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On the Super Saint Petersburg Paradox Tom Cover Stanford February 24, 2012 Tom Cover On the Super Saint Petersburg Paradox The St. Petersburg Paradox Daniel Bernoulli (1738): X =2 k , with prob. 2 -k , k =1, 2,... x : 2 4 8 16 32 ... p (x ): 1 2 1 4 1 8 1 16 1 32 ... Pay c . Receive X . Note EX = . Tom Cover On the Super Saint Petersburg Paradox

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Page 1: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

On the Super Saint Petersburg Paradox

Tom Cover

Stanford

February 24, 2012

Tom Cover On the Super Saint Petersburg Paradox

The St. Petersburg Paradox

Daniel Bernoulli (1738):

X = 2k ,with prob. 2−k , k = 1, 2, . . .

x : 2 4 8 16 32 . . .p(x) : 1

214

18

116

132 . . .

Pay c .

Receive X .

Note EX =∞.

Tom Cover On the Super Saint Petersburg Paradox

Page 2: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

The St. Petersburg Paradox

X = 2k ,with prob. 2−k , k = 1, 2, . . .

x : 2 4 8 16 32 . . .p(x) : 1

214

18

116

132 . . .

Are you willing to pay any price c to receive X − c?

Some say No.

That’s clearly wrong since E (X − c) =∞.

But would you pay c = $1, 000, 000?

So the answer should be Yes. But why?

What do we mean?

Tom Cover On the Super Saint Petersburg Paradox

Previous Resolutions of the Paradox

Bernoulli Utility u(X ). Let u(X ) = lnX .Now Eu(X ) = E lnX .Maximize utility.

Samuelson (1977): Objection to arbitrariness of log utility.

Not enough money in world.

Fellern∑

i=1

Xi − nC

But who is offering repeated bets? And where is nC comingfrom?

Menger, Super Saint Petersburg paradox,P(X = 22

k−1) = 2−k .

Tom Cover On the Super Saint Petersburg Paradox

Page 3: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

New ingredients in discussion

Maximum growth rate

Relative growth rate (for Super St. Pete)

Competitive optimality

Investment always positive.

Fast drop-off of investment for high costs c .

Resolution of Super St. Pete.

Eyeball test

Tom Cover On the Super Saint Petersburg Paradox

Growth rate optimality

Kelly (1956)

Horse race:

Horses V : 1, 2, . . . , mProbabilities p : p1, p2, . . . , pm

Odds (m for 1) o : m, m, . . . , mBets b : b1, b2, . . . , bm, bi ≥ 0,

∑i bi = 1

Wealth factor: S = bim, with probability pi .

Sn =n∏

j=1

S(Vj) = exp(n∑

j=1

lnS(Vj))

= exp(n1

n

n∑j=1

lnS(Vj))

→ exp(nE lnS(V ))

Tom Cover On the Super Saint Petersburg Paradox

Page 4: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Growth rate optimality

Kelly(1956):

We have

W ∗ = maxbi≥0,

∑bi=1

m∑i=1

pi lnmbi

= lnm − H(p), b∗ = p

Thus W ∗ + H(p) = lnm

Side information Y : p(x)p(y |x)Increase in growth rate:

∆W = I (X ;Y ) =∑x ,y

p(x , y) logp(x , y)

p(x)p(y)

Optimal investment:

b∗(v |y) = p(v |y)

Tom Cover On the Super Saint Petersburg Paradox

Growth rate optimality

Shannon (1966, MIT): Invest (b1, b2, . . . , bm) in stocks(X1,X2, . . . ,Xm) to maximize

W ∗ = maxb

E ln btX

Breiman (1961) b∗ maximizes growth rate W and minimizestime for Sn ≥ A.

Tom Cover On the Super Saint Petersburg Paradox

Page 5: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Growth rate optimality

Samuelson(1977):“Our analysis enables us to dispel a fallacy that has beenborrowed into portfolio theory from information theory of theShannon type”

Samuelson(1979): “Why we should not make mean log ofwealth big though years to act are long”

Shannon (1986, Brighton) Meetings with Paul Samuelson in1960’s.

Samuelson vs TC and vs Ziemba (personal correspondence).

Tom Cover On the Super Saint Petersburg Paradox

Figure: Samuelson Correspondence.

