portrait of Nicolas Bernoulli ( 1723 ) The **St. Petersburg paradox** or **St. Petersburg lottery** [ 1 ] is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches eternity but however seems to be worth merely a very modest measure to the participants. The St. Petersburg paradox is a situation where a naive decisiveness criterion which takes only the expected value into account predicts a naturally of carry through that presumably no actual person would be will to take. It is related to probability and decisiveness hypothesis in economics. several resolutions to the paradox have been proposed.

Reading: St. Petersburg paradox – Wikipedia

The problem was invented by Nicolas Bernoulli, [ 2 ] who stated it in a letter to Pierre Raymond de Montmort on September 9, 1713. [ 3 ] [ 4 ] however, the paradox takes its name from its analysis by Nicolas ‘ cousin Daniel Bernoulli, erstwhile resident of the eponymous russian city, who in 1738 published his thoughts about the problem in the *Commentaries of the Imperial Academy of Science of Saint Petersburg*. [ 5 ]

## The St. Petersburg Game [edit ]

The St. Petersburg paradox is typically framed in terms of gambles on the result of fair mint tosses. A casino offers a crippled of gamble for a unmarried player in which a fair coin is tossed at each stage. The initial venture begins at 2 dollars and is doubled every time heads appears. The inaugural time tails appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if tails appears on the first flip, 4 dollars if heads appears on the foremost flip and tails on the moment, 8 dollars if heads appears on the first two tosses and tails on the third gear, and then on. Mathematically, the actor wins 2 thousand + 1 { \displaystyle 2^ { k+1 } } dollars, where k { \displaystyle kilobyte } is the number of consecutive head tosses. What would be a honest price to pay the casino for entering the crippled ? To answer this, one needs to consider what would be the expected payout at each stage : with probability 1/2, the player wins 2 dollars ; with probability 1/4 the musician wins 4 dollars ; with probability 1/8 the player wins 8 dollars, and so on. Assuming the game can continue a long as the coin toss results in heads and, in particular, that the casino has inexhaustible resources, the expect value is thus

E = 1 2 ⋅ 2 + 1 4 ⋅ 4 + 1 8 ⋅ 8 + 1 16 ⋅ 16 + ⋯ = 1 + 1 + 1 + 1 + ⋯ = ∞. { \displaystyle { \begin { aligned } E & = { \frac { 1 } { 2 } } \cdot 2+ { \frac { 1 } { 4 } } \cdot 4+ { \frac { 1 } { 8 } } \cdot 8+ { \frac { 1 } { 16 } } \cdot 16+\cdots \\ & =1+1+1+1+\cdots \\ & =\infty \, .\end { aligned } } } This union grows without bound, and so the expect acquire is an infinite sum of money. [ 6 ]

## The Paradox [edit ]

Considering nothing but the expected value of the net change in one ‘s monetary wealth, one should therefore play the game at any price if offered the opportunity. Yet, Daniel Bernoulli, after describing the game with an initial stake of one ducat, stated, “ Although the standard calculation shows that the value of [ the player ‘s ] expectation is infinitely great, it has … to be admitted that any reasonably reasonable man would sell his gamble, with capital pleasure, for twenty ducats. ” [ 5 ] Robert Martin quotes Ian Hacking as saying, “ few of us would pay flush $ 25 to enter such a game, ” and he says most commentators would agree. [ 7 ] The paradox is the discrepancy between what people seem will to pay to enter the game and the countless expected value. [ 5 ] The St. Petersburg Paradox is a veridical paradox but not a compression, as no delusive statement is derived. It is a counterexample against the principle of maximizing the expected value .

## Solutions [edit ]

several approaches have been proposed for solving the paradox .

### Expected utility hypothesis [edit ]

portrayal of Daniel Bernoulli ( 1720-1725 ) The authoritative solution of the paradox involved the denotative insertion of a utility function, an have a bun in the oven utility guess, and the presumption of diminishing bare utility of money. In Daniel Bernoulli ‘s words :

The decision of the value of an detail must not be based on the monetary value, but rather on the utility program it yields … There is no doubt that a reach of one thousand ducat is more significant to the pauper than to a rich serviceman though both gain the like come .

