Perfect Prediction Equilibrium

Perfect Prediction Equilibrium

An Intriguing Paradox, A Prediction Model, And Its Application To Game Theory

(Systems Group Lunch Seminar, ETH Zürich, June 2011)

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Ghislain Fourny

June 03, 2011
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  1. Friday, June 3rd, 2011 The Systems Group at ETH Zurich

    Perfect Prediction Equilibrium An intriguing paradox, a prediction model, and its application to game theory Ghislain Fourny (ETH Zurich) Stéphane Reiche (Mines ParisTech) Jean-Pierre Dupuy (Stanford University) Systems Group Lunch Seminar
  2. Friday, June 3rd, 2011 Systems Group Lunch Seminar NEWCOMB‘S PARADOX

    No cat will be harmed in this talk. 2
  3. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox

    3
  4. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox

    4 $ 1,000 $ 1,000,000 or $ 0
  5. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    choice 1 5 $ 1,000 $ 1,000,000 or $ 0
  6. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    choice 2 6 $ 1,000 $ 1,000,000 or $ 0
  7. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    the catch 7 Long, long, long ago ? ?
  8. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    the catch 8 $ 1,000,000 Long, long, long ago
  9. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    the catch 9 $ 0 Long, long, long ago
  10. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    you choose! $ 1,000 $ 1,000,000 or $ 0
  11. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    One-boxers‘ reasoning $ 1000
  12. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    One-boxers‘ reasoning $ 0 $ 1000 $ 1,000 $ 1,000,000 + = = $ 1,000,000
  13. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    One-boxers‘ reasoning $ 0 $ 1000 $ 1,000 $ 1,000,000 + = = $ 1,000,000
  14. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    Two-boxers‘ reasoning $ 1000
  15. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    Two-boxers‘ reasoning $ x $ x $ 1000
  16. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    Two-boxers‘ reasoning $ x $ x $ 1000 $ x +1,000 = = + $ x
  17. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    Two-boxers‘ reasoning $ x $ x $ 1000 = + $ x +1,000 = $ x
  18. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    Two-boxers‘ reasoning $ 0 $ 0 $ 0 $ 1000 $ 1,000 = +
  19. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    Compatibilism §  Three topics relevant to this paradox: §  Free will §  Perfect Prediction §  Fixity of the past
  20. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox

    §  Free will §  1. The player could have acted otherwise. t2 Player does A Player does B Possible World 1 Possible World 2
  21. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox

    §  Perfect Prediction §  2. In each possible world, the prediction is true. t2 Player does C Player does D Possible World 1 Possible World 2 t1 Predictor predicts that Player will do C Predictor predicts that Player will do D
  22. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox

    §  Fixity of the past §  3. There is nothing that the player can do at t2 such that, if he were to do it, P would not have happened. t2 Player does A Player does B Possible World 1 Possible World 2 t1 P P
  23. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox

    §  You can have at most two of these three (Ockham): §  Free will §  Perfect Prediction §  Fixity of the past
  24. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    No game at all §  Free will §  Perfect Prediction §  Fixity of the past t2 Player takes n boxes Only possible world t1 Predictor predicts that Player will take n boxes
  25. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    Two-boxers §  Free will §  Perfect Prediction §  Fixity of the past t2 Player chooses 2 boxes Player choose 1 box Actual world Other Possible World t1 Predictor predicts that Player will choose 2 boxes Predictor predicts that Player will choose 2 boxes
  26. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    One-boxers §  Free will §  Perfect Prediction §  Fixity of the past t2 Player chooses 1 box Player choose 2 boxes Actual world Other Possible World t1 Predictor predicts that Player will choose 1 box Predictor predicts that Player will choose 2 boxes
  27. Friday, June 3rd, 2011 Systems Group Lunch Seminar Newcomb‘s Paradox:

    One-boxers §  Free will §  Perfect Prediction §  Fixity of the past t2 Player chooses 1 box Player choose 2 boxes Actual world Other Possible World t1 Predictor predicts that Player will choose 1 box Predictor predicts that Player will choose 2 boxes Past counterfactually dependent on the future.
  28. Friday, June 3rd, 2011 Systems Group Lunch Seminar PERFECT PREDICTION

    EQUILIBRIUM Game Theory and One-Boxers 28
  29. Friday, June 3rd, 2011 Systems Group Lunch Seminar Game Theory:

    meet the players. 29 Peter Mary (They are our Alice and Bob)
  30. Friday, June 3rd, 2011 Systems Group Lunch Seminar Game Theory

    30 Assumption: both are rational. And know they are. And know they know they are. And...
  31. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    31 0,0 -1,2 1,1
  32. Friday, June 3rd, 2011 Systems Group Lunch Seminar 0,0 Nash

    Equilibrium 32 -1,2 1,1 What would Mary do if she were to play?
  33. Friday, June 3rd, 2011 Systems Group Lunch Seminar 0,0 Nash

