How to Gamble Against All Odds Gilad Bavly 1 Ron Peretz 2 1 Bar-Ilan - - PowerPoint PPT Presentation

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How to Gamble Against All Odds Gilad Bavly 1 Ron Peretz 2 1 Bar-Ilan - - PowerPoint PPT Presentation

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How to Gamble Against All Odds Gilad Bavly 1 Ron Peretz 2 1 Bar-Ilan University 2 London School of Economics Heidelberg, June 2015 How to Gamble Against All Odds 1 Preface starting with an algorithmic randomness problem transformed to a

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How to Gamble Against All Odds

Gilad Bavly1 Ron Peretz2

1Bar-Ilan University 2London School of Economics

Heidelberg, June 2015

How to Gamble Against All Odds 1

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Preface

starting with an algorithmic randomness problem transformed to a similar game, without computability

How to Gamble Against All Odds 2

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

How to Gamble Against All Odds 3

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

bets some money on each bit

How to Gamble Against All Odds 3

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

bets some money on each bit cannot borrow

How to Gamble Against All Odds 3

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

bets some money on each bit cannot borrow chooses initial wealth

How to Gamble Against All Odds 3

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

bets some money on each bit cannot borrow chooses initial wealth

can be represented by a martingale M : {h, t}∗ → R+ s.t. M(σ) = M(σh)+M(σt)

2

for any history σ ∈ {h, t}∗

How to Gamble Against All Odds 3

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

bets some money on each bit cannot borrow chooses initial wealth

can be represented by a martingale M : {h, t}∗ → R+ s.t. M(σ) = M(σh)+M(σt)

2

for any history σ ∈ {h, t}∗ A sequence is computably random if no computable martingale succeeds on it.

How to Gamble Against All Odds 3

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

bets some money on each bit cannot borrow chooses initial wealth

can be represented by a martingale M : {h, t}∗ → R+ s.t. M(σ) = M(σh)+M(σt)

2

for any history σ ∈ {h, t}∗ A sequence is computably random if no computable martingale succeeds on it.

success: unbounded gains

How to Gamble Against All Odds 3

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Unpredictability

betting strategy over an infinite casino sequence ∈ {h, t}N

bets some money on each bit cannot borrow chooses initial wealth

can be represented by a martingale M : {h, t}∗ → R+ s.t. M(σ) = M(σh)+M(σt)

2

for any history σ ∈ {h, t}∗ A sequence is computably random if no computable martingale succeeds on it.

success: unbounded gains

For A ⊂ R+, A-valued random if limiting wagers to A ∀σ |M(σh) − M(σ)| ∈ A

How to Gamble Against All Odds 3

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

How to Gamble Against All Odds 4

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

1 choose a computable A-strategy How to Gamble Against All Odds 4

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

1 choose a computable A-strategy 2 competes against the whole set of computable B-strategies How to Gamble Against All Odds 4

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

1 choose a computable A-strategy 2 competes against the whole set of computable B-strategies 3 choose a casino sequence How to Gamble Against All Odds 4

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

1 choose a computable A-strategy 2 competes against the whole set of computable B-strategies 3 choose a casino sequence

Can we choose s.t. only the A-strategy succeeds?

How to Gamble Against All Odds 4

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

1 choose a computable A-strategy 2 competes against the whole set of computable B-strategies 3 choose a casino sequence

Can we choose s.t. only the A-strategy succeeds? A = B, A ⊆ B

How to Gamble Against All Odds 4

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

1 choose a computable A-strategy 2 competes against the whole set of computable B-strategies 3 choose a casino sequence

Can we choose s.t. only the A-strategy succeeds? A = B, A ⊆ B A = {1, 2}, B = {1/2, 1}

How to Gamble Against All Odds 4

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Comparing sets

Given A, B, is there a sequence that is B-random, but not A-random?

1 choose a computable A-strategy 2 competes against the whole set of computable B-strategies 3 choose a casino sequence

Can we choose s.t. only the A-strategy succeeds? A = B, A ⊆ B A = {1, 2}, B = {1/2, 1} Countably many B-strategies

How to Gamble Against All Odds 4

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The game

1 Gambler 0 announces her A-strategy How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies 3 casino chooses a sequence How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies 3 casino chooses a sequence

Goal: “home team” (gambler 0 + casino) wins if player 0 succeeds and the others don’t.

