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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Utility-Based Decisions

Mathematicians estimate money in proportion to its quantity, and
men of good sense in proportion to the usage that they may make of it
.
Gabriel Cramer   (1704-1752)
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Related Links (Outside this Site)

Exponential Utility Function   |   Decision Analysis Society.

Wikipedia :   Utility  |  Decision theory  |  Prospect theory  |  Loss aversion  |  Risk aversion (or risk tolerance)

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How to make better decisions  "Prospect Theory"   (BBC Horizon)
 
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Utility Functions and Decision Analysis


(2001-07-04)   Rational decision analysis and the concept of  utility...
How do utilities differ from expectations?  How are utilities used?

A utility is a numerical rating assigned to every possible outcome a decision maker may be faced with.  (In a choice between several alternative prospects, the one with the highest utility is always preferred.)  To qualify as a true utility scale however, the rating must be such that the utility of any uncertain prospect is equal to the expected value (the mathematical expectation) of the utilities of all its possible outcomes (which could be either "final" outcomes or uncertain prospects themselves).

When decisions are made by a so-called rational agent (if A is preferred to B and B to C, then A must be preferred to C), it should be clear that some 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 process but it's not theoretically indispensable.  What may be less clear, however, is how to devise such a rating system so that it would possess the above fundamental property required of a utility scale.

One theoretical way to do so is to compare prospects and/or final outcomes to tickets entitling the holder to a chance at winning some jackpot, which is at least as valuable as any outcome under consideration.  A ticket with a face value of  75% means a chance of winning the jackpot with a probability of 0.75 and it will be assigned a utility 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 thing is completely equivalent (think long and hard about this) to having a chance to win the jackpot with probability 0.3 ´ 0.6  +  0.7 ´ 0.2  =  0.32 .  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 a mathematical expectation.  It would be so in any other case involving either lottery tickets or things/situations previously assigned a utility (by direct or indirect comparisons with such tickets).  Halmos Symbol
(QED)

The type of utilities 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 a utility scale without affecting at all the decisions it implies:  Each side of every comparison is translated or rescaled the same way and it does not affect inequalities as long as the scaling factor is positive.

In particular, we may keep the same utility scale if we're faced with an outcome more valuable than whatever jackpot we first considered.  If that jackpot is estimated to be just as desirable as a chance of winning the bigger prize with probability p, we may assign a utility 1/p to the bigger prize  (and that, of course, is larger than 1).  Similarly, the original "ticket" scale may have to be extended to assign negative utilities to certain undesirable situations.  Considering such a situation "in context", as an outcome of a prospect whose other outcomes are quite positive, allows the semi-direct use of the "ticket" scale to evaluate its negative utility.

Even when there is no such thing as a "top prize", the utilities of all prospects must be bounded.  (Recall the difference between a maximum, which is achieved in at least one case, and an upper limit, which may not be.  Utilities have an upper limit, not necessarily a maximum.) This may be visualized by considering that the utility function of money, which is normally nondecreasing, may either have an asymptote or be constant above a certain point.  For a proof 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 that the mathematical expectation of a monetary gamble need not be the proper utility measure to use.  The monetary expectation 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 for an (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 situation where 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" carefully when it comes to actual possible outcomes:  If your current wealth is  W,  what would be the exact 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 simple mathematical form under certain common assumptions (see below).

One caveat is that nonmonetary gratifications often play a role in actual choices which seem based solely on monetary exchanges:  There's some playful element in any lottery, which increases the appeal of purchasing a lottery ticket.  Lottery operators know this very well and they design their lottery "games" with this in mind.  Note that it's always the  entire  situation which is assigned a  utility  rating, not its separate components  (money, health, happiness, etc.).

Now, if you assume that your  attitude  towards money does  not depend 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 proper utility function must be bounded when the stakes are potentially unbounded.)  Mathematically, this assumption states that your utility 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] and only if [tougher] the function  u  is either linear [ruled out] or of the following 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 therefore irrelevant 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 utility function  (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 the risk tolerance.  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 rescaling of 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:

  1. EITHER:   When prospects are analyzed, the  risk tolerance  used in the analysis of future uncertainty is the  constant  corresponding to 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. 
  2. OR:   The utility function used to make  strategical  decisions incorporates the future variability of the investor's risk 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 that k x dy + y dx = 0 (or k dy/y + dx/x = 0), which means that y is proportional to x-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 companies behaves as if k=1/6 (risk tolerance = one sixth of equity).  With the second of the above interpretations, this would mean the utility function of a major corporation (unless it's close to bankruptcy) would typically be -1/x5.  Rather surprisingly, interviews of experienced corporate decision makers seem to be consistent with this...


 Remond de Montmort (2001-07-04)   The Saint-Petersburg Game
A fair coin is tossed until heads appears.  If the game lasts for n+1 tosses, the player receives 2n dollars.  Namely: $1 if heads appears first,  Bernoulli coat-of-arms $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":  The mathematical expectation of the above  Saint-Petersburg Game  is infinite, since it would be the sum of the following divergent series:

(1/2)(1) + (1/4)(2) + (1/8)(4) + (1/16)(8) + ...   =   1/2 + 1/2 + 1/2 + 1/2 + ...