Tom Cover On the Super Saint Petersburg Paradox

Page 6: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Tom Cover On the Super Saint Petersburg Paradox

Growth rate optimality

Investment is exchanging one random variable for another.

Invest proportion b of current wealth in Saint Petersburg. Keepthe rest in cash.

1→ 1− b +bX

c

Choose b yielding highest growth rate.Define

W (b, c) = E log(b +bX

c)

ThenSn

.= 2nW (b,c)

Tom Cover On the Super Saint Petersburg Paradox

Page 7: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Growth rate optimality

Sn.

= 2nW (b,c)

Maximize over 0 ≤ b ≤ 1. Then

S∗n.

= 2nW (b∗(c),c)

Maximizing investment b:

W (b, c) = E ln(b +bX

c)

∂W (b, c)

∂b= E

Xc − 1

b + bXc

= 0

Put proportion b∗ of wealth in St. Pete.

Optimal investment b∗ : E1

b + bXc

= 1 (∗)

Tom Cover On the Super Saint Petersburg Paradox

Growth rate optimality

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c

b*

Figure: The growth optimal proportion b∗ of wealth invested in the SaintPetersburg gamble as a function of the cost c .

Tom Cover On the Super Saint Petersburg Paradox

Page 8: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Growth rate optimality

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

c

W*(b

*, c)

Figure: Growth rate W ∗(b∗, c) as a function of c .

Tom Cover On the Super Saint Petersburg Paradox

Growth rate optimality

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

c

2W

*(b

*,

c)

Figure: The wealth factor 2W ∗(b∗,c) as a function of c . Note that 2W ∗is

not much greater than 1 for costs c ≥ 4

Tom Cover On the Super Saint Petersburg Paradox

Page 9: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Eye ball test

Cash:

(1 1 1 1 . . .12

14

18

116 . . .

)0 1

0

1

Figure: cash

St. Pete:

(2 4 8 16 . . .12

14

18

116 . . .

)0 1

0

1

Figure: All St. Pete.

Mixture: S(b) = 1− b + bXc(

b + 2bc b + 4b

c . . .12

14 . . .

)Cash= S(0), and St. Pete= S(1).

Tom Cover On the Super Saint Petersburg Paradox

Example

c = 3Then b∗(c) = 1, W ∗ = 0.4150, 2W

∗= 1.3333

Cash =

(1 1 1 1 . . .12

14

18

116 . . .

)0 1

0

1

Figure: Cash

All St. Pete/c =

(23

43

83

163 . . .

12

14

18

116 . . .

)0 1

0

1

Figure: All St. Pete/3

S(b∗) =

(23

43

83

163 . . .

12

14

18

116 . . .

)0 1

0

1

Figure: Optimal mixture: c=3,b∗ = 1, 2W ∗

= 1.3333

Tom Cover On the Super Saint Petersburg Paradox

Page 10: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Example

c = 4Then b∗(c) ∼= 0.384, 2W

∗= 1.1086

Cash: S(0) =

(1 1 1 1 . . .12

14

18

116 . . .

)All St. Pete/4:S(1) =

(12 1 2 4 . . .12

14

18

116 . . .

)S(b∗) =

(0.85 1 1.3 1.9 3.1 . . .12

14

18

116

132 . . .

)

Tom Cover On the Super Saint Petersburg Paradox

Example

0 10

1

Figure: Cash

0 10

1

Figure: All St. Pete/4

0 10

1

Figure: Optimal mixture: c=4, b∗ = 0.38, 2W ∗= 1.1086

Tom Cover On the Super Saint Petersburg Paradox

Page 11: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Example

C=4 (continued)

S(b∗) =(0.85 1 1.3 1.9 3.1 . . .12

14

18

116

132 . . .

)0 1

0

1

Figure: Optimal mixture: c=4

Growth optimal b∗(c) is somewhat conservative.P(S(b∗) = 0.85) = 1

2 vs P(S(1) = 12) = 1

2 .

Tom Cover On the Super Saint Petersburg Paradox

Example

c = 8Then b∗(c) ∼= 0.02484, S∗ = 0.97516 + 0.02484X

8All St. Pete/8:

S(1) =

(14

12 1 2 . . .