How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies 3 casino chooses a sequence

Goal: “home team” (gambler 0 + casino) wins if player 0 succeeds and the others don’t. pure strategies, no probabilities

How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies 3 casino chooses a sequence

Goal: “home team” (gambler 0 + casino) wins if player 0 succeeds and the others don’t. pure strategies, no probabilities “passing” is allowed

How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies 3 casino chooses a sequence

Goal: “home team” (gambler 0 + casino) wins if player 0 succeeds and the others don’t. pure strategies, no probabilities “passing” is allowed If home team can win, we say that A evades B.

How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies 3 casino chooses a sequence

Goal: “home team” (gambler 0 + casino) wins if player 0 succeeds and the others don’t. pure strategies, no probabilities “passing” is allowed If home team can win, we say that A evades B. “not evade” is a preorder

How to Gamble Against All Odds 5

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The game

1 Gambler 0 announces her A-strategy 2 Gamblers 1, 2, 3 . . . announce their B-strategies 3 casino chooses a sequence

Goal: “home team” (gambler 0 + casino) wins if player 0 succeeds and the others don’t. pure strategies, no probabilities “passing” is allowed If home team can win, we say that A evades B. “not evade” is a preorder “not evade each other” is an equivalence relation

How to Gamble Against All Odds 5

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Example

Against a single player, A = {1, 2}, B = {1}

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Example

Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time

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Example

Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time until you are richer (cheating)

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Example

Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time until you are richer (cheating) 2nd phase: bet 1 on heads all the time

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Example

Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time until you are richer (cheating) 2nd phase: bet 1 on heads all the time casino chooses heads on 1st phase; fails the opponent on the 2nd

How to Gamble Against All Odds 6

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Example

Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time until you are richer (cheating) 2nd phase: bet 1 on heads all the time casino chooses heads on 1st phase; fails the opponent on the 2nd Either the opponent is bankrupt, or he stops betting.

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Example

Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time until you are richer (cheating) 2nd phase: bet 1 on heads all the time casino chooses heads on 1st phase; fails the opponent on the 2nd Either the opponent is bankrupt, or he stops betting. (fix cheating) casino chooses tails to signal phase change

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A Is it also a sufficient condition?

How to Gamble Against All Odds 7

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A Is it also a sufficient condition? It is for finite A, B (Chalcraft, Dougherty, Freiling, Teutsch, 2012)

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A Is it also a sufficient condition? It is for finite A, B (Chalcraft, Dougherty, Freiling, Teutsch, 2012) {1, 2, 3} and {5, 7, 9, 10} evade each other

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A Is it also a sufficient condition? It is for finite A, B (Chalcraft, Dougherty, Freiling, Teutsch, 2012) {1, 2, 3} and {5, 7, 9, 10} evade each other What about general sets A, B ?

How to Gamble Against All Odds 7

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A Is it also a sufficient condition? It is for finite A, B (Chalcraft, Dougherty, Freiling, Teutsch, 2012) {1, 2, 3} and {5, 7, 9, 10} evade each other What about general sets A, B ? Any set is equivalent to its closure

How to Gamble Against All Odds 7

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A Is it also a sufficient condition? It is for finite A, B (Chalcraft, Dougherty, Freiling, Teutsch, 2012) {1, 2, 3} and {5, 7, 9, 10} evade each other What about general sets A, B ? Any set is equivalent to its closure

From here on A, B are closed

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Containing a Multiple

A evades B = ⇒ B does not contain a multiple of A Is it also a sufficient condition? It is for finite A, B (Chalcraft, Dougherty, Freiling, Teutsch, 2012) {1, 2, 3} and {5, 7, 9, 10} evade each other What about general sets A, B ? Any set is equivalent to its closure

From here on A, B are closed

still not sufficient in general; but is sufficient for some classes of A, B

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Bounded and bounded away

1 [1, ∞) evades N How to Gamble Against All Odds 8

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Bounded and bounded away

1 [1, ∞) evades N 2 {1 + 1

n : n ∈ N} evades N

How to Gamble Against All Odds 8

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Bounded and bounded away

1 [1, ∞) evades N 2 {1 + 1

n : n ∈ N} evades N

3 { 1

n : n ∈ N} evades [1, ∞)

How to Gamble Against All Odds 8

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Bounded and bounded away

1 [1, ∞) evades N 2 {1 + 1

n : n ∈ N} evades N

3 { 1

n : n ∈ N} evades [1, ∞)

from Peretz (2013)

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Bounded and bounded away

1 [1, ∞) evades N 2 {1 + 1

n : n ∈ N} evades N

3 { 1

n : n ∈ N} evades [1, ∞)

from Peretz (2013) {1, π} evades N

How to Gamble Against All Odds 8

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Bounded and bounded away

1 [1, ∞) evades N 2 {1 + 1

n : n ∈ N} evades N

3 { 1

n : n ∈ N} evades [1, ∞)

from Peretz (2013) {1, π} evades N Theorem If A is bounded, 0 ∈ B \ {0}, and B does not contain a multiple of A, then A evades B.

