Rate distortion function in bettinggame system

Masayuki KumonAssociation for Promoting Quality Assurance in Statistics

1

Abstract

Among various aspects of game theoretic

probability, when exploring

mathematical structure of the optimal

strategies in betting games,

Kullback-Leibler divergence is naturally

derived as the optimal exponential growth

rate of the betting capital process.

2

This structure had been obtained by Prof.

Takeuchi nearly fifty years ago.

Inspired by Claude Shannon’s Information

Theory, an optimizing betting strategy was

also pioneered by John Larry Kelly Jr. in

A New Interpretation of Information

Rate. Bell System Technical

Journal, Vol.35, 917-926, 1956.

3

The optimalities of Kelly’s strategy

• Minimal expected time property

• Asymptotic largest magnitude property

were investigated by Leo Breiman in

Optimal Gambling Systems for

Favorable Games. Fourth Berkeley

Symposium on Probability and

Statistics I, 65-78, 1961.

4

The historical reviews and the recent

developments concerning Kelly’s strategy

such as T. M. Cover’s Universal Portfolio

are presented in

L. C. MacLean, E. O. Thorp, W. T.

Ziemba eds. The Kelly Capital

Growth Investment Criterion :

Theory and Practice. Handbook in

Finantial Economic Series, Vol.3,

World Scientific, London, 2010.

5

In this talk, the following are addressed.

• Game mutual information which

measures information transmission

between betting games is introduced.

• Two characteristics Game channel

capacity and Game rate distortion

function are defined from the mutual

information, and these meanings are

explained.

6

• The effect of the optimal strategy in

conditional betting game is verified by

using real stock price data.

• As an application of Game rate

distortion function, an efficient lossy

source coding scheme based on the

optimal conditional betting strategy is

proposed.

7

1. Mutual information in betting game

system

1.1 Definition of mutual information

¥ Mutual information in information

theory

X ∼ PX(x) Y ∼ PY (y) (X, Y ) ∼ PXY (x, y)

H(X) = −EPX[log PX(X)] etc.

8

• Shannon’s source coding theorem :

Entropy H(X) is the nearly achievable

lower bound on the average length of the

shortest description of the random

variable X.

I(X; Y ) = H(X) − H(X|Y ) = H(Y ) − H(Y |X)

= H(X) + H(Y ) − H(X, Y )

= D(PXY ‖PXPY ) ≥ 0

I(X; Y ) = 0 ⇔ PXY (x, y) = PX(x)PY (y)

9

Y)|H(X X)|H(Y

H(X) H(Y)

Y)H(X,

Y)I(X;

10

¥ mutual information in betting games

A, B ∼ two betting games

C ∼ joint betting game of A and B

PA, PB, PC : empirical prob. of Reality

QA, QB, QC : risk neutral prob. of Forecaster

µA := D(PA‖QA) : quantity of the game A

µB := D(PB‖QB) : quantity of the game B

µC := D(PC‖QC) : quantity of the game C

11

A B

B|A AB|

C

B)I(A;

12

I(A; B) := µB|A − µB = µA|B − µA

= µC − (µA + µB)

∵ µC = µA + µB|A = µB + µA|B (additivity)

µB|A := µC − µA = D(PB|A‖QB|A|PA)

µB|A : quantity of the conditional betting game

B|A given A

D(PB|A‖QB|A|PA) :

conditional K-L divergence between

PB|A and QB|A given PA

13

¥ Decomposition of I(A; B)

I(A; B) = I1(A; B) − I2(A; B)

I1(A; B) = D(PC‖PAPB) ≥ 0 :

usual mutual information between PA and PB

I2(A; B) = EPC

[log

QC(X, Y )

QA(X)QB(Y )

]

