Implementation of periodic sequences Call a periodic sequence ˜ x of actions of players 1 and 2 ( m 1 , m 2 ) -implementable if ∃ A 1 , A 2 ∈ Σ m 1 × Σ m 2 that do not observe player 3’s actions and generate ˜ x . pure correlation – p. 11/24
Implementation of periodic sequences Call a periodic sequence ˜ x of actions of players 1 and 2 ( m 1 , m 2 ) -implementable if ∃ A 1 , A 2 ∈ Σ m 1 × Σ m 2 that do not observe player 3’s actions and generate ˜ x . Thus, all m -periodic sequences are ( m, m ) -implementable, and that an ( m 1 , m 2 ) -implementable sequence is at most m 1 m 2 -periodic. pure correlation – p. 11/24
Periods of implementable sequences pure correlation – p. 12/24
Periods of implementable sequences Proposition: Let δ ∈ ∆( X − 3 ) be rational with full support. Let ˜ x be random n -periodic with n first elements i.i.d. ∼ δ . pure correlation – p. 12/24
Periods of implementable sequences Proposition: Let δ ∈ ∆( X − 3 ) be rational with full support. Let ˜ x be random n -periodic with n first elements i.i.d. ∼ δ . Then ∃ C such that n ≤ Cm ln m implies x is ( m, m ) -implementable ) → 1 P (˜ pure correlation – p. 12/24
Periods of implementable sequences Proposition: Let δ ∈ ∆( X − 3 ) be rational with full support. Let ˜ x be random n -periodic with n first elements i.i.d. ∼ δ . Then ∃ C such that n ≤ Cm ln m implies x is ( m, m ) -implementable ) → 1 P (˜ Hence, a pair of automata of size m can jointly implement almost every Cm ln m periodic sequences. pure correlation – p. 12/24
Proof of the main result from the prop. pure correlation – p. 13/24
Proof of the main result from the prop. Let m = min( m 1 , m 2 ) . pure correlation – p. 13/24
Proof of the main result from the prop. Let m = min( m 1 , m 2 ) . 1. Choose n such that m ln m ≫ n ≫ m 3 ln m 3 . pure correlation – p. 13/24
Proof of the main result from the prop. Let m = min( m 1 , m 2 ) . 1. Choose n such that m ln m ≫ n ≫ m 3 ln m 3 . 2. Approximate an optimal correlated strategy of players 1 and 2 in G by δ rational with full support. pure correlation – p. 13/24
Proof of the main result from the prop. Let m = min( m 1 , m 2 ) . 1. Choose n such that m ln m ≫ n ≫ m 3 ln m 3 . 2. Approximate an optimal correlated strategy of players 1 and 2 in G by δ rational with full support. 3. Draw ˜ x n -periodic, with n first coordinates i.i.d. ∼ δ . pure correlation – p. 13/24
Proof of the main result from the prop. Let m = min( m 1 , m 2 ) . 1. Choose n such that m ln m ≫ n ≫ m 3 ln m 3 . 2. Approximate an optimal correlated strategy of players 1 and 2 in G by δ rational with full support. 3. Draw ˜ x n -periodic, with n first coordinates i.i.d. ∼ δ . Then for ε > 0 pure correlation – p. 13/24
Proof of the main result from the prop. Let m = min( m 1 , m 2 ) . 1. Choose n such that m ln m ≫ n ≫ m 3 ln m 3 . 2. Approximate an optimal correlated strategy of players 1 and 2 in G by δ rational with full support. 3. Draw ˜ x n -periodic, with n first coordinates i.i.d. ∼ δ . Then for ε > 0 x, A 3 ) < min P (min A 3 γ (˜ x 3 E δ g − ε ) → 0 x is ( m, m ) -implementable ) P (˜ → 1 pure correlation – p. 13/24
Proof of the main result from the prop. Let m = min( m 1 , m 2 ) . 1. Choose n such that m ln m ≫ n ≫ m 3 ln m 3 . 2. Approximate an optimal correlated strategy of players 1 and 2 in G by δ rational with full support. 3. Draw ˜ x n -periodic, with n first coordinates i.i.d. ∼ δ . Then for ε > 0 x, A 3 ) < min P (min A 3 γ (˜ x 3 E δ g − ε ) → 0 x is ( m, m ) -implementable ) P (˜ → 1 In particular, there exist ( m, m ) -implementable sequences that guarantee min x 3 E δ g − ε . pure correlation – p. 13/24
Implementation of sequences pure correlation – p. 14/24
Implementation of sequences Let ˜ x be n -periodic. We construct an automaton of player 1 that follows ˜ x as long as the other player does. pure correlation – p. 14/24
