Mixed Integer Programming (MIP) for Daily Fantasy Sports, Statistics and Marketing

Transcription

1 Mixed Integer Programming (MIP) for Daily Fantasy Sports, Statistics and Marketing Juan Pablo Vielma Massachusetts Institute of Technology AM/ES 121, SEAS, Harvard. Boston, MA, November, 2016.

2 MIP & Daily Fantasy Sports

3 Example Entry

4 100% of the money in the top 20% lineups 26% of the money in the top 10 lineups (0.04%)

5 Building a Lineup

6 Integer Programming Formulation We will make a bunch of lineups consisting of 9 players each Use an integer programming approach to find these lineups Decision variables

7 Basic Feasibility 9 different players Salary less than $50,000 Basic constraints

8 Position Feasibility Between 2 and 3 centers Between 3 and 4 wingers Between 2 and 3 defensemen 1 goalie Position constraints

9 Team Feasibility At least 3 different NHL teams Team constraints

10 Maximize Points Forecasted points for player p: Score type Points Goal 3 Assist 2 Shot on Goal 0.5 Blocked Shot 0.5 Short Handed Point Bonus (Goal/Assist) 1 Shootout Goal 0.2 Hat Trick Bonus 1.5 Win (goalie only) 3 Save (goalie only) 0.2 Goal allowed (goalie only) -1 Shutout Bonus (goalie only) 2 Table 1 Points system for NHL contests in DraftKings. Points Objective Function

11 Lineup Projections: $9500 $2700 $4600 $3800 $4600 $6400 $5200 $5100 $8000 W UTIL D D C C W W G 23 points on average

12 Need > 38 points for a chance to win

13 Increase variance to have a chance

14 Structural Correlations - Teams

15 Structural Correlations - Lines Goal = 3 pt, assist = 2 pt

16 Structural Correlations Lines = Stacking At least 1 complete line (3 players per line) At least 2 partial lines (at least 2 players per line) 1 complete line constraint 2 partial lines constraint

17 Structural Correlations Goalie Against Opposing Players

18 Structural Correlations Goalie Against Skaters No skater against goalie No skater against goalie constraint

19 Good, but not great chance

20 Play many diverse Lineups Make sure lineup l has no more than g players in common with lineups 1 to l-1 Diversity constraint

21 Were we able to do it? 200 lineups

22 Policy Change 200 lineups -> 100 lineups

23 Were we able to continue it? December 12, 2015 > $15K 100 lineups

24 How can you do it? Download Code from Github:

25 Performance Time < 30 Minutes Runtime (minutes) CBC GLPK Gurobi CBC GLPK Gurobi Solver

26 MIP and Statistics: Inference for the Chilean Earthquake

27 The 2010 Chilean Earthquake

28 6 th Strongest in Recorded History (8.8)

29 Impact on Educational Achievement? PSU = SAT

30 Earthquake Intensity + Great Demographic Info

31 Randomized experiment Treatment / control have similar characteristics (covariates). Treatment( Control(

32 Covariate Balance Important for Inference Dose Covariate 1 2 Gender Male Female School SES Low Mid-low Medium Mid-high High Mother s education Primary Secondary Technical College Missing

33 Observational Study: e.g. After Earthquake Treatment / control can have different characteristics. Treatment( Control( Solution = Matching?

34 Matching Treated Units: T = {t 1,...,t T } Control Units: C = {c 1,...,c C } Observed Covariates: P = {p 1,...,p P } Covariate Values: x t = x t p p2p, x c = x c p p2p, t 2 T c 2 C

35 Nearest Neighbor Matching minimize m subject to 0 apple m t,c apple 1 X t2t X c2c X t2t X c2c t,cm t,c + X i2i m t,c = m, t 2 T 1 m t,c apple 1, c 2 C! i µ i (m m t,c 2 {0, 1}, t 2 T, c 2 C i (m) apple " i, i 2 I e.g. t,c = x t x c 2 Easy to solve

36 Balance Before After Matching SIMCE school (decile) SIMCE student (decile) GPA ranking (decile) Attendance (decile) Rural school Catholic school High SES school Mid High SES school Mid SES school Mid Low SES school Public School Voucher School

