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Generalized n-Paul paradox
ARTICLE IN PRESS
Statistics & Probability Letters 77 (2007) 1043–1049 www.elsevier.com/locate/stapro
Generalized n-Paul paradox$ Pe´ter Kevei Bolyai Institute, University of Szeged, Aradi ve´rtanu´k tere 1, Szeged 6720, Hungary Received 6 June 2006; received in revised form 9 July 2006; accepted 28 August 2006 Available online 28 March 2007 Dedicated to Professor Sa´ndor Cso¨rgo+ for his 60th birthday
Abstract The paradoxical results of Cso¨rgo+ and Simons for mutually beneficial sharing among any fixed number of St. Petersburg gamblers are extended to games played by a possibly biased coin, with p as the probability of ‘heads.’ The extension is not straightforward because, unlike in the classical case with p ¼ 1=2, admissibly pooled winnings generally fail to stochastically dominate individual ones for more than two gamblers. Best admissible pooling strategies are determined when p is rational, and the algebraic depth of the problem for an irrational p is illustrated by an example. r 2007 Elsevier B.V. All rights reserved. MSC: primary 60E99; secondary 60G50 Keywords: Generalized St. Petersburg games; Comparison of infinite expectations; Pooling strategies
1. Results and discussion Peter offers to let Paul toss a possibly biased coin until it lands heads and pays him rk ducats if this happens on the kth toss, k 2 N ¼ f1; 2; . . .g, where r ¼ 1=q for q ¼ 1 p and p 2 ð0; 1Þ is the probability of ‘heads’ at each throw. This is the generalized St. PetersburgðpÞ game, in which PfX ¼ rk g ¼ qk1 p, k 2 N, for Paul’s gain X . Following Cso¨rgo+ and Simons (2002, 2006), we assume that Peter plays exactly one such game with each of nX2 players, Paul1 ; Paul2 ; . . . ; Pauln , whose independent individual winnings are X 1 ; X 2 ; . . . ; X n . The n players may agree to use a pooling strategy pn ¼ ðp1;n ; p2;n ; . . . ; pn;n Þ, before they play, where P p1;n ; p2;n ; . . . ; pn;n X0 and nj¼1 pj;n ¼ 1. Under this strategy Paul1 receives p1;n X 1 þ p2;n X 2 þ þ pn;n X n , Paul2 receives pn;n X 1 þ p1;n X 2 þ þ pn1;n X n , Paul3 receives pn1;n X 1 þ pn;n X 2 þ p1;n X 3 þ þ pn2;n X n ; . . ., and Pauln receives p2;n X 1 þ p3;n X 2 þ þ pn;n X n1 þ p1;n X n ducats. This strategy is fair to every Paul in the sense that their winnings are equally distributed and each receives the same added value equal to Z 1 Ap ðpn Þ ¼ E½p1;n X 1 þ þ pn;n X n ; X 1 ¼ ½Pfp1;n X 1 þ þ pn;n X n 4xg PfX 1 4xg dx, (1) 0 $ Work supported in part by the Hungarian Scientific Research Fund, Grants T–034121 and T–048360, and carried out within the Analysis and Stochastics Research Group of the Hungarian Academy of Sciences. E-mail address: [email protected].
0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.08.027
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whenever the integral is defined, so that comparison is possible. We refer to Cso¨rgo+ and Simons (2002, 2006) for a detailed exposition and discussion of the comparison operator E½; . We call a strategy pn ¼ ðp1;n ; . . . ; pn;n Þ admissible if each of its components is either zero or a nonnegative integer power of q ¼ 1 p. Individualistic strategies ð1; 0; . . . ; 0Þ are thus admissible for each p, otherwise P the powers in nonzero components are positive integers. The entropy of a pooling strategy is H r ðpn Þ ¼ nj¼1 pj;n logr 1=pj;n , where logr denotes the base r logarithm and 0logr 1=0 ¼ 0. We say that the random variable U is stochastically larger than the random variable V , written as UXD V , if PfU4xgXPfV 4xg for all x 2 R. Theorem 1. For any p 2 ð0; 1Þ and n 2 N, the added value Ap ðpn Þ exists as an improper Riemann integral if and only if pn is admissible, in which case Ap ðpn Þ ¼ p=qH r ðpn Þ. Cso¨rgo+ and Simons (2006) proved this theorem for the classical St. Petersburgð1=2Þ game, played with an unbiased coin. However, in that case they proved the following stronger result: the independent St. Petersburgð1=2Þ variables X 1 ; . . . ; X n can be defined on a rich enough probability space that carries, for each admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ, a St. Petersburgð1=2Þ random variable X pn and a nonnegative random variable Y pn such that T pn ¼ p1;n X 1 þ þ pn;n X n ¼ X pn þ Y pn almost surely. This implies the stochastic inequality T pn XD X 1 . Hence the integrand in A1=2 ðpn Þ is nonnegative and thus A1=2 ðpn Þ is trivially finite as a Lebesgue integral. As the next result shows, stochastic dominance is preserved for two players for an arbitrary St. Petersburg parameter p 2 ð0; 1Þ. Theorem 2. For any p 2 ð0; 1Þ, if p2 ¼ ðqa ; qb Þ is an admissible pooling strategy for some a; b 2 N, then T p2 ¼ qa X 1 þ qb X 2 is stochastically larger than X 1 . Surprisingly, however, for nX3 gamblers stochastic dominance generally fails to hold for admissible strategies. Our example to demonstrate this is when p ¼ ðn 1Þ=n, q ¼ 1 p ¼ 1=n, so that r ¼ 1=q ¼ n is also the number of Pauls. Then PfX ¼ nk g ¼ ðn 1Þ=nk , k 2 N, and the averaging pooling strategy pn ¼ pn ¼ ð1=n; 1=n; . . . ; 1=nÞ is admissible. For this strategy the weighted sum is T pn ¼ ðX 1 þ þ X n Þ=n, so that for n ¼ 2 Theorem 2 says in particular that S 2 ¼ 2T p ¼ X 1 þ X 2 is stochastically larger than 2X 1 . This is not 2 true for nX3. Theorem 3. If p ¼ ðn 1Þ=n, q ¼ 1=n and nX3, then neither S n ¼ X 1 þ þ X n nor nX 1 is stochastically larger than the other. In view of Theorem 2, the integrand in (1) is nonnegative whenever p2 is admissible, so that the integral Ap ðp2 Þ described in Theorem 1 strengthens to that of a Lebesgue integral when n ¼ 2. While the same conclusion holds for nX3, Theorem 3 rules out so simple a line of reasoning. Theorem 4. For every p 2 ð0; 1Þ and every admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ the integral Ap ðpn Þ in (1) is finite as a Lebesgue integral. Theorem 1 characterizes the pooling strategies that yield added values. However, admissible strategies do not exist for all, in fact, for most parameters p 2 ð0; 1Þ. Call a parameter p admissible, if for p there exists an admissible strategy which is not individualistic. Theorem 1 then says that p is admissible if and only if for q ¼ 1 p there exist positive integers a1 Xa2 X Xak , for some k 2 N, such that qa1 þ qa2 þ þ qak ¼ 1. In this case, r ¼ 1=q is an algebraic integer. If a1 4a2 , then q is also an algebraic integer, thus q is an algebraic unit. The set of algebraic numbers is countable, so there are at most a countable number of admissible parameters p. When q ¼ 1 p is rational for an admissible p 2 ð0; 1Þ, the equation implies q ¼ 1=m for some integer mX2. Thus the set of rational admissible parameters is fðm 1Þ=m : mX2g. In particular, it is interesting that the classical p ¼ 1=2 is the smallest such St. Petersburg parameter. It follows that the set of all admissible parameters p is countable. Nevertheless, it can be shown that this set is dense in the interval ð0; 1Þ. When a given number of our Pauls happen to have admissible strategies, a natural question is: Which is the best? In the latter rational case when p ¼ ðm 1Þ=m for some integer mX2, and so r ¼ 1=q ¼ mX2 is an integer, the answer is given by the next result, in which bxc ¼ maxfk 2 Z : kpxg is the integer part, dxe ¼ minfk 2 Z : kXxg is the integer ceiling and hxi ¼ x bxc ¼ x þ dxe is the fractional part of a number x 2 R.
