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The Game Inside the Game

The Game Inside the Game. Karl Lieberherr based on Master Thesis of Anna Hoepli at ETH Zurich in 2007 (communicated by Emo Welzl). SDG Partial Satisfaction Game Theoretic View. 2 person game, Bob and Alice. Unsatisfiable CSP Formula F is the “board” of the game. Bob chooses a constraint C.

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The Game Inside the Game

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  1. The Game Inside the Game Karl Lieberherr based on Master Thesis of Anna Hoepli at ETH Zurich in 2007 (communicated by Emo Welzl)

  2. SDG Partial SatisfactionGame Theoretic View • 2 person game, Bob and Alice. Unsatisfiable CSP Formula F is the “board” of the game. • Bob chooses a constraint C. • Alice chooses an assignment A. • If A satisfies C, Alice wins; otherwise Bob.

  3. Traditional Game • F must be unsatisfiable, otherwise Bob would not have a chance if Alice knows a satisfying assignment. • The unsatisfiable constraint is not allowed to be in F. (Relation 0). • Both play simultaneously. • We play with symmetric formulas.

  4. T = {(1,1) (2,0)} • Two constraint types: 1. A / 2. !A or !B • Assume F is symmetric. • Bob chooses a constraint of type 1 with prob. m1 and a constraint of type 2 with probability m2=1-m1.

  5. Game matrix sik for symmetric F variables set 1 relation R b(n,k) = binomial(n,k)

  6. Game matrix sik for symmetric F:General Case variables set 1 relation R b(n,k) = binomial(n,k)

  7. General Formula • http://www.ccs.neu.edu/research/demeter/papers/evergreen/cp07-submission.pdf • page 6

  8. Linear program (Alice maximizes t) • max t (t = satisfaction ratio) • s10*l0+s11*l1+ … +s1n*ln≥ t • s20*l0+s21*l1+ … +s2n*ln≥ t • l0+l1+ … +ln= 1 • li≥ 0 for all i in [0,1, … ,n]

  9. Linear program (Alice maximizes t) • max t (t = satisfaction ratio) • -s10*l0-s11*l1- … -s1n*ln+t ≤ 0 • -s20*l0-s21*l1- … -s2n*ln+t ≤ 0 • l0+l1+ … +ln= 1 • li≥ 0 for all i in [0,1, … ,n]

  10. Dual linear program(Bob minimizes t) • min t (t = satisfaction ratio) • s10*m1+s20*m2≤ t • s11*m1+s21*m2≤ t • … • s1n*m1+s2n*m2≤ t • m1+m2= 1 • mi≥ 0 for all i in [1,2]

  11. Mechanical way of finding best price and worst raw material • Given derivative d = ((R1, … ), p?,seller) • Choose n, e.g. n = 20. • Generate matrix sik. It has one row per relation and n columns. • Create input to LP solver using dual program, because we need the the mi for the raw materials. The minimum is the break-even price.

  12. SDG classic tpred = lim n -> ∞ min all raw materials rm of size n satisfying predicate pred max all finished products fp produced for rm q(fp) Seller approximates minimum efficiently Buyer approximates Max efficiently

  13. 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)

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