37 Maximum Cardinality Matching max s.t. X t2t p,k X X t2t c2c m t,c X m t,c apple 1, t2t X m t,c apple 1, c2c X c/2c p,k m t,c = X t/2t p,k X c2c p,k m t,c 8c 2 C 8t 2 T 8p 2 P,k 2 K(p) m t,c 2 {0, 1} 8t 2 T, c 2 C. K(p) ={x c p} c2p [ {x t p} t2t C p,k = {c 2 C : x c p = k} T p,k = {t 2 T : x t p = k} Very hard to solve ( and very hard to understand! )

38 Advanced Maximum Cardinality Matching max s.t. X t2t X t2t X x t t2t p,k x t = x t = X c2c y c, X Matching without matching variables K(p) ={x c p} c2p [ {x t p} t2t C p,k = {c 2 C : x c p = k} c2c p,k y c, 8p 2 P, k 2 K(p) x t 2 {0, 1} 8t 2 T y c 2 {0, 1} 8c 2 C. T p,k = {t 2 T : x t p = k} Easy to solve: Small, but inherits matching properties

39 Balance Before After Cardinality Matching SIMCE school (decile) SIMCE student (decile) GPA ranking (decile) Attendance (decile) Rural school Catholic school High SES school Mid High SES school Mid SES school Mid Low SES school Public School Voucher School

40 Can Also do Multiple Doses Dose 1. No quake 2. Medium quake 3. Bad quake Dose Covariate Gender Male Female School SES Low Mid-low Medium Mid-high High Mother s education Primary Secondary Technical College Missing

41 Relative (To no Quake) Attendance Impact 3 Doses Attendance (%) Medium quake Bad quake

42 Relative (To no Quake) PSU Score Impact 3 Doses PSU Score !? 2 3 Medium quake Bad quake

43 MIP and Marketing: Chewbacca or BB-8?

44 Adaptive Preference Questionnaires Feature SX530 RX100 Zoom 50x 3.6x Prize $ $ Weight ounces 7.5 ounces Prefer Feature TG-4 G9 Waterproof Yes No Prize $ $ Weight 7.36 lb 7.5 lb Prefer We recommend: Feature TG-4 Galaxy 2 Waterproof Yes No Prize $ $ Viewfinder Electronic Optical Prefer

45 Choice-based Conjoint Analysis (CBCA) Feature Chewbacca BB-8 Wookiee Yes No Droid No Yes Blaster Yes No 0 1 1A = x 2 0 I would buy toy Product Profile x 1 x 2

46 Preference Model and Geometric Interpretation Utilities for 2 products, d features, logitmodel U 1 = U 2 = x = X d i=1 i x 1 i + 1 x = X d i=1 i x 2 i + 2 part-worths product profile noise (gumbel) Utility maximizing customer Geometric interpretation of preference for product 1 without error 2 x 1 x 2 0 x 1 x 2, U 1 U 2 1

47 Next Question = Minimize (Expected) Volume Good estimator Question? for? Center Postchoice Choice of ellipsoid balance symmetry

48 With Error = Volume of Ellipsoid f x 1,x 2 Prior distribution Answer likelihood Posterior distribution Prior ellipsoid Question/Answer Posterior ellipsoid

49 Rules of Thumb Still Good For Ellipsoid Volume ( µ) 0 1 ( µ) apple r Choice balance: Minimize distance to center µ x 1 x 2 µ Postchoice symmetry: Maximize variance of question x 1 x 2 0 x 1 x 2

50 Simple Formula for Expected Volume Expected Volume = Non-convex function distance: d := µ x 1 x 2 f(d, v) variance: v := x 1 x 2 0 x 1 x 2 of Can evaluate f(d, v) with 1-dim integral!