12

14

18

116 . . .

)All cash:

S(0) =

(1 1 1 1 . . .12

14

18

116 . . .

)Growth optimal mixture:

S(b∗) =

(0.978 0.984 1 1.032 1.096 1.224

12

14

18

116

132

164

1.480 1.992 3.016 5.064 . . .1

1281

2561

5121

1024 . . .

)2W

∗= 1.004278.

Tom Cover On the Super Saint Petersburg Paradox

Page 12: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Example

0 10

1

Figure: Cash

0 10

1

Figure: All St. Pete/8

0 10

1

Figure: Optimal mixture: c=8, b∗ = 0.024, 2W ∗= 1.0043

Tom Cover On the Super Saint Petersburg Paradox

Example

c=8. Growth optimal mixture:

S(b∗) =

(0.978 0.984 1 1.032 1.096 1.224

12

14

18

116

132

164

1.480 1.992 3.016 5.064 . . .1

1281

2561

5121

1024 . . .

)

0 10

1

Figure: Optimal mixture: c=8

b∗ = 0.024, 2W∗

= 1.004278.Tom Cover On the Super Saint Petersburg Paradox

Page 13: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Critical costs and r-th mean

r-th mean: µr = (E |X |r )1r .

Arithmetic mean: µ1 = EX .Geometric mean: µ0 = exp (E lnX ).Harmonic mean: µ−1 = 1

E( 1X).

Laws of Large numbers

1

n

n∑i=1

Xi → µ1

(n∏

i=1

Xi

) 1n

→ µ0

0µ−1

µ0 µ1 c

Figure: Various cases for the cost c

Tom Cover On the Super Saint Petersburg Paradox

Critical Costs

For what costs c do you want some or all of Saint Petersburg?

∂W (b, c)

∂b= E

Xc − 1

b + bXc

All: b∗ = 1 :∂W (b, c)

∂b> 0, if c <

1

E ( 1X )

Some: b∗ = 0 :∂W (b, c)

∂b> 0, if c < EX

Tom Cover On the Super Saint Petersburg Paradox

Page 14: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Competitive Optimality

Stocks: X ∼ F (x), x ∈ Rm.Portfolio: b = (b1, b2, . . . , bm), bi ≥ 0,

∑bi = 1.

b∗ : maxb E ln btX .

S∗ = b∗tX

S = btX

Theorem (Bell and TC, 1980):

b∗ : EXi

b∗tX= 1, i = 1, . . .m

E (S

S∗ ) ≤ 1,∀b,

Pr {S ≥ tS∗} ≤ 1

t

Pr{S ≥ U[0,2]S∗} ≤ 1

2

Tom Cover On the Super Saint Petersburg Paradox

Competitive optimality

Thus no other portfolio b can beat Ub∗ with probability greaterthan 1

2 .Therefore, b∗ is both growth-rate optimal andcompetitively optimal.

Tom Cover On the Super Saint Petersburg Paradox

Page 15: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Relative Growth Rate and Super Saint Petersburg

Super Saint Petersburg

P(X = 22k−1) = 2−k x : 2 8 128 32768 . . .

p(x) : 12

14

18

116 . . .

Even when Sn(b)↗∞ for any proportion b ∈ [0, 1], there is arationale for not choosing b = 1.Compare investor (b, b) with investor (12 ,

12).

Then,

Sn(b)

Sn(12)= exp

(nE ln

b + bXc

12 + 1

2Xc

)Define the relative growth rate

∆(b, b) = E lnb + bX

c¯b + bX

c

Tom Cover On the Super Saint Petersburg Paradox

Relative Growth Rate and Super Saint Petersburg

Note:

∆(b1, b2) = W (b1)−W (b2) when W (b) is finite.

∆ is always finite.

∆(b1, b2) = ∆(b1, b3) + ∆(b3, b2)

maxb ∆(b1, b2) is achieved by b∗(c), the solution to

E1

b + bXc

= 1. (∗)

NOTE: Same equation for b∗ as before.

In repeated plays,

Sn(b∗(c))

Sn(12)

.= exp

(n∆(b∗,

1

2)

)↗∞

Also note that b∗(c) = 1, for all prices c ≤ µ0.

