THE NBA LOTTERY AND GAME THEORY

Executive Summary

Sports are used as prime examples where game theory is prevalent. In the NBA, a very special draft system is often used to determine which team gets what collegiate player. This has created a chicken game of sorts, where the some teams have a distinct preference to either win a championship or to lose games intentionally in order to select the best collegiate player. The problem with this is that it has created a skewed model where there are two distinct games present at the same time. Similarly, because this is a repeating simultaneous game, teams often find themselves stuck in the middle over a long period of time thus neither getting the best player nor truly competing for a championship.

Our paper studies the game theory aspects behind this as well as the often irrational drive for an immediate championship compared to a future championship. We also provide a quantitative data that analyses the true value of each position using the most recent APBR metrics statistics. Finally, we have recommended a few courses that teams ought to take if they find themselves in the middle, as well as recommendations for the leagues draft policies to ensure a reduction in purposely losing games.

Introduction

Every year towards the end of the regular NBA season, the majority of teams begin making a run at the NBA playoffs. It’s a very exciting time for fans as their favorite teams begin strategizing how to position themselves to have a favorable spot in the playoff seeding. The NBA is divided into two conferences, the Eastern and Western. Both conferences take the 8 teams with the best Win-Loss records and send them to the playoffs; the winner becomes the NBA champion. But while most teams battle for a favorable spot in the playoffs, there are some that fall into temptation to “tank” (lose games) in order to have a favorable spot in the NBA lottery. The NBA lottery is the system that rewards the worst teams in the league with favorable positioning to select a top player, new entrant into the league, during the NBA draft.

This is how it works. At the end of the season the NBA holds a lottery in which the participants are the 14 teams that do not make the playoffs. Each team, starting with the one who has the worst record, is given a number from 1000 possible permutations. The team that has the lowest season record receives a number with 250 possible permutations of those 1000, and so on. For example, as the team with the poorest record in the league, The Orlando Magic would be given a number that makes up 25% of the possible 1000 permutations. The team’s number is represented by a ping pong ball, one for each team, that is placed in a drum where the lottery then takes place and ping pong balls are drawn to determine which teams get the top picks (see exhibit 1). After the first three picks though, the lottery is finished and the remaining teams are assigned their position based on their win-loss record.

While there is speculation at the end of every season about this happening, the 1984 season is one that exemplifies tanking best. The Chicago Bulls went on to lose 14 of there last 15 regular season games and with a favorable draft position selected Michael Jordan. The Houston Rockets also tanked by losing 14 of their last 17 games and were rewarded with Hakeem Olajuwon, one of the greatest centers to ever play.[1]  

Such strategies by NBA teams to secure a favorable position at the end of the season have even led noted social theorist Malcom Gladwell to state, “The draft structure rewards risk...” it’s a “moral hazard…”[2]

And we can infer that it’s true, if you are a team toward the bottom of the NBA rankings what incentive is there to try and make the playoffs? You will be playing against the number 1 team in your conference whose overall chances of winning the championship are approximately 32%. You will play a team that has better resources (players, coaches), win-loss record, and home court advantage (eg, in a series of 7 games, 4 of those games are played at the arena of the team with the best record). And to top it off, you also lose a favorable spot in the draft. This would be a dominated strategy, yet, most NBA teams play this strategy; being caught in the “middle;” not quite high enough to have a great chance at winning the NBA championship, and not quite low enough to acquire a great new player to help your chances the following year at a better record.

Much to the confusion of their fans, understanding motivation and incentive for losing is something NBA franchises deal with every year. Yet, as previously illustrated with the problem of being stuck in the middle, by trying to do the right thing and fulfill the purpose of the game (win, a dominated strategy), a team may lose greater payoffs in the end. The whole idea of rewarding failure is one the most concrete criticisms of the NBA lottery.  But without it, teams may remain in the middle year over year since playing a dominated strategy (trying to win) as a team ranked in the middle ensures that it will result in neither winning the NBA championship or having a favorable position in the NBA lottery to draft a star player.

Lastly, the NBA Lottery situation appears to have elements of similarity to the situation Grenada and Barbados found themselves in at the Shell Caribbean Cup in 1994.[3] The rules of the game presented a situation where both teams favored scoring own goals at the end of regulation in order to advance further in the tournament. This seemingly irrational behavior can be perceived as confusing and self-dominating, but in fact, like tanking, it’s a dominant strategy.

