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
§ 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
§ 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
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
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
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
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.
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.
§ 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.
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
(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!
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
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
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
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.
§ 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