Using Artificial Markets to Teach Computer Science Through Trading Robots

1. Using Artificial Markets to Teach Computer Science Through Trading Robots How to get students interested in algorithms, combinatorial optimization and software development Karl Lieberherr, Northeastern University, Boston SDG Mittagsseminar

2. Outline • Specker Derivative Game (SDG) • history • example, bottom up • top-down • derivatives, raw materials, finished products • Risk analysis for a derivative • Problem reductions – noise elimination • SDG(MAX-SAT): risk analysis using polynomials • Conclusions SDG Mittagsseminar

3. History • Around 1975: working on non-chronological backtracking for MAX-SAT for my PhD with Erwin Engeler. • Ernst Specker analyzed MAX-SAT which lead to the Golden Ratio Result: joint FOCS 79 and JACM 1981 paper. Ideas applicable to MAX-CSP. • 2006: sabbatical at Novartis reactivated my interest in MAX-SAT. • 2007: Turned Golden Ratio Result into a game SDG(Max) parameterized by a maximization problem Max. • 2007/2008: Taught SDG to students who had a lot of fun trying to produce a winning robot (class size 30). SDG Mittagsseminar

4. SDG Example • derivative(CNF{(2,0),(1,1)}, 0.70) • 4 variables maximum • would you like to buy it? • you will get two rights • you will receive a CNF R of the given type. • if you can satisfy fraction q of clauses in R, I will pay back q to you. CNF = SEQUENCE of clauses SDG Mittagsseminar

5. CNF{(2,0),(1,1)} raw materials 2: a 2: b 2: c 1: !a !b 1: !b !c 2: a 2: b 3: c 1: d 3: !a !b 9: !a !c 7: !b !c 1: !a !d 6: !b !d 6: !c !d 2: a 3: b 1: c 8: !a !b 6: !b !c Find best satisfaction ratio 1: a 1: b 8: !a !b SDG Mittagsseminar

6. CNF{(2,0),(1,1)} raw materials,finished products 2: a 2: b 2: c 1: !a !b 1: !b !c 2: a 2: b 3: c 1: d 3: !a !b 9: !a !c 7: !b !c 1: !a !d 6: !b !d 6: !c !d 6/8=0.75 2: a 3: b 1: c 8: !a !b 6: !b !c 17/20=0.85 price of 0.7 seems fair!? 1: a 1: b 8: !a !b 9/10=0.9 35/40 = 0.875 SDG Mittagsseminar

7. Oops • Our analysis was not thorough enough! • 2 kinds of uncertainty: • worst formula? • best assignment? SDG Mittagsseminar

8. Playing with the weights x: a x: b x: c x: d y: !a !b y: !a !c y: !b !c y: !a !d y: !b !d y: !c !d x =1, y=1 best assignment a=1, b=0, c=0, d=0: (1+6)/10=7/10=0.7 x = 2, y=1 best assignment a=1, b=1, c=0, d=0: (4+5)/14=9/14=0.64 derivative(CNF{(2,0),(1,1)}, 0.70) LOSS: 0.06 SDG Mittagsseminar

9. The SDG Game: Two high level views • Financial: Implement trading robots that survive in an artificial derivative market through offering derivatives, and buying and processing derivatives produced by other trading robots. • Biological: Implement organisms that survive in an artificial world through offering outsourced services, and consuming and processing outsourced services produced by other organisms. SDG Mittagsseminar

10. Derivative: (pred, p, s) bought by b • Max is an NP-hard combinatorial maximization problem with objective function range [0,1]. • Buyer b buys derivative at price p. • Seller s delivers raw material R (instance of Max) satisfying predicate pred. • Raw material R is finished by buyer with outcome O of quality q and seller pays q to buyer. • Buyer only buys if she thinks q > p. • Uncertainty for buyer: which raw material R will I get? Only know the predicate! What is the quality of the solution of Max I can achieve for R? SDG Mittagsseminar

11. Buyer b Seller s (pi,p,s) SDG Mittagsseminar

12. Buyer makes profit of 0.8 - 0.7 = 0.1 0.8 (p2,0.9,s) Buyer b Seller s O 0.8 O 0.8 R R (p1,0.7,s) sold Derivatives R satisfies pi 0.7 SDG Mittagsseminar

13. Artificial markets • Trading Robots that survive in a virtual world of an artificial market of financial derivatives. • Trading Robots that don’t follow the world rules don’t survive. • Trading Robots are ranked based on their bank account. • Teaches students about problem solving, software development, analyzing and approximating combinatorial maximization problems, game design and financial derivatives. SDG Mittagsseminar

14. Survive in an artificial market • Each robot contains a: • Derivative buying agent • Derivative offering agent • Raw material production agent • Finished product agent (solves Max) • Winning in robot competitions strongly influences the final grade. • Game is interesting even if robots are far from perfect. • Focus today: how to play the game perfectly (never losing) SDG Mittagsseminar

15. To play well: solve min max maximum solutions instances selected by predicate (an infinite set) 0.91 0.62 0.8 0.618 0.619 minimum Analysis for one Derivative SDG Mittagsseminar