A common utility program model, suggested by Daniel Bernoulli, is the logarithmic function *U* ( *w* ) = ln ( *w* ) ( known as *log utility* ). It is a affair of the gambler ‘s sum wealth *w*, and the concept of diminishing bare utility of money is built into it. The ask utility guess posits that a utility officiate exists that provides a good criterion for real people ‘s behavior ; i.e. a affair that returns a positive or negative value indicating if the bet is a good gamble. For each possible event, the change in utility ln ( wealth after the event ) − ln ( wealth before the consequence ) will be weighted by the probability of that event occurring. Let *c* be the cost charged to enter the game. The expect incremental utility program of the lottery nowadays converges to a finite value :

Δ E ( U ) = ∑ k = 1 + ∞ 1 2 kilobyte [ ln ( watt + 2 potassium − deoxycytidine monophosphate ) − ln ( w ) ] < + ∞. { \displaystyle \Delta E ( U ) =\sum _ { k=1 } ^ { +\infty } { \frac { 1 } { 2^ { kelvin } } } \left [ \ln \left ( w+2^ { k } -c\right ) -\ln ( w ) \right ] < +\infty \ ,. } This formula gives an implicit kinship between the gambler ‘s wealth and how much he should be will to pay ( specifically, any *c* that gives a positive change in expected utility ). For model, with natural log utility, a millionaire ( $ 1,000,000 ) should be will to pay up to $ 20.88, a person with $ 1,000 should pay up to $ 10.95, a person with $ 2 should borrow $ 1.35 and pay up to $ 3.35. Before Daniel Bernoulli published, in 1728, a mathematician from Geneva, Gabriel Cramer, had already found parts of this idea ( besides motivated by the St. Petersburg Paradox ) in stating that

the mathematicians estimate money in proportion to its measure, and men of good sense in proportion to the custom that they may make of it.

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He demonstrated in a letter to Nicolas Bernoulli [ 8 ] that a feather root function describing the diminishing marginal benefit of gains can resolve the trouble. however, unlike Daniel Bernoulli, he did not consider the total wealth of a person, but only the amplification by the lottery. This solution by Cramer and Bernoulli, however, is not wholly satisfying, as the lottery can well be changed in a way such that the paradox reappears. To this calculate, we merely need to change the game so that it gives even more quickly increasing payoffs. For any boundless utility function, one can find a lottery that allows for a variant of the St. Petersburg paradox, as was first pointed out by Menger. [ 9 ] recently, expected utility hypothesis has been extended to arrive at more behavioral decision models. In some of these newfangled theories, as in accumulative candidate theory, the St. Petersburg paradox again appears in certain cases, even when the utility function is concave, but not if it is bounded. [ 10 ]

### probability weighting [edit ]

Nicolas Bernoulli himself proposed an alternate mind for solving the paradox. He conjectured that people will neglect improbable events. [ 4 ] Since in the St. Petersburg lottery only improbable events yield the gamey prizes that lead to an infinite expected value, this could resolve the paradox. The mind of probability weighting resurfaced a lot later in the work on expectation hypothesis by Daniel Kahneman and Amos Tversky. Paul Weirich similarly wrote that Risk Aversion could solve the paradox. Weirich went on to write that increasing the choice actually decreases the find of person paying to play the game, stating “ there is some number of birds in bridge player worth more than any number of birds in the bush ”. [ 11 ] however, this has been rejected by some theorists because as they point out, some people enjoy the risk of gambling and because it is confused to assume that increasing the loot will lead to more risks. accumulative candidate theory is one democratic abstraction of expected utility theory that can predict many behavioral regularities. [ 12 ] however, the overweighting of modest probability events introduced in accumulative expectation theory may restore the St. Petersburg paradox. accumulative prospect theory avoids the St. Petersburg paradox alone when the power coefficient of the utility routine is lower than the ability coefficient of the probability weighting serve. [ 13 ] intuitively, the utility function must not plainly be concave, but it must be concave proportional to the probability weighting function to avoid the St. Petersburg paradox. One can argue that the formula for the view theory are obtained in the region of less than $ 400. [ 12 ] This is not applicable for infinitely increasing sums in the St. Petersburg paradox .

### Finite St. Petersburg lotteries [edit ]

The classical St. Petersburg game assumes that the casino or banker has infinite resources. This assumption has farseeing been challenged as unrealistic. [ 6 ] [ 14 ] Alexis Fontaine des Bertins pointed out in 1754 that the resources of any potential angel of the game are finite. [ 15 ] More importantly, the expect measure of the game lone grows logarithmically with the resources of the casino. As a result, the expected value of the game, even when played against a casino with the largest bankroll realistically conceivable, is quite modest. In 1777, Georges-Louis Leclerc, Comte de Buffon calculated that after 29 rounds of play there would not be enough money in the Kingdom of France to cover the count. [ 16 ] If the casino has finite resources, the game must end once those resources are exhausted. [ 14 ] Suppose the total resources ( or maximum pot ) of the casino are *W* dollars ( more broadly, *W* is measured in units of half the game ‘s initial venture ). then the maximum issue of times the casino can play before it no longer can in full cover the future count is *L* = floor ( log2 ( *W* ) ). [ 17 ] Assuming the game ends when the casino can nobelium long cover the bet, the expected value *E* of the lottery then becomes : [ 17 ]

