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When NBA teams don’t want to win

GAMES

TO LOSE

Team X Stefano Bertani Federico Fabbri Jorge Machado Scott Shapiro MBA 211 Game Theory, Spring 2010

Games to Lose – MBA 211 Game Theory

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Games to lose – When NBA teams don’t want to win

1. Introduction ................................................................................................................................................. 3

1.1 Situation ................................................................................................................................................ 3

1.2 NBA Structure ........................................................................................................................................ 3

1.3 NBA Playoff Seeding ............................................................................................................................... 4

1.4 NBA Playoff Tournament ........................................................................................................................ 4

1.5 Home Court Advantage .......................................................................................................................... 5

1.6 Structure of the paper ............................................................................................................................ 5

2. Situation analysis ......................................................................................................................................... 6

2.1 Scenario analysis .................................................................................................................................... 6

2.2 Calculating winning probabilities ............................................................................................................ 9

2.3 Payoffs calculation ............................................................................................................................... 10

3. Dissecting Popovich’s move ....................................................................................................................... 12

3.1 Playing to win or playing to lose? ......................................................................................................... 12

3.2 Payoff when both teams play to lose .................................................................................................... 13

3.3 Calculating the equilibrium solution ..................................................................................................... 14

3.4 Signaling by sitting the best players ...................................................................................................... 15

4. Conclusions ................................................................................................................................................ 18

4.1 What can NBA do to change the incentives to lose? ............................................................................. 18

4.2 The next chapter of the story ............................................................................................................... 19

5. Appendix ................................................................................................................................................... 21

5.1 Calculating the winning probabilities .................................................................................................... 21

5.2 Calculating the payoffs of a war of attrition .......................................................................................... 24


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1. Introduction

1.1 Situation

On Wednesday, April 14th, 2010, the San Antonio Spurs were heading to Dallas for the last game of

the NBA regular season. While both the Spurs and Dallas Mavericks were guaranteed spots in the

playoffs, the outcome of this game would have a far reaching impact on each team’s seed in the

playoffs. If San Antonio won, they could become a 6th seed, but if they lost, they would be a 7th

seed. San Antonio coach Greg Popovich submitted his team roster earlier that day. As usual, the

starting lineup included his two best players: Tim Duncan and Manu Ginobili.

Minutes before the 8pm tipoff, coach Popovich publicly instructed the leaders of his team, Duncan

and Ginobili, to sit this game out. In fact, they were dressed in business suits throughout the game

and were not even given a chance to play. So in this all-important final game of the season, why

would a coach bench his two best players? Perhaps he had developed a mental model of his

opponent and the entire NBA Western conference and determined that his expected payoff of

future winnings would increase by not playing his two stars against Dallas.

1.2 NBA Structure

The NBA playoffs are a single elimination tournament consisting of several “Best of 7” series. The

playoffs commence following the NBA regular season. In the NBA regular season there are 30 teams

who each play 82 games. These teams are split into two 15-team conferences: East and West. Each

conference is further divided into three divisions.


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1.3 NBA Playoff Seeding

The top eight teams (by overall winning percentage) in each conference are seeded one through

eight at the conclusion of the regular season. The team with the best record in each division is

named a division champion and guaranteed a seed of four or better. For example, let’s envision a

situation in which four teams in the Western Conference Pacific Division have the top four records in

the NBA Western Conference. All of these four teams are guaranteed playoff spots, but only the

division champion and the second best record holder will end up in the top 4. The third and fourth

seeds are allocated to the champions of the Southwest and Northwest divisions even though Pacific

division teams may have better records. So the fifth through eighth seeds are allocated by remaining

best records, starting with the teams that finished third and fourth place in the Pacific division.

If two teams are tied for the same record, the team with the better regular season record against the

opponent gets the higher seed. If they are tied in regular season playing, then the team with the

best record against teams within its own division gets the higher seed. If their division records are

tied then the team with the best winning percentage against teams in its conference gets the higher

seed. This continues with three more levels of tiebreaking potential1.

