Mathematicians estimate money in proportion to its quantity,and men of good sense in proportion to the usage that they may make of it. GabrielCramer (1704-1752)
(2001-07-04) How do utilities differ from expectations? How are utilities used?
Autility is a numerical rating assigned to every possible outcomea decision maker may be faced with. (In a choice between several alternative prospects,the one with the highestutility is always preferred.) To qualify as a trueutility scale however, the rating must be such thattheutility of any uncertain prospect is equal to the expected value(the mathematicalexpectation) of theutilities of all its possible outcomes(which could be either "final" outcomes or uncertain prospects themselves).
When decisions are made by a so-calledrational agent(if A is preferred to B and B to C, then A must be preferred to C), it should beclear thatsome numerical scale can be devised to rate any possible outcome"simply" by comparing and ranking these. Determining equivalence in money terms may be helpful in such a systematic processbut it's not theoretically indispensable. What may be less clear, however, is how to devise such a rating system so thatit would possess the above fundamental property required of autility scale.
One theoretical way to do so is to compare prospects and/or final outcomesto tickets entitling the holder to a chance at winning somejackpot,which is at least as valuable as any outcome under consideration. A ticket with a face value of 75% means a chance of winning thejackpotwith a probability of 0.75 and it will be assigned autility of 0.75. Anything which is estimated to be just as valuable as such a ticket (no more, no less)will be assigned a utility of 0.75 as well.
The scale so defined does have the property required of utility scales.Consider, for example, a prospect which may have one of two outcomes:
The first outcome has a probability of 0.3 and a utility of 0.6 (it could be a ticket with a 60% face value).
The second outcome has a probability of 0.7 and a utility of 0.2 (it could be a ticket with a 20% face value).
When these two outcomes actually consist of lottery tickets, the whole thingis completely equivalent(think long and hard about this) to having a chanceto win thejackpot with probability. The prospect has therefore, by definition, a utility of 0.32,and we do observe that the result has been computed with the same rule as amathematical expectation. It would be so in any other case involving either lottery tickets orthings/situations previously assigned autility(by direct or indirect comparisons with such tickets).
The type ofutilities introduced above are between 0 and 1,but no such restriction is in fact required. The key observation is that we may either translate or rescale autility scalewithout affecting at all thedecisions it implies: Each side of every comparison is translated or rescaled the same way and it doesnot affect inequalitiesas long as the scaling factor is positive.
In particular, we may keep the same utility scale if we're faced with an outcomemore valuable than whateverjackpot we first considered. If thatjackpot is estimated to be just as desirable as a chance ofwinning the bigger prize with probability p, we may assign a utility 1/p to thebigger prize (andthat, of course, is larger than 1). Similarly, the original "ticket" scale may have to be extended to assignnegative utilities to certain undesirable situations. Considering such a situation "in context",as an outcome of a prospect whose other outcomes are quite positive, allowsthe semi-direct use of the "ticket" scale to evaluate itsnegative utility.
Even when there is no such thing as a "top prize",theutilities of all prospects must be bounded. (Recall the difference between amaximum, which is achieved in at least onecase, and anupper limit, which may not be. Utilities have an upper limit, notnecessarily a maximum.)This may be visualized by considering that theutility function of money,which is normally nondecreasing, may either have an asymptote or be constant abovea certain point. For aproof that utilities must be bounded,see our discussion of the St. Petersburg's Paradox...
In real life, utilities are not linearly related to money values(or else the lotteries would go out of business), which is another way to say thatthemathematical expectation of a monetary gamble need not be the properutility measure to use. The monetaryexpectation is only a special example of a utility, which is mathematically acceptable but not at all realistic. It is, unfortunately, given in elementary texts (which do not introduce the utility concept) as the sole basis for a rational analysis of gambling decisions. This is clearly not so in practice:
For example, you may be willing to pay one dollar foran (unfair) chance in 2000 000 at $1000 000,but very few people (if any) would pay$499 999 for a chance in two at $1000 000. However, someone could take the latter bet in a very special situationwhere an immediate gain of $1000 000 would make a critical difference,whereas the loss of even half a million might not be crucial...
