Many competitions in life—from innovation races and the assignment of bonuses in the workplace to college admissions and sports—are organized as tournaments. Participants expend effort or other resources and are rewarded based on the ranking of their performance. Those trying the hardest do not always win. While ability certainly plays a role, a nontrivial component of success is luck. Competition is often the most heated among equally able contestants using similar strategies, in which case luck becomes the key ingredient of success. As D.G. Faust, Harvard president at the time, famously stated in 2014, “We could fill our class twice over with valedictorians.” Thus, perhaps controversially, while tournaments undoubtedly incentivize a great deal of effort, in settings where contestants are similar in ability, tournaments essentially reward luck. How should luck be rewarded?

To make the question more precise, suppose a tournament organizer has a fixed prize budget, say, $100 (feel free to multiply all the numbers by a thousand, a million, or a billion if you prefer). Should the winner of the tournament—the participant with the best performance—receive $100? Or should perhaps the prize money be divided so that the winner gets, say, $60, the second place gets $40, and the third place gets $20? Of course, there are many other ways to divide the prize money as well. Which of them will incentivize participants the most?

It may seem that a single prize awarded to the top performer—the winner-take-all prize structure—should be optimal when players do not care about risk. Indeed, sharing the prize among multiple contestants reduces the incentives to strive for the top, by providing some insurance in case it does not happen—which appears useless if risk is not a concern. More generally, making prizes more equitable seems like a bad idea. Yet, this intuition turns out to be incorrect. Winning a tournament, especially when there are many contestants, requires a lot of good luck. We can think of luck as noise—a random shock to performance that is beyond the individual’s control. A burst of good luck can be, for example, a rich and agreeable client suddenly walking into a car dealership, and a salesperson who happens to be nearby closes a great sale. Or, a researcher has ten different ideas, all looking equally promising, and randomly decides to pursue the one that leads to a breakthrough discovery. Of course, similar instances of bad luck can happen as well. However, winners are the lucky ones, and incentives, and the optimal prize structure, are determined by the properties of the largest bursts of good luck, known in statistics as the upper tail of noise.

Statisticians broadly categorize noise depending on the properties of its upper tail into two classes: light-tailed and heavy-tailed. Light-tailed shocks are predominantly moderate in size, and the likelihood of a very large positive shock is extremely small. The most common example of light-tailed shocks is the normal distribution that is perhaps the most studied and widely used type of noise across disciplines. When noise is light-tailed, the second largest shock will be close to the largest one, and hence, each contestant has a strong incentive to increase effort in case she will have the second highest shock. Hence, the most intense competition is between the luckiest players, and it is optimal to award only one prize to the very best player.

But when noise is heavy-tailed, following a distribution such as a power law, or the famous Pareto distribution, the situation is quite different. In this case, large positive shocks are quite likely, and the largest positive shock—the “white swan” (in contrast to “black swans,” or large negative shocks discussed in Nassim Taleb’s 2007 book “The black swan: The impact of the highly improbable”)—can be so large that even the second luckiest player has no chance to catch up with the winner. Therefore, giving the prize only to the top performer is not very useful: players might as well do nothing and wait for a good shock, because if you do not have the best shock there is nothing you can do to win anyway. Then it makes sense to depart from the winner-take all principle and give prizes to lower ranks, more so the heavier the tail of noise. In fact, the case of Pareto distribution is so extreme that it is optimal to give the same prizes to all contestants but the very last, which is the complete opposite of the winner-take-all structure.

These results can be interpreted in a broader sense as providing a neo-classical justification for prize sharing as an alternative to winner-take-all incentives in competitive environments with a substantial luck component. Winner-take-all incentives have become prevalent in many sectors, exacerbating the rise in inequality and fall in social mobility, which can have negative consequences for long-term economic growth and the stability of democratic institutions. The justification stems from the changing nature of fluctuations people in the modern, post-industrial world face. Indeed, while disasters and catastrophes we face may be heavy-tailed “black swans,” the winners of tournaments in various domains rely increasingly on the heavy-tailed “white swans” of percolating innovation and creative ideas, and more and more markets are characterized by the economics of superstars. It is then suggested by our results that prize sharing among winners is a win-win approach because it simultaneously provides stronger incentives and reduces inequality.

Source:

Drugov, Mikhail, and Dmitry Ryvkin. “Tournament rewards and heavy tails.” Journal of Economic Theory 190 (2020): 105116, https://doi.org/10.1016/j.jet.2020.105116