1.4 NBA Playoff Tournament

Once the 1st through 8th seeds are determined for each conference, the first round of the playoffs

beings with teams matched in a single elimination bracket. This first round consists of a best-of-

seven series such that the first team to win four matches moves on to the next round. In this round,

the first seed plays the eighth seed, second plays seventh, third plays sixth and fourth plays fifth. In

theory, the first seed, by having the best record in the conference, should enter the first round of the

1 NBA playoff seed tiebreaking rules http://www.cbssports.com/nba/story/7164305

http://www.cbssports.com/nba/story/7164305


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playoffs with the highest likelihood of advancing to the second round. By playing the eighth seed,

which has the worst record of all teams in the top 8 in that conference, the first seed should breeze

through to the second round.

1.5 Home Court Advantage

Unlike some other sporting leagues, all NBA games (except for the All-star game and certain

exhibition games) are played at one of the participating team’s venues. Games are never played at a

neutral 3rd party venue. It has also been shown that the home team is more likely to win a game, all

things being equal, than the visiting team. In a particular matchup in the playoffs, the team with the

better regular season record receives home court advantage according to NBA rules. Home court

advantage is manifested in the number of games played at a specific venue during a best-of-seven

series. The team with the better record and thus home court advantage will play in its arena during

four of the seven possible games (encounters 1, 2, 5 and 7). This rule is intended to motivate teams

to have the best regular season record possible so that they can maximize the amount of games they

play at home during the playoffs.

1.6 Structure of the paper

In chapter 2, we will analyze the scenarios from the game and the outcomes for Dallas and San

Antonio depending on the result of the matches. In chapter 3 we will describe the mental model of

San Antonio’s Coach Popovich and why he decided to sit the best players. Finally in chapter 4 we will

discuss what the NBA can do to change this negative incentive to lose.


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2. Situation analysis

2.1 Scenario analysis

On April 13th, before the last day of the regular season, the 8 teams in the western conference

heading to the playoffs were already determined. However, the final rankings and thus the resulting

playoff bracket were still open to wide swings. As it is shown in Table 1, Dallas, Denver, Utah and

Phoenix were all in a 1-point interval (according to regular season wins). Portland and San Antonio

were tied for the 6th position.

Los Angeles and Oklahoma were the only two teams locked into a position, respectively 1st and 8th.

All the other teams could still move in the ranking, depending on the results of the matches being

played in the last day of the regular season. Phoenix, for example, could improve its ranking by as

much as 3 slots, from 5th to 2nd (in case of a tie, the conference results are taken into account).

Table 1: Ranking before the last day of the regular season

Team Wins Ranking

Los Angeles Lakers 57 1

Dallas Mavericks 54 2

Denver Nuggets 53 T-3

Utah Jazz 53 T-3

Phoenix Suns 53 T-3

Portland Trailblazers 50 T-6

San Antonio Spurs 50 T-6

Oklahoma Thunder 49 8


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On April 14th 2010 at 8:00pm EST, the last turn of regular season offered four games that apparently

had a direct effect on the final ranking, specifically 1) San Antonio at Dallas, 2) Phoenix at Utah, 3)

San Francisco at Portland and 4) Memphis at Oklahoma.

The resulting 16 possible combinations of outcomes define the whole space of possible scenarios for

playoff brackets. Analyzing the situation, we firstly notice that the result of game 4) does not

influence the playoff seeding. Thus considering only the first three matches, we obtain 8 possible

combinations, some of which lead to the same scenario.

The 5 unique scenarios are summarized in Table 2 and 3, along with the different combinations of

results that lead to those scenarios. It is worth noticing that, for example, different combinations can

generate scenario 4, while if Dallas and Utah win, we will end up in scenario 5 no matter what

happens in the match between San Francisco and Portland.

