What is the significance of the st petersburg paradox

The process produces CBD without producing any of the unwanted side effects that are commonly produced by the typical chemical methods.

This is especially important because the government is not requiring commercialization of this extract under the Controlled Substances Act, which regulates the manufacture and distribution of prescription drugs. Currently there are no clinical trials that have been approved by the FDA to evaluate the safety and effectiveness of CBD. The research being done by the University of Mississippi will not be limited to the medical marijuana industry however. The study is currently ongoing, so the possibilities for CBD products are endless.

These products can include oils, capsules, pills, and even lotions. That's because it has been shown to help reduce anxiety, increase memory recall and enhance focus and is popular with teenagers. Since CBD is derived from hemp plants, it does not pose the same health risks as illegal substances such as cannabis and does not produce the same negative side effects like smoking marijuana does. If and when they decide to do so, it will be in a more concentrated form that will not cause the same level of euphoria or mood altering effects that are often associated with prescription drugs.

There are other uses for CBD that we do not currently have access to. One of the most promising uses of CBD is to help treat and possibly prevent the occurrence of seizures in those suffering from epilepsy. Since hemp oil and CBD are in its most natural form, it is safe for use in food, cosmetics and dietary supplements.

This can be seen from a consideration of the finite variant of the St. Petersburg lottery:. If the total resources of the casino are W dollars, then the maximum payoff and therefore the maximum number of rounds is "capped", and the expected value of the lottery becomes. L is the maximum number of times the casino can play before it can no longer cover the next bet.

The floor function gives the greatest integer less than or equal to its argument. The logarithm function becomes infinite as its argument becomes infinite, but does so very, very slowly. This logarithmic growth is the inverse behavior of exponential growth. A typical graph of average winnings over one course of a St. Petersburg Paradox lottery shows how occasional large payoffs lead to an overall very slow rise in average winnings.

After 20, gameplays in this simulation the average winning per lottery was just under 8 dollars. The graph encapsulates the paradox of the lottery: The overall upward slope in the average winnings graph shows that average winnings tend upward to infinity, but the slowness of the rise in average winnings a rise that becomes yet slower as gameplay progresses indicates that a tremendously huge number of lottery plays will be required to reach average winnings of even modest size.

The following table shows the expected value of the game with various potential backers and their bankroll:. An average person might not find the lottery worth even the modest amounts in the above table, arguably showing that the naive decision model of the expected return causes the same problems as for the infinite lottery, however the possible discrepancy between theory and reality is far less dramatic.

The assumption of infinite resources can produce other apparent paradoxes in economics. See martingale roulette system and gambler's ruin. Players may assign a higher value to the game when the lottery is repeatedly played. This can be seen by simulating a typical series of lotteries and accumulating the returns, compare the illustration right. Petersburg paradox and the theory of marginal utility have been highly disputed in the past. For an interesting but not always sound contribution from the point of view of a philosopher, see Martin, February The use of unbounded utility functions in expected-utility maximization: Response.

Quarterly Journal of Economics 88 1 : — Handle: RePEc:tpr:qjecon:vyip Lousie Sommer. January Exposition of a New Theory on the Measurement of Risk. Econometrica 22 1 : 22— Petersburg Paradox ".

Edward N. Stanford, California : Stanford University. ISSN Retrieved on Cumulative prospect theory and the St. It's of course true that any real game would impose some upper limit on L, and thus a finite number of possible consequences of the game; but this does not solve the St. Petersburg puzzle because it does not show that the expected value of the game as described is not infinite. After all, any game with a limit L is not the game we have been talking about. Our question was about the St.

Petersburg game, not about its L-limited relative. Do these realistic considerations show that the genuine St. Petersburg game—exactly as originally described—can never be encountered in real life? Petersburg paradox consists in the remark that anyone who offers to let the agent play the St. Petersburg game is a liar, for he is pretending to have an indefinitely large bank. It can be quibbled that Jeffrey is not exactly right: that someone can offer a game even though he is aware that there's a possibility that this offer involves the possibility of requiring consequences he cannot fulfill.

Compare my offer to drive you to the airport tomorrow. I realize that there's a small possibility my car will break down between now and then, and thus that I'm making an offer I might not be able to fulfill. But the conclusion is not that I'm not really offering what I appear to be.

If someone invites you to play St. Petersburg, we can't conclude that he's in fact not offering the St. Petersburg game, that he's really offering some other game. Real casinos right now play games that offer the extremely remote possibility of continuing too long for anyone to complete, or of prizes too large to be managed. Casinos can go ahead and play these games anyway, confident that the risk of running into an impossible situation is very very small. They need not lose any sleep worrying about incurring a debt they can't manage.

