## NBA Draft Lottery

The National Basketball Association (NBA) uses an amateur draft, in which teams are allowed to select collegiate and international players that are new to the league. The teams select in reverse order of the previous year’s record, from worst to first. Essentially, the draft allows the bad teams an opportunity to improve in the future, which creates greater parity among the teams. This draft, along with a salary cap, is in place to help the NBA achieve one of its primary goals: establish competitive balance, and as a result produce compelling a product as possible. However, the draft has become incentive enough to entice some teams to intentionally lose, which is commonly referred to as “tanking,” which in turn creates a less compelling product for the NBA.

Why tank? It seems counterintuitive, as an NBA team should want to field as competitive a team as possible, to drive up revenue from broadcasting, ticket and paraphernalia sales. However, while tanking reduces the odds today of winning, it can greatly increase the odds of future winning, often enough to justify tanking. Additionally, in the NBA it is often only the very best teams that have a chance of winning a championship each year, and the majority of championship caliber teams are lead by one of the Top 10 players in the league, future Hall of Famers. For small market teams especially, who have little hope of luring a star free agent or trading from a superstar, the draft is often the only way to have one of these generational players play for their squad.

The Philadelphia 76ers, a team that has obviously and intentionally embraced tanking, have incentivized winning championships above all else, and are trying to find a future Hall of Famer in the draft. Using the 76ers as a case study, we can create a simplified game of sorts, giving them two options: Tank, or Not tank. If they decide to tank, they will win 15 games this year, and have the first pick in the draft. If they decide not to tank, they will field a below average team and win 35 games and have the fourteenth pick in the draft. A heuristic we can use to use to value the picks will be the number of hall of famers taken at that pick in the past. We can say that the 76ers value the chance at a hall of famer 5 times more than current wins, so we find:

Value for 76ers = wins + 5 x No. of HOFers at pick.

The NBA wishes to field a competitive product, so it is wants to disincentive tanking. Thus, in 1984 it introduced a lottery, which gives odds that each team will pick at each spot, and the worst teams have the best odds. For example, the first pick has approximately a 25% of winning the pick 1, a 21% chance of 2, 18% chance of 3, and 36% chance of 4. We can update the value equation:

Value for 76ers = wins + 5 x sum of (chance of pick x No. of HOFers at pick.)

Still, with this system, the 76ers are tanking. As a result, the NBA is thinking of implementing a new system that more evenly distributes odds, which hopes to prevent tanking at the cost of making it harder for bad teams to get better. We can say that the NBA only values its product, so if teams aren’t tanking it gains a value of +1. The resulting game is:

NBA\76ers Not Tank Tank

Old Lotto 1, 80 0, 110

New Lotto 1, 90 0, 80

Now, this is not an ideal game: the events are not simultaneous, and the event only occurs once, and so some of the logic about game theory and Nash Equilibria may not apply ideally. In this heuristic, the NBA should implement the New Lottery and the 76ers should not tank, which is not a complex result. However, the complexity comes from the values at which the NBA should set its Lottery odds; it should set them such that Not Tank is Philadelphia’s Best response to the new Lottery.

Source: http://grantland.com/the-triangle/nba-lottery-reform-is-coming/