Table 2: Possible unique playoff scenarios

Scenario Playoffs seeds

1 8 4 5 3 6 7 2

1 LA Okl.C. Phoenix Denver Dallas San

Antonio Portland Utah

2 LA Okl.C. Phoenix Denver Dallas Portland San

Antonio Utah

3 LA Okl.C. Denver Utah Phoenix San

Antonio Portland Dallas

4 LA Okl.C. Denver Utah Phoenix Portland San

Antonio Dallas

5 LA Okl.C. Phoenix Denver Utah Portland San

Antonio Dallas


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Table 3: Combinations of results of last day of regular season leading to unique playoff scenarios

Scenario

Generating

combinations

San Antonio

@ Dallas

Phoenix

@ Utah

Golden State

@ Portland

1 1/8 San Antonio Utah Golden State

2 1/8 San Antonio Utah Portland

3 1/8 San Antonio Phoenix Golden State

4 1/8 2/8

San Antonio Dallas

Phoenix Phoenix

Portland Indifferent

5 2/8 Dallas Utah Indifferent

Figure 1: Example of the playoff lineup for one of the unique scenarios (scenario 5)

Focusing on the final game of the season between San Antonio and Dallas, we noticed that if Dallas

wins then the two teams would be guaranteed to meet again in the first round of the playoffs

(scenarios 4 or 5), regardless of the outcomes of other games. If instead San Antonio beats Dallas, it

could play with Dallas, Utah or Phoenix, depending on the results of the other games (scenario 1, 2

or 3), while Dallas would meet either San Antonio or Portland.

1

8

5

4

2

7

6

3


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2.2 Calculating winning probabilities

The structure of the game and the high number of matches in a season make the use of statistics

applied to basketball more precise and reliable than for most other team sports. As a matter of fact,

an enormous amount of historic data is available, for teams as well as for individual players.

In order to estimate the winning probabilities of each team versus the other teams, we created a

method that relies on two key elements:

a) A measure of absolute strength, calculated using Hollinger’s formula, which uses a number of

variables (win percentage, point differential, etc.) and returns a numerical value

b) A measure of relative strength, which relies on the head-to-head results in the regular

seasons.

A detailed explanation of this method is given in the appendix. The resulting winning probabilities

are reported in Table 4. The table has to be read horizontally, i.e. each number represents the

probability that the team in the row beats the team in the column. For simplicity’s sake, we did not

quantify the home advantage, which for a seven games series is quite small.

Table 4: Winning probabilities

Team S. Antonio Phoenix Utah Portland Denver Oklahoma LA Dallas

S. Antonio 49.6% 48.0% 55.4% 63.2% 66.0% 63.6% 68.3%

Phoenix 50.4% 51.7% 57.4% 64.4% 60.5% 59.8% 67.1%

Utah 52.0% 48.3% 62.4% 57.7% 58.0% 58.1% 68.7%

Portland 44.6% 42.6% 37.6% 50.3% 55.6% 54.9% 62.1%

Denver 36.8% 35.6% 42.3% 49.7% 52.8% 52.9% 55.1%

Oklahoma 34.0% 39.5% 42.0% 44.4% 47.2% 47.6% 56.5%

LA 36.4% 40.2% 41.9% 45.1% 47.1% 52.4% 56.4%

Dallas 31.7% 32.9% 31.3% 37.9% 44.9% 43.5% 43.6%


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2.3 Payoffs calculation

Given that each of the 8 teams are already qualified for the playoffs, they will play the last game of

the regular season with the goal of maximizing the probabilities of winning the subsequent first

round of the playoffs. Los Angeles and Oklahoma for example, being mathematically sure of their

final ranking, will likely leave their best players on the bench to avoid injuries and save energy for the

playoffs. The other teams however, will have to consider the possible future scenarios and play

accordingly in order to maximize their probabilities of having access to the second round of the

playoffs.

Now let’s focus again on the game between San Antonio and Dallas and analyze how favorable for

each team are the possible scenarios arising from a win or a loss. We have already noticed that if San

Antonio loses its last game of the season against Dallas, it will meet Dallas again in the first round of

the playoffs. San Antonio in that case has a 68.3% probability of winning that series.