16. To play well: solve min max maximum solutions instances selected by predicate (an infinite set) 0.91 0.62 0.8 0.618 0.62 0.619 minimum small subset of raw materials guaranteed to contain minimum of maxima Analysis for one Derivative Noise SDG Mittagsseminar

17. Raw material selected all possible finished products 0.7 0.7 0.7 small subset of finished products guaranteed to contain maximum Noise SDG Mittagsseminar

18. Risk analysis • Life cycle of a derivative (pred,p) • offer • risk high if I can find rm and fp with q(fp) > p • buy • risk high if I can find rm and fp with q(fp) < p • raw material (rm) • finished product (fp ,quality q(fp)) • Two uncertainties • raw material is not the worst (uncertainty_rm) • finished product is not the best (uncertainty_fp) SDG Mittagsseminar

19. To play SDG perfectlyeliminate risk • buy • break-even price • sell • break-even price • produce • efficiently find worst case example • process • efficiently achieve break-even quality SDG Mittagsseminar

20. Goal: never lose with offer/buy • Choose algorithms RM and FP SDG Mittagsseminar

21. Concerns for SDG(Max) • Robot communication • exchange language for derivatives, raw material, finished product • centralized game with administrator managing store of derivatives • decentralized game • Winning in the market • clever algorithm design SDG Mittagsseminar

22. Analysis of SDG(Max) tpred = inf all raw materials rm satisfying predicate pred max all finished products fp produced for rm q(fp) SDG Mittagsseminar

23. Analysis of SDG(Max) tpred = lim n -> ∞ min all raw materials rm of size n satisfying predicate pred max all finished products fp produced for rm q(fp) SDG Mittagsseminar

24. Spec for RM and FP tpred = lim n -> ∞ min all raw materials rm of size n satisfying predicate pred and having property WORST(rm) max small subset of all finished products fp produced for rm q(fp) SDG Mittagsseminar

25. Analysis of SDG(Max) tpred = lim n -> ∞ min all raw materials rm of size n produced by RM satisfying predicate pred max all finished products fp produced for rm by FP q(fp) SDG Mittagsseminar

26. Hope • Max is NP-hard • SDG(Max) simplifies Max if our goal is to never lose. SDG Mittagsseminar

27. SDG(MAX-SAT) • Predicates using clause types. • Example predicate PairSat = All CNFs with clauses of any length but clauses of length 1 must contain one positive literal. • What is the right price p for derivative (PairSat, p, Specker) SDG Mittagsseminar

28. SDG(MAX-SAT) • Predicate space: any subset of clause types of PairSat • t all PairSat = (√5 -1)/2 • t {(2,0),(1,1)} = (√5 -1)/2 • t {(100,50), (3,2), (2,0),(1,1)} = (√5 -1)/2 Noise for the purpose of constructing raw material. SDG Mittagsseminar

29. SDG(MAX-SAT) • t {(2,0),(1,1)} = t SYM{(2,0),(1,1)} = (√5 -1)/2 • SYM stands for Symmetrization: Idea: if you give me a CNF with a satisfaction ratio f, I give you a symmetric CNF with a satisfaction ratio <= f. • For a CNF in SYM{(2,0),(1,1)}, the MAX-SAT problem reduces to maximizing a polynomial. SDG Mittagsseminar

30. Students implement trading robots • Fall semester (undergraduates): SDG(MAX-SAT) • Spring semester (graduates): SDG(MAX-CSP) • Predicate space: Any subset of Boolean relations of rank 3 • CSP({22},4/9,robot1) • CSP({17,22},1/2,robot2) SDG Mittagsseminar

31. Opportunities for learningExample SDG(MAX-SAT) • Abstraction: What is important to play the game well. • Game reductions: To play game SDG(MAX-SAT) well, it is sufficient to play game SDG(X) well, where X is simpler than MAX-SAT. SDG Mittagsseminar

32. Complexity theory connection • Break-even prices are not only interesting for the SDG game. • They also have complexity-theoretic significance: they are critical transition points separating P from NP (for “most” predicates). SDG Mittagsseminar

33. General Dichotomy Theorem MAX-CSP(G,f): For each finite set G of relations there exists an algebraic number tG For f ≤ tG: MAX-CSP(G,f) has polynomial solution For f≥ tG+ e: MAX-CSP(G,f) is NP-complete, e>0. 1 hard (solid) NP-complete polynomial solution: Use optimally biased coin. Derandomize. P-Optimal. tG = critical transition point easy (fluid) Polynomial 0 due to Lieberherr/Specker (1979, 1982) SDG Mittagsseminar

34. Other break-even prices(Lieberherr/Specker (1982)) • G = {R0,R1,R2,R3}; Rj : rank 3, exactly j of 3 variables are true. tG= ¼ SDG Mittagsseminar