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Well-ordered

Theorem If B is well-ordered, namely ∀x ∈ R+ x ∈ B \ [0, x], and B does not contain a multiple of A, then A evades B.

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Well-ordered

Theorem If B is well-ordered, namely ∀x ∈ R+ x ∈ B \ [0, x], and B does not contain a multiple of A, then A evades B. B ⊆ N contains no ideal r · N = ⇒ N evades B

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Well-ordered

Theorem If B is well-ordered, namely ∀x ∈ R+ x ∈ B \ [0, x], and B does not contain a multiple of A, then A evades B. B ⊆ N contains no ideal r · N = ⇒ N evades B

N evades any 0-density subset

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Well-ordered

Theorem If B is well-ordered, namely ∀x ∈ R+ x ∈ B \ [0, x], and B does not contain a multiple of A, then A evades B. B ⊆ N contains no ideal r · N = ⇒ N evades B

N evades any 0-density subset N evades odds

How to Gamble Against All Odds 9

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Well-ordered

Theorem If B is well-ordered, namely ∀x ∈ R+ x ∈ B \ [0, x], and B does not contain a multiple of A, then A evades B. B ⊆ N contains no ideal r · N = ⇒ N evades B

N evades any 0-density subset N evades odds N evades N \ {n · φ(n)}∞

n=1

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Well-ordered

Theorem If B is well-ordered, namely ∀x ∈ R+ x ∈ B \ [0, x], and B does not contain a multiple of A, then A evades B. B ⊆ N contains no ideal r · N = ⇒ N evades B

N evades any 0-density subset N evades odds N evades N \ {n · φ(n)}∞

n=1

Evens evade odds, but not vice versa.

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Not Sufficient

R+ does not evade [0, 1].

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Not Sufficient

R+ does not evade [0, 1]. {. . . , 4, 2, 1, 1

2, 1 4 . . .} does not evade {1, 1 2, 1 4 . . .}.

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Not Sufficient

R+ does not evade [0, 1]. {. . . , 4, 2, 1, 1

2, 1 4 . . .} does not evade {1, 1 2, 1 4 . . .}.

For A, B, define for any x > 0 P(x) = PA,B(x) := {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} }

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Not Sufficient

R+ does not evade [0, 1]. {. . . , 4, 2, 1, 1

2, 1 4 . . .} does not evade {1, 1 2, 1 4 . . .}.

For A, B, define for any x > 0 P(x) = PA,B(x) := {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} } For any M ≥ 0 qM(x) := max(P(x) ∩ [0, M])

How to Gamble Against All Odds 10

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Not Sufficient

R+ does not evade [0, 1]. {. . . , 4, 2, 1, 1

2, 1 4 . . .} does not evade {1, 1 2, 1 4 . . .}.

For A, B, define for any x > 0 P(x) = PA,B(x) := {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} } For any M ≥ 0 qM(x) := max(P(x) ∩ [0, M]) Theorem If for some M, ∞

  • qM(x) dx = ∞ , then A does not evade B.

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Not Sufficient

R+ does not evade [0, 1]. {. . . , 4, 2, 1, 1

2, 1 4 . . .} does not evade {1, 1 2, 1 4 . . .}.

For A, B, define for any x > 0 P(x) = PA,B(x) := {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} } For any M ≥ 0 qM(x) := max(P(x) ∩ [0, M]) Theorem If for some M, ∞

  • qM(x) dx = ∞ , then A does not evade B.

Compare with previous example; P unbounded

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Example

A = N, B = N \ {n2}∞

n=1

How to Gamble Against All Odds 11

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Example

A = N, B = N \ {n2}∞

n=1

P(x) = {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} }

How to Gamble Against All Odds 11

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Example

A = N, B = N \ {n2}∞

n=1

P(x) = {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} } For any x, any prime larger than x is in P(x)

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Example

A = N, B = N \ {n2}∞

n=1

P(x) = {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} } For any x, any prime larger than x is in P(x) = ⇒ P(x) is unbounded

How to Gamble Against All Odds 11

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Example

A = N, B = N \ {n2}∞

n=1

P(x) = {r ≥ 0 : r · (A ∩ [0, x]) ⊆ B ∪ {0} } For any x, any prime larger than x is in P(x) = ⇒ P(x) is unbounded For any M ≥ 0 qM(x) = max(P(x) ∩ [0, M]) = 0