The rational basis for such choices is based on the utilities involved. Before you analyze choices, you have to determine the relevant "utility curve" carefullywhen it comes to actual possible outcomes: If your current wealth is W, what would be theexact utility rating to you of a total wealth equal to W, W-1, W-499999, or W+1000000? How does that compare to nonmonetary things like the loss of a limb? Above or below the knee? What's a relationship or a marriage worth? What about social status? Recognition? Public ridicule? Will you go out naked for $10 000, for $10,or would someone have to pay you not to expose yourself? Everything that carries any weight at all in your choices has to be assigned some utility on your own personal scale, which you may only build by introspection or,better, retrospection (recalling relevant past choices). In some cases, comparisons with the ubiquitous money scale may help. Although the so-called utility function (u) which gives utility as a function of money (total wealth) is normally not a linear function,it may have a simplemathematical form under certain common assumptions (see below).
One caveat is that nonmonetary gratifications often play a role in actual choices whichseem based solely on monetary exchanges: There's some playful element in any lottery, which increases the appeal ofpurchasing a lottery ticket. Lottery operators know this very well and they designtheir lottery "games" with this in mind. Note that it's always the entire situation which isassigned a utility rating, not its separate components (money, health, happiness, etc.).
Now, if you assume that your attitude towards money does notdepend on how much of it you have right now, then the monetary part of your own utility function u must be (up to irrelevant rescaling) an exponential function of your wealth. (It could also be linear, but this is usually disallowed on the ground that a properutility function must be bounded when the stakes are potentially unbounded.) Mathematically, this assumption states that yourutility function u is such that the quantity h is irrelevant in your preference between
something of utility p u(a+h) + (1-p) u(b+h) compared to
something of utility q u(c+h) + (1-q) u(d+h).
For now, we'll leave it up to the reader to show that this is true if [easy] andonly if [tougher] the function u is either linear [ruled out] or of thefollowing form, up to some irrelevant rescaling:
u(x) = 1 - exp( -x/r )
Although x should normally be equal to one's entire wealth, changing the"zero point" merely rescales linearly an exponential function and is thereforeirrelevant to decisions (as explained above). It's therefore customary, when using the exponential utility function,to consider that x is the amount to be gained or lost in a given gamble. Separate gambles can be analyzed separately with an exponential utilityfunction (that's not true for any other utility functions). In the above expression for an exponential utility function of money,the constant amount r (measured in the same money unit used for the variable x) is called therisk tolerance (or risk aversion). For more general utility functions that risk tolerance isn't a constant and may be defined at each point x of the utility curve as follows:
r (x) = -u' (x)/ u'' (x)
Notice that this definition is indeed independent of the allowed linear rescalingof the utility function. Portfolio managers will tell you that an investor's risk tolerance is roughly proportional to his assets (at least that's what most of them assume to be true). This may be interpreted in either one of two ways:
EITHER: When prospects are analyzed, the risk tolerance used in the analysis of future uncertainty is the constant correspondingto the current situation. At the next step, when certain events have actually come to pass,a different constant will be used to make a slightly different analysis,using the new risk tolerance corresponding to the new situation.
OR: The utility function used to make strategical decisions incorporates the future variability of the investor'srisk tolerance.
For example, if the risk tolerance is indeed proportional to wealth (r=kx),then the utility function is a solution of the differential equation:
k x u'' + u' = 0
Solve this by letting y be u'(x), so thatk x dy + y dx = 0 (ork dy/y + dx/x = 0), which meansthat y is proportional tox-1/k.Therefore, up to some irrelevant rescaling,the utility u is also a power of x, namely-x1-1/k.For this function to have an upper limit, the exponent should be negative.This is to say that we must have k<1.
The rule of thumb [that's all it is]in the corporate world seems to be that the management of most companiesbehaves as if k=1/6 (risk tolerance = one sixth of equity). With the second of the above interpretations, this would mean theutility function of a major corporation (unless it's close to bankruptcy)would typically be -1/x5. Rather surprisingly, interviews of experienced corporate decision makersseem to be consistent with this...
(2001-07-04) A fair coin is tossed until heads appears. If the game lasts for n+1 tosses, the player receives2 dollars. Namely: $1 if heads appears first,$2 if it takes two tosses, then $4, $8, $16, $32, etc. What's a decent price to pay for the privilege to play this game?