35. Other break-even prices(Lieberherr/Specker (1982)) • G(p,q) = {Rp,q = disjunctions containing at least p positive or q negative literals (p,q≥1)} • Let a be the solution of (1-x)p=xq in (0,1). tG(p,q)=1-aq SDG Mittagsseminar

36. Lessons learned from SDG • Developing trading robots and make them survive in an artificial market is very motivating to students • Students learn experientially about many important topics driven by the single goal of making their robots competitive • software development • problem solving by reduction (noise reduction) • combinatorial optimization • game design • sub-optimal playing is very educational too! SDG Mittagsseminar

37. Noise reduction: important topic • seen in solving minimization and maximization problems • To implement trading robots, we use a tool called DemeterF which is good at noise reduction during programming process: focus on important classes and eliminate noise classes SDG Mittagsseminar

38. Conclusions • SDG(Max) is an interesting tool for teaching a wide variety of topics. • It helps if you give your students a robot that knows the basic rules. Then the students can focus on improving the robots rather than getting all robots to communicate properly. SDG Mittagsseminar

39. Conclusions • SDG(Max) is an interesting tool for research. • Does it always turn an NP-hard maximization problem into a polynomial time approximation algorithm? SDG Mittagsseminar

40. References • Lieberherr/Specker (1979, 1981) FOCS and Journal of the ACM • Lieberherr (1982) Journal of Algorithms • Recent Workshop paper: • Christine D. Hang and Ahmed Abdelmeged and Daniel Rinehart and Karl J. Lieberherr, The Promise of Polynomial-based Local Search to Boost Boolean MAX-CSP Solvers, 2007, Proceedings of Fourth International Workshop on Local Search Techniques in Constraint Satisfaction, CP2007, Providence, Rhode Island. • DemeterF home page: http://www.ccs.neu.edu/research/demeter/DemeterF/ • SDG home page: http://www.ccs.neu.edu/home/lieber/evergreen/specker/sdg-home.html SDG Mittagsseminar

41. Obstacles to finding p • Try to find a CNF satisfying PairSat in which only a small percentage of the clauses can be satisfied. • Challenge of finding the worst case. • Even if we find the worst case, we might not find the maximum assignment for that case. SDG Mittagsseminar

42. CNF{(2,0),(1,1)} raw materials,finished products 2: a 2: b 2: c 1: !a !b 1: !b !c 2: a 2: b 3: c 1: d 3: !a !b 9: !a !c 7: !b !c 1: !a !d 6: !b !d 6: !c !d 6/8=0.75 2: a 3: b 1: c 8: !a !b 6: !b !c 17/20=0.85 price of 0.7 seems fair!? 1: a 1: b 8: !a !b 9/10=0.9 35/40 = 0.875 SDG Mittagsseminar

43. CNF{(2,0),(1,1)} raw materials,finished products s 2: a 2: b 3: c 1: d 3: !a !b 9: !a !c 7: !b !c 1: !a !d 6: !b !d 6: !c !d 35/40 = 0.875 SDG Mittagsseminar

44. CNF{(2,0),(1,1)} reduction T(s) no new variables 2: a !b !c 2: !a b !d 3: a c 1: c d 3: !a !b !c 9: !a !c !d 7: !b !c 1: !a !b !d 6: !b !d 6: !c !d To each interpretation I1 of T(s) corresponds an interpretation I of s which satisfies at least as many clauses in s as I1 in T(s). 35/40 = 0.875 SDG Mittagsseminar

45. CNF{(2,0),(1,1)} reduction no new variables 2: a !b !c 2: !a b !d 3: a c 1: c d 3: !a !b !c 9: !a !c !d 7: !b !c 1: !a !b !d 6: !b !d 6: !c !d To each interpretation I1 of T(s) corresponds an interpretation I of s which satisfies at least as many clauses in s as I1 in T(s). 35/40 = 0.875 SDG Mittagsseminar

46. CNF{(2,0),(1,1)} reduction with new variables! 2: a !e 2: b !f g 3: c e h i 1: d !e !g 3: !a !b !h !i 9: !a !c !f !g 7: !b !c !g 1: !a !d !f !i 6: !b !d !g !h 6: !c !d 35/40 = 0.875 SDG Mittagsseminar

47. SDG(Max) • Derivative = (Predicate, Price in [0,1], Player). • Players offer and buy derivatives. • Buying a derivative gives you the rights: • to receive raw material R satisfying the predicate. • upon finishing the raw material R at quality q (trying to find the maximum solution), you receive q in [0,1]. SDG Mittagsseminar

48. After talk • Discussion with Emo • break-even price as Nash equilibrium • 2 person zero sum game • distribution on clauses • distribution on assignments SDG Mittagsseminar

49. SDG(Max) • not percentage of all but percentage of maximum. • approximation: seller knows a good solution for raw material. If buyer achieves SDG Mittagsseminar

50. derivative = (predicate,p,robot) • p = price = estimate how close I can get to the maximum • problem = knowing exact maximum is hard • approximation: p = price = estimate how close I can get to the outcome the seller knows for an instance satisfying the predicate. SDG Mittagsseminar