Specker Derivative Game

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Specker Derivative Game

• Karl Lieberherr
• Alex Dubreuil
• Spring 2009

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Mega moves in classic and secret SDG

• White-black mega move
• white offer derivatives
• black buy derivatives or reoffer
• if bought then
• repeat r times for each bought derivative
• white deliver raw material with witness
  quality(S) of secret finished product S
• black deliver finished product FP
• white reveal secret S
• black check secret S against witness quality(S)
• win
• classic SDG satisfaction ratio sr(FP) wrt all.
  win if sr(FP) gt price 1.
• secret SDG satisfaction ratio sr(FP) wrt secret
  S (think of secret S as the maximum) win if
  sr(FP) gt price quality(S).
• pay for performance in raw material finishing
  aggregate wins

3

• derivative (CSP predicate)

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SDG Game Versions

• T Ball (one relation)
• Softball
• Slow Pitch (recognizing noise)
• one implication chain of any number of relations.
• Fast Pitch
• any number of relations
• Level k Independent (k independent relations with
  no implication relationship). Note Level 1
  Independent T Ball
• Level k Reduced (any number of relations that can
  be reduced to Level k Independent.) Note Slow
  Pitch is a special case of Level 1 Reduced.
• Baseball
• Classic and Secret
• CSP
• Any Combinatorial Maximization Problem

T Ball and Softball are based on CSP
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SDG Game Versions
Softball
T Ball Fast Pitch Level 1 Independent Slow
Pitch Special case of Fast Pitch Level 1 Reduced
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Level 3 15 (Implied by all)
7 11 13 14
Level 2
3 5 9 6 10 12
Level 1 odd
Level 1 even
Level 0
1 2 4 8
Null level 0 (Implies all)
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Derivative (1 2 3 10 12 7 11 13 14)
ARITY 2
Level 3 15 (Implied by all)
7 11 13 14
Level 2
3 5 9 6 10 12
Level 1 odd
Level 1 even
Level 0
1 2 4 8
Null level 0 (Implies all)
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Interpretation Arity 2

• One node per relation.
• Edges implication between relations.
• There are choose(4,2) independent relations for
  arity 2.
• choose(4,k), k 04, the maximum is at k4/22.
• choose(n,k), k 0n, the maximum is at k n/2.

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Arity 3

• choose(n,k), k 0n, the maximum is at k n/2.
• n8, choose(8,4)8765/(1234)70
• Note sumk0..n choose(n,k) 2n