QC(x, y) = QA(x)QB(y) ⇒ I2(A; B) = 0

14

movesReality'

channelerasuresymmetricBinary

1a

2a

1b

3b

p

qp1

qp1

A B2bp

q

q

15

moves'Forecaster

channelerasuresymmetricBinary

1

2

1

3

r

sr1

sr1

A B2r

s

s

16

PB|A(y|x) = δxy QB|A(y|x) = PB(y) X = Y⇒ D(PB|A‖QB|A|PA)

=∑

x∈XPA(x)

∑

y∈YPB|A(y|x) log

PB|A(y|x)

QB|A(y|x)

=∑

x∈XPA(x)

∑

y∈Yδxy log

δxy

PB(y)

= −∑

x∈XPA(x) log PA(x) = H(X)

17

movesReality'

channelEntropy

1a

2a

1b

A B2b

3a 3b

18

moves'Forecaster

channelEntropy

1

2

1

3

A B2

3

1b

2b

3b

19

1.2 Game channel capacity

¥ Channel capacity in information theory

C = supPX

I(X; Y ) : capacity of channel X ⇒ Y

• Shannon’s channel coding theorem :

Capacity C is the supremum of rates R at

which information can be sent with

arbitrarily low probability of error.

20

¥ Channel capacity in betting games

Cg := supPA,QA=PA

I(A; B) :

capacity of betting game channel A ⇒ B

I(A; B) = µB|A − µB = µA|B − µA

Cg = supPA

µA|B = supPA

D(PA|B‖QA|B|PB) ≥ 0

21

feedbackwithchannelionCommunicat

W W�

A B

)Y,X|P(Y1nn

n

nX nY

knY

Encoder Decoder

Delay

k

)Y,X|P(Y1nn)Y,X|P(Y )Y,X|P(Yn )Y,X|P(Y )Y,X|P(Y )Y,X|P(Y )Y,X|P(Y )Y,X|P(Y )Y,X|P(YEncoder

X)|H(YH(Y)supC

)P|Q||D(PsupC

X

A

P

BB|AB|AP

g

X)|H(YH(Y)supC

)P|Q||D(PsupC

X

A

PXX

BPPB|AB|AD(PD(PPAA

g

DecoderChannel

22

1.3 Game rate distortion function

¥ Rate distortion function in information

theory

R(D) = infPX|X :EP

XXd(X,X)≤D

I(X; X) :

Rate distortion function of transmission X ⇒ X

• Shannon’s rate distortion theorem :

Rate distortion function R(D) is the

infimum of rates R that asymptotically

achieve the distortion D.

23

¥ Rate distortion function in betting

games

Rg(D) := infPA|A:EP

AAd(X,X)≤D,QA|A:QA=PA

I(A; A) :

Rate distortion function of transmission A ⇒ A

I(A; A) = µA|A − µA = µA|A − µA

Rg(D) = infPA|A

µA|A = infPA|A

D(PA|A‖QA|A|PA) ≥ 0

24

dfeedforwarwithdistortionRate

nX

nX�

A A�

RRateEncoder Decoder

Delay

kn

Encoder

)X|H(XH(X)infR(D)

)P|Q||D(Pinf(D)R

X|X

A|A

P

AA|AA|APg

�

�

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)X))|H(XH(X)infR(D)

)P|Q||D(Pinf(D)R

X|X

A|A

PXX

APPA|AA|AD(PD(P

PAA

g

�

�

�

��

Decoder

25

2. Optimal conditional betting strategy

2.1 Optimal limit order strategy (cf. [6])

¥ Investor selects δ > 0 and sequentially

decides the trading times 0 < t1 < t2 < · · ·as follows.

