The Euler-MacLaurin formula applied to the St. Petersburg problem

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1 The Euler-MacLaurin formula applied to the St. Petersburg problem Herbert Müller, Switzerland Published in 2013 on Abstract In the St. Petersburg problem, a person has to decide whether he should or should not participate in a game with a participation fee and different possible outcomes and gains. How should he decide? Daniel Bernoulli s answer was: if the utility of the game to the player is positive (negative), he should (should not) participate. Bernoulli s utility-formula contains an infinite sum, which we evaluate with the Euler-MacLaurin formula. As a result we obtain a simple approximation formula for the boundary participation fee, for which the lost fee and the possible gains balance. 1 Introduction The St.Petersburg problem (or paradox) was stated for the first time in 1713 by Nicolas Bernoulli in a letter to the French mathematician Pierre Rémond Montmort [1, 2], and resolved independently in 1728 by Gabriel Cramer and in 1738 by Daniel Bernoulli [3]. The statement of the problem evolved in the course of the correspondence between Montmort, Cramer and the Bernoulli-cousins, see the excerpts given in the appendix. The St. Petersburg problem is a decision problem: a person must decide whether or not he should accept a proposition (or gamble, or risk) with different possible outcomes and profits. Section 2 contains the modern statement of the St. Petersburg problem and of the generalized decision problem, and the resolution of both problems by Daniel Bernoulli. Section 3 contains the result of this article: we use the Euler-MacLaurin formula to calculate the boundary participation fee, for which the lost fee and the possible gains balance. I am a single author, by we I mean the reader and me. 1

2 2 The St. Petersburg problem and its resolution The St. Petersburg problem is a Drosophila of probability theory, i. e. there are more articles on it than you or I can read in a lifetime (and now I add my drop to the ocean). Therefore I will be brief in the following exposition. The following statement of the St. Petersburg problem follows the wording of N. Bernoulli in his letter to D. Bernoulli in 1728 [2]. St. Petersburg Problem: A tosses a coin until the side of heads turns up. B (who is supposed to be wealthy) is engaged to give to him 1 ducat if heads turns up at the first toss; 2 ducats if it turns up at the second toss; 4 ducats at the 3rd toss; 8 at the 4th, etc.. What sum of ducats should A at most accept to pay to B to be allowed to play the game? The naïve answer A should pay less than the expected gain doesn t work here, for the expected gain is infinite (St. Petersburg Paradox). The St. Petersburg problem is an instance of the following more general Decision Problem: A has the choice to accept or not to accept a proposition with k = 1...K possible outcomes. The probability of outcome k is p k. A s profit in outcome k is Q k (Q k < 0 is possible). A s wealth is W. Should A accept the proposition? D. Bernoulli s solution to the decision problem [3]: The utility (to A) of outcome k is U k = ln(1 + Q k /W ). The average utility (to A) of the proposition is Ū = p k U k. The value V (to A) of the proposition is the profit which corresponds to this utility, i. e. V = W (exp Ū 1). A should accept the proposition when its value (or its expected utility) is positive. Remark: Inserting Ū = p k ln(1 + Q k /W ) into V = W (exp Ū 1) and Taylor-developing gives V = Q v Q /2W +..., where v Q = p k (Q k Q) 2 is the variance of the Q k. The variance term diminishes the value of the proposition below the expected profit Q! (The next term is s Q /3W 2 Qv Q /2W 2, where s Q = p k (Q k Q) 3 is the skew.) D. Bernoulli s solution to the St. Petersburg problem [3]: The St. Petersburg game has infinitely many possible outcomes: 1 toss, 2 tosses, 3 tosses... The probability of outcome k is p k = p k 1 (1 p), with p the probability for tails (= 0.5 in the St. Petersburg game) and 1 p the probability for heads. The gain of outcome k is p k+1, the participation fee (loss) is L. The average utility of the game (to A, who s wealth is W ) is therefore, after renumbering, ( ) Ū = (1 p) p k ln 1 + p k L W The quantity of interest is the participation fee ˆL = ˆL(W ) for which the expected utility is nil. L = ˆL or Ū = 0 represents the boundary between A s decision to participate (L < ˆL, Ū > 0) or not to participate (L > ˆL, Ū < 0) in the game. 2 (1)

3 3 The participation fee in the St. Petersburg game We first derive an approximation formula for the participation fee ˆL corresponding to the boundary between acceptance and refusal of the St. Petersburg game. Then we apply the formula to a historical numerical example given by Laplace. 3.1 Approximation formula The utility-formula (1) contains the three parameters p, W, L. The number of relevant terms in the sum is of the order ln W/ ln(1/p). Under normal circumstances this number is very large, and we can evaluate the sum on the RHS of (1) by the Euler-MacLaurin formula [5] K K f(k) = dk f(k) (f(0) + f(k)) + B ( 2m f (2m 1) (K) f (2m 1) (0) ) m=1 (2m)! The B 2m are the Bernoulli numbers. To simplify the calculation, we slightly rearrange the utility-formula (1) and express the sum on the RHS by an integral: ( Ū = ln 1 L ) 1/(W L) + (1 p) dt (1 + tp k ) W Applying the Euler-MacLaurin-formula to the last sum gives, after some calculation, (1 + tp k ) = ln 1 + t / ln 1 t p + 1 2(1 + t) +... Here we have omitted the small terms containing the Bernoulli numbers B 2, B 4,.... Inserting this expression into the previous eq. and integrating gives, after some calculation, ( Ū = ln 1 L ) + (1 p) W (1 + 1 ) ln(1 + 1 ) 1 ln( 1 ) W L W L W L W L ln ) ( (1 2 ln O ln 1 ) W L p p Usually we have: gain at first toss = 1 < L << W ; in this case, the above utility formula simplifies to Ū = 1 p W 1 + ln W ln L W p Setting Ū = 0 gives us the desired formula for the limiting fee ˆL(W ): 1 + ln W ˆL = (1 p) ln p (2) (3) The approximation formula (3) is the main result of this article. 3

