The St. Petersburg Game and the Paradox - Report Example

REPORT Proposal Request of a potential client to quote a spread for a single “St. Petersburg Game”. The St. Petersburg game and the paradox The game goes as follows. We toss a coin: if it comes up heads, we win £1 and the game ends; if tails comes up we toss the coin again; On the second toss of the coin, if it comes up heads we win £2 and the game ends; if tails comes up we toss the coin again; On the third toss of the coin, if it comes up heads we win £4 and the game ends; if tails comes up we toss the coin again; etc. The expected payoff of this game is in fact infinite (for each outcome the payoff is 2n with probability. However, there are infinitely many outcomes as well. But intuitively we can see that it would not be worth paying very much to take part in such a game. A Decision theory would apparently tell you that you should be prepared to pay any finite amount to play this game once. This seems absurd—and thus we have the St. Petersburg paradox. Whenever you toss tails the prize is doubled. After n tosses you get $2n if heads appear for the first time. The only catch is you have to pay to play the game. How much should you be willing to pay? Classical decision theory says that you should be willing to pay any amount up to the expected prize, the value of which is obtained by multiplying all the possible prizes by the probability that they are obtained and adding the resulting numbers. The chance of winning $2 is 1/2 (heads on the first toss); the chance of winning $4 is 1/4 (tails followed by heads); the chance of winning $8 is 1/8 (tails followed by tails followed by heads); and so on. Since the expected payoff of each possible consequence is $1 ($2 x 1/2, $4 x 1/4), etc) and there are an infinite number of them, the total expected payoff is an infinite sum of money. A simple way to describe the paradox is this. Suppose that you were offered a bet where if you flip a coin ten times and it comes up heads every time you win $1 billion. Otherwise, you lose $10,000. Would you take the bet? On average, you stand to win by taking the bet, but most people would not take it. Even if somebody raises the payoff for winning to $10 billion or $10 trillion, most people would not take the bet. Why not? The answer is that people mentally truncate the upper value of what you might win. It is hard to believe that somebody is really going to pay you $1 billion if you flip ten heads in a row. So you act as if they were offering you something lower. Challenges and Risks of Trading First and foremost we agree that the game is inconsistent and thus a paradox. Hence, it is an equally risky proportion to all parties concerned. It’s a game one wins and the other looses. Any trading for that matter has the reputation of being a highly risky endeavor and involves challenges as well. It is true that a high percentage of traders lose money as it is interesting to note that most people tend to think of the outcome as a relative change rather than the final status, they have different risk attitudes towards gains and losses, and they tend to overweight unlikely events but underweight highly possible events. But, most people are naturally risk averse. They dont like to take big risks, especially financial risks. Trading is a business of making and losing money. Any trade, no matter how well thought out, has a chance of becoming a loss. Many people think the best traders dont lose any money and have only winning trades. This is absolutely not true. The best traders lose a lot of money, but they eventually make even more over time. There is no point trading commodities if you cannot handle the psychological discomfort of making losing trades. While people tend to take losses personally as a sign of failure, good traders shrug them off. The best trading plans result in many losses. Because of the amount of randomness in market price action, such losses are inevitable. Profit enhancement is possible only through risk Suggestion: We need to trade like a technical analyst; by understanding the fundamentals behind an investment and follow ethical ways to achieve our desired objectives. When our fundamental and technical signals point to the same direction, we have a good chance to have a successful trade, especially with good money management skills. Rule of thumb: Should not ignore standard rules and trade against the trend; this will usually cause psychological worries, and frequently, losses. Before becoming too excited about the substantial returns possible from trading, it is a good idea to take a long, sober look at the risks. Reward and risk are always related. It is unrealistic to expect to be able to earn above-average investment returns without taking above-average risks as well. A rational gambler would enter a game if and only if the price of entry was less than the expected value. In the St. Petersburg game, any finite price of entry is smaller than the expected value of the game. Thus, the rational gambler would play no matter how large the entry price is! Most people would offer between £5 and £20 on the grounds that the chance of winning more than £4 is only 25% and the odds of winning a fortune are very small. And therein lies the paradox: If the expected payoff is infinite, why is no one willing to pay a huge amount to play? Questions 1 : Acceptance of the business proposal It is indeed a challenging decision to accept the proposal forwarded by this particular client. It is best to address certain issues related to the uncertainty and take an ideal decision accordingly. Ask yourself, "How much am I ready to lose?" When you are finished, scrutinize more to understand the risks and steps to be taken to avoid them? It is obvious that there are many important areas that