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BAMS 517 – 2011Decision Analysis -IIIUtility

Martin L. Puterman

UBC Sauder School of Business

Winter Term 2011

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A curious gamble

Gamble A; Flip a coin- Heads win $1; Tails lose $1

Gamble B; Flip a coin- Heads win $1 million; Tails lose $1 million

The expected value for each is $0. Which would you choose?

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Another curious gamble Consider the following game; flip a coin until you get a head. Payoff – head the first time $2, head the second time $4, head the third time

$8, … What is the expected value of this game?

This is called the St. Petersburg Paradox The paradox is named from Daniel Bernoulli's presentation of the problem and his

solution, published in 1738 in the Commentaries of the Imperial Academy of Science of Saint Petersburg. However, the problem was invented by Daniel's cousin Nicolas Bernoulli who first stated it in a letter to Pierre Raymond de Montmort of 9 September 1713. 

Of it, Daniel Bernoulli said“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.”

Gabriel Cramer, a Swiss Mathematician said;“mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it”.

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nXE )2

1(2][

1

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Basic idea

To avoid these situations we can work backwards through a

decision tree by replacing each gamble by our personal

assessment of its value

Shortcomings of this approach

It is tedious

May not be accurate

May be inconsistent

Sometimes difficult; especially if gamble has many possible

outcomes.

Another approach - replace outcomes by there utility.

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When is utility useful?

Monetary outcomes To systematically incorporate personal attitudes towards

consequences of decisions and risk Large losses may be catastrophic

To evaluate complex gambles systematically Win $207 with probability .17 and lose $114 with probability .83.

To evaluate decisions involving non-monetary outcomes Health outcomes – years of life vs. quality

To evaluate decisions with multiple dimensions to outcomes Wealth, happiness, …

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What is utility? Some preliminary definitions

Let {x, y, z, w} be possible outcomes of a decision problem. Then

1. x > y means "x is preferred to y" (also known as strict preference)

2. x ~ y means "x is viewed indifferently relative to y"3. x >~ y means "x is either preferred or viewed indifferently

relative to y" (also known as weak preference)4. (x,p,y) means a gamble (an uncertain outcome, or a lottery)

in which outcome x will be received with probability p, and outcome y will be received with probability 1-p.

Example: x is $150, y is a ticket to a Canucks game; z is a 50-50 lottery which either wins $200 or a 20 year old PC and w is one week of good health

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What is utility? Some consistency Axioms for outcomes and preferences

For any outcomes x,y,z,w and numbers p,q between 0 and 1 the following hold:

1. Weak ordering.(a) x >~ x. (Reflexivity)

(b) x >~ y or y >~ x. (Connectivity)

(c) x >~ y and y >~ z imply x >~ z. (Transitivity)

2. Reducibility. ((x,p,y),q,y) ~ (x,pq,y).

3. Independence. If (x,p,z) ~ (y,p,z), then (x,p,w) ~ (y,p,w).

4. Betweenness. If x > y, then x > (x,p,y) > y.

5. Solvability. If x > y > z, then there exists p such that y ~ (x,p,z).

Example: x is $150, y is a ticket to a Canucks playoff game, z is a 50-50 lottery which either wins $200 or a 20 year old PC, and w is one week of good health

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What is utility? A key theoretical result and interpretation

Theorem: (J. von Neumann and O. Morgenstern, Theory of Games and Economic

Behavior, 1944 (Also attributable to Ramsay (1931))

If Axioms 1-5 are satisfied for all outcomes, then there exists a real-valued utility function u(s) defined on outcomes, with the properties that:

1. x > y if and only if u(x) > u(y), and x ~ y if and only if u(x) = u(y);

2. u(x,p,y) = pu(x)+(1-p)u(y);

3. u is unique up to order preserving affine transformations; that is, if v is any other function satisfying 1. and 2. then there exist real numbers b, and a>0, such that v(x) = au(x)+b.