- E = ∑ k = 1 L 1 2 potassium ⋅ 2 potassium = L. { \displaystyle { \begin { aligned } E & =\sum _ { k=1 } ^ { L } { \frac { 1 } { 2^ { k } } } \cdot 2^ { thousand } =L\, .\end { aligned } } }

The following table shows the have a bun in the oven value *E* of the plot with versatile potential bankers and their bankroll *W* :

**Note:** Under game rules which specify that if the player wins more than the casino ‘s bankroll they will be paid all the casino has, the extra expected measure is less than it would be if the casino had enough funds to cover one more round, i.e. less than $ 1. For the player to win *W* he must be allowed to play round *L* +1. So the extra expected value is *W* /2 *L* +1. The premise of space resources produces a kind of apparent paradoxes in economics. In the dolphin striker bet system, a gambler betting on a discard coin doubles his bet after every loss so that an eventual winnings would cover all losses ; this system fails with any finite bankroll. The gambler ‘s downfall concept shows that a dogged gambler who raises his bet to a situate fraction of his bankroll when he wins, but does not reduce his bet when he loses, will finally and inevitably go broke—even if the game has a convinced expected value .

### rejection of mathematical arithmetic mean [edit ]

assorted authors, including Jean lupus erythematosus Rond d’Alembert and John Maynard Keynes, have rejected maximization of arithmetic mean ( evening of utility ) as a proper rule of conduct. [ 21 ] [ 22 ] Keynes, in particular, insisted that the *relative risk* [ *clarification needed* ] of an alternative could be sufficiently senior high school to reject it even if its arithmetic mean were enormous. [ 22 ] Recently, some researchers have suggested to replace the expected rate by the median as the fairly measure. [ 23 ] [ 24 ]

### ergodicity [edit ]

The paradox can be resolved by means of ergodicity economics, which besides rejects the manipulation of mathematical anticipation unless it can be justified by dynamic arguments. Ergodicity is a property which ensures that the expected prize of a fluctuate quantity is besides its long-time average. This property is believed to hold in equilibrium statistical mechanics, but it broadly does not hold for models of personal wealth. To solve the St. Petersburg paradox using the ergodicity concept, one computes the time-average growth rate of wealth which results from playing the lottery at a given price and wealth. This is different from the increase experienced by the ask rate for any dynamic which is not additive. The lottery tag price where the time-average growth rate is zero for naturalistic dynamics is a realistic measure for the maximal a player may be will to pay. An early resolution containing the essential mathematical arguments assuming multiplicative dynamics was put forward in 1870 by William Allen Whitworth. [ 25 ] An denotative link to the ergodicity problem was made by Peters in 2011. [ 26 ] These solutions are mathematically like to using the Kelly criterion or logarithmic utility. Conceptually, however, they are different because they emphasize the interpretation of expected rate as an ensemble average and identify as a more relevant criterion the growth of the actor ‘s wealth as experienced over clock. General dynamics beyond the strictly multiplicative case can correspond to non-logarithmic utility functions, as was pointed out by Carr and Cherubini in 2020. [ 27 ]

## holocene discussions [edit ]

Although this paradox is three centuries erstwhile, new arguments have still been introduced in recent years .

### chap [edit ]

A solution involving sampling was offered by William Feller. [ 28 ] Intuitively Feller ‘s answer is “ to perform this game with a big count of people and calculate the expect measure from the sample distribution extraction ”. In this method acting, when the games of infinite number of times are possible, the expected measure will be eternity, and in the case of finite, the expected value will be a much smaller value .

### Samuelson [edit ]

Paul Samuelson resolves the paradox [ 29 ] by arguing that, even if an entity had infinite resources, the game would never be offered. If the lottery represents an space expected gain to the player, then it besides represents an infinite expect personnel casualty to the server. No one could be observed paying to play the plot because it would never be offered. As Samuelson summarized the argument : “ Paul will never be willing to give ampere much as Peter will demand for such a sign ; and hence the indicate natural process will take place at the equilibrium tied of zero saturation. ”

Read more: Coin rotation paradox – Wikipedia

### Peters [edit ]

Ole Peters [ 26 ] resolved the paradox by computing the time-average performance of the lottery, arguing that the expected payout is irrelevant to an individual player because it corresponds to the average payout achieved by an countless corps de ballet of players which is not accessible to the individual .