SDG Mittagsseminar1 Using Artificial Markets to Teach Computer Science Through Trading Robots How to...

transcript

SDG Mittagsseminar 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

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

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 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

CNF{(2,0),(1,1)} raw materials

2: a2: b3: c1: d3: !a !b9: !a !c7: !b !c1: !a !d6: !b !d6: !c !d

2: a3: b1: c8: !a !b6: !b !c

1: a1: b8: !a !b

2: a2: b2: c1: !a !b1: !b !c

Find bestsatisfaction ratio

CNF{(2,0),(1,1)} raw materials,finished products

2: a3: b1: c8: !a !b6: !b !c

1: a1: b8: !a !b

17/20=0.85

9/10=0.9

35/40 = 0.875

2: a2: b2: c1: !a !b1: !b !c

6/8=0.75

price of 0.7seems fair!?

• Our analysis was not thorough enough!

• 2 kinds of uncertainty:– worst formula?– best assignment?

Playing with the weights

x: ax: bx: cx: dy: !a !by: !a !cy: !b !cy: !a !dy: !b !dy: !c !d

x =1, y=1best assignmenta=1, b=0, c=0, d=0: (1+6)/10=7/10=0.7

x = 2, y=1best assignmenta=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

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.

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?

Seller s Buyer b

(pi,p,s)

Seller s Buyer b

(p2,0.9,s)

(p1,0.7,s)sold

Buyer makes profit of 0.8 - 0.7 = 0.1

R satisfies pi

Derivatives

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.

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)

To play well: solve min max

instancesselected by predicate(an infinite set)

maximumsolutions

0.619 minimum

Analysis for one Derivative

To play well: solve min max

instancesselected by predicate(an infinite set)

maximumsolutions

0.619 minimum

Analysis for one Derivative

small subset of raw materialsguaranteed to containminimum of maxima

Raw material selected

all possible finished products

Noise small subset of finished productsguaranteed to contain maximum

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)

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

Goal: never lose with offer/buy

• Choose algorithms RM and FP

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

Analysis of SDG(Max)

tpred = infall raw materials rmsatisfying predicate pred maxall finished products fpproduced for rm q(fp)

tpred = lim n -> ∞ minall raw materials rm of size nsatisfying predicate pred maxall finished products fpproduced for rm q(fp)

Spec for RM and FP

tpred = lim n -> ∞ minall raw materials rm of size nsatisfying predicate pred andhaving property WORST(rm) maxsmall subset of all finished products fp produced for rm q(fp)

tpred = lim n -> ∞ minminallall raw materialss rm of size n produced by RMsatisfying predicate pred maxallall finished productss fp produced for rm by FP q(fp)

• Max is NP-hard

• SDG(Max) simplifies Max if our goal is to never lose.

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(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(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.

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)

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.

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).

General Dichotomy Theorem

MAX-CSP(G,f): For each finite set G of relationsthere exists an algebraic number tG

For f ≤ tG: MAX-CSP(G,f) has polynomial solutionFor f ≥ tG+ : MAX-CSP(G,f) is NP-complete,

tG critical transition pointeasy (fluid)Polynomial

hard (solid)NP-complete

due to Lieberherr/Specker (1979, 1982)

polynomial solution:Use optimally biased coin.Derandomize.P-Optimal.

Other break-even prices(Lieberherr/Specker (1982))

• G = {R0,R1,R2,R3}; Rj : rank 3, exactly j of 3 variables are true. tG= ¼

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

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!

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

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.

Conclusions

• SDG(Max) is an interesting tool for research.

• Does it always turn an NP-hard maximization problem into a polynomial time approximation algorithm?

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

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.

2: a3: b1: c8: !a !b6: !b !c

1: a1: b8: !a !b

17/20=0.85

9/10=0.9

35/40 = 0.875

2: a2: b2: c1: !a !b1: !b !c

6/8=0.75

price of 0.7seems fair!?