This means that if we believe the consistency axioms: There is a function u(s) that captures our preferences for outcomes; the higher the utility

the more we prefer the outcome. The utility of a lottery is the expected utility. The relative difference between outcomes measures our relative preference for

outcomes

The consequence of all this is that in a decision problem We value all outcomes by their utility We replace a lottery by its expected utility We choose decisions which maximize utility

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Using Utility: U(x) = ((x+2000)/10000).5

Choice B

Choice A

.5

.5

5000

-1000

1500

Utility

.8367

.3163

.5916

Choice A has EMV = $2000 and Expected Utility = .5764; so under EMV you prefer A and under Expected Utility you prefer B.

At what value for Choice B would you be indifferent? 1322.87 which is the certainty equivalent of A

Alberta Exploration Revisited

Suppose we revisit Alberta Exploration but use expected utility instead of expected monetary value as our optimality criterion How might our utility change.

For simplicity we assume an exponential utility function u(x) = xa normalized so that 0 ≤ u(x) ≤ 1. That is u(x) = ((x-b)/c)a

See Alberta Utility

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St. Petersburg Paradox Revisited

Bernoulli suggest used a utility function u(x) =ln(x) Thus the value of receiving 2n is ln(2n) = nln(2) so

that the E(u(X))= 2ln(2) = 1.3863 Hence the certainty equivalent of this gamble would

be e1.3863 = 4.

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Assessing a decision maker’s utility

Choice B

Choice A

1-p

p

Best outcome

Worst outcome

Specified intermediate outcome

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Assessing a decision maker’s utility

We can use a similar approach to that used for assessing probabilities. Take a reference lottery for which the utility of the two outcomes

are known and the probability of receiving the better one is p. We start by assigning utility of 1 to best outcome B and 0 to worst

outcome W. We compare it to a decision with no randomness and a fixed payoff

C. Two approaches;

Fix p and vary C Fix C and vary p (using spinner)

In the first case we find the value C that has utility p In the second case we find the utility p of receiving C for sure. Which is easier?

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Assessing a decision maker’s utility Iterative approach;

Set p = .5 and find C1 so that u(C1)=.5 Now consider a 50-50 lottery between C1 and B. Assign

utility .75 to the equivalent value C2. Now consider a 50-50 lottery between C1 and W. Assign

utility .25 to the equivalent value C3. Continue this process

Check for consistency and whether it agrees with our attitude towards risk.

Plot and smooth utility curve. Considerable behavioral research on doing this to avoid

bias Exercise

Find your utility curve for a decision problem with outcomes ranging from -5000 to 20000.

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Another interpretation of utility

Assume u(s) is normalized so it’s value falls between 0 and 1.

Then suppose we have a fixed outcome with utility q. Then this fixed outcomes is equivalent to a lottery with

outcomes B with probability q and w with probability 1-q. Thus we can reduce a decision problem to one in which

every endpoint is a lottery between B and W. Of course our preference is for lotteries with higher probabilities

of receiving B.

Thus we can think of the utility as the probability of receiving the best outcome.

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Shape of Utility Functions A utility function u(•) is concave if for any a satisfying 0 a 1

u(ax + (1-a)y) au(x) + (1-a) u(y) for any x and y in S.

A twice differentiable concave utility function has a non-positive second derivative (u’’ ≤ 0) Examples

u(x) =ln(x) u(x) = √x u(x) = 1-e-ax

An individual who possesses a strictly concave utility function is said to be risk averse.

By strictly concave , I mean it is not linear on any intervals. A key property of a strictly concave utility function is decreasing marginal utility. That

means as x increases the slope or value of an additional unit, u`(x), decreases. An individual with a linear utility function is said to be risk neutral.

This is appropriate when you Repeat the gamble many times Consequences are small (utility curve is approximately linear over small increments)

An individual with a convex utility function is risk seeking. Example u(x) = x2

Comments Often we normalize the utility function so that its maximum value is 1 and its minimum value

is 0. In decision problems we often define our utility for our total wealth, not the just the outcome

of the gamble.