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Theorem 5. If p ¼ ðr 1Þ=r and n ¼ rblogr nc þ ðr 1Þrn for some integers rX2 and 0prn prblogr nc 1, then Ap ðpn Þ ¼
p p H r ðpn Þp logr n dp ðnÞ¼:Ap;n q q
(2)
for every admissible strategy pn , where dp ðuÞ ¼ 1 þ ðr 1Þhlogr ui rhlogr ui , u40. Moreover, the bound Ap;n is attainable by means of the admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ ¼ ðrpn ; . . . ; rpn ; pn ; . . . ; pn Þ with pn ¼
1 , rdlogr ne
where the number of pn ’s and rpn ’s are, respectively, m1;p ðnÞ ¼
rn rdlogr ne r1
and
m2;p ðnÞ ¼
rdlogr ne n . r1
Apart from reorderings of the components of pn , the point of maximum is unique. It is easy to see that if n is not in the form rblogr nc þ ðr 1Þrn , then 0 must be included among the components of the strategy, which does not increase the entropy. So it is enough to investigate the number of players in the form above. The continuous function dp ðÞ is nonnegative; its maximum is given in formula (3.4) of Cso¨rgo+ and Simons (1996). Theorem 5 is not applicable for an irrational p. On the other hand, in every admissible situation Ap ðpn Þ ¼ ðr 1ÞH r ðpn Þ by Theorem 1, and the trivial upper bound H r ðpn Þplogr n ¼ H r ðpn Þ is valid for the entropy of every pn , where pn ¼ ð1=n; 1=n; . . . ; 1=nÞ. However equality cannot hold in general because pn is not admissible for every admissible parameter p. Apart from those cases which can be reduced to the rational case, that is, pffiffiffiffi when q ¼ 1 p ¼ 1= k m for some integers m; kX2, the problem of the best admissible strategy is unsolved. 2 For the pffiffiffi irrational case the simplest example is the equation q þ q ¼ 1, the solution of which is q ¼ t:¼ð 5p ffiffiffi 1Þ=2 0:618, the ratio of golden section. Thus, pertaining to the irrational parameter p% ¼ ð3 5Þ=2 0:382, the vector ðt2 ; tÞ is an admissible strategy for two players. From this strategy we can generate admissible strategies for an arbitrary number of players. Indeed, substituting t3 þ t2 ¼ t for t, and t4 þ t3 ¼ t2 for t2 , we obtain ðt3 ; t2 ; t2 Þ and ðt4 ; t3 ; tÞ, both admissible strategies for three Pauls. Continuing this algorithm, each time substituting tmþ2 þ tmþ1 for tm if the exponent m is present, after l steps we obtain admissible strategies for 2 þ l gamblers, l 2 N. However, even if we allow all possible branches generated by this algorithm, the result is incomplete in the sense that there are admissible strategies, such as ðt8 ; t5 ; t5 ; t5 ; t3 ; t3 ; t3 Þ for seven Pauls, that are avoided. Consider all the strategies that can be generated by the branching algorithm from ðt2 ; tÞ, and for every nX2 call the best among these conditionally optimal, denoted by p%n . Let f n be the nth Fibonacci number, so that with f 0 ¼ 1, f 1 ¼ 1 and f nþ1 ¼ f n1 þ f n , n 2 N. We can show that the conditionally optimal strategy for f n þ k players, k 2 f0; 1; . . . ; f n1 1g, each playing a St. Petersburgðp% Þ game, is pf%n þk ¼ ðtnþ1 ; . . . ; tnþ1 ; tn ; . . . ; tn ; tn1 ; . . . ; tn1 Þ, |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} k times
f n2 þk times
f n1 k times
with the corresponding added value Ap ðp%f n þk Þ ¼ tn ½kð2 tÞ þ nf n2 t þ ðn 1Þf n1 . Because of the inherent number-theoretic difficulties, we do not know whether these conditionally optimal strategies are optimal in general. Finally, we show that an extended form of our branching algorithm has an interesting property concerning stochastic domination. For any admissible parameter p 2 ð0; 1Þ, let ðqa1 ; qa2 ; . . . ; qan Þ and ðqb1 ; qb2 ; . . . ; qbm Þ be admissible strategies for n and m Pauls for any n; mX2. Substituting qak þb1 þ qak þb2 þ þ qak þbm ¼ qak for qak , where k 2 f1; . . . ; ng is arbitrary, we obtain a strategy ðqd 1 ; qd 2 ; . . . ; qd nþm1 Þ for n þ m 1 gamblers, where the sequence d 1 Xd 2 X Xd nþm1 is a nonincreasing rearrangement of the sequence a1 ; . . . ; ak1 ; ak þ b1 ; . . . ; ak þ bm ; ak þ 1; . . . ; an . We say that a strategy pn ¼ ðp1;n ; . . . ; pn;n Þ is stochastically dominant if p1;n X 1 þ þ pn;n X n XD X 1 . The last theorem states that the branching algorithm preserves stochastic dominance. Choosing first n ¼ m ¼ 2, it may be used in conjunction with Theorem 2 as a starting point.
