on the super saint petersburg paradox - stanford...

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 =∞.


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?


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 .


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


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 ))



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)



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.



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).


Figure: Samuelson Correspondence.




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)



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 (∗)



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 .



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 .



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


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).


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


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 . . .

)


Example

0 10

1

Figure: Cash

0 10

1


0 10

1

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


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 .


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.


Example

0 10

1

Figure: Cash

0 10

1


0 10

1

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


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


b∗ = 0.024, 2W∗

= 1.004278.Tom Cover On the Super Saint Petersburg Paradox

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


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


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


Competitive optimality

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

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


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



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.



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.


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



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.


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.


Acknowledgement

Kartik Venkat

Gowtham Kumar

Lei Zhao


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.


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.


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.


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