GPD GTP? Dusko Pavlovic Pennies Does God Play Dice Question in Game Theoretic Probability? Dusko Pavlovic University of Hawaii GTP , Guanajuato, 14/11/14
GPD GTP? Outline Dusko Pavlovic Pennies Matching Pennies Question Question
GPD GTP? Outline Dusko Pavlovic Pennies Matching Pennies Game No Winning Winning Game Question No Wealth by Matching Pennies Wealth by Matching Pennies Question
GPD GTP? Bimatrix presentation of 2-player games Dusko Pavlovic Pennies Game ◮ n = 2 No Winning Winning ◮ A 1 = { U , D } Question ◮ A 2 = { L , R } ◮ u = � u 1 , u 2 � : A 1 × A 2 → R × R L R u 2 ( U , L ) u 2 ( U , R ) u 1 ( U , L ) u 1 ( U , R ) U u 2 ( D , L ) u 2 ( D , R ) D u 1 ( D , L ) u 1 ( D , R )
GPD GTP? Matching Pennies Dusko Pavlovic Pennies Game ◮ players: A , B No Winning Winning ◮ moves: M A = M B = { H , T } Question ◮ u = � u A , u B � : M A × M B → R × R H T − 1 1 H 1 − 1 1 − 1 T − 1 1
GPD GTP? Matching Pennies Dusko Pavlovic Pennies Game No Winning Winning Strategy: Randomize! Question The only Nash equilibrium for Matching Pennies is the profile � a , b � where the players randomize p ( a = H ) = 1 2 = p ( b = H )
GPD GTP? Matching Pennies Dusko Pavlovic Pennies Game No Winning Winning Randomness: Strategy! Question The other way around, we can define that a sequence H , T , T , H , T . . . is random iff it is a strategy for Matching Pennies that does not lose against any opponent.
GPD GTP? Matching Pennies Dusko Pavlovic Pennies Game No Winning Winning Randomness: Strategy! Question The other way around, we can define that a sequence H , T , T , H , T . . . is random iff it is a strategy for Matching Pennies that does not lose against any opponent. [ Reason : If you can write a short program to predict the next move with probability > 1 2 , then you can win Matching Pennies.]
GPD GTP? Matching Pennies Dusko Pavlovic Pennies Game No Winning Winning Question Suspicion ◮ Is this a bit like Game Theoretic Probability?
GPD GTP? Matching Pennies Dusko Pavlovic Pennies Game No Winning Winning Question Suspicion ◮ Is this a bit like Game Theoretic Probability? ◮ Maybe not quite. . .
GPD GTP? Matching Pennies against Nature Dusko Pavlovic Pennies Game ◮ players: A , B , N No Winning Winning ◮ moves: M A = M B = { + , −} , M N = { 00 , 01 , 10 , 11 } Question ◮ u = � u A , B , u N � : M A , B × M N → R × R 00,01,10 11 − 1 1 ++, -- 1 − 1 1 − 1 +-, -+ − 1 1
GPD GTP? Matching Pennies against Nature Dusko Pavlovic Pennies Game No Winning Game protocol Winning Question ◮ N moves first with xy ∈ { 0 , 1 } 2 ◮ A sees x (not y or b ) and responds with a ∈ { + , −} ◮ B sees y (not x or a ) and responds with b ∈ { + , −}
GPD GTP? Matching Pennies against Nature Dusko Pavlovic Pennies Game No Winning Game protocol Winning Question ◮ N moves first with xy ∈ { 0 , 1 } 2 ◮ A sees x (not y or b ) and responds with a ∈ { + , −} ◮ B sees y (not x or a ) and responds with b ∈ { + , −} Remark They play a game of imperfect information.
GPD GTP? Coordinating pennies: Strategies Dusko Pavlovic Pennies Game No Winning Winning Question ◮ N ’s moves xy are random and uniformly distributed. ◮ A and B should coordinate to specify ◮ A ’s strategy: probability distribution p ( a | x ) ◮ B ’s strategy: probability distribution p ( b | y ) to maximize their payoffs.