If instead San Antonio wins, its playoffs opponent will depend upon the results of the other two

relevant matches, i.e. Phoenix at Utah and San Francisco at Portland. The results of these two games

can obviously give four possible combinations, of which two will lead again to a San Antonio-Dallas

playoffs match-up, one to San Antonio-Utah and one to San Antonio-Phoenix. In our calculations we

considered each of these 4 combinations to be equally probable. This choice not only allows us to

keep the model reasonably simple, but it actually fits well the mental model of the teams. In fact, as

we have previously described, each team will think strategically about scenarios arising from winning

or losing and will play accordingly. Thus, applying the probabilities obtained in Table 4 would not be

correct. Moreover, it is hard to imagine that each team takes into account the complex strategic

thinking of teams other than its direct opponent. For example, before playing Phoenix, Utah will


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likely not consider all the strategic thinking going on between San Antonio and Dallas that we are

analyzing in this paper, but instead assume that each outcome is equally probable.

The results are summarized in Tables 5 and 6, where we clearly see that both San Antonio and Dallas

would actually be better off by losing. In particular, if San Antonio loses, it would see its probabilities

of winning the first round of the playoffs increase by 9.8%. Similarly, Dallas would improve its

chances by 3.1% by losing this game.

Table 5: San Antonio’s payoff of losing expressed as increased winning probabilities at 1st round of playoffs

14th April

result

Playoffs

opponent

Scenario

likelihood

Winning

probabilities

San Antonio Win Dallas 50% 68.3%

Phoenix 25% 49.6%

Utah 25% 48.0%

Total 100% 58.6%

Lose Dallas 100% 68.3%

Total 100% 68.3%

Payoff of losing as increased winning probabilities at 1st playoffs round 9.8%

Table 6: Dallas’ payoff of losing expressed as increased winning probabilities at 1st round of playoffs

14th April

result

Playoffs

opponent

Scenario

likelihood

Winning

probabilities

Dallas Win San Antonio 100% 31.7%

Total 100% 31.7%

Lose San Antonio 50% 31.7%

Portland 50% 37.9%

Total 100% 34.8%

Payoff of losing as increased winning probabilities at 1st playoffs round 3.1%


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3. Dissecting Popovich’s move

3.1 Playing to win or playing to lose?

From the previous chapter, we concluded that both Dallas and San Antonio have a future benefit in

the playoffs from losing their match on the 14th of April. Independently of the result, both teams

have already qualified for the playoffs. On one side, the team that loses the match will more likely be

matched with a weaker team in the playoffs and its chances of winning in the playoff would be

higher. On the other side, the team that wins the game will gain glory and some more fans, but since

the number of games in a season is relatively high, this benefit is marginal and we won’t quantify it.

Given this situation, each coach of Dallas and San Antonio has to decide between two options. Either

they can instruct their team to play to win (giving a motivating speech, organizing the team tactics,

etc.) or they can instruct the team to lose the game (the players would intentionally miss most shots,

turnover the ball to the other team, etc.). This set of decisions corresponds to a simultaneous game,

and we will have the following four different scenarios.

1) Both Dallas and San Antonio play with full performance to win: In this case, according to

Table 4, Dallas will win with a probability of 31.7% and San Antonio with a probability of

68.3%. As payoffs, the losing team will get the future benefit of facing a weaker team in the

playoffs. So:

Payoff (San Antonio) = 68.3% * 0% + 31.7% * 9.8% = 3.1%

Payoff (Dallas) = 68.3% * 3.1% + 31.7% * 0% = 2.1%

2) Dallas plays to win and San Antonio plays to lose: In this case, it’s certain that Dallas will win

and San Antonio will lose. The audience won’t recognize that San Antonio is playing


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intentionally to lose, since they still would be scoring points, but not as much as Dallas. The

payoffs are:

Payoff (San Antonio) = 9.8%

Payoff (Dallas) = 0%

3) Dallas plays to lose and San Antonio plays to win: This is the opposite scenario of 2), and it’s

certain that San Antonio will win and Dallas will lose. Dallas will enjoy the incremental payoff

of winning in the playoffs:

Payoff (San Antonio) = 0%

Payoff (Dallas) = 3.1%

4) Both teams play to lose: Despite this situation might sound pathetic, it is feasible to happen.