Implementation of sequences Let ˜ x be n -periodic. We construct an automaton of player 1 that follows ˜ x as long as the other player does. For 1 ≤ l ≤ n , let φ be a permutation of X 2 , and let ˜ y n -periodic such that for 1 ≤ t ≤ n . � y t = ˜ ˜ if l does not divide t ; x t , x 1 x 2 y t = (˜ ˜ t , φ (˜ t )) if l divides t . pure correlation – p. 14/24
Implementation of sequences Let ˜ x be n -periodic. We construct an automaton of player 1 that follows ˜ x as long as the other player does. For 1 ≤ l ≤ n , let φ be a permutation of X 2 , and let ˜ y n -periodic such that for 1 ≤ t ≤ n . � y t = ˜ ˜ if l does not divide t ; x t , x 1 x 2 y t = (˜ ˜ t , φ (˜ t )) if l divides t . y 1 ˜ t is player 1’s action at stage t . pure correlation – p. 14/24
Implementation of sequences Let ˜ x be n -periodic. We construct an automaton of player 1 that follows ˜ x as long as the other player does. For 1 ≤ l ≤ n , let φ be a permutation of X 2 , and let ˜ y n -periodic such that for 1 ≤ t ≤ n . � y t = ˜ ˜ if l does not divide t ; x t , x 1 x 2 y t = (˜ ˜ t , φ (˜ t )) if l divides t . y 1 ˜ t is player 1’s action at stage t . y 2 ˜ t is player 1’s anticipation at stage t , it differs from the x 2 played action ˜ t of player 2 every l stages. pure correlation – p. 14/24
Implementation of sequences Let ˜ x be n -periodic. We construct an automaton of player 1 that follows ˜ x as long as the other player does. For 1 ≤ l ≤ n , let φ be a permutation of X 2 , and let ˜ y n -periodic such that for 1 ≤ t ≤ n . � y t = ˜ ˜ if l does not divide t ; x t , x 1 x 2 y t = (˜ ˜ t , φ (˜ t )) if l divides t . y 1 ˜ t is player 1’s action at stage t . y 2 ˜ t is player 1’s anticipation at stage t , it differs from the x 2 played action ˜ t of player 2 every l stages. y as the concatenation of words We write the first period of ˜ l in ( X − 3 ) l . r 1 . . . r n pure correlation – p. 14/24
Implementation of sequences Let ˜ x be n -periodic. We construct an automaton of player 1 that follows ˜ x as long as the other player does. For 1 ≤ l ≤ n , let φ be a permutation of X 2 , and let ˜ y n -periodic such that for 1 ≤ t ≤ n . � y t = ˜ ˜ if l does not divide t ; x t , x 1 x 2 y t = (˜ ˜ t , φ (˜ t )) if l divides t . y 1 ˜ t is player 1’s action at stage t . y 2 ˜ t is player 1’s anticipation at stage t , it differs from the x 2 played action ˜ t of player 2 every l stages. y as the concatenation of words We write the first period of ˜ l in ( X − 3 ) l . All words are i.i.d. ∼ ρ . r 1 . . . r n pure correlation – p. 14/24
Set of states pure correlation – p. 15/24
Set of states Let α > 1 . The set of states is a cycle z 1 , . . . , z m of elements of X − 3 such that for every r , N ( r ) = # { i, ( z i , . . . z i + l ) = r } ≥ αρ ( r ) n l pure correlation – p. 15/24
Set of states Let α > 1 . The set of states is a cycle z 1 , . . . , z m of elements of X − 3 such that for every r , N ( r ) = # { i, ( z i , . . . z i + l ) = r } ≥ αρ ( r ) n l Relying on DeBruijn sequences, we can construct such a cycle if m ≥ β n l for some β > 0 . pure correlation – p. 15/24
Programmation pure correlation – p. 16/24
Programmation If the anticipation is correct, go to the next state in the cycle. pure correlation – p. 16/24
Programmation If the anticipation is correct, go to the next state in the cycle. q 1 = i 1 such that ( z i 1 , z i 1 +1 , . . . , z i 1 + l − 1 ) = r 1 Start at ˆ pure correlation – p. 16/24
Programmation If the anticipation is correct, go to the next state in the cycle. q 1 = i 1 such that ( z i 1 , z i 1 +1 , . . . , z i 1 + l − 1 ) = r 1 Start at ˆ At z i 1 + l − 1 , if the action of 2 does not match the anticipation, go to i 2 such that ( z i 2 , z i 2 +1 , . . . , z i 2 + l − 1 ) = r 2 pure correlation – p. 16/24