every trader should cover both BEFORE and DURING a trade. There are all sorts of practical considerations which must be considered in making a real gambling decision. For example, in deciding whether to raise, see, fold, or cash in and go home, in a particular game, you must consider not only probability and expected value, but also the facts that its 5 A.M. and you are cross-eyed from fatigue and drink; but its not expected that classical decision theory has to deal with these. Quoting a Spread: The principle of spread is the same regardless of the types of markets involved in it. So there you go - it may seem complex, Spread for this particular game, especially being a single one, is relatively easy to pick up. Classical decision theory says that we should be willing to pay any amount up to the expected value of the wager. However, there is a warning! Accordingly we cannot invest money which we cannot afford to lose. So, there is significant risk in any deal. Any transaction involves risks including, but not limited to, the potential for changing political and/or economic conditions, that may substantially affect the organization and individuals. It is an unavoidable reality that any wrong decision will have an equally proportional effect on your funds. This may work against you, your organization, as well as your client. The possibility exists that you could sustain a total loss. Lets calculate the expected value, subsequently quote based on this theory: (here a spread can be incorpoarted) The probability of winning at step n is 2^-n, and the payoff at step n is 2^n, so the sum of the products of the probabilities and the payoffs is : E = sum over n (2^-n * 2^n) = sum over n (1) = infinity. This is called the "St. Petersburg Paradox. The classical solution to this problem was given by Bernoulli. He noted that peoples desire for money is not linear in the amount of money involved. In other words, people do not desire $2 million twice as much as they desire $1 million. Suppose, for example, that peoples desire for money is a logarithmic function of the amount of money. Then the expected VALUE of the game is : E = sum over n (2^-n * C * log(2^n)) = sum over n (2^-n * C * n) = C Here the Cs are constants that depend upon the risk aversion of the player, but at least the expected value is finite. However, it turns out that these constants are usually much higher than people are really willing to pay to play, and in fact it can be shown that any non-bounded utility function (map from amount of money to value of money) is prey to a generalization of the St. Petersburg paradox. So the classical solution of Bernoulli is only part of the story. The rest of the story lies in the observation that bankrolls are always finite, and this dramatically reduces the amount you are willing to bet in the St. Petersburg game. To figure out what would be a fair value to charge for playing the game we must know the banks resources. Assume that the bank has 1 million dollars (1*K*K = 2^20). I cannot possibly win more than $1 million whether I toss 20 tails in a row or 2000. Therefore my expected amount of winning is E = sum n up to 20 (2^-n * 2^n) = sum n up to 20 (1) = $20 And my expected value of winning is E= sum n up to 20 (2^-n * C * log(2^n)) = some small number Note that the expected value of the players profit is 0.2e. Now lets vary the banks resources and observe how e and p change. It will be seen that as e (and hence the expected value of the profit) increases, p diminishes. The more the game is to the players advantage in terms of expected value of profit, the less likely it is that the player will come away with any profit at all. This is mildly counter-intuitive The classical solution to this mystery, provided by Daniel Bernoulli and another Swiss mathematician, Gabriel Cremer, goes beyond probability theory to touch areas of psychology and economics. Accordingly, they pointed out that a given amount of money isnt always of the same use to its owner. For example, to a millionaire $1 is nothing, whereas to a beggar it can mean not going hungry. In a similar way, the utility of $2 million is not twice the utility of $1 million. Thus, the important quantity in the St. Petersburg game is the expected utility of the game (the utility of the prize multiplied by its probability) which is far less than the expected prize. This explanation forms the theoretical basis of the insurance business. A decision theory using only this expected value would therefore suggest that any fee, no matter how high, would be worth paying for this opportunity. Availability of data related to liquidity: There are various methods for measuring counterparty credit risk. In this case since that data pertaining to client’s liquidity is available it is always safe to accept the challenge and go ahead with it. We need to measure the probability distribution of loss due to counterparty default and rest, in part, on measurements of the potential future credit exposure to the counterparty, the future default probability of the counterparty and the uncontrollable - no single event, individual, or factor rules it. Please note that, just like any other speculative business, increased risk entails chances for a higher profit/loss. Instead of calculating just prices, Bernoulli would have modeled a decision-making process in which a psychological variable (the gambler’s utility) would account for the hedging of the value of the stake, in such a way as to render the price of the game reasonable. You will come out ahead in the long run, the idea being that on the very rare occasions when a large payoff comes along, it will far more than repay however much money you have paid to play. Ask price based on Value at Risk (VaR) Value at Risk is an estimate, with a predefined confidence interval, of how much one can lose from holding a position over a set horizon. Potential horizons may be one day for typical trading activities or a month or longer for portfolio management. The methods described in our documentation use historical returns to forecast volatilities and correlations that are then used to estimate the market risk. While measures like VAR play an important role in quantifying risk exposure, they comprise only one piece of the risk-management puzzle: probabilities. (Here ask price can be incorporated) In this particular case it is easy to go for an ask price based on “The Bank of International Settlements (BIS) requirements and applicable norms. i.e. 99% of 10-day VaR below 1/3 of the capital. Value of the game becoming a finite number The payoff of any conceivable game is always finite. 