They live, and prosper, on probabilities, not certainties. If these considerations are persuasive, then what Jeffrey gives is not a rebuttal of the paradox. In effect, he accepts the fact that the game offers the possibility of indefinitely large payoffs.

The reason the game is not offered by casinos is that they realize that sooner or later probably much later the game will bankrupt them. This is correct reasoning—but it is done using the ordinary, general theory of choice. When casinos reason about the game, they do not decide that, since ordinary theory shows that the game has infinite value, ordinary theory should be restricted to exclude its consideration. There are other reasons why we should not restrict theory to exclude consideration of the game.

This ruling, in order to be theoretically acceptable, ought not merely rule out the St. Petersburg game in particular, ad hoc; it ought to be general in scope. And if it is, it will also rule out perfectly acceptable calculations. What price should an insurance company charge for this policy? For simplicity, we shall ignore possible effects of inflation, and profits from investing the entry price. Standard empirically-based mortality charts give the chances of living another year at various ages.

Of course, they don't give the chances of surviving another year at age , because there's no empirical evidence available for this; but a reasonable function to extend the mortality curve indefinitely beyond what's provided by available empirical evidence can be produced; this curve asymptotically approaches zero. On this basis, ordinary mathematical techniques can give the expected value of the policy. But note that it promises to pay off without limit. If we think that, for each age, there is a large or small probability of living another year, then there are an indefinitely large number of consequences to be considered when doing this calculation, but mathematics can calculate the limit of this infinite series; and ignoring other factors an insurance company will make a profit, in the long run, buy charging anything above this amount.

There's no problem in calculating its expected value. This insurance policy call it Policy 1 offers an indefinite number of outcomes; but consider a different one call it Policy 2 which would truncate the series at age , and offer only outcomes. The probability of reaching age is so tiny that the difference in expected value between the two policies is negligible, a tiny fraction of 1 cent. If you don't like infinite lotteries, you might claim that Policy 1 is ill-formed, and suggest substitution of Policy 2, pointing out that the expected value of this one is, for all practical purposes, equal to that of Policy 1.

But note that your judgment that the two are virtually identical in expected value depends on your having calculated the expected value of Policy 1. So your statement presupposes that the expected value of Policy 1 is calculable, after all.

Imagine you were offered the following deal. For a price to be negotiated, you will be given permanent possession of a cash machine with the following unusual property: every time you punch in a dollar amount, that amount is extruded.

This is not a withdrawal from your account; neither will you later be billed for it. You can do this as often as you care to. Now, how much would you offer to pay for this machine? Do you find it impossible to perform this thought-experiment, or to come up with an answer? Perhaps you don't, and your answer is: any price at all. Provided that you can defer payment of the initial price for a suitable time after receiving the machine, you can collect whatever you need to pay for it from the machine itself.

Of course, there are practical considerations: how long would it take you to collect its enormous purchase price from the machine?

Would you or the machine be worn out or dead before you are finished? Any bank would be crazy to offer to sell you an infinite cash machine and unfortunately I seem to have lost the address of the crazy bank which has made this offer.

But so what? The point is that there appears to be nothing wrong with this thought experiment: it imagines an action buying the machine with no upper limit on expected value. We easily ignore practical considerations when calculating the expected value in this case, merely potential withdrawals minus purchase price , which is infinite. But the only difference between this machine and a single-play St. Petersburg game is that this machine guarantees an indefinitely large number of payouts, while the game offers a one-time lottery from among an indefinitely large number of possible payouts, each with a certain probability.

The expected value of both the St. Petersburg game and the infinite cash machine are both indefinitely large. You should offer any price at all for either.

These arguments appear to show that the notion of infinite expected value is perfectly reasonable. It's quite true that when infinities show up in certain considerations, nonsense results. Consider this example: I write down an integer, at random, and seal it in an envelope. You open the envelope and observe I've written down 8,, Given the infinity of integers I've had to choose from, the probability of my writing down this one is zero.

It's a miracle! Or maybe: a contradiction! The problem here is, of course, the incoherence of the idea of choice among literally an infinite number of integers. Doubts about the metaphysical reality of infinity, and about the proper rational employment of that concept have been raised throughout the history of philosophy and mathematics.

So it's tempting to attribute the paradox raised by St. Petersburg to merely another illegitimate use of infinity. But notice that we don't need to invoke infinity in describing the gamble or its consequences. The payoff of any conceivable game is always finite. So is the length of any conceivable game.

The paradoxical result can be put this way: no matter what finite entry price X is charged, it can be shown that the expected payoff of the game is larger than that, due to the very small possibility of the number of flips growing larger than X. Note similarly that the wonderful cash machine mentioned above is not contingent on the reality of any infinity: every payoff it can make is finite; and at any point, it has been used only a finite number of times.

If you see standard theory as normative, you can ignore objections of the first type.