This is called the "Saint-Petersburg Paradox": Themathematical expectation of the above Saint-Petersburg Game is infinite, since it would be the sum of the following divergent series:
Clearly however, nobody would ever pay more than a few dollars for ashot at this type of gamble... Why?
When the question was first posed,early in the 18th century, it was still believed that the value of a gamble should onlybe based on its "fair" price, which is another name for itsmathematical expectation. The fact that it clearly cannot be so with the abovegame ultimately led to the introduction of the modern concept of theutility of a prospect.
The discussion originated with acorrespondence between the Swiss mathematician, residing in Basel,Nicolas Bernoulli (1687-1759, not to be confused with his well-known father,also called Nicolas, 1662-1705), and Pierre Rémond de Montmort (1678-1719), in Paris. Montmort had authored a successful book entitledEssay d'analyse sur les jeux de hazard (Paris, 1708). Bernoulli was makingsuggestions for a future edition, focusing on a set of 5 problems to appear on page 402,including "Problem 5", which essentially describes aversion of the above Petersburg Game...
The very first letter from Bernoulli (dated September 9, 1713) mentions a dieinstead of a fair coin, but the lower probability (1/6) of terminating the game at each tossmakes the expectation series diverge even more rapidly. (Bernoulli introduces other payoff sequences which are not necessarily paradoxical,so that Montmort initially missed his point.) A few years later,Gabriel Cramer (1704-1752) was promptedto address the issue from London, in aletter to Bernoulli, dated May 21, 1728. (Since he turned 20, in 1724, Cramer had been sharing a chair of mathematics in Genevawith Giovanni Ludovico Calandrini , under an arrangement that called for one of them totravel while the other was teaching.) Cramer restated the game in its modern form, for the sake of simplicity,with a fair coin instead of a die. He went on to say that "mathematicians estimate money in proportion to its quantity,and men of good sense in proportion to the usage that they may make of it". Cramer quantified that statement in terms of what's now called a"utility function", which he dubbed a "moral value of goods".
Cramer's first example of autility functionwas simply proportional to the money amount up to a certain point(he used 2coins, for convenience)and constant thereafter. His second example was autility function of money proportional to the square rootof the amount of money. Either of these utility functions does assign a finite utilityto the original Petersburg game, but the second onewould fail to resolve the issue if the payoff sequence was increasing faster(for example, if the player was payed4 dollars for completing n+1 tosses).In fact,this very example may be used to show thatanyutility functionmust have an upper bound, or else one could exhibit an infinitesequence of prospects, the n-th of which having a utility at least equal to2.Offering the n-th such prospect as payoff for successfullycompleting n tosses in aPetersburg game would assign infinite "utility"to such a game, which is not acceptable. (The basic utility tenet is to assign afinite utility ratingto a single prospect, which is what the whole Petersburg game is.)
This revived the issue originally raised by Nicolas Bernoulli,who asked the opinion of his brilliant cousin,Daniel Bernoulli (1700-1782). At that time, Daniel was professor of mathematics in St. Petersburg,and his influential work on the subject would later be published (in 1738) by theSt. Petersburg Academy, which is how the paradox got its modern name.
Back in 1731, Daniel Bernoulli rediscovered (independently of Cramer)the modern notion ofutilities, which Nicolas Bernoulli kept rejecting... Daniel also made a point which Cramer had missed entirely, namely that it isgenerally crucial to consider only the entire wealth of the player and assignautility only to the whole thing, as the marginal utility ofan additional coin will depend on the rest of one's fortune. Bernoulli ventured the guess that the additional utility (du) of an additional dollar (dw)could be inversely proportional to one's entire wealth w.This assumption (du = k dw/w) makesutility (u) a logarithmic function of thetotalwealth (w). As we are free to rescale utilities, it may then be stated without loss of generalitythat this translates intou(w) = ln(w).However, this logarithmic "utility" function suffersfrom the same flaw as Cramer'ssquare root function of money, because it'snot bounded either: If a successful sequence of n+1 tosses was payedexp(k 2n), the game would still end up having aninfinite "utility", even for a small value of the parameter k. With a small value like k=0.01, there's an unattractive sequence of payoffs atfirst, then the growth becomes explosive:$1.01, $1.02, $1.04, $1.08, $1.17, $1.38, $1.90, $3.60, $12.94, $167.34, $28001.13,$784063053.14, ... This sequence of payoffs is clearly worth a substantial premium,but consider the related schedule where you get payed $1.00 forany successful sequence of less than 100 tosses and exp(k 2n-100) dollarsthereafter.That gamble is worth $1.00 to absolutelyanybody,in spite of the fact that its logarithmic "utility" is infinite...