4 3.2 Numerical example Laplace, in discussing the St. Petersburg problem [4], invents his own version of the game: the player gets 2 francs instead of 1 franc for heads at the first toss. Laplace then derives an implicit, infinite product formula for the mean utility, and gives the following numerical example (which was also discussed by Jaynes in [6]): Therefore, the physical wealth of B being initially [francs], he cannot reasonably stake in this game more than 8.78 francs. Let us check Laplace s result. The monetary unit is now 2 francs, and the initial wealth in this unit is Formula (3) with p = 0.5 and W = gives the boundary participation fee ˆL = 4.32, corresponding to 8.65 francs. The numerical evaluation of the initial sum formula (1) with Ū = 0, p = 0.5 and W = gives ˆL = 4.36, corresponding to 8.72 francs. Let us collect these results in the natural currency: W = ˆL = 4.36 (exact) 4.39 (Laplace) 4.32 (eq. (3)) The precision of our result is similar to Laplace s, and quite sufficient in this example. Appendix: Historical Wordings of the St. Petersburg Problem Nicolas Bernoulli to Montmort, Basel, 9 September 1713 [1], [2]... Fourth Problem. A promises to give a coin to B, if with an ordinary die he brings forth 6 points on the first throw, two coins if he brings forth 6 on the second throw, 3 coins if he brings forth this point on the third throw, 4 coins if he brings forth it on the fourth and thus in order; one asks what is the expectation of B? Fifth Problem. One asks the same thing if A promises to B to give him some coins in this progression 1, 2, 4, 8, 16 etc. or 1, 3, 9, 27 etc. or 1, 4, 9, 16, 25 etc. or 1, 8, 27, 64 instead of 1, 2, 3, 4, 5 etc. as beforehand.... 4

5 Cramer to Nicolas Bernoulli, London, 21 May 1728 [2]... I know not if I deceive myself, but I believe to hold the solution of the singular case that you have proposed to Mr. de Montmort in your letter of 9 September 1713, Prob. 5, page 402. In order to render the case more simple I will suppose that A throw in the air a piece of money, B undertakes to give him a coin, if the side of Heads falls on the first toss, 2, if it is only the second, 4, if it is the 3rd toss, 8, if it is the 4th toss, etc. The paradox consists in this that the calculation gives for the equivalent that A must give to B an infinite sum, which would seem absurd, since there is no person of good sense, who would wish to give 20 coins. One asks the reason for the difference between the mathematical calculation and the common value. I believe that it comes from this that the mathematicians value money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.... Nicolas Bernoulli to Daniel Bernoulli, Basel, 27 October 1728 [2]... I would well wish to know your sentiment on the 2nd and 5th Problems of those that I have formerly proposed to the late Mr. de Montmort, see the l Analysis sur les Jeux de hazard page 402, particularly concerning the last, of which Mr. Professor Cramer of Geneva has communicated to me a solution, which does not satisfy me entirely; in order to render the case more simple, one is able to suppose, that A throws into the air a piece of money, B is engaged to give to him a coin if the side of Heads falls on the first throw; 2, if it is only the second; 4, if it is the 3rd throw; 8, if it is the 4th throw, etc. The concern is to find what must be the equivalent that A must give to B; the calculation gives an infinite sum, that which is absurd, because there is no person of good sense who wished to give merely 20 coins.... Daniel Bernoulli, 1738 [3]... My most honourable cousin the celebrated Nicolas Bernoulli, Professor utriusque iuris at the University of Basle, once submitted five problems to the highly distinguished mathematician Montmort. These problems are reproduced in the work L analyse sur les jeux de hazard de M. de Montmort, p The last of these problems runs as follows: Peter tosses a coin and continues to do so until it should land heads when it comes to the ground. He agrees to give Paul one ducat if he gets heads on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul s expectation. My aforementioned cousin discussed this problem in a letter to me asking for my opinion. Although the standard calculation shows that the value of Paul s expectation is infinitely great, it has, he said, to be admitted that any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats.... 5

6 Pierre-Simon Laplace, 1812 [4]... The principle we used just now to calculate the moral expectation has been proposed by Daniel Bernoulli, in order to explain the difference between the result of probability theory and what our common sense tells us. Two players A and B play heads or tails, with the condition that A pays to B, 2 francs if heads comes up at the first toss; 4 francs if it comes up at the second toss, 8 francs if it comes up at the third toss, and so on until the nth toss. We ask what B must give to A at the beginning of the game.... References [1] Montmort, P. R., Essai d analyse sur les jeux de hazard, 1708, Paris, p [2] Pulskamp, R. J., Correspondence of Nicolas Bernoulli concerning the St. Petersburg Game, translated from Die Werke von Jakob Bernoulli Band 3, K9, 2013, Dept. of Mathematics and Computer Science, Xavier University, Cincinnati, OH. [3] Bernoulli, D., Exposition of a new Theory on the Measurement of Risk, 1738, Papers of the Imperial Academy of Sciences in Petersburg, Vol. V, pp , translated in Econometrica, Vol. 22, No. 1. (Jan., 1954), pp [4] Laplace, P.-S., Théory analytique des probabilités, 1812, Ve. Courcier, Paris, 2nd ed. pp [5] Arfken, G., Mathematical Methods for Physicists, 1970, Academic Press, New York, 2nd ed. p. 281f. [6] Jaynes, E. T., Probability Theory - The Logic of Science, 2003, Cambridge University Press, p. 399f. 6