Clearly however,  nobody  would ever pay more than a few dollars for a shot 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 only be based on its "fair" price, which is another name for its mathematical expectation.  The fact that it clearly cannot be so with the above game ultimately led to the introduction of the modern concept of the utility of a prospect.

The discussion originated with a correspondence 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 entitled Essay d'analyse sur les jeux de hazard (Paris, 1708).  Bernoulli was making suggestions for a future edition, focusing on a set of 5 problems to appear on page 402, including "Problem 5", which essentially describes a version of the above  Petersburg Game...

The very first letter from Bernoulli (dated September 9, 1713) mentions a die instead of a fair coin, but the lower probability (1/6) of terminating the game at each toss makes 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 prompted to address the issue from London, in a letter to Bernoulli, dated May 21, 1728.  (Since he turned 20, in 1724, Cramer had been sharing a chair of mathematics in Geneva with Giovanni Ludovico Calandrini , under an arrangement that called for one of them to travel 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 a utility function was simply proportional to the money amount up to a certain point (he used 224 coins, for convenience) and constant thereafter.  His second example was a utility function of money proportional to the square root of the amount of money.  Either of these utility functions does assign a finite utility to the original Petersburg game, but the second one would fail to resolve the issue if the payoff sequence was increasing faster (for example, if the player was payed 4n dollars for completing n+1 tosses). In fact, this very example may be used to show that any utility function must have an upper bound, or else one could exhibit an infinite sequence of prospects, the n-th of which having a utility at least equal to 2n. Offering the n-th such prospect as payoff for successfully completing n tosses in a Petersburg game would assign infinite "utility" to such a game, which is not acceptable.  (The basic utility tenet is to assign a finite utility rating to 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 the St. Petersburg Academy, which is how the paradox got its modern name.

Back in 1731, Daniel Bernoulli rediscovered (independently of Cramer) the modern notion of utilities, which Nicolas Bernoulli kept rejecting...  Daniel also made a point which Cramer had missed entirely, namely that it is generally crucial to consider only the entire wealth of the player and assign a utility only to the whole thing, as the marginal utility of an 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) makes utility (u) a logarithmic function of the total wealth (w).  As we are free to rescale utilities, it may then be stated without loss of generality that this translates into u(w) = ln(w). However, this logarithmic "utility" function suffers from the same flaw as Cramer's square root function of money, because it's not bounded either: If a successful sequence of n+1 tosses was payed exp(k 2n), the game would still end up having an infinite "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 at first, 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 for any successful sequence of less than 100 tosses and exp(k 2n-100) dollars thereafter. That gamble is worth $1.00 to absolutely anybody, 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 an upper limit (in practical situations, potential payoffs are always bounded, which makes the exact mathematical form of the utility function irrelevant beyond 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 the Petersburg 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-consistent analysis 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-called risk tolerance)!  Read on:

There's an hidden assumption in this and other similar theoretical puzzles, which we must make explicit in order to solve the riddle: The question is asked out of context and must be answered likewise if it is to be answered at all.  We are not to involve sordid details about the rest of the player's life (size of bank accounts, mortgages, etc.).  That approach is logically consistent only with the assumption of an exponential utility function of money, which is the only type of utility function where decisions about a particular prospect are not influenced by the rest of one's situation...  It does not make sense to analyze an isolated gamble except by assuming an exponential utility function, since no other utility function of money even permits such isolation.  This is a theoretical argument, of course, but it's clearly appropriate for a theoretical question like the one at hand...

Since we must, we shall happily assume that the player's utility function of money is of the form 1-exp(-x/r) for some parameter r (which is a dollar amount, usually called risk tolerance).  In this, x should generally be the player's total wealth, but the unique properties of the exponential function allow us to consider that x is simply the amount gained or lost in the gamble(s) at hand (since changing the zero point on the money scale merely rescales exponential utilities without affecting the comparisons relevant for decisions).  We do not have such freedom with a more general utility function, as Daniel Bernoulli first recognized.  Also, since additive and/or (positive) multiplicative constants in the utility function do not affect decisions, we may as well use u(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 longer have a utility of 0 for a gain of 0.)

It's interesting to observe that the exponential utility function (with a positive risk tolerance) does not have a lower bound.  Therefore it could not be used to analyze a gamble with unbounded negative payoffs (or fees). This is not surprising in view of the fact that such a gamble is clearly a major decision which cannot be considered independently of the rest of the player's situation, because the entire wealth of the player (and more) is at risk.  Everybody's actual overall utility function is bounded on both sides (if you can't possibly repay a huge debt, it makes very little difference if it is $100,000,000 or $200,000,000). The decision of whether to play the Petersburg Game is a minor one for which the exponential utility function is entirely appropriate.  The decision to bankroll such a game would be a major one, even for a risk-loving entity (if one was ever foolish enough to be attracted by the tiny fees ordinary players are willing to risk).