Tom Cover On the Super Saint Petersburg Paradox

Page 16: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Relative Growth Rate and Super Saint Petersburg

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c

b*

Figure: Optimal portfolio b∗ as a function of c for Super SaintPetersburg.

Tom Cover On the Super Saint Petersburg Paradox

Example: Super St. Pete

cost c = 8.Super St. Pete (2, 8, 128, . . . )

Super St. Pete/8:

S(1) =(14 1 16 4096 . . .12

14

18

116 . . .

)b∗(8) = 0.4238Growth optimal mixture:

S(b∗) =(0.682 1 7.356 1736.44 . . .

12

14

18

116 . . .

)

0 10

1

Figure: Super St. Pete/8

0 10

1

Figure: Optimal mixture: c=8

Tom Cover On the Super Saint Petersburg Paradox

Page 17: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Conclusions

Let X be Saint Petersburg or Super Saint Petersburg.Let b(c) maximize relative growth rate

∆(b||12

) = E lnb + bX

c12 + 1

2Xc

Thus optimal proportion invested in X is b∗(c), solution to

E1

b + bXc

= 1.

Tom Cover On the Super Saint Petersburg Paradox

Conclusions

For Saint Petersburg and Super Saint Petersburg,

Want X at any price c , i.e. b∗(c) > 0,∀c .

E1

b + bXc

= 1.

S∗ = 1− b∗(c) + b∗(c)Xc is growth rate optimal.

S∗ is relatively growth rate optimal.

S∗ is competitively optimal in one-shot investment.S∗n

Sn↗∞ in repeated plays.

S∗ passes eyeball test.

Want X at any price, but less and less as price goes up.

Tom Cover On the Super Saint Petersburg Paradox

Page 18: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

Acknowledgement

Kartik Venkat

Gowtham Kumar

Lei Zhao

Tom Cover On the Super Saint Petersburg Paradox

References I

R.M. Bell and T.M. Cover, “Competitive optimality oflogarithmic investment,” Mathematics of Operations Research,5(2):161-166, May 1980.

R.M. Bell and T.M. Cover, “Game-Theoretic OptimalPortfolios,”Management Science, 34(6): 724-733, June 1988.

D. Bernoulli, “Exposition of a new theory on the measurementof risk,” Econometrica, 22:22-36, 1954. Originally published in1738; translated by L. Sommer.)

L. Breiman, “Optimal gambling systems for favorable games,”Proceedings of the Fourth Berkeley Symposium on Math,Statitsic and Probability, 1:65-78, University of CaliforniaPress, Berkeley, 1961.

T.M. Cover and J. Thomas, Elements of Information Theory,Second Edition, Wiley, 2006. Problem 6.17 pp.181-182,problem 6.18 p.182, chapter 6, chapter 16.

Tom Cover On the Super Saint Petersburg Paradox

Page 19: On the Super Saint Petersburg Paradox - Stanford Universityweb.stanford.edu/group/isl/public/documents/st_pete_stanfordstats_talk_print.pdfRelative Growth Rate and Super Saint Petersburg

References II

S. Csorgo and R. Dodunekova, “Limit theorems for thePetersburg game,” Sums, Trimmed Sums and Extremes,Progress in Probability, M.G. Hahn, D.M. Mason and D.C.Weiner, eds., 23:285-315, Birkhauser, Boston, 1991.

W. Feller, “Note on the law of large numbers and ’fair’games,” Annals of Mathematical Statistics, 16:301-304, 1945.

L.C. MacLean, E.O. Thorp, and W.T. Ziemba, eds., The KellyCapital Growth Investment Criterion, World ScientificHandbook in Financial Economic Series, vol. 3, 2011.

A. Martin-Lof, “A limit theorem which clarifies the ‘Petersburgparadox’,” Journal of Applied Probability, 22:634-643, 1985.

Samuelson, P. A., “Saint Petersburg paradoxes: defanged,dissected, and historically described,” Journal of EconomicLiterature, 1977, 15 (1): 2455.

Tom Cover On the Super Saint Petersburg Paradox

References III

L. Gyorfi and P. Kevei,“On the rate of convergence of theSaint Petersburg game,”Periodica Mathematica Hungarica,Vol. 62(1), 2011,pp. 13-37.

Tom Cover On the Super Saint Petersburg Paradox