35/40 = 0.875

CNF{(2,0),(1,1)} reduction

2: a !b !c2: !a b !d3: a c1: c d3: !a !b !c9: !a !c !d7: !b !c1: !a !b !d6: !b !d6: !c !d

35/40 = 0.875

no new variables

To each interpretation I1 of T(s) corresponds an interpretationI of s which satisfies at least as many clauses in s as I1 in T(s).

2: a !b !c2: !a b !d3: a c1: c d3: !a !b !c9: !a !c !d7: !b !c1: !a !b !d6: !b !d6: !c !d

35/40 = 0.875

no new variables

To each interpretation I1 of T(s) corresponds an interpretationI of s which satisfies at least as many clauses in s as I1 in T(s).

2: a !e2: b !f g3: c e h i1: d !e !g3: !a !b !h !i9: !a !c !f !g7: !b !c !g 1: !a !d !f !i6: !b !d !g !h6: !c !d

35/40 = 0.875

with newvariables!

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].

• 18. Juli 18.00 Steinbruechelstr. 45

• Edi Zehnder

After talk

• Discussion with Emo

• break-even price as Nash equilibrium

• 2 person zero sum game– distribution on clauses– distribution on assignments

SDG(Max)

• not percentage of all but percentage of maximum.

• approximation: seller knows a good solution for raw material. If buyer achieves

• 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.

• payoff: seller achieves with outcome O quality q(O).

• I achieve

• seller: 18, price 15/18

• I: 15: paid back 15/18

• I: 16: paid back 14/18

• I: 100: paid back 100/18

• still flavor of derivatives

• (pred,p,r) p is estimate how well I can solve an instance satisfying pred compared to how well r can solve it. r has more time available. Here don’t need assumption that objective function in [0,1].

• now: p is an estimate how well I can solve an instance satisfying pred compared to 1.

cryptography

• Emo’s satisfiability class: 4 algorithms for 2-Sat. For my algorithms class?

• r knows good solution. If I would know this good solution, could break system.

• any approximation would not be useful. Price would have to be 1

• SDG(CNF-ALL)• SDG(CNF-MAX) uninteresting: eval is

exponential• SDG(CNF-SECRET)

– owner delivers with rm a number f.– seller, after receiving fp, delivers outcome satisfying f– if buyer achieves f * price or higher, buyer receives 1,

otherwise 0. all-or-nothing. Can buyer discover the secret that was put into instance by seller?

– problem: price of 1 not interesting. Discount? Buyer only pays 0.9 * f.

discount idea

• works also with SDG(CNF-ALL)

• price is p, but pay only 0.9*p. More realistic. Pay only 0.9, 0.1 is for effort.

• Use all or nothing. discount = d

• Trading Robot development for artificial markets (to illustrate good separation of concerns from design to implementation using a traversal-based, functional implicit invocation architecture)– why trading robot: industrial IT– why good separation of concerns: always helps.– why traversal-based, functional implicit invocation

architecture: built-in variability (class graph generic), support for parallelism: exploiting multi-core architecture

• Example: Shannon decomposition, testing history of generic SDG game.

derivative[CSP] = (pred, approx (all, max, secret), robot)

price = (1-d)*approx• global discount d: pay only (1-d)*approx.

• if achieve approx of (all, max, secret): receive 1 otherwise 0.

• secret requires extra protocol step.

derivative[CSP] = (pred, approx (all, secret), robot)

• if achieve approx of (all, secret): receive 1 otherwise 0.

• secret requires extra protocol step.

• (delete max, because evaluation is costly. Game becomes less interesting.)

• good for variant of SDG game

derivative[BooleanCSP] = (pred, approx, robot)

• d is compensation for effort to produce finished product

• if achieve approx of all or more: receive 1 otherwise 0.

• good for talk: quick intro.

• Use BooleanCSP because it is clear what all means.

SDG Mittagsseminar1 Using Artificial Markets to Teach Computer Science Through Trading Robots How to...

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