Saint Petersburg paradox

In probability theory and decision theory, the Saint Petersburg paradox is a paradox involving a gambling game with infinite expected value. In this situation, decision theory recommends that any bet, no matter how high, be accepted, an action that no rational person would follow.

History

The original formulation of the paradox appears in a letter sent by Nicolaus Bernoulli to Pierre de Montmort, dated September 9, 1713. After this, Nicolaus spent some time trying to find the solution to the problem that he himself had posed., but finally in the year 1715 he chose to consult his brother Daniel, whom he recognized as having a mathematical capacity superior to his own. At that time Daniel Bernoulli was in Saint Petersburg, attracted along with other great scientists and thinkers of the time by the magnificent living and working conditions offered by Peter the Great to make that city the greatest focus of knowledge in all of Europe. After his first answer, Daniel spent a few years reflecting on the problem raised, publishing his analysis and his proposed solution in 1738 in the Proceedings of the Academy of Sciences of Saint Petersburg, the city from which the name of the paradox comes.

Formulation

The standard formulation of the St. Petersburg paradox is as follows: the player has to pay a bet to participate in the game. Then it makes successive releases of a coin until cross comes out for the first time. Then stop the game, count the number of releases that have occurred, and the player gets 2ⁿ coins (euros for example). If it comes out cross the first time the player wins 21=2{displaystyle 2^{1}=2} euro; if the cross leaves in the second release wins 22=4{displaystyle 2^{2}=4} euros; if it comes out in the third 23=8{displaystyle 2^{3}=8}if in the room 24=16{displaystyle 2^{4}=16}How much would the reader be willing to pay to play this game?

Analysis

In decision theory, the sum of the prizes (g1, g2, g3... gn) associated with each of the n possible outcomes of the game (r1) is called mathematical expectation (EM) or expected gain of a game, r2, r3... rn), weighted by the probability of each of these outcomes occurring (p1,p2, p3...pn): MS = p1•g1 + p2•g2 + p3•g3 + …… + pn•gn

Thus, in a game based on the throw of a dice, where you win 20 euros if the 6 comes up, 8 euros if the 5 comes up, and - 1 euro (one euro is paid in addition to the initial bet) if from 1 to 4, the expected profit is, counting with a probability of 1/6 for each of the possible results: MS = 1/6 • 20 + 1/6 • 8 + 1/6 • -1 + 1/6 • -1 + 1/6 • -1 + 1/6 • -1 = 4 euros.

The rational player should accept a game proposition if the expected payoff (the average amount of money he would get by playing that game many times) is greater than the amount required to enter the game, and reject the proposition when the expected payoff is is less than that sum. Therefore, if the amount required to participate in the previous game is less than 4 euros, the rational player must bet (the bet is favorable), if it is greater than 4 euros, he should not bet (the bet is unfavorable), and if it is exactly 4 euros may or may not bet (the bet is even).

What is the expected payoff in the St. Petersburg Paradox game?

Before starting the game there is an infinite number of possible results: that the first cross comes out on the 1st flip, on the 2nd flip, on the 3rd, on the 4th… The probability that the first "tails" appear at launch k is from:

pk=12k{displaystyle p_{k}={frac {1}{2^{k}}}}}

and the gain is 2^k,

Leading tails for the first time in the 1st has a payoff of 2¹ and a probability of 1/2; Leading tails for the first time in the 2nd has a payoff of 2² and a probability of 1/2²; Leading tails for the first time in the 3rd has a payoff of 2³ and a probability of 1/2³…, and so on indefinitely.

This means that when playing we have a probability of 1/2 of winning 2 euros, but also a probability of 1/4 of winning 4, and a probability of 1/8 of winning 8... and a probability of 1/64 of winning 64... and a probability of 1/4,096 of winning 4,096... Therefore, when calculating the expected profit of the game by adding the profits of all possible outcomes weighted by the probability that they occur (1/2 • 2 + 1/4 • 4 + 1/8 • 8 + 1/16 • 16 + 1/32 • 32 +.....+... = 1 + 1 + 1 + 1 + 1 +...+.....) results in an infinite value:

E=␡ ␡ k=1∞ ∞ pk2k=␡ ␡ k=1∞ ∞ 1=∞ ∞ .{displaystyle E=sum _{k=1}^{infty }p_{k}2^{k}=sum _{k=1}^{infty }{1}=infty. !