There is no way around it. Utilities are always bounded. If we're presented with a theoretical problem where payoffs are unbounded,as they are in the Petersburg Gamble, then the utility function itself must have anupper limit (in practical situations, potential payoffs are always bounded,which makes the exact mathematical form of the utility function irrelevantbeyond a certain point and the issue does not arise because of such practical limits). If a tool, like Bernoulli's logarithmic utilities, fails to make sense of thePetersburg Gamble for some particular payoff schedule,then it clearly cannot be trusted to analyze any other schedule. It turns out that only very few utility functions allow a self-consistentanalysis fully compatible with the nature of the question we are asked. In fact, we only have the freedom to choose a single scalar parameter(the player's so-calledrisk tolerance)! Read on:
There's an hidden assumption in this and other similar theoretical puzzles,which wemust make explicit in order to solve the riddle:The question is askedout of context and must be answered likewise ifit is to be answered at all. We are not to involve sordid details about therest of the player's life (size of bank accounts, mortgages, etc.). That approach is logically consistent only with the assumption of anexponentialutility function of money, which is the only type of utility functionwhere decisions about a particular prospect are not influenced by therest of one's situation... It does not make sense to analyze anisolated gamble exceptby assuming an exponential utility function, since no otherutility function of money even permits such isolation. This is atheoretical argument, of course, but it's clearly appropriatefor atheoretical question like the one at hand...
Since we must, we shall happily assume that the player'sutility function of money isof the form1-exp(-x/r) for some parameterr(which is a dollar amount, usually calledrisk tolerance). In this, x should generally be the player's total wealth, but theunique propertiesof the exponential function allow us to consider that x is simply theamount gained or lost in the gamble(s) at hand(since changing thezero point on the money scale merely rescalesexponentialutilities without affecting the comparisons relevant for decisions). We do not have such freedom witha more general utility function, as Daniel Bernoulli first recognized. Also, since additive and/or (positive) multiplicative constants in the utility functiondo not affect decisions, we may as well useu(x) = -exp(-x/r)as the utility of gaining (or losing) x dollars in the gamble at hand.(The only aesthetic thing lost in the rescaling is that we no longerhave a utility of 0 for a gain of 0.)
After this long preamble, the rest is easy.Let's call u(x) the utility of having x more dollars than initially. If you pay y dollars for the privilege to play, theutility of playing thePetersburg game is clearly u(2n-y) / 2n+1and the gamble should be accepted if and only if this is greater thanu(0).
u(2n) / 2n+1andu(y), namely the utility of thefree gambleand the utility of a so-calledcertainty equivalent (CE). The CE is whatever (minimum) amount of money we would be willing toreceive as acompensation for giving up the right to gamble. It may not be quite the same as the (maximum) price we're willing to pay to acquirethat right! Only in the case of the exponential (or linear) utility function are these two amounts alwaysequal. The CE is the quantity actually computed inCramer's original text based on asquare root utility function. It was probably silently assumed at the time that the CE would not be too different fromthe price one would be willing to pay. However, rigorously speaking, the minimum acceptable selling price (the CE) and the maximumacceptable buying price are only equal in the case of the exponential(or linear) utility function!
All told, if a player has an exponential utility function with arisk tolerance equal tor (expressed in dollars),the highest price (y) s/he will be willing to pay for a shot at the Petersburg gameis given by the relation:
exp(-y/r) =
exp(-2n/r) / 2n+1
Once we evaluate the sum on the RHS, this is easy to solve for y(just take the natural logarithms of both sides and multiply by -r).The computation is best done numerically (see table below) for midrange values ofr,but we may also want to investigate what happens whenris very large or very small:
For educational purposes, we've included what a similar analysis would entailfor a nonexponential utility function (last two columns of the above table). The utility function chosen is such that wealth (or equity) is 6 times therisk tolerance appearing in the first column. The entire fortune of the player is thus taken intoaccount (something we avoided with the exponential function). Note that the price for which the player is willing to sell a right to play(the CE, orcertainty equivalent) is different from the price he would bewilling to pay to acquire such a right, although this is only significant at lowlevels ofrisk tolerance (both prices are always equal for an exponential utility).At a zero risk tolerance, it's the buying pricewhich is equal to $1 (since we're guaranteed to get $1 back, no matter what), whereasthe selling price may be significantly greater(the value is ½ 631/5or about $1.145086 in this particular case). That's because a nonexponential utility function integrates future variations of therisk tolerance and this influences the decision, which is not solely basedon the currentinstantaneous value of the player's risktolerance -u'(x)/u"(x)...