    Equilibrium 33 -1,2 1,1 If Peter trusted Mary, he would get -1.
  34. Friday, June 3rd, 2011 Systems Group Lunch Seminar Nash Equilibrium

    34 0,0 -1,2 1,1 By not trusting her, he gets 0. This is the Nash Subgame Perfect Equilibrium.
  35. Friday, June 3rd, 2011 Systems Group Lunch Seminar 0,0 Backward

    Induction Paradox 35 -1,2 1,1 Being here contradicts Peter's rationality...
  36. Friday, June 3rd, 2011 Systems Group Lunch Seminar Backward Induction

    Paradox 36 0,0 -1,2 1,1 ... because we showed that, if Peter is rational, only the Nash Equilibrium (0,0) can be reached.
  37. Friday, June 3rd, 2011 Systems Group Lunch Seminar 0,0 Backward

    Induction Paradox 37 -1,2 1,1 And Mary knows it. It cannot possibly be her turn.
  38. Friday, June 3rd, 2011 Systems Group Lunch Seminar 0,0 Backward

    Induction Paradox 38 -1,2 1,1 Reasoning here implies some opacity in Mary's knowledge.
  39. Friday, June 3rd, 2011 Systems Group Lunch Seminar Game Theory

    39 Assumption: both are rational. And know they are. And know they know they are. And... Can we achieve total transparency?
  40. Friday, June 3rd, 2011 Systems Group Lunch Seminar Total transparency:

    the main idea 40 If there were an equilibrium which is totally transparent to itself, what would it look like? Total transparency implies that both players are perfect predictors (remember? one-boxers!) Two principles can be derived from this assumption. These principles lead to a unique equilibrium that the players can compute and which is stable against their knowledge of it.
  41. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    §  Perfect transparency implies that the players are perfect predictors (Dupuy, 2000) 41 Peter predicts Mary does A. Peter predicts Mary does B. Mary does A. Mary does B.
  42. Friday, June 3rd, 2011 Systems Group Lunch Seminar Dupuy's Projected

    Time §  It follows that the past is counterfactually dependent on the future. 42
  43. Friday, June 3rd, 2011 Systems Group Lunch Seminar Dupuy's Projected

    Time §  It follows that the past is counterfactually dependent on the future. §  And the future is, as usual, causally dependent on the past. 43
  44. Friday, June 3rd, 2011 Systems Group Lunch Seminar Dupuy's Projected

    Time: Preemption §  What if some hypothetical future event (counterfactually) brings about a past event, and this past event causes a different future to happen (a.k.a. grandfather's paradox)? 44
  45. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    45 0,0 -1,2 1,1
  46. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    46 0,0 -1,2 1,1
  47. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    47 0,0 -1,2 1,1
  48. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    48 0,0 -1,2 1,1 ≠
  49. Friday, June 3rd, 2011 Systems Group Lunch Seminar Two principles

    of choice (simplified) §  First Principle: Outcomes and nodes preempted by the past they bring about cannot be chosen. 49
  50. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    50 0,0 -1,2 1,1
  51. Friday, June 3rd, 2011 Systems Group Lunch Seminar Projected time

    (Dupuy) §  Past counterfactually dependent on the future §  Future causally dependent on the past §  A fixpoint problem
  52. Friday, June 3rd, 2011 Systems Group Lunch Seminar Projected time

    (Dupuy) §  Past counterfactually dependent on the future §  Future causally dependent on the past §  A fixpoint problem §  The players know about this fixpoint problem and this is why they are perfect predictors!
  53. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    53 0,0 -1,2 1,1
  54. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    54 0,0 -1,2 1,1
  55. Friday, June 3rd, 2011 Systems Group Lunch Seminar Two principles

    of choice (simplified) §  First Principle: Outcomes and nodes preempted by the past they bring about cannot be chosen. §  Second Principle: Among the remaining outcomes and nodes, the player chooses the one (s)he prefers. 55
  56. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    56 0,0 -1,2 1,1
  57. Friday, June 3rd, 2011 Systems Group Lunch Seminar Promise Game

    57 0,0 -1,2 1,1 Perfect Prediction Equilibrium
  58. Friday, June 3rd, 2011 Systems Group Lunch Seminar Another example:

    Binary tree 58 0,1 2,2 1,4 3,3
  59. Friday, June 3rd, 2011 Systems Group Lunch Seminar Nash Equilibrium

    59 0,1 2,2 1,4 3,3
  60. Friday, June 3rd, 2011 Systems Group Lunch Seminar Nash Equilibrium

    60 0,1 2,2 1,4 3,3 What would Mary do if she were to play here? ... and here?
  61. Friday, June 3rd, 2011 Systems Group Lunch Seminar Nash Equilibrium

    61 0,1 2,2 1,4 3,3
  62. Friday, June 3rd, 2011 Systems Group Lunch Seminar Nash Equilibrium

    62 0,1 2,2 1,4 3,3 Knowing this, what does Peter do?
  63. Friday, June 3rd, 2011 Systems Group Lunch Seminar Nash Equilibrium