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Theorem 6. If the admissible strategies ðqa1 ; qa2 ; . . . ; qan Þ and ðqb1 ; qb2 ; . . . ; qbm Þ are both stochastically dominant, then so is the generated strategy ðqd 1 ; qd 2 ; . . . ; qd nþm1 Þ. All our results here are for fixed numbers of players. Cso¨rgo+ and Simons (2005) proved for an arbitrary sequence of strategies pn ¼ ðp1;n ; . . . ; pn;n Þ that ðp1;n X 1 þ þ pn;n X n Þ=H r ðpn Þ converges in probability to p=q, as n ! 1, whenever H r ðpn Þ ! 1. 2. Proofs The first two lemmas are needed for the proof of Theorem 1, while the third lemma is used in the proof of Theorem 5. These lemmas are the generalizations of Lemmas 3–5 in Cso¨rgo+ and Simons (2006) (hereafter abbreviated as Cs–S). Lemma 1. If X 1 ; X 2 are independent St. PetersburgðpÞ random variables and c1 and c2 positive constants, then Eðminðc1 X 1 ; c2 X 2 ÞÞo1. Proof. We know from Cs–S (1996, 2005) that 1 F p ðxÞ ¼ PfX 4xg ¼ qblogr xc ¼ rhlogr xi =x for all xXr, and 1 otherwise. Hence, if xXr maxðc1 ; c2 Þ, then the inequality R 1Pfminðc1 X 1 ; c2 X 2 Þ4xg ¼ Pfc1 X 1 4xgPfc2 X 2 4xg oc1 c2 r2 =x2 holds and, therefore, Eðminðc1 X 1 ; c2 X 2 ÞÞ ¼ 0 Pfminðc1 X 1 ; c2 X 2 Þ4xg dxo1. & Lemma 2. If X is a St. PetersburgðpÞ random variable and bX1, then Z b PfX 4xg dx ¼ ðr 1Þblogr bc þ rhlogr bi ¼ 1 þ ðr 1Þlogr b dp ðbÞ, 0
where the function dp ðÞ is defined in Theorem 5. Notice that 1 ¼ 1 F p ðxÞ ¼ PfX 4xg ¼ qblogr xc even for x 2 ½1; rÞ. So what we need to prove is that RProof. b blogr xc dx ¼ ðr 1Þblogr bc þ rhlogr bi 1 for b41. If c ¼ logr b40, then 1 q Z rc Z c Z c Z 1 Z c blogr xc byc y hyi y hyi q dx ¼ q r log r dy ¼ ðlog rÞ r dy ¼ ðlog rÞ bcc r dy þ r dy 0
1
Z
¼ bccðr 1Þ þ ðlog rÞ
0
0
bcc
hci
ry dy ¼ ðr 1Þbcc þ rhci 1,
0
where log ¼ loge is the natural logarithm, which is the desired equation. Lemma 3. If r 2 f2; 3; . . .g, then the number of the smallest strictly positive components of an admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ is divisible by r. P Proof. Let the smallest strictly positive component be 1=rk for some k 2 N. Since nj¼1 pj;n rk ¼ rk , the sum must be divisible by r, so the number of terms equal to 1 in the sum, which is the number of the components 1=rk in pn , is also divisible by r. & Proof of Theorem 1. With the extended Lemmas 1 and 2, the proof is an easy generalization of that in the classical case p ¼ 1=2 in Cs–S (2006), so we only sketch the differences. For a given strategy pn ¼ ðp1;n ; . . . ; pn;n Þ, the integral Ap ðpn Þ in (1) is defined in the improper Riemann sense if and only if Ap ðpn ; bÞ ! Ap ðpn Þ as b ! 1, where Z b Ap ðpn ; bÞ ¼ ½Pfp1;n X 1 þ þ pn;n X n 4xg PfX 1 4xg dx. 0
It can Pbe shown that Ap ðpn ; bÞ ¼ ðr 1ÞH r ðpn Þ þ Rr ðpn ; bÞ þ oð1Þ as b ! 1, where Rr ðpn ; bÞ ¼ dp ðbÞ nj¼1 pj;n dp ðb=pj;n Þ. Notice that dp ðurk Þ ¼ dp ðuÞ, u40, for every k 2 Z. Thus if pn is admissible,
ARTICLE IN PRESS P. Kevei / Statistics & Probability Letters 77 (2007) 1043–1049
then Rr ðpn ; bÞ ¼ dp ðbÞ
n X j¼1
pj;n dp
b pj;n
! ¼ dp ðbÞ
n X
1047
pj;n dp ðbÞ ¼ 0,
j¼1
and hence Ap ðpn ; bÞ ¼ ðr 1ÞH r ðpn Þ þ oð1Þ as b ! 1, which is the ‘if part’ of the theorem. Conversely, suppose that Ap ðpn Þ in (1) exists, so that Ap ðpn ; bÞ ! Ap ðpn Þ as b ! 1. Using the above periodicity property of dp ðÞ, we get Rr ðpn ; rk bÞ ¼ Rr ðpn ; bÞ for every k 2 Z. Fixing b40 and letting k ! 1, so that rk b ! 1, we get Rr ðpn ; bÞ ¼ Ap ðpn Þ ðr 1ÞH r ðpn Þ. Let D ¼ Dþ D , where Dþ and D are the rightside and left-side differential operators, respectively. Then one can prove that 8 < r 1 for s ¼ rk when k 2 Z; rk Ddp ðsÞ ¼ :0 for all other s40; from which, for all j 2 f1; . . . ; ng for which pj;n 40, we find that ! 8r 1 < b for b ¼ rk pj;n when k 2 Z; Dpj;n dp ¼ rk : pj;n 0 for all other b40: Consequently, we have DRr ðpn ; bÞjb¼1 ¼ r 1 ðr 1Þ
X
pj;n ,
j2A
where A is the set of indicesPj 2 f1; . . . ; ng for which pj;n is an integer power of r. Since, on the other hand, DRr ðpn ; bÞ ¼ 0, this implies j2A pj;n ¼ 1, and thus completes the proof. & Proof of Theorem 2. Let qa þ qb ¼ 1 for some a; b 2 N. Then PfX 1 prk g ¼ F p ðrk Þ ¼ 1 qk for every k 2 N. We estimate the probability PfT 2 prk g, where T 2 ¼ qa X 1 þ qb X 2 . If T 2 prk , then (1) X 1 ; X 2 prk , or (2) X 1 ¼ rkþ1 ; . . . ; rkþa1 and X 2 prk1 , or (3) X 2 ¼ rkþ1 ; . . . ; rkþb1 and X 1 prk1 . We obtain PfT 2 prk gpð1 qk Þ2 þ ð1 qk1 Þqk ð1 qa1 Þ þ ð1 qk1 Þqk ð1 qb1 Þ 2 1 k 2 k1 k k1 2k 1 1 . ¼ ð1 q Þ þ ð1 q Þq 2 þq ¼1q q q Since the distribution function of X 1 jumps only in the points x ¼ rk , it is enough to show that PfT 2 ork gpPfX 1 ork g ¼ PfX 1 prk1 g ¼ 1 qk1 . This is true, because PfT 2 ork g ¼ PfT 2 prk g PfT 2 ¼ rk g 2 k1 2k 1 1 PfX 1 ¼ rk ; X 2 ¼ rk g p1 q þq q ¼ 1 qk1 , completing the proof.
&
Proof of Theorem 3. We prove that stronger statement that the graphs of the distribution functions of Sn and nX 1 cross each other infinitely often. More exactly we show that both PfnX 1 pnk g4PfS n pnk g and PfnX 1 onk goPfS n onk g hold whenever kX3.