In this scenario, as the game progresses, it would be more evident by the audience that both

teams would be intentionally playing to lose. While both teams are in this scenario, we can

consider that the reputation of each team will suffer a marginal cost for each additional

action taken to lose. This series of moves would end when one team decides to change its

strategy and play to win. Indeed, independently of the coach’s orders, in a basketball match

one team has to win and the other has to lose. This scenario can be effectively described as a

war of attrition, and the payoffs depend on the mental model of each team.

3.2 Payoff when both teams play to lose

To find the equilibrium of the game we have first to figure out what the payoffs are when both

teams play to lose. More precisely, we are interested in understanding what both teams believe

their own payoffs would be at the end of the war of attrition they would engage on. We thus assume

to express costs and payoffs in terms of monetary value and we define the following variables:


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W is the benefit of winning the war of attrition (by losing the basketball game), converted

to monetary value. This number can be calculated based on the increase in winning

probability in the playoff game;

MC is the marginal cost per unit of time of playing to lose, converted to monetary value;

Q = W/MC is the time that the cost of playing to lose (MC t) equals the benefit that the

team gets by losing ( W).

Then, as proven in the appendix, the payoff in engaging in a war of attrition for both teams depends

on their mental model about the rival. Conceptually, the payoff depends strongly on the comparison

between QD and QSA, or in other words, for how long a team is willing to play the war of attrition. So

we can write:

SA

SA

D

DDDallas

MC

W

MC

Wprob attritionof war Payoff f

D

D

SA

SASAAntonio San

MC

W

MC


Assuming that both teams are optimistic about engaging in a war of attrition, they each believe that

they each are willing to stay longer in the war of attrition than the rival, and consequently their

expected payoffs are positive.

3.3 Calculating the equilibrium solution

Given the four scenarios described, and since both teams expect some benefit if they engage in a

war of attrition, then we will have a simultaneous game that resembles a prisoner’s dilemma, as

represented in Table 7. Both teams have a dominant best response of playing to lose, and that’s the

only Nash equilibrium of this game. Notice that the equilibrium estimated payoffs include the


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expected costs of playing to lose and thus should be converted to a common unity of measure, e.g.

monetary value. However, as it appears in Table 7, for a correct modeling of the game it is enough to

assume that the estimated equilibrium payoffs are >0, whatever unity of measure we choose.


Dallas

Play to win Play to lose

San

An

ton

io

Play to win 3.1%, 2.1% 0.0%, 3.1%

Play to lose 9.8%, 0%

3.4 Signaling by sitting the best players

Given this mental model, we can now understand why San Antonio’s coach Popovich changed the

team formation only some minutes before the game. If he really didn’t want to get his players tired,

as he mentioned after the game, he could have announced that days before the game, and could

only have sat them in uniform on the bench. However, he communicated it just minutes before and

placed them in the public without uniforms. In fact, he is a perfect game theorist, and by doing this

he signaled to Dallas that they wanted to lose but also made it more costly for Dallas to play to lose.

By sitting his best players in the stands, the coach transmitted an effective signal that he didn’t want

to win the game, and that San Antonio has high incentive to lose (high payoff and thus high

willingness to “pay” a high cost in terms of reputation and supporters satisfaction). In fact this

follows the three main characteristics of an effective signal:

Visible: Dallas’s coach understood immediately that Popovich was leaving his two best

players out of the game;

> 0, > 0


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Costly: since he is sitting the players in the audience without their uniforms, not being able to

join the game in the middle;

Differential: by benching the two best players, San Antonio became a much weaker team

with lower chances of winning. If Popovich was playing to win, this would be a meaningful

cost in team performance. Since he is showing that reducing the team performance is not

valuable to him, he is suggesting that he wants to lose.