The paradox then arises because although following the guidelines of the decision theory we should bet any sum that is required of us, no matter how high it may seem (since the bet will always be favourable), people considered reasonable are generally not willing to do so. bet more than 10, 15 or 20 coins.

Solution proposals

Since its formulation, the St. Petersburg paradox has witnessed many attempts at a solution, some focusing more on the player's decision and others more on the game structure itself.

Among the first are the considerations on the crucial difference between monetary gain and the utility of that gain, already pointed out in Daniel Bernoulli's first analysis ("any increase in wealth, no matter how insignificant, will always result in an increase in utility that is inversely proportional to the amount of goods already owned"). The idea is that although the monetary profit can increase indefinitely, the utility of that profit does not increase in a parallel way. In addition to this basic fact, there are authors who argue that many people are not willing to make high bets because they are risk averse.

Other attempts at a solution focus on the very structure of the game, arguing on the one hand that in practice there would be moves that would never be carried out, for example that a move where the first cross does not appear until 300 is not conceivable and on the other hand, for there to be a reliable betting game, a bank with enough money to cover the maximum prize is needed, and in this world there are no banks capable of paying a trick where the cross came out for the first time in a number as small as 50 launch (prize of 2⁵⁰ > 300 billion euros).

Recently, an analysis of the paradox has been proposed (Luis Cañas, 2008) which seems to be a step forward towards solving this problem. The idea is to decompose the paradox game into a set of SP ordered games such that

SP1: when tossing a coin the player wins 2¹ coins if tails, and wins 0 if no tails; game over (mathematical expectation: 1/2 • 2 = 1).

SP2: The player wins 2¹ coins if tails the first time, and it's game over. If not, it rolls again and wins 2² coins if tails and 0 if no tails; game over (EM: 1/2 • 2 + 1/4 • 4 = 1 + 1 = 2).

SPn: the player tosses a coin as many times as it takes to get tails for the first time, up to a maximum of n tosses. When a cross appears or n throws have been made, the game ends. The number of tosses, j, it took to get tails is counted, and the player wins 2^j coins; If no tails have come up, the player wins 0 coins (EM: 1/2 • 2¹ + 1/2² • 2² + 1/2³ • 2³ +.....+ 1/2^j • 2^j +......1/2ⁿ • 2ⁿ = 1 + 1 + 1 +........= n).

All SP games have an expectation (EM), and therefore an equal bet, equal to their order number, and a maximum prize (MP) of 2^n:

SP	Hope Mathematics	Prize Maximum
1{displaystyle 1}	1{displaystyle 1}	2{displaystyle 2}
2{displaystyle 2}	2{displaystyle 2}	4{displaystyle 4}
8{displaystyle 8}	8{displaystyle 8}	256{displaystyle 256}
15{displaystyle 15}	15{displaystyle 15}	32.768{displaystyle 32.768}
40{displaystyle 40}	40{displaystyle 40}	1,099.511× × 106{displaystyle 1.099.511times 10^{6}}
n{displaystyle n}	n{displaystyle n}	2n{displaystyle 2^{n}}
Русский Русский 0{displaystyle aleph _{0}}	Русский Русский 0{displaystyle aleph _{0}}	2Русский Русский 0{displaystyle 2^{aleph _{0}}}}

Each of these games can now be tackled without problems by using the expected profit, accepting favorable or fair bets (thus, in SP8 you could bet 5, 6, 7 and up to 8 euros, with possible prizes of 0, 2, 4 … up to 256 euros and an expected profit of 8 euros). Only from a certain n variable for each player (15, 20, 30...) do the above considerations fit, the decreasing utility of money, risk aversion, and the suspicion that the bank does not have sufficient funds to face the payment of the maximum prize.

And what's the St. Petersburg paradox game now? In this game there is no limitation for the number of releases until a cross comes out, and therefore corresponds to a SP game with infinite order number. This is precisely the SP gameРусский Русский 0{displaystyle aleph _{0}} (the symbol Русский Русский 0{displaystyle aleph _{0}}, corresponds to the smallest infinity, the cardinal number of the set of natural numbers), with an equitable bet and a mathematical hope of infinite value, but with a maximum value 2^Русский Русский 0{displaystyle aleph _{0}} which corresponds to an infinite greater than Русский Русский 0{displaystyle aleph _{0}}an infinite not numberable. The solution to St.Petersburg's paradox would then be that in the paradox game a non-counterable infinity of coins is assumed as possible, and that same idea is incompatible with the very concept of money.