51 Optimization Model min f(d, v) s.t. µ x 1 x 2 = d x 1 x 2 0 x 1 x 2 = v Formulation trick: linearize x k i xl j A 1 x 1 + A 2 x 2 apple b x 1 6= x 2 x 1,x 2 2 {0, 1} n Experimental Design with MIP 50 / 27

52 Technique 2: Piecewise Linear Functions D-efficiency = Non-convex function distance: d := µ x 1 x 2 variance: v := x 1 x 2 0 x 1 x 2 of f(d, v) Can evaluate f(d, v) with 1-dim integral! Piecewise Linear Interpolation MIP formulation

53 Computational Performance Advanced formulations provide an computational advantage Advantage is significantly more important for free solvers State of the art commercial solvers can be significantly better that free solvers Still, free is free! Time [s] Time [s] Simple CPLEX GLPK Advanced Simple Advanced

54 Summary and Main Messages Always choose Chewbacca! How to YOU use MIP / Optimization / OR / Analytics? Study for the 2 nd midterm! Use JuMP and Julia Opt. How about grad school down the river? Masters of Business Analytics / OR Ph.D. in Operations Research

55 How Hard is MIP?

56 How hard is MIP: Traveling Salesman Problem? Paradoxes, Contradictions, and the Limits of Science Many research results define boundaries of what cannot be known, predicted, or described. Classifying these limitations shows us the structure of science and reason. Noson S. Yanofsky S A computer would have to check all these possible routes to find the shortest one.

57 MIP = Avoid Enumeration Number of tours for 49 cities Fastest supercomputer Assuming one floating point operation per tour: > years times the age of the universe! How long does it take on an iphone? Less than a second! 4 iterations of cutting plane method! = 48!/ flops Dantzig, Fulkerson and Johnson 1954 did it by hand! For more info see tutorial in ConcordeTSP app Cutting planes are the key for effectively solving (even NPhard) MIP problems in practice.

58 50+ Years of MIP = Significant Solver Speedups Algorithmic Improvements (Machine Independent): CPLEX v1.2 (1991) v11 (2007): 29,000x speedup Gurobi v1 (2009) v6.5 (2015): 48.7x speedup Commercial, but free for academic use (Reasonably) effective free / open source solvers: GLPK, COIN-OR (CBC) and SCIP (only for non-commercial) Easy to use, fast and versatile modeling languages Julia based JuMP modelling language

59 Technique 1: Binary Quadratic x 1,x 2 2 {0, 1} n x 1 x 2 0 x 1 x 2 = v X l i,j apple x l i, X l i,j apple x l j, X l i,j x l i + x l j 1, X l i,j 0 apple X l i,j = x l i x l j (l 2 {1, 2}, i,j 2 {1,...,n}) : W i,j = x 1 i x 2 j : W i,j apple x 1 i, W i,j apple x 2 j, W i,j x 1 i + x 2 j 1, W i,j 0 nx i,j=1 X 1 i,j + X 2 i,j W i,j W j,i i,j = v

60 Technique 1: Binary Quadratic x 1,x 2 2 {0, 1} n x 1 6= x 2, kx 1 x 2 k Xi,j l = x l i x l j (l 2 {1, 2}, i,j 2 {1,...,n}) : Xi,j l apple x l i, Xi,j l apple x l j, Xi,j l x l i + x l j 1, Xi,j l 0 W i,j = x 1 i x 2 j : W i,j apple x 1 i, W i,j apple x 2 j, W i,j x 1 i + x 2 j 1, W i,j 0 nx i,j=1 X 1 i,j + X 2 i,j W i,j W j,i 1

61 f(d 3 ) f(d 2 ) f(d 5 ) f(d 1 ) f(d 4 ) Simple Formulation for Univariate Functions z = f(x) d 1 d 2 d 3 d 4 d 5 Size = O (# of segments) x z Non-Ideal: Fractional Extreme Points = X 5 1= X 5 j=1 j=1 dj f(d j ) j j, j 0

62 f(d 3 ) f(d 2 ) f(d 5 ) f(d 1 ) f(d 4 ) Advanced Formulation for Univariate Functions z = f(x) x z d 1 d 2 d 3 d 4 d 5 Size = O (log 2 # of segments) Ideal: Integral Extreme Points = X 5 1= X 5 j=1 j=1 y 2 {0, 1} 2 dj f(d j ) j j, j 0