If your browser can run JavaScript (which is probably the case),you may obtain nontabulated values by enteringeither the risk toleranceor the exponential CE at the top of their respective columns (in some cases,you may not get more than 7 or 8 significant digits from the script,whereas the tabulated valuesare correct within half a unit of the last digit displayed). You may wish to use the table backwards: Determine by introspection what thePetersburg Gamble is worth to youand you will know roughly what your risk tolerance is. For example, if you decide that a Petersburg game is worth $6,your risk tolerance is $1872.28. The method may not be very accurate becauseyou are essentially guessing on a logarithmic scale which amplifies errors(estimating the game to be worth $6.05 would correspond to arisk tolerance of $2008.07). However, it's only the order of magnitude of yourrisk tolerancewhich counts for many decisions, and the Petersburg game will allowyou to evaluatethat.
(2016-07-12) That's puzzling only if one misuses the concept of random variable.
You are presented with two indistinguishable envelopes,knowing only that one of them contains twice as much money as the other. After you've made your choice, you're given an opportunity to swap. Should you do so?
Common sense (correctly) says that it doesn't make any difference. However, there's a popular fallacious argument associated with this problem whichwould seem to indicate that the other envelope is always preferablebecause whatever the actual value X of your envelope maybe, the expectedvalue of the other envelope is supposedly 25% larger, based on the followingequation:
½ [ X / 2 ] + ½ [ 2 X ] = 1.25 X
This misguiding tautology is certainly not a correct way to computethe expected value of the second envelope! The fallacy is simply that X is a random variable, which cannot be used as if it were an ordinary variable (i.e., the unknown value of some fixed parameter, not subject to chance).
The only legitimate parameter in this problem determines the unrevealed amount of money in the envelopes (a in one, 2a in the other). The fact that the parameter a is hidden is just acircumstance which makes it impossible to knowwhich one of the following events has occured, even if you can peek insideyour envelope before deciding to swap or not:
X =a and the value of the other envelope is 2 X = 2a.
X = 2a and the value of the other envelope is X/2 =a.
The expected value of the amount in either envelope is clearly equal to:
½ [a ] + ½ [ 2a ] = 1.5a
That expression is utterly irrelevant in practice, since you don't know the value of a at the time you are given the opportunity to swap. You simply know that it will be the same value regardless of your choice.
However, being allowed to look inside your chosen envelope before deciding to swap may definitely influence your decision: Are you willing to gamble half of your winnings for an 50% chance at doubling your money? Well, as the rest of this page goes to show, it all depends on what amount of money is actually at stake.
A rational decision will depend on how much money you had before the deal unless your utility function isexponential. Let's consider only that case. If r is your risk aversion (risk tolerance) then the (normalized) utility you assign toa gain of x amount of money is:
u(x) = 1 - exp ( -x / r )
Thus, if you see an amount x in your envelope, you should risk to lose x/2 for a 50% chanceto gain x only if the utility of swapping is positive:
0 < ½ [ 1 - exp ( x / 2r ) ] + ½ [ 1 - exp ( -x / r ) ]
Introducing the variable y = exp ( x / 2r ) > 1 the above reads:
0 < 2 - y - 1 / y2 or 0 < 2 y2 - y3 - 1 = (1-y)(y2-y-1)
Therefore, the last bracketed polynomial must be negative. That quadratic polynomial has a negative root -1/ and a positive root (thegolden ratio). Our inequality is thus satisfied for y > 1 if and only if y exceedsthe latter quantity. For the amount of money x this translate into this condition:
x < 2 r Log ( ) = (0.96242365...) r
In other words, rational players should swap envelopes iff they find less than about 96.24% of their own risk aversion inside their first envelope.
