Basketball fans know the problem of “tanking” all too well. Early in the season, teams compete fiercely to win a position in the playoffs, and coaches try to use their star players to best effect. Later on, however, it becomes obvious that some teams have no hope of making the playoffs, and a different incentive kicks in for the coaches of those teams – one that seems to spoil the fun.
The NBA uses a weighted lottery system for drafting new players, one that was intended to give the weaker teams a greater chance at picking the most promising stars from the firmament of collegiate basketball. The team that loses the most games in the season has the best odds of winning first pick in the draft; the second-worst team has the second-best chance, and so on. Although the first three picks are assigned on the basis of a random lottery and any team can win first pick, having more losses is like having more lottery tickets. So once a coach has decided that his team has no hope of making the playoffs, his best strategy is to try to lose as many of the remaining games as possible, thus improving his odds of getting the most promising rookie for the next season. He need not cajole his players into throwing a game; keeping the best ones on the bench at critical times is enough.
That’s tanking, and it frustrates not only fans who want to see their favorite teams play to win, but also NBA officials who never intended this outcome. Over the years they have tinkered with the system to try to fix it, but the problem remains. How can the lottery be biased to favor genuinely weaker teams without creating a perverse incentive to lose late in the season?
An elegant solution has come to me, not through any flash of personal brilliance, but through e-mail from a brilliant friend. I first got to know Danny Kleinman – a professional bridge player, musician, and amateur mathematician – when he replied to a piece I wrote about solving difficult political problems by getting the incentives right. This conversation led to another article on the perverse incentives that have run amok in our nation’s electoral system and how we can fix them. Many seemingly intractable problems can be solved in a freely evolving system – like biology or a free-market economy – simply by getting the incentives right. Noteworthy examples include runaway healthcare costs, excessive executive compensation, our nation’s budget deficit, and, yes, tanking in the NBA.
Kleinman calls his solution to the tanking problem Silver points in honor of Esther Silver, an especially competent administrator and friend from his youth (and no relation, as far as I know, to NBA Commissioner Adam Silver). Rather than doling out lottery tickets according to the total number of losses for the season, Kleinman suggests they be distributed by the number of Silver points each team has won. So what is a Silver point?
In essence, the big idea behind Silver points is to count losses differently depending on when in the season they occur. Losses early in the season count more than later ones, because the incentive to win is greatest in the early games. This idea alone would greatly reduce the incentive for tanking, but Kleinman’s solution does more. It also incorporates the concept of the transition, late in the season, when the weaker teams discover they have no hope of entering the playoffs. This is when the incentive flips under the current lottery system. To eliminate this change in incentive, Kleinman flips the awarding of Silver points at the transition. Any losses that occur after the transition count as negative Silver points. They subtract from the team’s previously awarded Silver points and diminish the team’s chances in the draft lottery, so there is never an incentive to lose.
“But wait!” you may object. “If losing late-season games hurts a team’s Silver point score, then the weakest team might not have the most Silver points. Don’t we want the weakest team to have the best chance in the lottery? And how do we know exactly when in the season the transition in incentive occurs?”
It turns out that these objections are not all that serious. The first problem – that the genuinely weakest team will tend to get negative Silver points after the transition – is probably negligible. The transition happens late in the season, so the number of negative silver points accumulated during that phase will be small compared to the number of positive ones acquired before the transition. More importantly, there is no guarantee, even under the current system, that the team with the most losses wins first pick in the draft: the randomness of the lottery eliminates that certainty. All we really care about is distinguishing genuinely weak teams from strong ones and allocating lottery tickets roughly in proportion to weakness.
The second problem – trying to guess exactly when the transition in incentive occurs – is also less important than it might seem. All that matters is that our guess be about right, somewhere well past the middle of the season and before that point in seasons past when the most obvious signs of tanking have appeared. Losses near the transition have little effect, because the absolute number of Silver points awarded for a loss (either positive or negative) is in direct proportion to how many games before or after the transition the loss occurred. A loss in the first game of the season awards the most Silver points, a loss in the last game subtracts the most, and losing the game that happens to fall exactly on the guessed transition has no effect at all on the team’s Silver point score. This means that small errors in guessing the transition point have little effect on the incentive to win. Kleinman suggests setting the transition at game number 64 out of 82 games total.
Having ranked the teams by their net total Silver point scores, the NBA could allocate lottery chances according to the somewhat arcane formula they currently use. But Kleinman suggests yet another improvement here: allocate lottery tickets in direct proportion to the Silver points themselves. This would further diminish the effects of any small error in guessing the transition point.
My goal here has been merely to convey the essence of Kleinman’s elegant idea for fixing the tanking problem. Should the NBA choose to try it, they will need a more mathematically precise statement of it. They can find that at my website, where I have posted Kleinman’s original description of Silver points, complete with the necessary equations. Both he and I would be happy to answer any questions the NBA may have.
And although the tanking problem may seem trivial in the grand scheme of things, there’s a broader lesson here for life in general and the thriving of democracy in particular. In Danny’s words, “Incentives and disincentives are essential in life as well as games, and they reduce reliance on power and control.”
Danny Kleinman can be reached at:
John C. Wathey is an author and computational biologist whose research interests include protein folding, evolutionary algorithms, and the biological forces behind religion. Learn more at www.watheyresearch.com.