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Notice that the inequality Sn pnk holds if and only if X 1 pnk1 ; X 2 pnk1 ; . . . ; X n pnk1 . This implies for arbitrary kX2 that ( ) n \ n1 n1 n1 n 1 n k k1 þ 2 þ þ k1 PfS n pn g ¼ P fX j pn g ¼ ¼ 1 k1 . n n n n j¼1 Clearly, PfnX 1 pnk g ¼ PfX 1 pnk1 g ¼ 1 1=nk1 , so PfnX 1 pnk g4PfSn pnk g. Now consider the probabilities PfnX 1 onk g and PfS n onk g. When k ¼ 2, both of them are zero. So, assume kX3. Noticing that PfSn onk g ¼ PfSn pnk g PfS n ¼ nk g, n1 n k k1 k1 k1 PfS n ¼ n g ¼ PfX 1 ¼ n ; X 2 ¼ n ; . . . ; X n ¼ n g ¼ , nk1 and PfnX 1 onk g ¼ PfnX 1 pnk1 g ¼ PfX 1 pnk2 g, we have 1 n n1 n 1 PfS n onk g ¼ 1 k1 41 k2 ¼ PfnX 1 onk g, n nk1 n where the inequality holds for nX3 and kX3 by elementary calculations.
&
Proof of Theorem 4. Let nX2 be the number of Pauls. By Theorem 1, for every admissible strategy q ¼ 1 p satisfies the equation qa1 þ qa2 þ þ qam ¼ 1, where a1 ; a2 ; . . . ; am 2 N and m 2 f2; 3; . . . ; ng. Without loss of generality we assume that the zeros, if P any, are the last components of the strategy, so that pn ¼ ðqa1 ; qa2 ; . . . ; qam ; 0; . . . ; 0Þ. Then T m :¼ nj¼1 pj;n X j ¼ qa1 X 1 þ þ qam X m . We estimate the probability PfT m prk g. If the event fT m prk g occurs, then we must have all the inequalities X 1 prkþa1 1 ; X 2 prkþa2 1 ; . . . ; X m prkþam 1 . Hence, PfT m prk gpð1 qkþa1 1 Þð1 qkþa2 1 Þ ð1 qkþam 1 Þ ¼ 1 qk ðqa1 1 þ qa2 1 þ þ qam 1 Þ þ q2k C 2 þ þ qmk C m ¼ 1 qk1 þ q2k C 2 þ þ qmk C m , where the constants C 2 ; C 3 ; . . . ; C m doR not depend on k. 1 Since pn is admissible, the integral 0 ½PfT m 4xg PfX 1 4xg dx exists as an improper Riemann integral. Hence it suffices to show that the integral of the negative part g m ðxÞ of the function gm ðxÞ:¼PfT m 4xg PfX 1 4xg is finite. Notice that gm ðxÞ ¼ PfX 1 pxg PfT m pxgXPfX 1 oxg PfT m pxg¼:hm ðxÞ for all x40. Clearly, the function hm ðxÞ takes a minimum value on the interval ðrk1 ; rk at x ¼ rk , for which the estimate above yields hm ðrk Þ ¼ PfX 1 ork g PfT m prk gX1 qk1 ð1 qk1 þ q2k C 2 þ þ qmk C m Þ ¼ ðq2k C 2 þ þ qmk C m Þ. R1 Therefore, setting C 1 ¼ 0 h m ðxÞ dx, we obtain Z 1 Z 1 1 Z X gm ðxÞ dxp hm ðxÞ dxpC 1 þ 0
0
k¼1
rk
rk1
ðq2k jC 2 j þ þ qmk jC m jÞ dx
1 X 1 k ¼ C1 þ r 1 ðq2k jC 2 j þ þ qmk jC m jÞ r k¼1 ¼ C 1 þ ð1 qÞ
1 X
ðqk jC 2 j þ þ qðm1Þk jC m jÞo1,
k¼1
which proves the theorem.
&
Proof of Theorem 5. This is based on the proof of Theorem 2 in Cs–S (2006), so we skip the details. Without loss of generality we assume that pn is ordered: p1;n Xp2;n X Xpn;n . The proof is by induction on rn .
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For rn ¼ 0 the statement is true. Now suppose that all the statements of the theorem hold for rn 1X0, and consider the case n ¼ rblogr nc þ ðr 1Þrn . If pn;n ¼ 0, then we have at least r 1 zeros. Deleting them, we get a strategy p^ nðr1Þ , and we are done in view of the fact that the bound Ap;n in (2) is nondecreasing in n. In the other case, when pn;n ¼ 1=rk for some k 2 N, we have at least r of these smallest components by Lemma 3. Changing r of these to a single component 1=rk1 , we obtain a strategy p^ nðr1Þ for which H r ðpn Þ H r ð^pnðr1Þ Þ ¼ 1=rk1 . Using the induction hypothesis and the formula Ap;n ¼ ðr 1Þblogr ncþ ðr 1Þrn =rblogr nc , the proof can be completed as in Cs–S (2006). The uniqueness assertion of the theorem follows by the same induction argument. & Lemma 4. If U; V ; W are independent random variables and UXD V , then U þ W XD V þ W . Proof. Let F ; G and H be the distribution functions of U; V and W , respectively. By assumption, F ðxÞpGðxÞ for all x 2 R. The random variables U þ W and V þ W have the distribution functions F HðÞ and G HðÞ, where denotes Lebesgue–Stieltjes convolution. Thus Z 1 Z 1 F HðxÞ ¼ F ðx yÞ dHðyÞp Gðx yÞ dHðyÞ ¼ G HðxÞ, 1
1
which proves the statement. & Proof of Theorem 6. Let Y 1 ; . . . ; Y m ; X 1 ; . . . ; X n be independent St. PetersburgðpÞ variables. From the assumption we get qak þb1 Y 1 þ qak þb2 Y 2 þ þ qak þbm Y m XD qak X k . By Lemma 6 this implies qa1 X 1 þ þ qak1 X k1 þ qak þb1 Y 1 þ þ qak þbm Y m þ qakþ1 X kþ1 þ þ qan X n XD qa1 X 1 þ þ qak1 X k1 þ qak X k þ qakþ1 X kþ1 þ þ qan X n . Now the assumption and obvious transitivity together imply the theorem. & Acknowledgments It is my pleasure to thank Professor Sa´ndor Cso¨rgo+ for introducing me to research through the problems treated here, and for his help in writing the paper. I also thank Professor Gordon Simons for his remarks on the manuscript. References + S., Simons, G., 1996. A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games. Cso¨rgo, Statist. Probab. Lett. 26, 65–73. + S., Simons, G., 2002. The two-Paul paradox and the comparison of infinite expectations. In: Berkes, I., Csa´ki, E., Cso¨rgo, + M. Cso¨rgo, (Eds.), Limit Theorems in Probability and Statistics, vol. I. Ja´nos Bolyai Mathematical Society, Budapest, pp. 427–455. + S., Simons, G., 2005. Laws of large numbers for cooperative St. Petersburg gamblers. Period. Math. Hungar. 50, 99–115. Cso¨rgo, + S., Simons, G., 2006. Pooling strategies for St. Petersburg gamblers. Bernoulli 12, 971–1002. Cso¨rgo,
Statistics & Probability Letters 77 (2007) 1043–1049 www.elsevier.com/locate/stapro
Generalized n-Paul paradox$ Pe´ter Kevei Bolyai Institute, University of Szeged, Aradi ve´rtanu´k tere 1, Szeged 6720, Hungary Received 6 June 2006; received in revised form 9 July 2006; accepted 28 August 2006 Available online 28 March 2007 Dedicated to Professor Sa´ndor Cso¨rgo+ for his 60th birthday
Abstract The paradoxical results of Cso¨rgo+ and Simons for mutually beneficial sharing among any fixed number of St. Petersburg gamblers are extended to games played by a possibly biased coin, with p as the probability of ‘heads.’ The extension is not straightforward because, unlike in the classical case with p ¼ 1=2, admissibly pooled winnings generally fail to stochastically dominate individual ones for more than two gamblers. Best admissible pooling strategies are determined when p is rational, and the algebraic depth of the problem for an irrational p is illustrated by an example. r 2007 Elsevier B.V. All rights reserved. MSC: primary 60E99; secondary 60G50 Keywords: Generalized St. Petersburg games; Comparison of infinite expectations; Pooling strategies