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, the utility of playing the Petersburg game is clearly å u(2n-y) / 2n+1 and the gamble should be accepted if and only if this is greater than u(0).

In the particular case where u is exponential, this is equivalent to comparing å u(2n) / 2n+1 and u(y), namely the utility of the free gamble and the utility of a so-called certainty equivalent (CE).  The CE is whatever (minimum) amount of money we would be willing to receive as a compensation for giving up the right to gamble.  It may not be quite the same as the (maximum) price we're willing to pay to acquire that right!  Only in the case of the exponential (or linear) utility function are these two amounts always equal.  The CE is the quantity actually computed in Cramer's original text based on a square root utility function.  It was probably silently assumed at the time that the CE would not be too different from the price one would be willing to pay.  However, rigorously speaking, the minimum acceptable selling price (the CE) and the maximum acceptable 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 a risk tolerance equal to r (expressed in dollars), the highest price (y) s/he will be willing to pay for a shot at the Petersburg game is given by the relation:

¥
exp(-y/r)  =   å   exp(-2n/r) / 2n+1
n=0

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 of r, but we may also want to investigate what happens when r is very large or very small:

       For large values of r, we may observe that, when r is much larger than 2n, each term of the sum is roughly equal to 1/2n+1-1/2r.  This near-constancy goes on for a number of terms roughly equal to the base-2 logarithm of r, after which the terms vanish exponentially fast.  (Notice how the exponential utility function turns out to behave very much like the original "moral value" function proposed by Cramer in 1728; proportional to the money at first, then nearly constant after a certain threshold.) We may thus expect the RHS to be equal to about k-ln(r)/(2r ln(2)) for some constant k, which turns out to be equal to 1.  The natural logarithm of that, for large values of r, would therefore be -ln(r)/(r ln(4)), so that y is roughly equal to ln(r)/ln(4) for large values of r (actually, it's about 0.5549745 above that).
      On the other hand, when r is very small, the sum on the RHS essentially reduces to its first term, so that y is extremely close to 1+r ln(2) . The rest of the expansion is smaller than any power of r, since the leading term equals (-r /2)exp(-1/r). In particular, a player with a risk tolerance of zero (r = 0) will only pay $1 for the gamble, since this is the amount s/he is guaranteed to get back...
The last two columns give the buy and sell thresholds for the gamble at the given level of risk tolerance (-u'/u") for a nonexponential utility.
Risk Tolerance
(r, in $)
Value (CE in $) of the
St. Petersburg Game
for u(x)=1-e-x/r
u(x) = -1/x5     [x=6r]


Buy below Sell above
(CE)

                 0
                 1
                10
               100
              1000
            10 000
           100 000
          1000 000
        10 000 000
       100 000 000
      1000 000 000
    10 000 000 000
   100 000 000 000
  1000 000 000 000
10 000 000 000 000

  1
  1.513746140959
  2.536882900027
  3.956652946627
  5.553508801015
  7.201448100212
  8.860189589670
 10.520814323633
 12.181730279322
 13.842687696625
 15.503650974340
 17.164615000601
 18.825578890826
 20.486542978191
 22.147507133452

  1
  1.5421
  2.5879
  4.0163
  5.6150
  7.2632
  8.9220
 10.5826
 12.2435
 13.9045
 15.5654
 17.2264
 18.8874
 20.5483
 22.2093

  1.1451
  1.6268
  2.6116
  4.0208
  5.6156
  7.2633
  8.9220
 10.5826
 12.2435
 13.9045
 15.5654
 17.2264
 18.8874
 20.5483
 22.2093
r » log4(r) + 0.555 » log4(r) + 0.61675

For educational purposes, we've included what a similar analysis would entail for a nonexponential utility function (last two columns of the above table).  The utility function chosen is such that wealth (or equity) is 6 times the risk tolerance appearing in the first column.  The entire fortune of the player is thus taken into account (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, or certainty equivalent) is different from the price he would be willing to pay to acquire such a right, although this is only significant at low levels of risk tolerance (both prices are always equal for an exponential utility). At a zero risk tolerance, it's the buying price which is equal to $1 (since we're guaranteed to get $1 back, no matter what), whereas the selling price may be significantly greater (the value is ½ 631/5 or about $1.145086 in this particular case).  That's because a nonexponential utility function integrates future variations of the risk tolerance and this influences the decision, which is not solely based on the current instantaneous value of the player's risk tolerance -u'(x)/u"(x)...

If your browser can run JavaScript (which is probably the case), you may obtain nontabulated values by entering either the risk tolerance or 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 values are correct within half a unit of the last digit displayed).  You may wish to use the table backwards:  Determine by introspection what the Petersburg Gamble is worth to you and 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 because you are essentially guessing on a logarithmic scale which amplifies errors (estimating the game to be worth $6.05 would correspond to a risk tolerance of $2008.07).  However, it's only the order of magnitude of your risk tolerance which counts for many decisions, and the Petersburg game will allow you to evaluate that.

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