26

S(t) > 0 : continuous asset price of Market

ti+1 : first time after ti such that

S(ti+1)

S(ti)= 1 + δ or

1

1 + δ

⇔ log S(ti+1) − log S(ti) = η or − η

η = log(1 + δ)

27

0 200 400 600 800 1000

0.00

0.02

0.04

0.06

0.08

0.10

Limit order for dlog S

Time

LS

t1 t2 t3 t4 t5 t6

��1==

��2

28

Embedded Coin-Tossing Game

K0 := 1

FOR n = 1, 2, . . . :

Investor announces αn ∈ RMarket announces xn ∈ {0, 1}Kn = Kn−1(1 + αn(xn − ρ))

END FOR

ρ =1

2 + δ: risk neutral prob. set by Investor

29

• Notations

χn1 , χn

0 : number of xi = 1, 0 (i = 1. . . . , n)

P (xn) =B(χn

1 + c1, χn0 + c0)

B(c1, c0)xn = x1 · · · xn

B(c1, c0) =Γ(c1)Γ(c0)

Γ(c1 + c0)c1, c0 > 0 :

beta binomial distribution modeled by Investor

30

Maximize EP [log Kn] ⇒ {αi}ni=1

αi =pi − ρ

ρ(1 − ρ)i = 1, . . . , n

pi = P (xi = 1|xi−1) =χi−1

1 + c1

i − 1 + c1 + c0

The optimal capital process of Investor is

expressed as the likelihood ratio

Kn =P (xn)

Q(xn)=

B(χn1 + c1, χn

0 + c0)/B(c1, c0)

ρχn1 (1 − ρ)χn

0

31

From the Stirling’s formula

log Kn = nD (pn‖q) − 1

2log n + O(1)

pn =

(χn

1

n,χn

0

n

): empirical prob. by Market

q = (ρ, 1 − ρ) : risk neutral prob. by Investor

D (pn‖q) : empirical K-L divergence

32

2.2 Optimal conditional limit order

strategy (cf. [7])

¥ Investor determines the betting ratios

αn ∈ R of conditional betting game B|Agiven A as follows.

α1 = 0, αn =

α+n if xn = 1

α−n if xn = 0

n = 2, 3, . . .

33

• Notations

χnx1, χn

x0 : number of xi = 1, 0 (i = 1. . . . , n)

χn11, χn

10, χn01, χn

00 :

number of (xi, yi) = (1, 1), (1, 0), (0, 1), (0, 0)

(i = 1, . . . , n)

P +(yn|xn) =B(χn

11 + c1, χn10 + c0)

B(c1, c0)

P −(yn|xn) =B(χn

01 + c1, χn00 + c0)

B(c1, c0):

beta binomial distribution modeled by Investor

35

Maximize EP [log Kn] P = P + × P − ⇒ {α±i }n

i=2

α+i =

p+i − ρ

ρ(1 − ρ)α−

i =p−

i − ρ

ρ(1 − ρ)

p+i = P +(yi = 1|xi−1) =

χi−111 + c1

χi−111 + χi−1

10 + c1 + c0

p−i = P −(yi = 1|xi−1) =

χi−101 + c1

χi−101 + χi−1

00 + c1 + c0

36

The optimal capital process of Investor is

expressed as the likelihood ratio

Kn = K+n × K−

n ξn = (x1, y1) · · · (xn, yn)

K+n =

P +(ξn)

Q+(ξn)=

B(χn11 + c1, χn

10 + c0)/B(c1, c0)

ρχn11(1 − ρ)χn

10

K−n =

P −(ξn)

Q−(ξn)=

B(χn01 + c1, χn

00 + c0)/B(c1, c0)

ρχn01(1 − ρ)χn

00

37

log Kn = nD(pn,y|x‖q|pn,x

)

−1

2

(log χn

x1 + log χnx0

)+ O(1)

pn,y|1 =

(χn

11

χnx1

,χn

10

χnx1

)pn,y|0 =

(χn

01

χnx0

,χn

00

χnx0

):

empirical conditional prob. by Market

q = (ρ, 1 − ρ) :

risk neutral prob. by Investor

D(pn,y|x‖q|pn,x

)pn,x =

(χn

x1

n,χn

x0

n

):