1. Results and discussion Peter offers to let Paul toss a possibly biased coin until it lands heads and pays him rk ducats if this happens on the kth toss, k 2 N ¼ f1; 2; . . .g, where r ¼ 1=q for q ¼ 1 p and p 2 ð0; 1Þ is the probability of ‘heads’ at each throw. This is the generalized St. PetersburgðpÞ game, in which PfX ¼ rk g ¼ qk1 p, k 2 N, for Paul’s gain X . Following Cso¨rgo+ and Simons (2002, 2006), we assume that Peter plays exactly one such game with each of nX2 players, Paul1 ; Paul2 ; . . . ; Pauln , whose independent individual winnings are X 1 ; X 2 ; . . . ; X n . The n players may agree to use a pooling strategy pn ¼ ðp1;n ; p2;n ; . . . ; pn;n Þ, before they play, where P p1;n ; p2;n ; . . . ; pn;n X0 and nj¼1 pj;n ¼ 1. Under this strategy Paul1 receives p1;n X 1 þ p2;n X 2 þ þ pn;n X n , Paul2 receives pn;n X 1 þ p1;n X 2 þ þ pn1;n X n , Paul3 receives pn1;n X 1 þ pn;n X 2 þ p1;n X 3 þ þ pn2;n X n ; . . ., and Pauln receives p2;n X 1 þ p3;n X 2 þ þ pn;n X n1 þ p1;n X n ducats. This strategy is fair to every Paul in the sense that their winnings are equally distributed and each receives the same added value equal to Z 1 Ap ðpn Þ ¼ E½p1;n X 1 þ þ pn;n X n ; X 1 ¼ ½Pfp1;n X 1 þ þ pn;n X n 4xg PfX 1 4xg dx, (1) 0 $ Work supported in part by the Hungarian Scientific Research Fund, Grants T–034121 and T–048360, and carried out within the Analysis and Stochastics Research Group of the Hungarian Academy of Sciences. E-mail address: [email protected].
0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.08.027
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whenever the integral is defined, so that comparison is possible. We refer to Cso¨rgo+ and Simons (2002, 2006) for a detailed exposition and discussion of the comparison operator E½; . We call a strategy pn ¼ ðp1;n ; . . . ; pn;n Þ admissible if each of its components is either zero or a nonnegative integer power of q ¼ 1 p. Individualistic strategies ð1; 0; . . . ; 0Þ are thus admissible for each p, otherwise P the powers in nonzero components are positive integers. The entropy of a pooling strategy is H r ðpn Þ ¼ nj¼1 pj;n logr 1=pj;n , where logr denotes the base r logarithm and 0logr 1=0 ¼ 0. We say that the random variable U is stochastically larger than the random variable V , written as UXD V , if PfU4xgXPfV 4xg for all x 2 R. Theorem 1. For any p 2 ð0; 1Þ and n 2 N, the added value Ap ðpn Þ exists as an improper Riemann integral if and only if pn is admissible, in which case Ap ðpn Þ ¼ p=qH r ðpn Þ. Cso¨rgo+ and Simons (2006) proved this theorem for the classical St. Petersburgð1=2Þ game, played with an unbiased coin. However, in that case they proved the following stronger result: the independent St. Petersburgð1=2Þ variables X 1 ; . . . ; X n can be defined on a rich enough probability space that carries, for each admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ, a St. Petersburgð1=2Þ random variable X pn and a nonnegative random variable Y pn such that T pn ¼ p1;n X 1 þ þ pn;n X n ¼ X pn þ Y pn almost surely. This implies the stochastic inequality T pn XD X 1 . Hence the integrand in A1=2 ðpn Þ is nonnegative and thus A1=2 ðpn Þ is trivially finite as a Lebesgue integral. As the next result shows, stochastic dominance is preserved for two players for an arbitrary St. Petersburg parameter p 2 ð0; 1Þ. Theorem 2. For any p 2 ð0; 1Þ, if p2 ¼ ðqa ; qb Þ is an admissible pooling strategy for some a; b 2 N, then T p2 ¼ qa X 1 þ qb X 2 is stochastically larger than X 1 . Surprisingly, however, for nX3 gamblers stochastic dominance generally fails to hold for admissible strategies. Our example to demonstrate this is when p ¼ ðn 1Þ=n, q ¼ 1 p ¼ 1=n, so that r ¼ 1=q ¼ n is also the number of Pauls. Then PfX ¼ nk g ¼ ðn 1Þ=nk , k 2 N, and the averaging pooling strategy pn ¼ pn ¼ ð1=n; 1=n; . . . ; 1=nÞ is admissible. For this strategy the weighted sum is T pn ¼ ðX 1 þ þ X n Þ=n, so that for n ¼ 2 Theorem 2 says in particular that S 2 ¼ 2T p ¼ X 1 þ X 2 is stochastically larger than 2X 1 . This is not 2 true for nX3. Theorem 3. If p ¼ ðn 1Þ=n, q ¼ 1=n and nX3, then neither S n ¼ X 1 þ þ X n nor nX 1 is stochastically larger than the other. In view of Theorem 2, the integrand in (1) is nonnegative whenever p2 is admissible, so that the integral Ap ðp2 Þ described in Theorem 1 strengthens to that of a Lebesgue integral when n ¼ 2. While the same conclusion holds for nX3, Theorem 3 rules out so simple a line of reasoning. Theorem 4. For every p 2 ð0; 1Þ and every admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ the integral Ap ðpn Þ in (1) is finite as a Lebesgue integral. Theorem 1 characterizes the pooling strategies that yield added values. However, admissible strategies do not exist for all, in fact, for most parameters p 2 ð0; 1Þ. Call a parameter p admissible, if for p there exists an admissible strategy which is not individualistic. Theorem 1 then says that p is admissible if and only if for q ¼ 1 p there exist positive integers a1 Xa2 X Xak , for some k 2 N, such that qa1 þ qa2 þ þ qak ¼ 1. In this case, r ¼ 1=q is an algebraic integer. If a1 4a2 , then q is also an algebraic integer, thus q is an algebraic unit. The set of algebraic numbers is countable, so there are at most a countable number of admissible parameters p. When q ¼ 1 p is rational for an admissible p 2 ð0; 1Þ, the equation implies q ¼ 1=m for some integer mX2. Thus the set of rational admissible parameters is fðm 1Þ=m : mX2g. In particular, it is interesting that the classical p ¼ 1=2 is the smallest such St. Petersburg parameter. It follows that the set of all admissible parameters p is countable. Nevertheless, it can be shown that this set is dense in the interval ð0; 1Þ. When a given number of our Pauls happen to have admissible strategies, a natural question is: Which is the best? In the latter rational case when p ¼ ðm 1Þ=m for some integer mX2, and so r ¼ 1=q ¼ mX2 is an integer, the answer is given by the next result, in which bxc ¼ maxfk 2 Z : kpxg is the integer part, dxe ¼ minfk 2 Z : kXxg is the integer ceiling and hxi ¼ x bxc ¼ x þ dxe is the fractional part of a number x 2 R.