Explanation of the Approach

To begin, it is necessary to explain the approach as well as the numbers that we took both in determining what game this was as well as the appropriate payoffs.  Basketball, inherently, is a very complex game with many integrated parts.  Therefore, statistically it is more difficult to isolate the value of a player when compared to games such as baseball, where there is limited activity.  Thus, we acknowledge this inability to judge a single player while trying to use the most advanced statistics possible in order to approximate it.  In this paper, we have chosen to use “Win Shares” due to the preponderance of data as well as the relatively simple concept behind it.  Essentially, every basketball activity is combined into segments to determine its value compared to other actions, and then each player receives a combined number based on their actions.  This number is then formed to mimic the number of actions needed for a team to win one game relative to base performance.  Therefore, through using this number, we were able to determine the value of a general spot in the draft in terms of wins that said player will create during their career.

Beyond this, we had to figure out the various numbers to include within a team’s chance of winning the championship based on playoff seeding as well as how to calculate the value of the draft position.  It is nearly impossible to be able to determine a team’s championship hopes solely based on regular season record; however we can approximate it assume there is no extreme outlying information.  First, we relied on historical data, which indicates that the top six teams overall have won every championship but one.  Given that the current playoff structure has only existed for around 30 years, we also used data from this season to determine each current playoff team’s chance of winning the playoffs.  We included both statistical measures as well as the public’s perception via prediction markets.  All of these were nearly identical to the aforementioned historical data.  In order to determine the value of draft positions, we used the current NBA’s draft lottery odds with the estimated win share values provided by basketball reference.  Similarly, we used the average regular season wins per lottery and playoff seed to determine how difficult it would be to move up and down during a season.

Incentives for Winning or Losing

Basketball is quite layered, and the incentives for each team differ based on their current economic status and the owner’s wishes.  However, in general, people see the NBA championship as what each team strives to get.  Only one team can win it every year though, which creates a lot of losing teams along the way.  Therefore, it is reasonable to assume that there ought to be some balance between trying to win this year and trying to win it in future years by receiving good players in the collegiate draft.  The time value of these championships tends to emphasize the current year over future years, but nevertheless there ought to be a relatively disperse spread between those trying to win this year compared to those stocking up for the future.  The NBA championship is a good incentive beyond fame alone, as teams that win it often see both current ticket sales as well as the financial value of the team as a whole increase due to a prestigious tradition of success.

The Chicken Game

Within the league, there is a constant focus on the 8th spot in each conference (15 teams).  This is because the 8th seed enters the playoffs, while the 9th team is regulated to the lottery.  Often, the emphasis is on how lucky the 8th team is since they will have an opportunity to compete for the championship.  By focusing on this battle, it is easy to use the chicken game to mimic how both teams would be better off if they used divergent strategies.  First, there is a strong and a weak strategy currently, and most teams would prefer to be strong and get in the playoffs over weak and being delegated to the lottery.  Second, there are two Nash equilibriums where neither team regrets their action nor would change it.  And last, the worst outcome for both teams is for both teams to enact a similar strategy.  This may seem counterintuitive, as in most seasons both teams do go for the tough strategy; however in actuality this is by far the worst outcome.

As previously mentioned, there is a well defined tough and weak position in the minds of the teams.  Often the payoffs are exaggerated in the minds of the teams, as both fans and players often think that the reward of the 8th seed brings a 1/16th shot of winning the championship.  This is not the case though, as both scientific and public opinion shows that the actual chance of an 8th seed winning the tournament is around .1% (APPENDIX).  Similarly, many teams end up fighting quite heavily over the 8th and final seed.  Usually, what occurs in the NBA is that many of the teams on the cusp of the playoffs end up trading for or signing mediocre players.  They do this with the hope that this mediocre player could put them over the edge and help them reach the playoffs.  Naturally, this creates a situation where every team on the boundary engages in this action, and all teams are worse off without having created separation.  This would be fine, except that often they trade valuable future assets away or that they sign these mediocre playoffs to expensive long-term contracts.  Therefore, you can see elements of a price war within this chicken game.

