How to Gamble Against All Odds
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1 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
2 Preface starting with an algorithmic randomness problem transformed to a similar game, without computability How to Gamble Against All Odds 2
3 Unpredictability betting strategy over an infinite casino sequence {h, t} N How to Gamble Against All Odds 3
4 Unpredictability betting strategy over an infinite casino sequence {h, t} N bets some money on each bit How to Gamble Against All Odds 3
5 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
6 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
7 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
8 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
9 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
10 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
11 Comparing sets Given A, B, is there a sequence that is B-random, but not A-random? How to Gamble Against All Odds 4
12 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
13 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
14 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
15 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
16 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
17 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
18 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
19 The game 1 Gambler 0 announces her A-strategy How to Gamble Against All Odds 5
20 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
21 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
22 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
23 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
24 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
25 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
26 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
27 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
28 Example Against a single player, A = {1, 2}, B = {1} How to Gamble Against All Odds 6
29 Example Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time How to Gamble Against All Odds 6
30 Example Against a single player, A = {1, 2}, B = {1} 1st phase: bet 2 on heads each time until you are richer (cheating) How to Gamble Against All Odds 6
31 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 How to Gamble Against All Odds 6
32 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
33 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. How to Gamble Against All Odds 6
34 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 How to Gamble Against All Odds 6
35 Containing a Multiple A evades B = B does not contain a multiple of A How to Gamble Against All Odds 7
36 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
37 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) How to Gamble Against All Odds 7
38 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 How to Gamble Against All Odds 7
39 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
40 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
41 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 How to Gamble Against All Odds 7
42 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 How to Gamble Against All Odds 7
43 Bounded and bounded away 1 [1, ) evades N How to Gamble Against All Odds 8
44 Bounded and bounded away 1 [1, ) evades N 2 {1 + 1 n : n N} evades N How to Gamble Against All Odds 8
45 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
46 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) How to Gamble Against All Odds 8
47 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
48 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) Theorem {1, π} evades N If A is bounded, 0 B \ {0}, and B does not contain a multiple of A, then A evades B. How to Gamble Against All Odds 8
49 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. How to Gamble Against All Odds 9
50 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 How to Gamble Against All Odds 9
51 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 How to Gamble Against All Odds 9
52 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
53 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 How to Gamble Against All Odds 9
54 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. How to Gamble Against All Odds 9
55 Not Sufficient R + does not evade [0, 1]. How to Gamble Against All Odds 10
56 Not Sufficient R + does not evade [0, 1]. {..., 4, 2, 1, 1 2, } does not evade {1, 1 2, }. How to Gamble Against All Odds 10
57 Not Sufficient R + does not evade [0, 1]. {..., 4, 2, 1, 1 2, } does not evade {1, 1 2, }. For A, B, define for any x > 0 P(x) = P A,B (x) := {r 0 : r (A [0, x]) B {0} } How to Gamble Against All Odds 10
58 Not Sufficient R + does not evade [0, 1]. {..., 4, 2, 1, 1 2, } does not evade {1, 1 2, }. For A, B, define for any x > 0 P(x) = P A,B (x) := {r 0 : r (A [0, x]) B {0} } For any M 0 q M (x) := max(p(x) [0, M]) How to Gamble Against All Odds 10
59 Not Sufficient R + does not evade [0, 1]. {..., 4, 2, 1, 1 2, } does not evade {1, 1 2, }. For A, B, define for any x > 0 P(x) = P A,B (x) := {r 0 : r (A [0, x]) B {0} } For any M 0 q M (x) := max(p(x) [0, M]) Theorem If for some M, o q M (x) dx =, then A does not evade B. How to Gamble Against All Odds 10
60 Not Sufficient R + does not evade [0, 1]. {..., 4, 2, 1, 1 2, } does not evade {1, 1 2, }. For A, B, define for any x > 0 P(x) = P A,B (x) := {r 0 : r (A [0, x]) B {0} } For any M 0 q M (x) := max(p(x) [0, M]) Theorem If for some M, o q M (x) dx =, then A does not evade B. Compare with previous example; P unbounded How to Gamble Against All Odds 10
61 Example A = N, B = N \ {n 2 } n=1 How to Gamble Against All Odds 11
62 Example A = N, B = N \ {n 2 } n=1 P(x) = {r 0 : r (A [0, x]) B {0} } How to Gamble Against All Odds 11
63 Example A = N, B = N \ {n 2 } n=1 P(x) = {r 0 : r (A [0, x]) B {0} } For any x, any prime larger than x is in P(x) How to Gamble Against All Odds 11
64 Example A = N, B = N \ {n 2 } 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
65 Example A = N, B = N \ {n 2 } 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 q M (x) = max(p(x) [0, M]) = 0 How to Gamble Against All Odds 11
66 {1, π} against N Gambler 0 alternates 1 and π How to Gamble Against All Odds 12
67 {1, π} against N Gambler 0 alternates 1 and π The casino: against a single opponent, ratio-minimize How to Gamble Against All Odds 12
68 {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
69 {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 How to Gamble Against All Odds 12
70 {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 How to Gamble Against All Odds 12
71 {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
72 {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
73 {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
74 {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
75 Not evade f(x) 0 S X n + 1 S(X n) M(X n) M(X n + 1) x How to Gamble Against All Odds 13
76 A repeated game 1 Gambler 0 places a bet How to Gamble Against All Odds 14
77 A repeated game 1 Gambler 0 places a bet 2 Gamblers 1, 2, 3... place bets How to Gamble Against All Odds 14
78 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
79 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 How to Gamble Against All Odds 14
80 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 B i to each opponent How to Gamble Against All Odds 14
81 Future Does N evade {1/n : n N}? How to Gamble Against All Odds 15
82 Future Does N evade {1/n : n N}? Does R +? How to Gamble Against All Odds 15
83 Future Does N evade {1/n : n N}? Does R +? Does {1/n : n N} evade {1/2 n : n N}? How to Gamble Against All Odds 15
84 Future Does N evade {1/n : n N}? Does R +? Does {1/n : n N} evade {1/2 n : n N}? Probabilistic martingales How to Gamble Against All Odds 15
85 The End How to Gamble Against All Odds 16
86 References I Bienvenu, L., Stephan, F., and Teutsch, J. (2012). How Powerful Are Integer-Valued Martingales? Theory of Computing Systems, 51(3): Buss, S. and Minnes, M. (2013). Probabilistic Algorithmic Randomness. Journal of Symbolic Logic, 78(2): Chalcraft, A., Dougherty, R., Freiling, C., and Teutsch, J. (2012). How to Build a Probability-Free Casino. Information and Computation, 211: How to Gamble Against All Odds 17
87 References II Downey, R. G. and Riemann, J. (2007). Algorithmic Randomness. Scholarpedia 2(10):2574, 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): How to Gamble Against All Odds 18
88 References III Peretz, R. (2013). Effective Martingales with Restricted Wagers. Schnorr, C. P. (1971). A unified approach to the definition of random sequences. Mathematical Systems Theory 5(3): Teutsch, J. (2014). A Savings Paradox for Integer-Valued Gambling Strategies. International Journal of Game Theory 43(1): V yugin, V. (2009). A On Calibration Error of Randomized Forecasting Algorithms Theoretical Computer Science 410: How to Gamble Against All Odds 19
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