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Theorem 5. If p ¼ ðr 1Þ=r and n ¼ rblogr nc þ ðr 1Þrn for some integers rX2 and 0prn prblogr nc 1, then Ap ðpn Þ ¼
p p H r ðpn Þp logr n dp ðnÞ¼:Ap;n q q
(2)
for every admissible strategy pn , where dp ðuÞ ¼ 1 þ ðr 1Þhlogr ui rhlogr ui , u40. Moreover, the bound Ap;n is attainable by means of the admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ ¼ ðrpn ; . . . ; rpn ; pn ; . . . ; pn Þ with pn ¼
1 , rdlogr ne
where the number of pn ’s and rpn ’s are, respectively, m1;p ðnÞ ¼
rn rdlogr ne r1
and
m2;p ðnÞ ¼
rdlogr ne n . r1
Apart from reorderings of the components of pn , the point of maximum is unique. It is easy to see that if n is not in the form rblogr nc þ ðr 1Þrn , then 0 must be included among the components of the strategy, which does not increase the entropy. So it is enough to investigate the number of players in the form above. The continuous function dp ðÞ is nonnegative; its maximum is given in formula (3.4) of Cso¨rgo+ and Simons (1996). Theorem 5 is not applicable for an irrational p. On the other hand, in every admissible situation Ap ðpn Þ ¼ ðr 1ÞH r ðpn Þ by Theorem 1, and the trivial upper bound H r ðpn Þplogr n ¼ H r ðpn Þ is valid for the entropy of every pn , where pn ¼ ð1=n; 1=n; . . . ; 1=nÞ. However equality cannot hold in general because pn is not admissible for every admissible parameter p. Apart from those cases which can be reduced to the rational case, that is, pffiffiffiffi when q ¼ 1 p ¼ 1= k m for some integers m; kX2, the problem of the best admissible strategy is unsolved. 2 For the pffiffiffi irrational case the simplest example is the equation q þ q ¼ 1, the solution of which is q ¼ t:¼ð 5p ffiffiffi 1Þ=2 0:618, the ratio of golden section. Thus, pertaining to the irrational parameter p% ¼ ð3 5Þ=2 0:382, the vector ðt2 ; tÞ is an admissible strategy for two players. From this strategy we can generate admissible strategies for an arbitrary number of players. Indeed, substituting t3 þ t2 ¼ t for t, and t4 þ t3 ¼ t2 for t2 , we obtain ðt3 ; t2 ; t2 Þ and ðt4 ; t3 ; tÞ, both admissible strategies for three Pauls. Continuing this algorithm, each time substituting tmþ2 þ tmþ1 for tm if the exponent m is present, after l steps we obtain admissible strategies for 2 þ l gamblers, l 2 N. However, even if we allow all possible branches generated by this algorithm, the result is incomplete in the sense that there are admissible strategies, such as ðt8 ; t5 ; t5 ; t5 ; t3 ; t3 ; t3 Þ for seven Pauls, that are avoided. Consider all the strategies that can be generated by the branching algorithm from ðt2 ; tÞ, and for every nX2 call the best among these conditionally optimal, denoted by p%n . Let f n be the nth Fibonacci number, so that with f 0 ¼ 1, f 1 ¼ 1 and f nþ1 ¼ f n1 þ f n , n 2 N. We can show that the conditionally optimal strategy for f n þ k players, k 2 f0; 1; . . . ; f n1 1g, each playing a St. Petersburgðp% Þ game, is pf%n þk ¼ ðtnþ1 ; . . . ; tnþ1 ; tn ; . . . ; tn ; tn1 ; . . . ; tn1 Þ, |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} k times
f n2 þk times
f n1 k times
with the corresponding added value Ap ðp%f n þk Þ ¼ tn ½kð2 tÞ þ nf n2 t þ ðn 1Þf n1 . Because of the inherent number-theoretic difficulties, we do not know whether these conditionally optimal strategies are optimal in general. Finally, we show that an extended form of our branching algorithm has an interesting property concerning stochastic domination. For any admissible parameter p 2 ð0; 1Þ, let ðqa1 ; qa2 ; . . . ; qan Þ and ðqb1 ; qb2 ; . . . ; qbm Þ be admissible strategies for n and m Pauls for any n; mX2. Substituting qak þb1 þ qak þb2 þ þ qak þbm ¼ qak for qak , where k 2 f1; . . . ; ng is arbitrary, we obtain a strategy ðqd 1 ; qd 2 ; . . . ; qd nþm1 Þ for n þ m 1 gamblers, where the sequence d 1 Xd 2 X Xd nþm1 is a nonincreasing rearrangement of the sequence a1 ; . . . ; ak1 ; ak þ b1 ; . . . ; ak þ bm ; ak þ 1; . . . ; an . We say that a strategy pn ¼ ðp1;n ; . . . ; pn;n Þ is stochastically dominant if p1;n X 1 þ þ pn;n X n XD X 1 . The last theorem states that the branching algorithm preserves stochastic dominance. Choosing first n ¼ m ¼ 2, it may be used in conjunction with Theorem 2 as a starting point.
