great for either team. A bunch of assumptions were made in this example, though, so it’s probably not too informative right now. If a probability model is used to generate the two sets of rankings, then the pair of percentiles of the jointly optimal trade, (p_1,p_2), would be a random variable of interest.

7 thoughts on “Quantitative Methods for Negotiating Trades in Pro Sports

  1. Congratulations Kenny, you have put yourself in good company. Your proposal is similar to that put forth by Shapley and Gale [Gale, David and Lloyd Shapley [1962], "College Admissions and the Stability of Marriage," American Mathematical Monthly, 69, 9-15]

    and is at the basis of much of the matching work done in the economics of game theory. Alvin Roth at Harvard's economics department has written a lot on this subject.

    I have not previously heard of your "statistical question of interest", so that might be novel, but I'm only passingly familiar with this topic.

    Regards,

    Bruce

    BD McCullough

    Professor of Decision Sciences

    Drexel University

  2. Soccer leagues outside the US solve this kind of problem by the use of transfer fees. Dollar-denominated trades are probably easier than negotiations involving items with a fuzzy value metric (e.g., people).

  3. I don't know anything about the literature on stuff like this, but the way you describe negotiation sounds odd. Really, it sounds like once both parties construct their ranks there's an algorithm that doesn't require negotiation at all: alternately remove the lowest ranked trades from both lists until one trade remains.

    I would think each team would estimate rankings for themselves and the other team. The algorithm you describe would lead to their proposal to the other team. If they disagreed on the best trade, then negotiation would begin, where one party would try and convinve the other that their rankings are flawed.

    Just an idea.

  4. I think "approximate" does too much work in "approximate inverse" for this method to work consistently. Often enough the lists aren't going to have a mutual median, and even when they do there's a potential for skipping over very positive-sum trades earlier in the lists.

  5. I see some practical difficulties with this idea because of the following.

    The list of all possible trades to be made between two teams is infinite. Maybe non-countable!
    To instruct each team to rank these possible trades is not factible because of the pevious item. Also, it can be necessary an automatic method to do it without ethernal disagreement.
    As dr.zeuss said, if the negotiations consist of each team taking turns eliminating the lowest-ranked trade from their respecitve lists, then you can reach different conclusions depending on what team starts the negotiations. For instance, suppose team T1 ranks ABCDEF and team T2 ranks FEDBCA. If T1 starts the negotiations, C is the best trade. If T2 starts the negotiations, D is the best trade.

    Can you imagine the problems arising with more than 2 teams and new conditions along the time?

    I'm not sure that there exists mach theory that deals with these problems.

  6. Thanks for the interesting comments and references so far. A few additional thoughts based on some comments:

    Dr. Z: True, a mutual median does not always exist. But any trade based on the sum of the teams' rankings doesn't account for which team is getting what. You might find the trade that increases overall productivity the most, but it might give almost all of that increased productivity to one team. I think positive-sum trades early in one list would be late in the other list, and therefore one team would be very unlikely to agree to the deal. The method of successively eliminating the worst trades protects each team from getting screwed, I think.

    GT: Good points. In my initial study, though, I got the list of possible trades down to 15,000 by doing the following things: Don't include more than two teams, draft picks, or cash considerations. Now, with 15 players on each roster, the max number of trades is about (2^15-1) * (2^15-1) = 1 Billion, where any subset of one team's players can be traded for any subset of the other team's players. Trading all 15 guys is kind of crazy, though, so if you cap the # to be traded from each team to 4 or 5 players, account for salary cap restrictions, and require Kobe to be traded and Deng to stay in Chicago (a key consideration in my example) then you come up with *only* 15,000 possible trades.

    To rank them, I just used a weighted sum of each player's previous year's statistics. The weights can be adjusted by each team depending on how that team values different things like points, rebounds, assists, hair color, etc… With this ranking procedure, I ignore things like character, marketability, the length of one's contract, etc. These are road bumps, for sure.

    Lastly, I wasn't clear earlier about the elimination process: Teams dont' take turns, they would eliminate their worst-ranked trades at the same time in each successive step – so that usually two trades are eliminated in each step. You're right, if they took turns, the order would impact the results.

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