We can also see that some teams end up playing a dominated strategy despite the numbers saying that they shouldn’t.  Obviously when only focusing on the 8th and 9th spot in a conference, both teams are relatively equal in stature and the payoffs are relatively similar.  You can argue, as we do, that given the benefits of entering the draft verse the chances of winning a championship, teams ought to actually lose rather than win.  However, based on the relatively small nature of the difference in payoffs, it is understandable given other factors that teams do fight to win rather than to lose.  When choosing though between going for the 8th spot verse trying to lose for the best chance in the lottery, the payoffs are vastly different and teams usually take the dominated strategy of trying to win over the dominant strategy of having the best future player.  This irrational behavior therefore requires a different model once the number of players is fully taken into account.

Reverse Median Voter Model

Oddly enough, there are parallels between the NBA situation and the median voter model.  In both models, you are relying on the actions of the other users in order to determine what the best course is for yourself.  Also, because there are too many players, you have use a “best response” technique rather than focus simply on your own preferences.  The vast majority of the benefits for NBA teams lie within the extremes: winning a championship or receiving the best possible player for the future.  This creates a system where naturally teams would prefer to be at either end rather than stuck in the middle.  Overall, like in the chicken game, most teams favor winning the championship, but this is not a likely possibility for most teams.  Therefore, teams ought to understand what their chance is of winning verse their chance of losing.  Thankfully, seasons are repeatable tests with rather predictable data, and therefore ought to allow each team to know their relative position within the league.  However, even with knowing their approximate spot, teams that have a best response of trying to tank often instead try to win every game.  Not only is this playing the incorrect approach, but it actually ignores the relative ease at which teams can lose.

In our model (APPENDIX), you can see that it does indeed skew; however what you neglect to see is that it is much easier to lose currently in the NBA than it is to win.  In basketball, a specific few players are so good that they can more-or-less guarantee their team a good shot at the championship.  If you do not have these players though, it is nearly impossible to win.  Therefore, despite the expensive free agents and potentially future-threatening trades, mediocre teams often can’t improve themselves enough to win.  This is not true when it comes to losing.  Some very smart general managers, most notably Sam Presti of the Oklahoma City Thunder, gutted their teams in order to lose as many games as possible in order to have high draft picks.  They were able to achieve this quite easily and actually saved a lot of money on salaries in the process.  Because the current structure emphasizes winning, the Thunder constantly lost and ended up with franchise players such as Kevin Durant, Russell Westbrook, and James Harden.  Now, because they were able to receive top-quality players in the draft through losing constantly, they are able to compete for championships on a yearly basis.  Thus, they were able to win because of taking advantage of the aversion against losing.

Advised Strategy

The correct strategy for each team depends on their payoffs from the chicken game as well as the extraneous factors that may influence the general manager or owner.  Because this is a multiplayer game that relies on the best response, information is vital.  You have to realistically be able to know your chances and approximate spot.  Similarly, you shouldn’t be too blinded by assuming that your team is an exception to historical precedence and statistical accuracy.  Often teams that are even in the 5th or 6th seed in their conference have no real shot of winning the championship, and therefore would benefit more from receiving a high draft pick instead of losing in the first round of the playoffs.  While the season is long, teams too often wait too long before deciding which direction they want to go towards.  Owners and general managers think that they need more information before being informed enough to make a choice, and therefore make the decision to tank or go after the championship well after they can actually make enough changes to benefit their decision. 

Once making their decision, teams ought to act in ways to achieve their aims by deterring other teams from tanking or from going for the championship.  Threats are possible, but often ignored.  Similarly, promises are often seen as collusion in sporting scenarios and thus are avoided.  However, commitment and signaling can work just as easily in the NBA as they can in the business world.  For example, if you have decided to tank for the year, you can choose to send your best player to the disabled list.  This would act as a “throwing away the steering wheel” example since it is a public and irreversible show that you plan on losing many games this year.  You can also signal your intention by stating publically that you are hoping to rebuild the team rather than win instantly.  This shows other teams your intentions.  Also, by choosing such tactics, you can effectively change a simultaneous game into a sequential one, which is extremely beneficial in a chicken game scenario.  By moving first through deciding your course early, you can force others to react to your plans and thus enact a policy of mediocrity rather than trying for one of the two extremes.