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Theorem 6. If the admissible strategies ðqa1 ; qa2 ; . . . ; qan Þ and ðqb1 ; qb2 ; . . . ; qbm Þ are both stochastically dominant, then so is the generated strategy ðqd 1 ; qd 2 ; . . . ; qd nþm1 Þ. All our results here are for fixed numbers of players. Cso¨rgo+ and Simons (2005) proved for an arbitrary sequence of strategies pn ¼ ðp1;n ; . . . ; pn;n Þ that ðp1;n X 1 þ þ pn;n X n Þ=H r ðpn Þ converges in probability to p=q, as n ! 1, whenever H r ðpn Þ ! 1. 2. Proofs The first two lemmas are needed for the proof of Theorem 1, while the third lemma is used in the proof of Theorem 5. These lemmas are the generalizations of Lemmas 3–5 in Cso¨rgo+ and Simons (2006) (hereafter abbreviated as Cs–S). Lemma 1. If X 1 ; X 2 are independent St. PetersburgðpÞ random variables and c1 and c2 positive constants, then Eðminðc1 X 1 ; c2 X 2 ÞÞo1. Proof. We know from Cs–S (1996, 2005) that 1 F p ðxÞ ¼ PfX 4xg ¼ qblogr xc ¼ rhlogr xi =x for all xXr, and 1 otherwise. Hence, if xXr maxðc1 ; c2 Þ, then the inequality R 1Pfminðc1 X 1 ; c2 X 2 Þ4xg ¼ Pfc1 X 1 4xgPfc2 X 2 4xg oc1 c2 r2 =x2 holds and, therefore, Eðminðc1 X 1 ; c2 X 2 ÞÞ ¼ 0 Pfminðc1 X 1 ; c2 X 2 Þ4xg dxo1. & Lemma 2. If X is a St. PetersburgðpÞ random variable and bX1, then Z b PfX 4xg dx ¼ ðr 1Þblogr bc þ rhlogr bi ¼ 1 þ ðr 1Þlogr b dp ðbÞ, 0
where the function dp ðÞ is defined in Theorem 5. Notice that 1 ¼ 1 F p ðxÞ ¼ PfX 4xg ¼ qblogr xc even for x 2 ½1; rÞ. So what we need to prove is that RProof. b blogr xc dx ¼ ðr 1Þblogr bc þ rhlogr bi 1 for b41. If c ¼ logr b40, then 1 q Z rc Z c Z c Z 1 Z c blogr xc byc y hyi y hyi q dx ¼ q r log r dy ¼ ðlog rÞ r dy ¼ ðlog rÞ bcc r dy þ r dy 0
1
Z
¼ bccðr 1Þ þ ðlog rÞ
0
0
bcc
hci
ry dy ¼ ðr 1Þbcc þ rhci 1,
0
where log ¼ loge is the natural logarithm, which is the desired equation. Lemma 3. If r 2 f2; 3; . . .g, then the number of the smallest strictly positive components of an admissible strategy pn ¼ ðp1;n ; . . . ; pn;n Þ is divisible by r. P Proof. Let the smallest strictly positive component be 1=rk for some k 2 N. Since nj¼1 pj;n rk ¼ rk , the sum must be divisible by r, so the number of terms equal to 1 in the sum, which is the number of the components 1=rk in pn , is also divisible by r. & Proof of Theorem 1. With the extended Lemmas 1 and 2, the proof is an easy generalization of that in the classical case p ¼ 1=2 in Cs–S (2006), so we only sketch the differences. For a given strategy pn ¼ ðp1;n ; . . . ; pn;n Þ, the integral Ap ðpn Þ in (1) is defined in the improper Riemann sense if and only if Ap ðpn ; bÞ ! Ap ðpn Þ as b ! 1, where Z b Ap ðpn ; bÞ ¼ ½Pfp1;n X 1 þ þ pn;n X n 4xg PfX 1 4xg dx. 0
It can Pbe shown that Ap ðpn ; bÞ ¼ ðr 1ÞH r ðpn Þ þ Rr ðpn ; bÞ þ oð1Þ as b ! 1, where Rr ðpn ; bÞ ¼ dp ðbÞ nj¼1 pj;n dp ðb=pj;n Þ. Notice that dp ðurk Þ ¼ dp ðuÞ, u40, for every k 2 Z. Thus if pn is admissible,
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then Rr ðpn ; bÞ ¼ dp ðbÞ
n X j¼1
pj;n dp
b pj;n
! ¼ dp ðbÞ
n X
1047
pj;n dp ðbÞ ¼ 0,
j¼1
and hence Ap ðpn ; bÞ ¼ ðr 1ÞH r ðpn Þ þ oð1Þ as b ! 1, which is the ‘if part’ of the theorem. Conversely, suppose that Ap ðpn Þ in (1) exists, so that Ap ðpn ; bÞ ! Ap ðpn Þ as b ! 1. Using the above periodicity property of dp ðÞ, we get Rr ðpn ; rk bÞ ¼ Rr ðpn ; bÞ for every k 2 Z. Fixing b40 and letting k ! 1, so that rk b ! 1, we get Rr ðpn ; bÞ ¼ Ap ðpn Þ ðr 1ÞH r ðpn Þ. Let D ¼ Dþ D , where Dþ and D are the rightside and left-side differential operators, respectively. Then one can prove that 8 < r 1 for s ¼ rk when k 2 Z; rk Ddp ðsÞ ¼ :0 for all other s40; from which, for all j 2 f1; . . . ; ng for which pj;n 40, we find that ! 8r 1 < b for b ¼ rk pj;n when k 2 Z; Dpj;n dp ¼ rk : pj;n 0 for all other b40: Consequently, we have DRr ðpn ; bÞjb¼1 ¼ r 1 ðr 1Þ
X
pj;n ,
j2A
where A is the set of indicesPj 2 f1; . . . ; ng for which pj;n is an integer power of r. Since, on the other hand, DRr ðpn ; bÞ ¼ 0, this implies j2A pj;n ¼ 1, and thus completes the proof. & Proof of Theorem 2. Let qa þ qb ¼ 1 for some a; b 2 N. Then PfX 1 prk g ¼ F p ðrk Þ ¼ 1 qk for every k 2 N. We estimate the probability PfT 2 prk g, where T 2 ¼ qa X 1 þ qb X 2 . If T 2 prk , then (1) X 1 ; X 2 prk , or (2) X 1 ¼ rkþ1 ; . . . ; rkþa1 and X 2 prk1 , or (3) X 2 ¼ rkþ1 ; . . . ; rkþb1 and X 1 prk1 . We obtain PfT 2 prk gpð1 qk Þ2 þ ð1 qk1 Þqk ð1 qa1 Þ þ ð1 qk1 Þqk ð1 qb1 Þ 2 1 k 2 k1 k k1 2k 1 1 . ¼ ð1 q Þ þ ð1 q Þq 2 þq ¼1q q q Since the distribution function of X 1 jumps only in the points x ¼ rk , it is enough to show that PfT 2 ork gpPfX 1 ork g ¼ PfX 1 prk1 g ¼ 1 qk1 . This is true, because PfT 2 ork g ¼ PfT 2 prk g PfT 2 ¼ rk g 2 k1 2k 1 1 PfX 1 ¼ rk ; X 2 ¼ rk g p1 q þq q ¼ 1 qk1 , completing the proof.
&
Proof of Theorem 3. We prove that stronger statement that the graphs of the distribution functions of Sn and nX 1 cross each other infinitely often. More exactly we show that both PfnX 1 pnk g4PfS n pnk g and PfnX 1 onk goPfS n onk g hold whenever kX3.