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. It is easy to apply the widely-accepted principle of Bernoulli that money has a decreasing marginal utility, and suggested that a realistic measure of the utility of money might be given by the logarithm of the money amount. Here are the first few lines in the table for this gamble if utiles = log($): n P(n) Prize Utiles Expected Utility 1 1/2 $2 0.301 0.1505 2 1/4 $4 0.602 0.1505 3 1/8 $8 0.903 0.1129 4 1/16 $16 1.204 0.0753 5 1/32 $32 1.505 0.0470 6  1/64 $64 1.806 0.0282 7 1/128 $128 2.107 0.0165 8 1/256 $256 2.408 0.0094 9 1/512 $512 2.709 0.0053 10 1/1024 $1024 3.010  0.0029 The sum of expected utilities is not infinite: it reaches a limit of about 0.60206 utiles (worth $4.00). The rational gambler, then, would pay any sum less than $4.00 to play. Marginal utility of a good or service is its utility in its least urgent use of the most-desired available uses, in other words, the use that is just in the margin. The same object may have Law of diminishing marginal utility. The determination of the value of an item must not be based on the price, but rather on the utility it yields…. There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount. One might imagine that some people have an upper limit on the utility they can enjoy - people who have a finite number of desires, and whose desires can each be completely satisfied by some finite state. For these people, the utility of prizes does not increase without limit, and the St. Petersburg game has some finite expected utility. Do such people exist? This is an empirical question. In any case, there surely are some people with some ‘the-more-the-better’ desires, and the theory of rational choice ought not to be restricted by the empirical and doubtful propositions that there arent any, and that value cannot increase without limit. And these propositions are surely insufficiently well-founded to serve as solutions to the St. Petersburg paradox. Many authors have pointed out that, practically speaking, there must be some point at which a run of heads would be truncated without a final tail. Any of these limits produces a finite expected value for the game, but sets an L which is higher than 25; what, then, explains Hackings $25 intuition? Another fact that would set a limit on L is the finitude of the bankroll necessary to fund the game. Any casino that offers the game must be prepared to truncate any run that, were it to continue, would cost them more than the total funds they have available for prizes. A run of 25 would require a prize of a mere $33,554,432, possibly within the reach of a larger casino. A run of 40 would require a prize of about 1.1 trillion dollars. Other facts make an upper limit L plausible, such as the limit on the amount of money available in the world. Perhaps all these financial limits can be overridden if we conceive of the games being offered by a state capable of printing all the money it wanted to. This state could pay any prize whatever; but printing up and handing out a huge amount of cash would create havoc with any economy, so no rational state would. For one thing, a factor for risk-aversion is not a generally applicable consideration in making rational decisions, because some people are not risk averse. In fact, some people may enjoy risk. This sort of risk-aversion, when generally applied, would paralyze anyone. It is central to rationality that one take account of the actual risks, and run suitably small ones. We easily ignore practical considerations when calculating the expected value (in this case, merely potential withdrawals minus purchase price), St. Petersburg paradox can be resolved by allowing only i.e., under the assumption of a finite expectation value, a (not necessarily strictly) concave utility function is sufficient to guarantee that the expected utility is finite. Any theoretical model is an idealization, leaving aside certain practicalities. "From the mathematical and logical point of view," observes Resnick, "the St. Petersburg paradox is impeccable." But this is the point of view to be taken when evaluating a theory per se (though not the only point of view ever to be taken). Utility function as solution to the Paradox The St. Petersburg game is sometimes dismissed because it has infinite expected value, which is thought not merely practically impossible, but theoretically objectionable - beyond the reach even of thought-experiment. Most of the realistic considerations show that St. Petersburg game - exactly as originally described - can never be encountered in real life? It is interesting to note an observation which says: Put briefly and crudely, our rebuttal of the St. 