Prospect Theory

One way in which we can also view the payoffs is in the prism of prospect theory.  In regards to how fans and players feel about losses, you could argue that the difference between barely missing the playoffs and drastically missing the playoffs is not substantially different.  If this is the case, then the extraneous factors may not matter as much as the raw tradeoff between future wins and current wins.  Thus, prospect theory would indicate that we ought to follow game theory alone and not play dominated strategies.  You could also use the prospect theory in events such as signaling.  Instead of having one major signal of your intentions, such as signing a high-profile free agent or placing one important player on the disabled list, you should instead do many tiny actions instead.  These mini-signals would be more effective in getting other actors to do what you wish. 

Behavioral Decision Making

Quite often teams get stuck in the middle.  They want to try hard out of principle and in the hope of winning the championship, but at the same time they are not good enough to do so.  This creates a vicious cycle where some teams spend decades straddling the line between playoffs and late lottery picks.  Therefore, by following a dominated strategy, these middle teams are guaranteeing that they won’t be relevant today or even in the future.

Although these teams would be doing well and maximize their payoffs by taking part in lottery and selecting the best players, they end up striving hard to win the championship. The exhibit enclosed depicts the clear picture of payoffs and how the payoffs show a positive movement given that these teams play a dominant strategy. In the present situation context, these teams have a better chance of winning the lottery and moving towards a dominated strategy reduces their chances of winning the lottery. This is also amplified by the fact that no matter how hard they try, they will not be able to win the championship.  There are numerous reasons which motivate these teams to assume this dominated strategy rationality.

Still, this neglects the fact that owners are often billionaires who run the franchise not for monetary reasons, but personal ones.

Nevertheless, there are ways of limiting your victories while still maintaining the integrity of the game.

  Therefore there is a Principal-Agent problem here since it’d be better for these teams to tank than to do well.  Through long, guaranteed contracts, you can convince a coach to do what is best for the team rather than what is best for his / her immediate future.

The teams specifically at the top of the non-playoffs teams, motivated by above reasons, would like to be part of playoffs and have a better chance of winning the championships. For doing this, they would follow a strategy of resorting to trading potentially good players in the future for presently mediocre players so as to be part of the playoffs and move up the order. Similarly, the team who is to be excluded from the playoffs because of this trade will likely have to make a similar trade in order to maintain the status quo.  However playing this so called Shubik game does not do well for any of the teams.  A few of the teams with eventually be excluded from the playoffs anyways, despite trading for good players by using future assets. Hence in this way, these teams end up giving more for gaining something worth less.

Conclusion

In the context of the details and facts put forward, the non-playoffs teams have more payoffs for selecting the best pick and the playoffs teams have dominant strategy of winning the championships. However by playing a chicken game, these teams make a have sub-optimal winning strategy when the teams behave in the same way.  Although the best response for non-playoffs team is to tank and select the best players, they end up going against the tide to strive hard for winning the championship. However the question to be asked is whether tanking is the right game to be played for non-playoffs teams/ bottom-end of playoff teams.

In this context, it is reasonable to conclude that tanking could have a detrimental effect on the quality of the league and thus a better approach is required to prevent teams from tanking and breaking the spirit of the game.

In order to prevent tanking by teams, there is a need for changing the way the game is being played and develops a ‘best system’. The following can be reasonable solutions to change the game:

It is reasonable to conclude that although tanking is existent, the very fact that more teams do not tank shows that the ethics of fair play are still valued highly.  Therefore although teams can and do tank for many seasons to build up quality teams, like the aforementioned Oklahoma City Thunder, most choose to simply win as many games as possible regardless of the consequences.  Once teams act in accordance with the payoffs however, it might be beneficial to change the draft policy to be better in line with the incentives at hand.

[1] http://www.nydailynews.com/sports/basketball/nfl-back-study-nba-1984-draft-debacle-tanking-fallout-article-1.972775

[2] http://bleacherreport.com/articles/177927-ziller-vs-gladwell

[3] http://www.guardian.co.uk/football/2011/may/25/the-greatest-runners-up-ever?INTCMP=SRCH

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*Paper written in collaboration with Jason Sigler, Adytia Anupkumar and Sushant Kumar.