Programmation If the anticipation is correct, go to the next state in the cycle. q 1 = i 1 such that ( z i 1 , z i 1 +1 , . . . , z i 1 + l − 1 ) = r 1 Start at ˆ At z i 1 + l − 1 , if the action of 2 does not match the anticipation, go to i 2 such that ( z i 2 , z i 2 +1 , . . . , z i 2 + l − 1 ) = r 2 At z i 2 + l − 1 , if the action of 2 does not match the anticipation, go to i 3 such that ( z i 3 , z i 3 +1 , . . . , z i 3 + l − 1 ) = r 3 pure correlation – p. 16/24
Programmation If the anticipation is correct, go to the next state in the cycle. q 1 = i 1 such that ( z i 1 , z i 1 +1 , . . . , z i 1 + l − 1 ) = r 1 Start at ˆ At z i 1 + l − 1 , if the action of 2 does not match the anticipation, go to i 2 such that ( z i 2 , z i 2 +1 , . . . , z i 2 + l − 1 ) = r 2 At z i 2 + l − 1 , if the action of 2 does not match the anticipation, go to i 3 such that ( z i 3 , z i 3 +1 , . . . , z i 3 + l − 1 ) = r 3 . . . pure correlation – p. 16/24
Size pure correlation – p. 17/24
Size When can we apply the construction? pure correlation – p. 17/24
Size When can we apply the construction? Two different transitions after after two incorrect anticipations must lead to two different states. pure correlation – p. 17/24
Size When can we apply the construction? Two different transitions after after two incorrect anticipations must lead to two different states. We thus need ∀ r, # { j, r j = r } ≤ N ( r ) pure correlation – p. 17/24
Size When can we apply the construction? Two different transitions after after two incorrect anticipations must lead to two different states. We thus need ∀ r, # { j, r j = r } ≤ N ( r ) This holds if ∀ r, # { j, r j = r } ≤ αρ ( r ) n l pure correlation – p. 17/24
Size When can we apply the construction? Two different transitions after after two incorrect anticipations must lead to two different states. We thus need ∀ r, # { j, r j = r } ≤ N ( r ) This holds if ∀ r, # { j, r j = r } ≤ αρ ( r ) n l Computation shows that this has probability close to one if l = γ ( α ) ln n . pure correlation – p. 17/24
Size When can we apply the construction? Two different transitions after after two incorrect anticipations must lead to two different states. We thus need ∀ r, # { j, r j = r } ≤ N ( r ) This holds if ∀ r, # { j, r j = r } ≤ αρ ( r ) n l Computation shows that this has probability close to one if l = γ ( α ) ln n . β Hence m ≥ β n n l = ln n , or for some C : γ ( α ) n ≤ Cm ln m pure correlation – p. 17/24
Length of implementable sequences pure correlation – p. 18/24
Length of implementable sequences What is the order of magnitude of n ( m ) such that the set of n ( m ) periodic ( m, m ) -implementable sequences has large probability? pure correlation – p. 18/24
Length of implementable sequences What is the order of magnitude of n ( m ) such that the set of n ( m ) periodic ( m, m ) -implementable sequences has large probability? We have proven the existence of C such that n ( m ) ≥ Cm ln m pure correlation – p. 18/24
Length of implementable sequences What is the order of magnitude of n ( m ) such that the set of n ( m ) periodic ( m, m ) -implementable sequences has large probability? We have proven the existence of C such that n ( m ) ≥ Cm ln m We also know that if n ( m ) ≫ m 3 ln m 3 then V p ( m, m, m 3 ) → v c . pure correlation – p. 18/24
Length of implementable sequences What is the order of magnitude of n ( m ) such that the set of n ( m ) periodic ( m, m ) -implementable sequences has large probability? We have proven the existence of C such that n ( m ) ≥ Cm ln m We also know that if n ( m ) ≫ m 3 ln m 3 then V p ( m, m, m 3 ) → v c . Thus we do not have n ( m ) ≫ m ln m pure correlation – p. 18/24