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. The concept of utility function also doesn’t come up with a concrete solution. It is all about an idea used by Bernoulli, which is incomplete and does not on any ground serve as as a solution to the paradox. Other observations and summarization Any theoretical model is an idealization, leaving aside certain practicalities. "From the mathematical and logical point of view," observes Resnick, "the St. Petersburg paradox is impeccable." But this is the point of view to be taken when evaluating a theory per se (though not the only point of view ever to be taken). By analogy, the aesthetic evaluation of a movie does not take into account the facts that the only local showing of the movie is far away, and that finding a baby sitter will be impossible at this late hour. If aesthetic theory tells you that the movie is wonderful, but other considerations show you that you shouldnt go, this isnt a defect in aesthetic theory. Similarly, the mathematical/logical theory for explaining ordinary casino games is not defective because it ignores practicalities such as a particular limit on a casinos bankroll, or on participants patience. Probability and Desirability : In a paradox, we should make probability and desirability work in tandem to ensure that all desirability come out finite—that’s surely the right way to go, and I think that it’s really what the idea of Dick * They could work to ensure, for example, that necessarily inflation puts a rein on the desirability of ever-increasing monetary amounts (risk aversion), and that the corresponding probabilities decay sufficiently quickly to offset the corresponding growth in desirability. Similar remarks would presumably apply to things that money cannot buy. Now you might worry that this is not so ecumenical. It is always possible, especially irrational thinkers, offer a game even though he is aware that theres 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 theres a small possibility my car will break down between now and then, and thus that Im making an offer I might not be able to fulfill. But the conclusion is not that Im not really offering what I appear to be. If someone invites you to play St. Petersburg, we cant conclude that hes in fact not offering the St. Petersburg game, that hes really offering some other game. Theory and Practicality There are several 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 to game, ad hoc; it ought to be general in scope. And if it is, it will also rule out perfectly acceptable calculations. Michael Resnik* (1987) notes that utility theory "is easily extended to cover infinite lotteries, and it must be in order to handle more advanced problems in statistical decision theory" but he gives no examples. If you see standard theory as normative, you can ignore objections of the first type. People are not always rational, and some people are rarely rational, and an adequate descriptive theory must take into account the various irrational ways people really do make decisions. Its no surprise that the classical rather a-prioristic theory fails to be descriptively adequate, and to criticize it on these grounds rather misses its normative point. The objections from standpoint need to be taken more seriously; and we have been treating the responses to St. Petersburg as cases of this sort. Various sorts of "realistic" considerations have been adduced to show that the result the theory draws in the St. Petersburg game about what a rational agent should do is incorrect. Its concluded that the unrestricted theory must be wrong, and (sometimes) that it must be restricted to exclude consideration of the game as invented. Well now consider the general plausibility of restricting the theory in these ways. When considering the plausibility of restricting expected value calculations in various ways that would rule out the St. Petersburg calculation, Amos Nathan (1984) remarks, "it ought, however, to be remembered that important and less frivolous application of such games have nothing to do with gambling and lie in the physical world where practical limitations may assume quite a different dimension." The St. Petersburg game commits participants to doing what we know they will not. The casino may have to pay out more than it has. The player may have to flip a coin longer than physically possible. But this may not show a defect with choice theory. Classical unrestricted theory is still serving its purpose, which is modeling the abstract ideal rational agent. It tells us that no amount is too great to pay as a ideally rationally acceptable entrance fee, and this may be right. From this point of view, the St. Petersburg ‘paradox’ does not, after all, point out any defect with classical decision theory, and is not a paradox after all. There are all sorts of practical considerations which must be considered in making a real gambling decision. For example, in deciding whether to raise, see, fold, or cash in and go home, in a particular poker game, you must consider not only probability and expected value, but also the facts that its 5 A.M. and you are cross-eyed from fatigue and drink; but its not expected that classical decision theory has to deal with these. Conclusion Probability, Price and Preferences form the basis of a systematic approach to rational decision-making in an uncertain world and these three P’s are central to Total Risk Management: prices, in considering how much one must pay for hedging various risks; probabilities, for assessing the likelihood of those risks; and preferences, for deciding how much risk to bear and how much to hedge. Reference Links 1. http://plato.stanford.edu/archives/fall2004/entries/paradox-stpetersburg/ 2. http://brainyplanet.com/index.php/St.%20P%20Solution?PHPSESSID=637426c5a7a97485a79acd0b89fcb568 3. http://everything2.com/index.pl?node_id=569520 4. http://thesaurus.maths.org/mmkb/entry.html;jsessionid=C4BA7EBFCE34DF89102DE86CB437BCCF?action=entryById&id=2512 Bibliography 1. Dick : The Logic of Decision Read More