San Antonio, by signaling that they have a high incentive to lose, are suggesting to Dallas that they

have a higher willingness to incur costs in playing to lose. This lowers Dallas’s estimated payoff

engaging in a War of Attrition:

SA

SA

D

DDallas

MC

W

MC


Another side of his strategy was the timing of the message. By taking action just some minutes

before the start of the game, Popovich is increasing Dallas’s marginal cost of playing to lose. Dallas

would have to quickly switch the team formation to a weaker team by benching also his best players.

However Dallas is playing at home and its fans bought tickets and are sitting in their seats. They

would be disappointed and angry at the coach. By using appropriate timing, San Antonio is

increasing the marginal cost of playing to lose, and lowering Dallas’s payoff:

SA

SA

D

DDallas

MC

W

MC


By doing this strategy, San Antonio’s coach is signaling to Dallas that they don’t have any chance to

“win” the war of attrition, and they would just be incurring in a negative payoff:


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Dallas

Play to win Play to lose

San

An

ton

io

Play to win 3.1%, 2.1% 0.0%, 3.1%

Play to lose

With his clever strategy, San Antonio’s coach effectively moved the equilibrium to the bottom left

quadrant, and forced Dallas to play to win the game (Table 8). As a final result of the match, as

expected, Dallas won and San Antonio lost.

9.8%, 0%

> 0%, < 0%

> 0, < 0


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4. Conclusions

4.1 What can NBA do to change the incentives to lose?

The situation faced by San Antonio in this season is not the first in the history of NBA. In several

situations, basketball teams end up tied in the last round of qualification and with a chance of

manipulating the playoff assignation. Table 9 shows that there has been an average of two ties per

NBA season of teams who are headed for the playoffs (note that for each tie, there are two or three

teams tied). On several occasions, three teams are tied for the same spot. This serves to show how

clustered it can be at the “top” of the league and that winning or losing a final game can make a big

difference in playoff opponents.

Table 9: Number of Playoff Seed Ties per NBA Season

The regular season ranking is not a perfect measure of the strength of a team. Moreover, each team

can suffer more or less the characteristics of a specific opponent. Situations like the one we have

described in this paper occur because the incentive to end up in a higher position in the standings

(getting more home court advantage) is not as strong as the incentive to match with a weaker rival in

the playoffs.


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For the NBA to reverse the trend of “playing to lose” it needs to institute policies that make every

game one that a team would want to win. One tactic has been to schedule games at the same time.

It is unknown whether games are scheduled at the same time intentionally to provoke winning (in

our case 4 games all tipping off at 8pm EST), or not. If the final few games were sequential, for

example, then the last game played would have a much higher probability of being moot. Having

simultaneous games reduced mootness.

One possibly tactic to increase the uncertainty of the first round playoff matches is randomness in

first round playoff assignments. This randomness would be similar to the NBA draft lottery, which

was instituted in 19902 to prevent “playing to lose.” Rather than guaranteeing that the first seed

play the eighth seed, second play seventh, and so on, the NBA should consider randomizing bracket

assignments in the first round of the playoffs. For example, at the end of the regular season a lottery

could be used to determine if the match-ups will be 1st- 8th and 2nd - 7th or instead 1st - 7th and 2nd -

8th , assigning probability p and 1-p to these two possible scenarios.

The same dynamic would apply to the third and fourth seeds with the fifth and sixth seeds,

respectively. In the same way the NBA draft brings attention and TV presence to the NBA, the

league can similarly benefit by creating hoopla around the “first round bracket lottery.”

