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Using Artificial Markets to Teach Computer Science Through Trading Robots

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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

- 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

- 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

- 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

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

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

- Our analysis was not thorough enough!
- 2 kinds of uncertainty:
- worst formula?
- best assignment?

SDG Mittagsseminar

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

- 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

- 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

Buyer b

Seller s

(pi,p,s)

SDG Mittagsseminar

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

- 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

- 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

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

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

all possible finished products

0.7

0.7

0.7

small subset of finished products

guaranteed to contain maximum

Noise

SDG Mittagsseminar

- 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))

- offer
- Two uncertainties
- raw material is not the worst (uncertainty_rm)
- finished product is not the best (uncertainty_fp)

SDG Mittagsseminar

- buy
- break-even price

- sell
- break-even price

- produce
- efficiently find worst case example

- process
- efficiently achieve break-even quality

SDG Mittagsseminar

- Choose algorithms RM and FP

SDG Mittagsseminar

- 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

tpred =

inf

all raw materials rm

satisfying predicate pred

max

all finished products fp

produced for rm

q(fp)

SDG Mittagsseminar

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

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

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

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

SDG Mittagsseminar

- 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

- 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

- 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

- 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)

- Predicate space: Any subset of Boolean relations of rank 3

SDG Mittagsseminar

- 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

- 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

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

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

SDG Mittagsseminar

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

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

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

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

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

- 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

- Discussion with Emo
- break-even price as Nash equilibrium
- 2 person zero sum game
- distribution on clauses
- distribution on assignments

SDG Mittagsseminar

- not percentage of all but percentage of maximum.
- approximation: seller knows a good solution for raw material. If buyer achieves

SDG Mittagsseminar

- 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

- 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

SDG Mittagsseminar

- 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.

SDG Mittagsseminar

- 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 Mittagsseminar

- 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.

SDG Mittagsseminar

- 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

SDG Mittagsseminar

- 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.

SDG Mittagsseminar

- global discount d: pay only (1-d)*approx.
- if achieve approx of (all, max, secret): receive 1 otherwise 0.
- secret requires extra protocol step.

SDG Mittagsseminar

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

SDG Mittagsseminar

- global discount d: pay only (1-d)*approx.
- 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 Mittagsseminar