empirical conditional K-L divergence38

¥ In the following figures

SA(t) : daily closing prices of Nikkei 225

SB(t) : daily opening prices of

Toyota Sony Nintendo

2003/1/6 - 2007/9/28

Kn : capital process of Investor

39

pn1 = pn,1|1 pn0 = pn,0|0 :

empirical conditional prob. by Market

MDIV = D(pn,y|x‖q|pn,x

):

empirical conditional K-L divergence

mLKn =log Kn

n:

empirical exponential growth rate of Kn

40

0 200 400 600 800 1000 1200

5000

10000

15000

Toyota Opening & Nikkei Closing Prices

Days

0 200 400 600 800 1000 1200

5000

10000

15000

03/1/6-07/9/28

ToyotaNikkei

41

0 200 400 600 800 1000 1200

5000

10000

15000

Sony Opening & Nikkei Closing Prices

Days

0 200 400 600 800 1000 1200

5000

10000

15000

03/1/6-07/9/28

SonyNikkei

42

0 200 400 600 800 1000 1200

10000

20000

30000

40000

50000

60000

Nintendo Opening & Nikkei Closing Prices

Days

0 200 400 600 800 1000 1200

10000

20000

30000

40000

50000

60000

03/1/6-07/9/28

NintendoNikkei

43

0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

Capital Process for Toyota 826 rounds

Rounds

k = 6.7

a1==2 a2==2

p0 = 0.498

Kn d = 0.01*2 k

K(T) = 0.01

44

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

Conditional Prob. of Toyota & Nikkei

Rounds

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

pn1pn0

45

0 100 200 300 400

0.0

0.2

0.4

0.6

0.8

1.0

Conditional Prob. of Sony & Nikkei

Rounds

0 100 200 300 400

0.0

0.2

0.4

0.6

0.8

1.0

pn1pn0

46

0 100 200 300 400 500 600

0.0

0.2

0.4

0.6

0.8

1.0

Conditional Prob. of Nintendo & Nikkei

Rounds

0 100 200 300 400 500 600

0.0

0.2

0.4

0.6

0.8

1.0

pn1pn0

47

Nikkei&Toyota

48

Nikkei&Sony

49

Nikkei&Nintendo

50

0 100 200 300 400 500

0.0 e+00

5.0 e+13

1.0 e+14

1.5 e+14

2.0 e+14

2.5 e+14

Capital Process for Toyota & Nikkei 500 rounds

Rounds

k = 6.5

p1 = 0.71 p0 = 0.67

Kn d = 0.01*2 k

K(T) = 5.64e+13

51

0 100 200 300 400

010000

20000

30000

40000

50000

60000

Capital Process for Sony & Nikkei 432 rounds

Rounds

k = 6

p1 = 0.64 p0 = 0.62

Kn d = 0.01*2 k

K(T) = 14449

52

0 100 200 300 400 500 600

010

2030

4050

Capital Process for Nintendo & Nikkei 597 rounds

Rounds

k = 6.4

p1 = 0.61 p0 = 0.53

Kn d = 0.01*2 k

K(T) = 20

53

3. Source coding and betting strategy

3.1 Lossy source coding with feedforward

¥ Source coding model

Xn = (X1, . . . , Xn) ∈ X n : source sequence

Xn = (X1, . . . , Xn) ∈ X n : estimated sequence

dn : X n × X n → R+ : distortion measure

54

dfeedforwarwithsystemcodingSource

nX

nX�

A A�

RRateEncoder Decoder

Delay

sequenceonreproducti:)X,...,X(X

functionsdecodingofsequence

:n,...1,iXX}2,...2,{1,:g

functionencoding:}2,...2,{1,:f

n1

n

kinR

i

nRn

n

���

�

Encoder Decoder

sequenceonreproducti:)X,...,X(X

functionsdecodingofsequence

:n,...1,iXX}2,...2,{1,:g

functionencoding:}2,...2,{1,:f

n1

n

kinR

i

nRn

nff

���

�

nfn

1ii }{gDelay

kn

55

• (2nR, n) code with k-delayed feedforward

fn : X n → {1, 2, . . . , 2nR} : encoding function

gi : {1, 2, . . . , 2nR} × X i−k → X i = 1, . . . , n :

sequence of decoding functions

fn(Xn) = w ∈ {1, 2, . . . , 2nR}gi(w, Xi−k) = Xi i = 1, . . . , n

Dn = EXn

[dn(Xn, Xn)