4.2 The next chapter of the story

While our analysis took place during the first round of the NBA playoffs, we closely monitored the

action to test our hypothesis. San Antonio (the 7th seed) ended up beating Dallas (the 2nd seed) in

the first round of the playoffs. However, San Antonio lost to Phoenix (4-0) in the 2nd round of the

playoffs. By losing to San Antonio on the last game of the season on April 14, San Antonio was able

2 http://en.wikipedia.org/wiki/NBA_Draft_Lottery#1990.E2.80.93present:_Weighted_lottery_system


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to prolong the inevitable encounter with Phoenix or another team to which it was disadvantaged. As

a result, San Antonio benefited from reaching the second round of the playoffs in several ways: more

TV coverage, exposure to fans, ticket and broadcast revenues. Had San Antonio fought and won on

April 14th, it would have had a much lower probability of entering the 2nd round of the playoffs and

thus a lower expected benefit.


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5. Appendix

5.1 Calculating the winning probabilities

As explained in the second chapter, the methodology we developed to assess the winning

probability of each team versus each other team relies on a measure of absolute strength and a

measure of relative strength. The former is derived from the Hollinger formula, while the latter uses

the statistics of head-to-head match-ups in the regular season.

Absolute strength - Hollinger’s formula

This formula was developed by John Hollinger, an influential figure in the field of APBRmetrics, the

quantitative analysis of basketball. The formula, which returns a numerical rating of the absolute

strength of a team is the following:

RATING = (((SOS-0.5)/0.037)*0.67) + (((SOSL10-0.5)/0.037)*0.33) + 100 + (0.67*(MARG+(((ROAD-

HOME)*3.5)/(GAMES))) + (0.33*(MARGL10+(((ROAD10-HOME10)*3.5)/(10)))))

The input variables that appear in it are:

SOS = Season win/loss percentage of team's opponents, expressed as a decimal

SOSL10 = Season win/loss percentage of team's last 10 opponents, expressed as a decimal

MARG = Team's average scoring margin

MARGL10 = Team's average scoring margin over the last 10 games


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HOME = Team's home games

HOMEL10 = Team's home games over the last 10 games

ROAD = Team's road games

ROADL10 = Team's road games over the last 10 games

GAMES = Team's total games

We used the rating obtained with this formula to create the matrix shown in Table 10. The

percentage values are the difference in Hollinger’s rating between the team in the row and the team

in the column, divided by the maximum rating difference (i.e., San Antonio and Dallas, which have a

difference in rating of 4.155 )

Table 10: Normalized differential Hollinger’s ratings

did S.Antonio Phoenix Utah Portland Denver Okl. LA Dallas

Holl. Rating

106.55 106.288 105.936 104.397 103.81 103.75 103.732 102.395

S.Antonio 106.55 0% 6% 15% 52% 66% 67% 68% 100%

Phoenix 106.288 -6% 0% 8% 46% 60% 61% 62% 94%

Utah 105.936 -15% -8% 0% 37% 51% 53% 53% 85%

Portland 104.397 -52% -46% -37% 0% 14% 16% 16% 48%

Denver 103.81 -66% -60% -51% -14% 0% 1% 2% 34%

Okl. City 103.75 -67% -61% -53% -16% -1% 0% 0% 33%

LA 103.732 -68% -62% -53% -16% -2% 0% 0% 32%

Dallas 102.395 -100% -94% -85% -48% -34% -33% -32% 0%


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Relative strength - Seasonal match-ups

While Hollinger’s formula is quite a good estimator of the strength of a team, it does not take into

account that a team can suffer against the particular characteristics of another team, resulting in a

bad track record in the direct matches. This actually happens quite often in the NBA and we decided

to take it into account. Table 11 shows the winning percentages in the regular season in the head-to-

head matches for every possible pairing.

