MARKETS, MECHANISMS, MACHINES University of Virginia, Spring 2019 Class 9: Applications of Linear Programming 12 February 2019 cs4501/econ4559 Spring 2019 David Evans and Denis Nekipelov https://uvammm.github.io Zinedine Zidane’s penalty in 2006 World Cup Final

What is the “Programming” in “Linear Programming”?

No content

Linear Program 3 ) subject to ≤

Possibilities 4 ) subject to ≤ Infeasible: no solution satisfies all the inequalities

Possibilities 5 ) subject to ≤ Infeasible: no solution satisfies all the inequalities 6 subject to 6 ≤ 5 −6 ≤ −7

Possibilities 6 ) subject to ≤ Unbounded: no limit on maximized value 6 subject to −6 ≤ −7

Brewer’s Problem 7 Robert Sedgewick and Kevin Wayne, Princeton Course Limited Resources Available Recipes and Outputs How much of each should we brew to maximize total profits?

8 Robert Sedgewick and Kevin Wayne, Princeton Course

9 Robert Sedgewick and Kevin Wayne, Princeton Course Maxmimize = 13 + 23 subject to constraints: 5 + 15 ≤ 480 (corn) 4 + 4 ≤ 160 hops 35 + 20 ≤ 1190 (malt) ≥ 0, ≥ 0

10 Robert Sedgewick and Kevin Wayne, Princeton Course Maxmimize = 13 + 23 subject to constraints: 5 + 15 ≤ 480 (corn) 4 + 4 ≤ 160 hops 35 + 20 ≤ 1190 (malt) ≥ 0, ≥ 0 Feasible Region (convex polygon)

11 Robert Sedgewick and Kevin Wayne, Princeton Course Maxmimize = 13 + 23 subject to constraints: 5 + 15 ≤ 480 (corn) 4 + 4 ≤ 160 hops 35 + 20 ≤ 1190 (malt) ≥ 0, ≥ 0

12 Robert Sedgewick and Kevin Wayne, Princeton Course Optimal solution must be at an extreme point (intersection of constraints)

see Jupyter Notebook 13

Dual Problem 14 Brewer Entrepreneur = 13 + 23 subject to constraints: 5 + 15 ≤ 480 (corn) 4 + 4 ≤ 160 hops 35 + 20 ≤ 1190 (malt) ≥ 0, ≥ 0 = 480 + 160ℎ + 1190 , ℎ, = unit prices for ingredients 5 + 4ℎ + 35 ≥ 13 (ale) 15 + 4ℎ + 20 ≥ 23 beer ≥ 0, ℎ ≥ 0, ≥ 0

Dual Theorem 15 ) subject to ≤ ≥ 0 ) subject to ) ≥ ≥ 0 primal problem dual problem T = T Proof sketch: roughly – show that simplex algorithm for both problems produces same result. if both feasible:

see Jupyter Notebook 16

Example: Network Flow 17 Source: Avrim Blum, Manual Blum, CMU course: https://www.cs.cmu.edu/~avrim/451f11/lectures/lect1101.pdf

Example: Network Flow 18 Source: Avrim Blum, Manual Blum, CMU course: https://www.cs.cmu.edu/~avrim/451f11/lectures/lect1101.pdf hi ≤ 4 hj ≤ 2 ik ≤ 3 jk ≤ 2 kj ≤ 1 kl ≤ 2 ml ≤ 4 maximize kl + ml subject to:

Example: Network Flow 19 Source: Avrim Blum, Manual Blum, CMU course: https://www.cs.cmu.edu/~avrim/451f11/lectures/lect1101.pdf hi ≤ 4 hj ≤ 4 ik ≤ 3 jk ≤ 2 kj ≤ 1 kl ≤ 2 jm ≤ 3 ml ≤ 4 maximize kl + ml subject to: hi = ik kl + kj = ik hj + kj = jm + jk ml = jm

see Jupyter Notebook 20

Minimize Cost: Network Flow 21 Source: Avrim Blum, Manual Blum, CMU course: https://www.cs.cmu.edu/~avrim/451f11/lectures/lect1101.pdf $2 $2

Penalty Kicks 22

Penalty Kicks Model Goalkeeper Direction Left Center Right Kicker Direction Left Center Right 23

Penalty Kicks Model Goalkeeper Direction Left Center Right Kicker Direction Left 0.20 0.90 0.90 Center 0.99 0.01 0.99 Right 0.98 0.98 0.40 24 two-player, zero-sum game

Goalkeeper Direction Left Center Right Kicker Direction Left 0.20 0.90 0.90 Center 0.99 0.01 0.99 Right 0.98 0.98 0.40 25 Kicker’s goal: maximize minimum payoff (probability to score assuming goalkeeper behaves optimally)

Goalkeeper Direction Left Center Right Kicker Direction Left 0.20 0.90 0.90 Center 0.99 0.01 0.99 Right 0.98 0.98 0.40 26 Kicker’s goal: maximize minimum payoff (probability to score assuming goalkeeper behaves optimally) Variables: opqr , sptrpu , uvwxr , v ≥ 0 and ∑ v = 1 v Objective: maximize Constraints: | v v} ≥ v

see Jupyter Notebook 27

28

Charge If you don’t have a team for Project 3, see us now 29