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Notice that the inequality Sn pnk holds if and only if X 1 pnk1 ; X 2 pnk1 ; . . . ; X n pnk1 . This implies for arbitrary kX2 that ( ) n \ n1 n1 n1 n 1 n k k1 þ 2 þ þ k1 PfS n pn g ¼ P fX j pn g ¼ ¼ 1 k1 . n n n n j¼1 Clearly, PfnX 1 pnk g ¼ PfX 1 pnk1 g ¼ 1 1=nk1 , so PfnX 1 pnk g4PfSn pnk g. Now consider the probabilities PfnX 1 onk g and PfS n onk g. When k ¼ 2, both of them are zero. So, assume kX3. Noticing that PfSn onk g ¼ PfSn pnk g PfS n ¼ nk g, n1 n k k1 k1 k1 PfS n ¼ n g ¼ PfX 1 ¼ n ; X 2 ¼ n ; . . . ; X n ¼ n g ¼ , nk1 and PfnX 1 onk g ¼ PfnX 1 pnk1 g ¼ PfX 1 pnk2 g, we have 1 n n1 n 1 PfS n onk g ¼ 1 k1 41 k2 ¼ PfnX 1 onk g, n nk1 n where the inequality holds for nX3 and kX3 by elementary calculations.
&
Proof of Theorem 4. Let nX2 be the number of Pauls. By Theorem 1, for every admissible strategy q ¼ 1 p satisfies the equation qa1 þ qa2 þ þ qam ¼ 1, where a1 ; a2 ; . . . ; am 2 N and m 2 f2; 3; . . . ; ng. Without loss of generality we assume that the zeros, if P any, are the last components of the strategy, so that pn ¼ ðqa1 ; qa2 ; . . . ; qam ; 0; . . . ; 0Þ. Then T m :¼ nj¼1 pj;n X j ¼ qa1 X 1 þ þ qam X m . We estimate the probability PfT m prk g. If the event fT m prk g occurs, then we must have all the inequalities X 1 prkþa1 1 ; X 2 prkþa2 1 ; . . . ; X m prkþam 1 . Hence, PfT m prk gpð1 qkþa1 1 Þð1 qkþa2 1 Þ ð1 qkþam 1 Þ ¼ 1 qk ðqa1 1 þ qa2 1 þ þ qam 1 Þ þ q2k C 2 þ þ qmk C m ¼ 1 qk1 þ q2k C 2 þ þ qmk C m , where the constants C 2 ; C 3 ; . . . ; C m doR not depend on k. 1 Since pn is admissible, the integral 0 ½PfT m 4xg PfX 1 4xg dx exists as an improper Riemann integral. Hence it suffices to show that the integral of the negative part g m ðxÞ of the function gm ðxÞ:¼PfT m 4xg PfX 1 4xg is finite. Notice that gm ðxÞ ¼ PfX 1 pxg PfT m pxgXPfX 1 oxg PfT m pxg¼:hm ðxÞ for all x40. Clearly, the function hm ðxÞ takes a minimum value on the interval ðrk1 ; rk at x ¼ rk , for which the estimate above yields hm ðrk Þ ¼ PfX 1 ork g PfT m prk gX1 qk1 ð1 qk1 þ q2k C 2 þ þ qmk C m Þ ¼ ðq2k C 2 þ þ qmk C m Þ. R1 Therefore, setting C 1 ¼ 0 h m ðxÞ dx, we obtain Z 1 Z 1 1 Z X gm ðxÞ dxp hm ðxÞ dxpC 1 þ 0
0
k¼1
rk
rk1
ðq2k jC 2 j þ þ qmk jC m jÞ dx
1 X 1 k ¼ C1 þ r 1 ðq2k jC 2 j þ þ qmk jC m jÞ r k¼1 ¼ C 1 þ ð1 qÞ
1 X
ðqk jC 2 j þ þ qðm1Þk jC m jÞo1,
k¼1
which proves the theorem.
&
Proof of Theorem 5. This is based on the proof of Theorem 2 in Cs–S (2006), so we skip the details. Without loss of generality we assume that pn is ordered: p1;n Xp2;n X Xpn;n . The proof is by induction on rn .
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For rn ¼ 0 the statement is true. Now suppose that all the statements of the theorem hold for rn 1X0, and consider the case n ¼ rblogr nc þ ðr 1Þrn . If pn;n ¼ 0, then we have at least r 1 zeros. Deleting them, we get a strategy p^ nðr1Þ , and we are done in view of the fact that the bound Ap;n in (2) is nondecreasing in n. In the other case, when pn;n ¼ 1=rk for some k 2 N, we have at least r of these smallest components by Lemma 3. Changing r of these to a single component 1=rk1 , we obtain a strategy p^ nðr1Þ for which H r ðpn Þ H r ð^pnðr1Þ Þ ¼ 1=rk1 . Using the induction hypothesis and the formula Ap;n ¼ ðr 1Þblogr ncþ ðr 1Þrn =rblogr nc , the proof can be completed as in Cs–S (2006). The uniqueness assertion of the theorem follows by the same induction argument. & Lemma 4. If U; V ; W are independent random variables and UXD V , then U þ W XD V þ W . Proof. Let F ; G and H be the distribution functions of U; V and W , respectively. By assumption, F ðxÞpGðxÞ for all x 2 R. The random variables U þ W and V þ W have the distribution functions F HðÞ and G HðÞ, where denotes Lebesgue–Stieltjes convolution. Thus Z 1 Z 1 F HðxÞ ¼ F ðx yÞ dHðyÞp Gðx yÞ dHðyÞ ¼ G HðxÞ, 1
1
which proves the statement. & Proof of Theorem 6. Let Y 1 ; . . . ; Y m ; X 1 ; . . . ; X n be independent St. PetersburgðpÞ variables. From the assumption we get qak þb1 Y 1 þ qak þb2 Y 2 þ þ qak þbm Y m XD qak X k . By Lemma 6 this implies qa1 X 1 þ þ qak1 X k1 þ qak þb1 Y 1 þ þ qak þbm Y m þ qakþ1 X kþ1 þ þ qan X n XD qa1 X 1 þ þ qak1 X k1 þ qak X k þ qakþ1 X kþ1 þ þ qan X n . Now the assumption and obvious transitivity together imply the theorem. & Acknowledgments It is my pleasure to thank Professor Sa´ndor Cso¨rgo+ for introducing me to research through the problems treated here, and for his help in writing the paper. I also thank Professor Gordon Simons for his remarks on the manuscript. References + S., Simons, G., 1996. A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games. Cso¨rgo, Statist. Probab. Lett. 26, 65–73. + S., Simons, G., 2002. The two-Paul paradox and the comparison of infinite expectations. In: Berkes, I., Csa´ki, E., Cso¨rgo, + M. Cso¨rgo, (Eds.), Limit Theorems in Probability and Statistics, vol. I. Ja´nos Bolyai Mathematical Society, Budapest, pp. 427–455. + S., Simons, G., 2005. Laws of large numbers for cooperative St. Petersburg gamblers. Period. Math. Hungar. 50, 99–115. Cso¨rgo, + S., Simons, G., 2006. Pooling strategies for St. Petersburg gamblers. Bernoulli 12, 971–1002. Cso¨rgo,