Build a better baseball team

Zach Hynes

2011 Free Agents

Player Value (WAR) Cost (Millions)

Albert Pujols 5.1 5

Brandon Phillips 6.0 2

Carlos Beltran 4.7 3

Grady Sizemore 0.2 1

Jose Reyes 6.2 3

Jimmy Rollins 3.8 2

Michael Cuddyer 3.1 2

Yadier Molina 4.7 1

John Henry (Owner of the Red Sox) gives you $7 million to spend in free agency. Pick the best set of players by value. IGNORE player position and the needs of the Red Sox.

How do we pick our team?   Strategies

How do we pick our team?   Strategies

  Pick all the best players   Pick the players that have the best value/cost ratio

How do we pick our team?   Strategies

  Pick all the best players   Pick the players that have the best value/cost ratio

  Those are good strategies, but…

How do we pick our team?   Strategies

  Pick all the best players   Pick the players that have the best value/cost ratio

  Those are good strategies, but…

“good” doesn’t cut it in the AL East

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina01234567

1

We’ve wasted all our money on John Lackey! We have no money to spend. How much value could you get with no money to spend? (This is not a trick question)

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.01234567

1

Suppose you had $1 million to spend, and Sizemore was the only player available.

S: Grady Sizemore: $1 Million, 0.2 WAR

What’s the best value for our money?

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.01234567

1

Suppose you had $1 million to spend, and Sizemore was the only player available.

S: Grady Sizemore: $1 Million, 0.2 WAR

Option 1: Pick no one

Option 2: Pick Sizemore

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.01 0.2234567

1

S: Grady Sizemore: $1 Million, 0.2 WAR

With $1 million to spend and only Sizemore available:

The best value for our money is 0.2.

Option 1: Pick no one

Option 2: Pick Sizemore

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.01 0.22 0.23 0.24 0.25 0.26 0.27 0.2

1

S: Grady Sizemore: $1 Million, 0.2 WAR

With more than $1 million to spend and only Sizemore available:

The best value for our money is still 0.2.

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.01 0.22 0.23 0.24 0.25 0.26 0.27 0.2

1

B: Carlos Beltran: $3 Million, 4.7 WAR

With Sizemore and Beltran available and 1 million to spend:

Option 1: Ignore Beltran

We don’t have enough money to sign Beltran!

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.23 0.24 0.25 0.26 0.27 0.2

1

With Sizemore and Beltran available and 3 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran

Option 2: Add Beltran

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.23 0.24 0.25 0.26 0.27 0.2

1

With Sizemore and Beltran available and 3 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran Option 2: Add Beltran

Spend our $3 million as if Sizemore was the only

player available. Spend $3 million on Beltran

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.23 0.24 0.25 0.26 0.27 0.2

1

With Sizemore and Beltran available and 3 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran Option 2: Add Beltran

Spend $3 million on Beltran Best($3 million, [Sizemore])

= 0.2

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.23 0.24 0.25 0.26 0.27 0.2

1

With Sizemore and Beltran available and 3 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran Option 2: Add Beltran

Best($4 million, [Sizemore])

= 0.2

Best($0 million,[Sizemore])+ Value(Beltran) = 0.0 + 4.7

= 4.7

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.2 0.23 0.2 4.74 0.25 0.26 0.27 0.2

1

With Sizemore and Beltran available and 3 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran

Option 2: Add Beltran

Beltran is more valuable than Sizemore!

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.2 0.23 0.2 4.74 0.25 0.26 0.27 0.2

1

B: Carlos Beltran: $3 Million, 4.7 WAR

With Sizemore and Beltran available and 4 million to spend:

Option 1: Ignore Beltran Option 2: Add Beltran

Spend our $4 million as if Sizemore was the only

player available. Spend $4 million on Beltran

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.2 0.23 0.2 4.74 0.25 0.26 0.27 0.2

1

With Sizemore and Beltran available and 3 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran Option 2: Add Beltran

Spend $3 million on Beltran Best($4 million, [Sizemore])

= 0.2

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.2 0.23 0.2 4.74 0.25 0.26 0.27 0.2

1

With Sizemore and Beltran available and 4 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran Option 2: Add Beltran

Best($4 million, [Sizemore])

= 0.2

Best($1 million,[Sizemore])+ Value(Beltran) = 0.2 + 4.7

= 4.9

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.01 0.2 0.22 0.2 0.23 0.2 4.74 0.2 4.95 0.26 0.27 0.2

1

With Sizemore and Beltran available and 4 million to spend:

B: Carlos Beltran: $3 Million, 4.7 WAR

Option 1: Ignore Beltran

Option 2: Add Beltran

We can afford both players!