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{1, π} against N

Gambler 0 alternates 1 and π

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize Later make him lose everything

How to Gamble Against All Odds 12

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize Later make him lose everything The money ratio weakly decreasing

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize Later make him lose everything The money ratio weakly decreasing = ⇒ It has a limit L

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize Later make him lose everything The money ratio weakly decreasing = ⇒ It has a limit L Lemma: The ratio of wagers is almost always close to L

How to Gamble Against All Odds 12

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize Later make him lose everything The money ratio weakly decreasing = ⇒ It has a limit L Lemma: The ratio of wagers is almost always close to L = ⇒ L = 0

How to Gamble Against All Odds 12

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize Later make him lose everything The money ratio weakly decreasing = ⇒ It has a limit L Lemma: The ratio of wagers is almost always close to L = ⇒ L = 0 Countably many opponents: one by one, with separate “accounts”

How to Gamble Against All Odds 12

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{1, π} against N

Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize Later make him lose everything The money ratio weakly decreasing = ⇒ It has a limit L Lemma: The ratio of wagers is almost always close to L = ⇒ L = 0 Countably many opponents: one by one, with separate “accounts” Attention to the smallest index active

How to Gamble Against All Odds 12

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Not evade

𝑁(𝑌|𝑜) 𝑁(𝑌|𝑜 + 1) 𝑔(𝑦) 𝑦

𝑇 𝑌 𝑜 + 1 − 𝑇(𝑌|𝑜)

How to Gamble Against All Odds 13

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A repeated game

1 Gambler 0 places a bet How to Gamble Against All Odds 14

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A repeated game

1 Gambler 0 places a bet 2 Gamblers 1, 2, 3 . . . place bets How to Gamble Against All Odds 14

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A repeated game

1 Gambler 0 places a bet 2 Gamblers 1, 2, 3 . . . place bets 3 Casino chooses h or t How to Gamble Against All Odds 14

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A repeated game

1 Gambler 0 places a bet 2 Gamblers 1, 2, 3 . . . place bets 3 Casino chooses h or t

The results apply to all variants

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A repeated game

1 Gambler 0 places a bet 2 Gamblers 1, 2, 3 . . . place bets 3 Casino chooses h or t

The results apply to all variants Also for different Bi to each opponent

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Future

Does N evade {1/n : n ∈ N} ?

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Future

Does N evade {1/n : n ∈ N} ?

Does R+ ?

How to Gamble Against All Odds 15

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Future

Does N evade {1/n : n ∈ N} ?

Does R+ ?

Does {1/n : n ∈ N} evade {1/2n : n ∈ N} ?

How to Gamble Against All Odds 15

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Future

Does N evade {1/n : n ∈ N} ?

Does R+ ?

Does {1/n : n ∈ N} evade {1/2n : n ∈ N} ? Probabilistic martingales

How to Gamble Against All Odds 15

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The End

How to Gamble Against All Odds 16

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References I

Bienvenu, L., Stephan, F., and Teutsch, J. (2012). How Powerful Are Integer-Valued Martingales? Theory of Computing Systems, 51(3):330–351. Buss, S. and Minnes, M. (2013). Probabilistic Algorithmic Randomness. Journal of Symbolic Logic, 78(2):579–601. Chalcraft, A., Dougherty, R., Freiling, C., and Teutsch, J. (2012). How to Build a Probability-Free Casino. Information and Computation, 211:160–164.

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References II

Downey, R. G. and Riemann, J. (2007). Algorithmic Randomness. Scholarpedia 2(10):2574, http://www.scholarpedia.org/article/algorithmic randomness. Hu, T. W. (2014). Unpredictability of complex (pure) strategies. Games and Economic Behavior 88:1–15. Hu, T. W. and Shmaya, E. (2013). Expressible inspections. Theoretical Economics 8(2):263–280.

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References III

Peretz, R. (2013). Effective Martingales with Restricted Wagers. http://arxiv.org/abs/1301.7465. Schnorr, C. P. (1971). A unified approach to the definition of random sequences. Mathematical Systems Theory 5(3):246–258. Teutsch, J. (2014). A Savings Paradox for Integer-Valued Gambling Strategies. International Journal of Game Theory 43(1):145–151. V’yugin, V. (2009). A On Calibration Error of Randomized Forecasting Algorithms Theoretical Computer Science 410: 1781– 1795.

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