GPD GTP? Coordinating pennies: Payoffs Dusko Pavlovic Pennies Game No Winning Winning Question 1 � � = E AB ( 00 ) + E AB ( 01 ) + E AB ( 10 ) − E AB ( 11 ) U AB 4 � E AB ( xy ) = a · b · p ( ab | xy ) a , b ∈ M AB where we muliply a , b ∈ { + , −} as if they are + 1 and − 1
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Game No Winning Theorem Winning Question If the mutual dependency of x and y is expressed by a variable λ ∈ Λ with density q : Λ → [ 0 , 1 ] , so that � p ( ab | xy ) = p ( a | x , λ ) · p ( b | y , λ ) · q ( λ ) d λ (1) Λ then 1 ≤ (2) U AB 2
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning Write Question � E A ( x , λ ) = a · p ( a | x , λ ) a ∈ M A � E B ( y , λ ) = b · p ( b | y , λ ) b ∈ M B E AB ( xy , λ ) = E A ( x , λ ) · E B ( y , λ )
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning Then Question � = U AB ( λ ) · q ( λ ) d λ U AB Λ for 1 � U AB ( λ ) = E A ( 0 , λ ) · E B ( 0 , λ ) + E A ( 0 , λ ) · E B ( 1 , λ ) + 4 � E A ( 1 , λ ) · E B ( 0 , λ ) − E A ( 1 , λ ) · E B ( 1 , λ )
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning Then Question � = U AB ( λ ) · q ( λ ) d λ U AB Λ for 1 � U AB ( λ ) = E A ( 0 , λ ) · E B ( 0 , λ ) + E A ( 0 , λ ) · E B ( 1 , λ ) + 4 � E A ( 1 , λ ) · E B ( 0 , λ ) − E A ( 1 , λ ) · E B ( 1 , λ ) 1 � � � = E A ( 0 , λ ) · E B ( 0 , λ ) + E B ( 1 , λ ) + 4 �� � E A ( 1 , λ ) · E B ( 0 , λ ) − E B ( 1 , λ )
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning Since − 1 ≤ E A ( 0 , λ ) , E A ( 1 , λ ) ≤ 1 Question 1 �� � � � � � + U AB ( λ ) ≤ � E B ( 0 , λ ) + E B ( 1 , λ ) � E B ( 0 , λ ) − E B ( 1 , λ ) � � � � � 4
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning Since − 1 ≤ E A ( 0 , λ ) , E A ( 1 , λ ) ≤ 1 Question 1 �� � � � � � + U AB ( λ ) ≤ � E B ( 0 , λ ) + E B ( 1 , λ ) � E B ( 0 , λ ) − E B ( 1 , λ ) � � � � � 4 � � If E B ( 0 , λ ) ≥ max 0 , E B ( 1 , λ ) , then it follows that 1 � � U AB ( λ ) E B ( 0 , λ ) + E B ( 1 , λ ) + E B ( 0 , λ ) − E B ( 1 , λ ) ≤ 4 1 � � = E B ( 0 , λ ) + E B ( 0 , λ ) 4 1 ≤ 2 since 0 ≤ E B ( 0 , λ ) ≤ 1.
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning Since − 1 ≤ E A ( 0 , λ ) , E A ( 1 , λ ) ≤ 1 Question 1 �� � � � � � + U AB ( λ ) ≤ � E B ( 0 , λ ) + E B ( 1 , λ ) � E B ( 0 , λ ) − E B ( 1 , λ ) � � � � � 4 If 0 ≥ E B ( 0 , λ ) ≥ E B ( 1 , λ ) , then it follows that 1 � � U AB ( λ ) − E B ( 0 , λ ) − E B ( 1 , λ ) + E B ( 0 , λ ) − E B ( 1 , λ ) ≤ 4 1 � � = − E B ( 1 , λ ) − E B ( 1 , λ ) 4 1 ≤ 2 since 0 ≥ E B ( 1 , λ ) ≥ − 1.
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning Since − 1 ≤ E A ( 0 , λ ) , E A ( 1 , λ ) ≤ 1 Question 1 �� � � � � � + U AB ( λ ) ≤ � E B ( 0 , λ ) + E B ( 1 , λ ) � E B ( 0 , λ ) − E B ( 1 , λ ) � � � � � 4 If E B ( 0 , λ ) ≤ E B ( 1 , λ ) , then the two analogous cases again give 1 U AB ( λ ) ≤ 2
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Proof Game No Winning Winning In all cases Question � � 1 1 = U AB ( λ ) · q ( λ ) d λ 2 q ( λ ) d λ = U AB ≤ 2 Λ Λ �
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Interpretation Game No Winning Winning Suppose that Question ◮ A , B and N repeat the game infinitely often, and ◮ A and B invest $ 1 2 each for every bet.
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Interpretation Game No Winning Winning Suppose that Question ◮ A , B and N repeat the game infinitely often, and ◮ A and B invest $ 1 2 each for every bet. Since N ’s moves are uniformly distributed, A and B ’s chances are 3 4 to win $ 1 ◮ 1 4 to lose $ 1 ◮ i.e. the expected winnings for each of them are 3 4 ($ 1 ) + 1 $ 1 4 ( − $ 1 ) = 2
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Interpretation Game No Winning Winning ◮ So if A and B randomize their moves uniformly, Question in the long run their wealth remains unchanged.
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Interpretation Game No Winning Winning ◮ So if A and B randomize their moves uniformly, Question in the long run their wealth remains unchanged. ◮ This is the Nash equilibrium of Matching Pennies.
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Interpretation Game No Winning Winning ◮ So if A and B randomize their moves uniformly, Question in the long run their wealth remains unchanged. ◮ This is the Nash equilibrium of Matching Pennies. ◮ The question is whether they can increase their wealth by coordinating. ◮ The answer suggested by the Theorem is NO .
GPD GTP? Hidden Variable Theorem Dusko Pavlovic Pennies Game No Winning Winning Question Another suspicion ◮ Is averaging out the hidden variable λ really the only way in which A and B can coordinate? ◮ Maybe not?
GPD GTP? Idea Dusko Pavlovic Pennies Game No Winning Winning Question E.g., they could also use entangled photons
GPD GTP? Idea Dusko Pavlovic Pennies Game No Winning Winning Question Plants extract their strategic advantage similarly: photosynthesis is a quantum effect!
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