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.0 0.0 0.0 0.01 0.2 0.2 0.2 0.2 0.22 0.2 0.2 3.1 3.8 3.83 0.2 4.7 4.7 4.7 4.74 0.2 4.9 4.9 6.9 6.95 0.2 4.9 7.8 8.5 8.56 0.2 4.9 8.0 8.77 0.2 4.9 8.0 11.6

1

U: Albert Pujols: $5 Million, 5.1 WAR

With Sizemore, Beltran, Cuddyer, Rollins, and Pujols available, 6 million to spend:

Option 1: Ignore Pujols Option 2: Add Pujols

Spend our $6 million on best combination of Sizemore,

Beltran, Cuddyer, and Rollins

Spend $5 million on Pujols, $1 million on the best combination of Sizemore,

Beltran, Cuddyer, and Rollins

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.0 0.0 0.0 0.01 0.2 0.2 0.2 0.2 0.22 0.2 0.2 3.1 3.8 3.83 0.2 4.7 4.7 4.7 4.74 0.2 4.9 4.9 6.9 6.95 0.2 4.9 7.8 8.5 8.56 0.2 4.9 8.0 8.77 0.2 4.9 8.0 11.6

1

U: Albert Pujols: $5 Million, 5.1 WAR

With Sizemore, Beltran, Cuddyer, Rollins, and Pujols available, 6 million to spend:

Option 1: Ignore Pujols Option 2: Add Pujols

Best($6M,[Sizemore, Beltran,Cuddyer,Rollins])

= 8.7

Best($1M,[Sizemore, Beltran,Cuddyer,Rollins]) + Value(Pujols)

= 0.2 + 5.1 = 5.3

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.0 0.0 0.0 0.01 0.2 0.2 0.2 0.2 0.22 0.2 0.2 3.1 3.8 3.83 0.2 4.7 4.7 4.7 4.74 0.2 4.9 4.9 6.9 6.95 0.2 4.9 7.8 8.5 8.56 0.2 4.9 8.0 8.7 8.77 0.2 4.9 8.0 11.6

1

U: Albert Pujols: $5 Million, 5.1 WAR

With Sizemore, Beltran, Cuddyer, Rollins, and Pujols available, 6 million to spend:

Option 1: Ignore Pujols Option 2: Add Pujols

We’re better off with Sizemore, Beltran, and Rollins

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.2 0.2 0.2 0.2 0.2 0.2 0.2 4.72 0.2 0.2 3.1 3.8 3.8 6.0 6.0 6.03 0.2 4.7 4.7 4.7 4.7 6.2 6.2 10.74 0.2 4.9 4.9 6.9 6.9 9.8 9.8 10.95 0.2 4.9 7.8 8.5 8.5 10.7 12.2 14.56 0.2 4.9 8.0 8.7 8.7 12.9 12.9 16.97 0.2 4.9 8.0 11.6 11.6 14.5 16.0

1

M: Yadier Molina: $1 Million, 4.7 WAR

With all players listed available, 7 million to spend:

Option 1: Ignore Molina Option 2: Add Molina

Spend $7 million on best combination of all but Molina

Spend $1 million on Molina, $6 million on best

combination of all but Molina

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.2 0.2 0.2 0.2 0.2 0.2 0.2 4.72 0.2 0.2 3.1 3.8 3.8 6.0 6.0 6.03 0.2 4.7 4.7 4.7 4.7 6.2 6.2 10.74 0.2 4.9 4.9 6.9 6.9 9.8 9.8 10.95 0.2 4.9 7.8 8.5 8.5 10.7 12.2 14.56 0.2 4.9 8.0 8.7 8.7 12.9 12.9 16.97 0.2 4.9 8.0 11.6 11.6 14.5 16.0