Table 11: Seasonal match-up score

Team S.Antonio Phoenix Utah Portland Denver Okl. LA Dallas

S.Antonio 33% 0% 0% 50% 75% 50% 33%

Phoenix 67% 50% 33% 75% 33% 25% 33%

Utah 100% 50% 100% 25% 25% 25% 67%

Portland 100% 67% 0% 25% 75% 67% 75%

Denver 50% 25% 75% 75% 75% 75% 33%

Oklahoma 25% 67% 75% 25% 25% 25% 50%

LA 50% 75% 75% 33% 25% 75% 50%

Dallas 67% 67% 33% 25% 67% 50% 50%

The final winning probability is then calculated as:

Pwin (A vs B) = 0.5 + (Normalized Hollinger rating differential / 5) + (Seasonal match-up / 10)

The winning probability is capped at 80%. However, in our case we never reach probabilities higher

than 70% (see Table 4). The numbers we obtained with this methodology are in most cases

satisfactorily coherent with experts’ predictions.


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5.2 Calculating the payoffs of a war of attrition

To analyze the payoffs of the war of attrition between two basketball teams playing to lose, we have

to start by defining the relevant variables:

W is the benefit of winning the war of attrition (by losing the basketball game), converted

to monetary value. This number can be calculated based on the increase in winning

probability in the playoff game;

MC is the marginal cost per unit of time of playing to lose, converted to monetary value;

Q = W/MC is the time that the cost of playing to lose (MC t) equals the benefit that the

team gets by losing ( W);

f is the probability distribution function that defines the expectation that a team has about

the valuation of the rival team (in the case that both teams are playing to lose). For example

f(QD) describes the expectation that San Antonio coach has regarding the value of Dallas to

lose. This curve is dependent on the mental model of the team.

Given these variables, we can calculate the payoffs that San Antonio and Dallas expect to get from

the war of attrition. Let’s assume that they are risk averse and willing to stay in the war of attrition

only until the cost gets similar to the benefit. In other words this means that there are “bidding”

their value for the war of attrition:

DD

SASA

Q)b(Q

Q)b(Q

So in this case the payoffs for San Antonio are (the payoffs for Dallas are the symmetrical solution):

SADSA

SADDSASA

SADSASA

SADDSASA

DSAQQ if W 0

QQ if )QQ(MC

QQ if QMC 0

QQ if QMC W )(QPayoff


25

SA

SA

Q

DDSA

Q

0

DDDSASA

0

DDDSADSA dQ )f(Q WdQ )f(Q )QMC W()dQf(Q )(QPayoff)](QE[Payoff

The expression above has a positive and a negative term. The first represents the gain that San

Antonio gets if they win the war of attrition, and the second the cost of taking the risk of engaging in

the war of attrition and losing. Depending on the mental model of the coach about the rival, one of

these terms can be dominant. In the following graphs we represent two different mental models,

and the implication in terms of the equation above:

0)dQf(Q )QQ(MC

)dQf(Q W)dQf(Q )QQ(MC]E[Payoff

SA

SA

SA

Q

0

DDDSASA

Q

DDSA

Q

0

DDDSASASA

0dQ )f(Q W

dQ )f(Q WdQ )f(Q )QQ(MC]E[Payoff

SA

SA

SA

Q

DDSA

Q

DDSA

Q

0

DDDSASASA

To summarize, the expected payoff for a team depends on its benefit of winning the war of attrition,

and on its mental model about the rival’s benefit. When the coach believes that his team’s benefit is

higher than the rival’s, then his expected payoff of engaging in the war of attrition is positive. Let’s

QD

f(QD)

E[QD] QSA

QD

f(QD)

E[QD] QSA


26

assume that both Dallas and San Antonio coaches believe that their respective benefits of losing the

game are higher than the other’s. Thus they both believe that their expected payoffs of engaging in a

war of attrition are positive, i.e.:

0)dQf(Q )QQ(MCdQ )(Qf WdQ )(Qf )QQ(MC]E[PayoffD

D

D Q

0

SASASADD

Q

SASASAD

Q

0

SASASASADDD

0dQ )(Qf )QQ(MCdQ )(Qf WdQ )(Qf )QQ(MC]E[PayoffSA

SA

SA Q

0

DDDDSASA

Q

DDDSA

Q

0

DDDDSASASA

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