]:

distortion associated with the (2nR, n) code

56

(R, D) : achievable

⇔ ∃(2nR, n) code with lim supn→∞

Dn ≤ D

• Rate distortion theorem

R ≥ Rkff(D) ⇒ (R, D) is achievable

Rkff(D) = lim inf

PXn|Xn ,Dn≤D

1

nIk(X

n → Xn) :

rate distortion function with k-delayed

feedforward

57

Ik(Xn → Xn) =

n∑

i=1

I(Xi+k−1; Xi|Xi−1)

= I(Xn; Xn) −n∑

i=k+1

I(Xi−k; Xi|Xi−1) :

directed information from Xn to Xn with

k-delayed feedforward

n∑

i=k+1

I(Xi−k; Xi|Xi−1) :

information quantity obtained for free

58

ontransmissisymmetricBinary

1a

2a

1a�

2a�

1

0

0

1A A�

distortion&movesReality'

59

1

2

1a�

2a�

1

1

A A�

moves'Forecaster

ontransmissisymmetricBinary60

¥ Binary symmetric transmission

Rg(D) = I(A; A)

⇔ ρ = PA|A(x2|x1) = PA|A(x1|x2) = D

σ = QA|A(x2|x1) = QA|A(x1|x2)

=α1 − a1 + D(α2 − α1)

a2 − a1

Rg(D) = D(PA|A‖QA|A|PA)

= D(ρ‖σ) − D(a1‖α1)

61

QC = QA × QB ⇔ σ = α1 = 0.5 QA|A = PA

Rg(D) = D(PA|A‖PA|PA)

= D(ρ‖0.5) − D(a1‖0.5)

= H(a1) − H(D) = R(D)

Rg(0) = D(δA|A‖PA|PA)

= D(0‖0.5) − D(a1‖0.5)

= H(a1) = R(0)

62

0.0 0.1 0.2 0.3 0.4

0.0

0.5

1.0

1.5

Rate distortion function Rg(D) in BST

D

Rg(D)

0.0 0.1 0.2 0.3 0.4

0.0

0.5

1.0

1.5

0.0 0.1 0.2 0.3 0.4

0.0

0.5

1.0

1.5 a1==0.3

��

1==0.6

��

1==0.5

��

1==0.4

63

¥ Conditional betting game A|A given A

• Notations

χnx1, χn

x0 : number of xi = 1, 0 (i = 1. . . . , n)

χnx1, χn

x0 : number of xi = 1, 0 (i = 1. . . . , n)

χn11, χn

10, χn01, χn

00 :

number of (xi−k, xi) = (1, 1), (1, 0), (0, 1), (0, 0)

(i = k + 1, . . . , n)

64

P ±A

(xn|xn) = P +A

(xn|xn) × P −A

(xn|xn)

P +A

(xn|xn) =B(χn

11 + c1, χn10 + c0)

B(c1, c0)

P −A

(xn|xn) =B(χn

01 + c1, χn00 + c0)

B(c1, c0):

conditional beta binomial distribution

modeled by Skeptic

65

QA(xn|xn) = PA(xn) =B(χn

x1 + c1, χnx0 + c0)