1

M: Yadier Molina: $1 Million, 4.7 WAR

With all players listed available, 7 million to spend:

Option 1: Ignore Molina Option 2: Add Molina

Best($7M,[All but Molina])

= 16.0

Best($6M,[All but Molina]) + Value(Molina)

= 12.9 + 4.7 = 17.6

Think like a small-market team

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.2 0.2 0.2 0.2 0.2 0.2 0.2 4.72 0.2 0.2 3.1 3.8 3.8 6.0 6.0 6.03 0.2 4.7 4.7 4.7 4.7 6.2 6.2 10.74 0.2 4.9 4.9 6.9 6.9 9.8 9.8 10.95 0.2 4.9 7.8 8.5 8.5 10.7 12.2 14.56 0.2 4.9 8.0 8.7 8.7 12.9 12.9 16.97 0.2 4.9 8.0 11.6 11.6 14.5 16.0 17.6

1

M: Yadier Molina: $1 Million, 4.7 WAR

With all players listed available, 7 million to spend:

Option 1: Ignore Molina Option 2: Add Molina

Molina’s valuable… and cheap

The best solutions… at any price

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina01 S S S S S S S M2 S S C R R P P P3 S B B B B SP SP PM4 S SB SB CR CR RP RP SPM5 S SB BC BR BR BP PY RPM6 S SB SBC SBR SBR CRP CRP PYM7 S SB SBC BCR BCR BRP RPY CRPM

1

Why do we need to build up the table?

Don’t want to miss anything!

Best($7M, All but Molina): Reyes, Phillips, Rollins

Best($7M, All): Cuddyer, Rollins, Phillips, Molina

The best solutions… at any price

Total $(Millions) Spent

Players

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina01 S S S S S S S M2 S S C R R P P P3 S B B B B SP SP PM4 S SB SB CR CR RP RP SPM5 S SB BC BR BR BP PY RPM6 S SB SBC SBR SBR CRP CRP PYM7 S SB SBC BCR BCR BRP RPY CRPM

1

Reyes is better than Cuddyer and Molina… but Cuddyer and Molina combined are better

Best($7M, All but Molina): Reyes, Phillips, Rollins

Best($7M, All): Cuddyer, Rollins, Phillips, Molina

Results   Our solution: 17.6 WAR (Cuddyer, Rollins, Phillips, Molina)   Picking the best value-to-cost players:

17.1 WAR (Reyes, Sizemore, Phillips, Molina)

Sizemore Beltran Cuddyer Rollins Pujols Phillips Reyes Molina0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.2 0.2 0.2 0.2 0.2 0.2 0.2 4.72 0.2 0.2 3.1 3.8 3.8 6.0 6.0 6.03 0.2 4.7 4.7 4.7 4.7 6.2 6.2 10.74 0.2 4.9 4.9 6.9 6.9 9.8 9.8 10.95 0.2 4.9 7.8 8.5 8.5 10.7 12.2 14.56 0.2 4.9 8.0 8.7 8.7 12.9 12.9 16.97 0.2 4.9 8.0 11.6 11.6 14.5 16.0 17.6

1

Going Beyond DP   Technique known as

“Dynamic Programming”   “It’s incredibly easy to play first base” –

Billy Beane, Moneyball   You can use other techniques from

computer science to develop more sophisticated models for optimizing value subject to certain constraints   5 starting pitchers   4 outfielders   One closer

Credits

  Theo Epstein, former Red Sox GM:http://www.bleachernation.com/wp-content/uploads/2011/09/theo-epstein.jpg

  Game 162 Drama: http://sportsillustrated.cnn.com/2011/writers/tom_verducci/09/29/game.162.drama/index.html

  Scott Hatteberg in Moneyball:http://twodollarcinema.blogspot.com/2012/03/how-can-you-not-be-romantic-about.html

  Inspiration from Introduction to Algorithms by Thomas Cormen, Charles Leiserson, Ron Rivest, and Clifford Stein (Modeled on Ex. 15-12 of 3rd edition)