    63 0,1 2,2 1,4 3,3
  64. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    Equilibrium 64 0,1 2,2 1,4 3,3
  65. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    Equilibrium 65 0,1 2,2 1,4 3,3
  66. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    Equilibrium 66 0,1 2,2 1,4 3,3
  67. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    Equilibrium 67 0,1 2,2 1,4 3,3
  68. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    Equilibrium 68 0,1 2,2 1,4 3,3
  69. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    Equilibrium 69 0,1 2,2 1,4 3,3
  70. Friday, June 3rd, 2011 Systems Group Lunch Seminar Perfect Prediction

    Equilibrium: main results §  Defined for games with no ties. §  Theorem (since 2004): §  It always exists and is unique. §  Its outcome is always Pareto-optimal. §  It is totally transparent to itself. 70
  71. Friday, June 3rd, 2011 Systems Group Lunch Seminar LINKS WITH

    QUANTUM INFORMATION (CONCLUSION) Newcomb's Cat 71
  72. Friday, June 3rd, 2011 Systems Group Lunch Seminar Links with

    Quantum Information §  Schrödinger's paradox "is interesting precisely because it blows up quantum consequences to real-life size" (Jon Lindsay, 1994) §  Newcomb's paradox "ties an elusive notion [prediction, determinism] to a real-life-sized fact [money in a box]" (J.-P. Dupuy, 2000) 72
  73. Friday, June 3rd, 2011 Systems Group Lunch Seminar Links with

    Quantum Information §  Free will as contingency of the choice. §  Quantum measurement as contingency of its outcome. 73
  74. Friday, June 3rd, 2011 Systems Group Lunch Seminar Grandfather Paradox

    74
  75. Friday, June 3rd, 2011 Systems Group Lunch Seminar Many-Worlds Interpretation

    75 David Deutsch, ' Quantum mechanics near closed timelike curves,' Phys. Rev. D 44, 3197 (1991).
  76. Friday, June 3rd, 2011 Systems Group Lunch Seminar Novikov Self-Consistency

    Principle 76
  77. Friday, June 3rd, 2011 Systems Group Lunch Seminar Total transparency

    in theoretical physics 77 If there were a theory which is deterministic and totally transparent to itself, what would it look like? Total transparency implies that scientists (we?) are perfect predictors. Does it imply quantum physics (contingency of measurements)? The actual world as a fixpoint. This leads to a unique possible actual world, which one can compute and is stable given its knowledge. It is transparent to itself.
  78. Friday, June 3rd, 2011 Systems Group Lunch Seminar Some References

    §  Gardner, M., 1973. Free Will Revisited, With a Mind-Bending Prediction Paradox by William Newcomb. Scientific American 229 §  Dupuy, J.-P., 2000. Philosophical Foundations of a New Concept of Equilibrium in the Social Sciences: Projected Equilibrium. Philosophical Studies 100, 323–356. §  Nash, J., 1951. Non-cooperative Games. Annals of Mathematics 54, 286 – 295. §  Perfect Prediction Equilibrium: to be submitted for publication §  Deutsch, D., 1991. Quantum Mechanics near Closed Timelike Curves, Physical Review D44. 3197-3217. 78
  79. Friday, June 3rd, 2011 Systems Group Lunch Seminar THANK YOU!

    Questions/Discussion 79
  80. Friday, June 3rd, 2011 Systems Group Lunch Seminar Another example:

    Take-or-Leave Game 80 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  81. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 81 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  82. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 82 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  83. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 83 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  84. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 84 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  85. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 85 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  86. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 86 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  87. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 87 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  88. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 88 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  89. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 89 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  90. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 90 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  91. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 91 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  92. Friday, June 3rd, 2011 Systems Group Lunch Seminar Take-or-Leave Game:

    Nash Equilibrium 92 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  93. Friday, June 3rd, 2011 Systems Group Lunch Seminar TOL Game:

    Perfect Prediction Equilibrium 93 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  94. Friday, June 3rd, 2011 Systems Group Lunch Seminar TOL Game:

    Perfect Prediction Equilibrium 94 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  95. Friday, June 3rd, 2011 Systems Group Lunch Seminar TOL Game:

    Perfect Prediction Equilibrium 95 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  96. Friday, June 3rd, 2011 Systems Group Lunch Seminar TOL Game:

    Perfect Prediction Equilibrium 96 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  97. Friday, June 3rd, 2011 Systems Group Lunch Seminar TOL Game:

    Perfect Prediction Equilibrium 97 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  98. Friday, June 3rd, 2011 Systems Group Lunch Seminar TOL Game:

    Perfect Prediction Equilibrium 98 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  99. Friday, June 3rd, 2011 Systems Group Lunch Seminar TOL Game:

    Perfect Prediction Equilibrium 99 1,0 0,2 3,0 0,4 5,0 0,6 7,0
  100. Friday, June 3rd, 2011 Systems Group Lunch Seminar 100 Pictures

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