1. SDG Mittagsseminar1 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

2. SDG Mittagsseminar2 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

3. SDG Mittagsseminar3 History Diplomarbeit with Ernst Specker on two-dimensional automata. 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.

4. SDG Mittagsseminar4 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.

5. SDG Mittagsseminar5 CNF{(2,0),(1,1)} raw materials 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 1: a 1: b 8: !a !b 2: a 2: b 2: c 1: !a !b 1: !b !c

6. SDG Mittagsseminar6 CNF{(2,0),(1,1)} raw materials, finished products 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 1: a 1: b 8: !a !b 17/20=0.85 9/10=0.9 35/40 = 0.875 2: a 2: b 2: c 1: !a !b 1: !b !c 6/8=0.75 price of 0.7 seems fair!?

7. SDG Mittagsseminar7 Oops Our analysis was not thorough enough! 2 kinds of uncertainty: –worst formula? –best assignment?

8. SDG Mittagsseminar8 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

9. SDG Mittagsseminar9 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.

10. SDG Mittagsseminar10 Derivative: (pred, p, s) bought by b Max is an NP-hard 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?

11. SDG Mittagsseminar11 Seller s Buyer b (pi,p,s)

12. SDG Mittagsseminar12 Seller s Buyer b (p2,0.9,s) (p1,0.7,s) sold R O 0.8 O 0.8 R 0.7 0.8 Buyer makes profit of 0.8 - 0.7 = 0.1 R satisfies pi Derivatives

13. SDG Mittagsseminar13 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 and analyzing and approximating combinatorial maximization problems.

14. SDG Mittagsseminar14 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)

15. SDG Mittagsseminar15 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].

16. SDG Mittagsseminar16 To play well: solve min max instances selected by predicate (an infinite set) maximum solutions 0.8 0.91 0.618 0.62 0.619 minimum Analysis for one Derivative

17. SDG Mittagsseminar17 To play well: solve min max instances selected by predicate (an infinite set) maximum solutions 0.8 0.91 0.618 0.62 0.619 minimum Analysis for one Derivative 0.62 Noise small subset of raw materials guaranteed to contain minimum of maxima

18. SDG Mittagsseminar18 Raw material selected 0.7 all possible finished products Noise small subset of finished products guaranteed to contain maximum 0.7

19. SDG Mittagsseminar19 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)

20. SDG Mittagsseminar20 To play SDG perfectly eliminate risk buy –break-even price sell –break-even price produce –efficiently find worst case example process –efficiently achieve break-even quality

21. SDG Mittagsseminar21 Goal: never lose with offer/buy Choose algorithms RM and FP

22. SDG Mittagsseminar22 Analysis of SDG(Max) t pred = inf all raw materials rm satisfying predicate pred max all finished products fp produced for rm q(fp)

23. SDG Mittagsseminar23 Analysis of SDG(Max) t pred = lim n -> ∞ min all raw materials rm of size n satisfying predicate pred max all finished products fp produced for rm q(fp)

24. SDG Mittagsseminar24 Spec for RM and FP t pred = 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)

25. SDG Mittagsseminar25 Analysis of SDG(Max)

26. SDG Mittagsseminar26 Hope Max is NP-hard SDG(Max) simplifies Max if our goal is to never lose.

27. SDG Mittagsseminar27 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)

28. SDG Mittagsseminar28 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.

29. SDG Mittagsseminar29 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.

30. SDG Mittagsseminar30 Students need to implement trading robots Fall semester (undergraduates): SDG(MAX-SAT) Spring semester (graduates): SDG(MAX- CSP)

31. SDG Mittagsseminar31 Opportunities for learning Example 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.

32. SDG Mittagsseminar32 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).

33. SDG Mittagsseminar33 General Dichotomy Theorem MAX-CSP(G,f): For each finite set G of relations there exists an algebraic number t G For f ≤ t G : MAX-CSP(G,f) has polynomial solution For f ≥ t G +  : MAX-CSP(G,f) is NP-complete,   t G  critical transition point easy (fluid) Polynomial hard (solid) NP-complete due to Lieberherr/Specker (1979, 1982) polynomial solution: Use optimally biased coin. Derandomize. P-Optimal. 

34. SDG Mittagsseminar34 Other break-even prices (Lieberherr/Specker (1982)) G = {R 0,R 1,R 2,R 3 }; R j : rank 3, exactly j of 3 variables are true. t G = ¼

35. SDG Mittagsseminar35 Other break-even prices (unpublished) (Lieberherr/Specker (1982)) G(p,q) = {R p,q = disjunctions containing at least p positive or q negative literals (p,q≥1)} –Let a be the solution of (1-x) p =x q in (0,1). t G(p,q) =1-a q

36. SDG Mittagsseminar36 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!

37. SDG Mittagsseminar37 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

38. SDG Mittagsseminar38 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.

39. SDG Mittagsseminar39 Conclusions SDG(Max) is an interesting tool for research. Does it always turn an NP-hard maximization problem into a polynomial time approximation algorithm?

40. SDG Mittagsseminar40 References Lieberherr/Specker (1979, 1981) FOCS and Journal of the ACM Lieberherr (1982) Journal of Algorithms Workshop paper: DemeterF home page

41. SDG Mittagsseminar41 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.