B(c1, c0):

beta binomial distribution modeled by

Forecaster

Maximize EPA± [log Kn] ⇒ {αA±i }n

i=1

αA±i = 0 1 ≤ i ≤ k

αA±i =

αA+i if xi−k = 1

αA−i if xi−k = 0

k + 1 ≤ i ≤ n

66

αA+i =

pP +

A

i − qQA

i

qQA

i (1 − qQA

i )αA−

i =p

P −A

i − qQA

i

qQA

i (1 − qQA

i )

pP +

A

i = P +A

(xi = 1|xi−k) =χi−1

11 + c1

χi−111 + χi−1

10 + c1 + c0

pP −

A

i = P −A

(xi = 1|xi−k) =χi−1

01 + c1

χi−101 + χi−1

00 + c1 + c0

qQA

i = QA(xi = 1|xi−k) =χi−1

x1 + c1

i − 1 + c1 + c0

67

The optimal capital process of Skeptic is

expressed as the likelihood ratio

KP ±A (xn|xn) =

n∏

i=1

(1 + αA±

i (xi − qQA

i ))

=n∏

i=1

P ±A

(xi|xi−k)

QA(xi|xi−k)=

P +A

(xn|xn) × P −A

(xn|xn)

QA(xn|xn)

68

log KP ±A (xn|xn) = nD

(pn,x|x‖qn,x|pn,x

)

−1

2

(log χn

x1 + log χnx0

)+ O(1)

pn,x|1 =

(χn

11

χnx1

,χn

10

χnx1

)pn,x|0 =

(χn

01

χnx0

,χn

00

χnx0

):

empirical conditional prob. of Reality

qn,x =

(χn

x1

n,χn

x0

n

):

empirical risk neutral prob. of Forecaster

D(pn,x|x‖qn,x|pn,x

):

empirical conditional K-L divergence69

3.2 Efficient source coding scheme

¥ Betting strategy and data compression

(a variant of the arithmetic coding)

• Encoding

xn = x1 . . . xn ∈ {0, 1}n ⇒ xn = x1 . . . xn ∈ {0, 1}n

such that PXn|Xn achieves Rkff(D)

x|x(n) = (x1|x1, . . . , xn|xn) :

observed sequence by the encoder

2n sequences {xn} : in lexicographical order

70

The encoder calculates the cumulative sum

G±A(x|x(n)) =

∑

x′|x(n)≤x|x(n):typical

R±A(x′|x(n))

R±A(x′|x(n)) =

1

KP ±A (x′|x(n))

=QA(x′|x(n))

P ±A

(x′|x(n))

x′|x(n) : typical

⇔∣∣∣∣1

nlog KP ±

A (x′|x(n)) − Rkff(D)

∣∣∣∣ < ε

G±A(x|x(n)) ∈ [0, 1] as n → ∞

71

` =⌈log KP ±

A (x|x(n))⌉

+ 1 m = dlog ne :

specifed numbers of bits⌊G±

A(x|x(n))

⌋= .c1c2 . . . c` ci ∈ {0, 1} :

binary decimal to ` place accuracy

χnx1 = d1d2 . . . dm di ∈ {0, 1} :

binary number to m digits

c(`) = (c1, c2, . . . , c`) d(m) = (d1, d2, . . . , dm) :

code sequences sent to the decoder

72

• Decoding

When there exists a feedforward X → X

and χnx1 is known, the decoder can also

sequentially calculate the cumulative sum

G±A(x|x(n)) =

∑

x′|x(n)≤x|x(n):typical

R±A(x′|x(n))

until G±A(x|x(n)) ≥ .c(`)

⇒ x|x(n) : the encoded sequence

73

From the expression

log KP ±A (xn|xn) = nD

(pn,x|x‖qn,x|pn,x

)

−1

2

(log χn

x1 + log χnx0

)+ O(1)

the required number of bits is

` + m =⌈log KP ±

A (x|x(n))⌉

+ 1 + dlog ne=

⌈nD

(pn,x|x‖qn,x|pn,x

)⌉+ O (log n)

74

The empirical codeword length L∗n = `+m

n

per source symbol is

L∗n =

` + m

n

≤ D(pn,x|x‖qn,x|pn,x

)+ O

(log n

n

)

→ Rkff(D) as n → ∞

75

References

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81