Any number of players Players { 1 , . . . , I } against player I + 1 . If min( m 1 . . . m I ) ≫ m I +1 and at least 2 players { 1 , . . . , I } have at least two actions, then { 1 , . . . , I } possess pure strategies that guarantee the correlated max min against I + 1 . pure correlation – p. 19/24
On the power of a team pure correlation – p. 20/24
On the power of a team One player of size m can implement all m -periodic sequences. pure correlation – p. 20/24
On the power of a team One player of size m can implement all m -periodic sequences. Two players of size m can implement almost all Cm ln m -periodic sequences. pure correlation – p. 20/24
On the power of a team One player of size m can implement all m -periodic sequences. Two players of size m can implement almost all Cm ln m -periodic sequences. More than two players cannot implement a large set of sequences of significantly larger period (or they could obtain v c against a player of the same size as theirs). pure correlation – p. 20/24
Correlated strategies 1 pure correlation – p. 21/24
Correlated strategies 1 We derive results from two player games. pure correlation – p. 21/24
Correlated strategies 1 We derive results from two player games. From Ben Porath (93): If ln m 3 ≪ m then V c ( m, m, m 3 ) → v c pure correlation – p. 21/24
Correlated strategies 1 We derive results from two player games. From Ben Porath (93): If ln m 3 ≪ m then V c ( m, m, m 3 ) → v c Furthermore, the same limit obtains when players 1 , 2 use oblivious strategies only. pure correlation – p. 21/24
Correlated strategies 1 We derive results from two player games. From Ben Porath (93): If ln m 3 ≪ m then V c ( m, m, m 3 ) → v c Furthermore, the same limit obtains when players 1 , 2 use oblivious strategies only. Over a period, each initial state of an automaton of player 3 can force a set of bounded probability of sequences to a significantly smaller payoff than E δ g − ε . pure correlation – p. 21/24
Correlated strategies 1 We derive results from two player games. From Ben Porath (93): If ln m 3 ≪ m then V c ( m, m, m 3 ) → v c Furthermore, the same limit obtains when players 1 , 2 use oblivious strategies only. Over a period, each initial state of an automaton of player 3 can force a set of bounded probability of sequences to a significantly smaller payoff than E δ g − ε . The asymptotic condition on m 3 and n is that this probability times the number m 3 of states for 3 goes to 0. pure correlation – p. 21/24
Correlated strategies improved pure correlation – p. 22/24
Correlated strategies improved Since two players of size m can implement a large set of sequences of size m ln m , applying the same method shows. pure correlation – p. 22/24
Correlated strategies improved Since two players of size m can implement a large set of sequences of size m ln m , applying the same method shows. If ln m 3 ≪ m ln m then V c ( m, m, m 3 ) → v c pure correlation – p. 22/24
Correlated strategies 2 pure correlation – p. 23/24
Correlated strategies 2 From Neyman (97): With K = ln | X 1 × X 2 | , if ln m 3 ≥ Km 1 m 2 then V c ( m 1 , m 2 , m 3 ) → v p pure correlation – p. 23/24
Correlated strategies 2 From Neyman (97): With K = ln | X 1 × X 2 | , if ln m 3 ≥ Km 1 m 2 then V c ( m 1 , m 2 , m 3 ) → v p There is a (mixed) strategy of player 3 that eventually plays a best response to almost all sequences of actions of players 1 and 2. pure correlation – p. 23/24
Correlated strategies 2 From Neyman (97): With K = ln | X 1 × X 2 | , if ln m 3 ≥ Km 1 m 2 then V c ( m 1 , m 2 , m 3 ) → v p There is a (mixed) strategy of player 3 that eventually plays a best response to almost all sequences of actions of players 1 and 2. This automaton is capable of finding which sequence of actions is implemented by players 1 and 2 with high probability. pure correlation – p. 23/24
Conjecture pure correlation – p. 24/24
Recommend
More recommend