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Decreasing marginal utility

It is not always reasonable to assume that you will value an additional $100 equally regardless of how much money you already have You would probably value an extra $100 less

if you were very wealthy than if you were very poor

This phenomenon of valuing an additional dollar less, the more money you already have, is known as decreasing marginal utility Decreasing marginal utility implies that the

utility function has a concave shape – see graph

The horizontal lines in the graph show the additional (marginal) utility of each additional $20

Note that the derivative of u is decreasing as the monetary value increases

Would this argument apply to an extra day of life?

u(20)

u(40)

u(60)

u(80)

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Risk-aversion: a graphical view

Consider the following options:1. A chance p of $50 and (1-p) of $102. A sure outcome of $[50p + 10(1-p)]

Line AB represents the expected utility of option 1 for any value of p Note this line lies under the curve It’s equation is pu(50)+(1-p)u(10)

For p = 0.5, the expected monetary value of either option is $30 The blue line represents the expected

utility of option 1 The red line to point C represents the

utility of option 2 The sure outcome will have a higher

expected value whenever the decision maker is risk averse.

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C

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Certainty equivalence of gambles Consider now a gamble (decision) that can

either result in having a dollars with probability p, or to b dollars with probability 1–p, where a > b

A is the expected utility of the gamble C is a dollar value such that you would be

indifferent between having $C for certain and accepting the gamble The utility of C for certain is equal to the

expected utility of the gamble u(C) = pu(a) +(1-p)u(b)

The value C is called the certainty equivalent value of the gamble

C = u-1(A) Certainty equivalent = inverse of u applied to

expected utility of gamble B = pa+(1-p)b is the expected monetary

outcome of the gamble which exceeds C. Why?

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b aC B

A

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An important decision problem; Insurance

Suppose you want to buy insurance against the theft of your car, which you value at $20,000. You currently have $40,000 in total assets

You assess the likelihood of your car being stolen in the next year at 0.5% If you don’t buy insurance, you will have $40,000 with 99.5% probability,

and $20,000 with probability 0.5% The expected monetary outcome is therefore .995($40,000)+.005($20,000) = $39,900

If you buy insurance at a price c, then you will have $40,000 – c, regardless of whether your car is stolen or not

You would buy insurance at price c ifu(40,000 – c) > 0.995u(40,000) + .005u(20,000)

If you are risk-neutral, you would be willing to spend up to $40,000 – $39,900 = $100 on car insurance

If you are risk-averse, would you be willing to spend more or less than $100?

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Insurance We plot the insurance-buying problem in

the graph at right, assuming risk-aversion (not to scale)

B represents the expected monetary value of not buying insurance = $39,900

A represents the expected utility of the decision not to buy insurance = 0.995u(40,000) + .005u(20,000)

C represents the monetary value at which u(C) = A (certainty equivalence)

You would pay up to $40,000 – C for insurance with this utility fn.

You would pay up to $40,000 – B if you were risk neutral

If you are risk-averse, you would be willing to pay more than the expected value for insurance

Why would anyone sell you insurance?$20K $40K C B

A

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Risk premiums

The previous examples show that for a risk-averse decision-maker, the value of a gamble (its certainty equivalent) will be less than its ‘fair value’ (expected monetary outcome) This follows from the concavity of the utility function Risk-averse people would refuse a ‘fair’ bet

The difference between the ‘fair value’ and the certain equivalence value of the gamble (between points B and C in the previous graphs) is known as the risk premium

If the risk premium is negative you would be risk seeking If this holds for all outcomes you would have a convex utility function. Recall roulette and horse race betting are “unfair games” and have negative risk

premiums

Note people may have a utility function that is risk seeking for gambles with positive payoffs and risk averse for negative consequences.

What would such a utility function look like?

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Risk premiums

The size of the risk premium can be used to measure the degree of risk-aversion to a particular gamble

Consider the two utility curves pictured at left. The red curve is ‘more concave’ in the region between the two outcomes and corresponds to a more risk-averse decision maker

The point D is the certainty-equivalent value for this more risk-averse decision-maker

The point C is the certainty-equivalent for the less risk averse decision maker

The risk premium B – D is greater than B-C for the risk averse decision maker.

$b $aC B

A

D

(Note: The two utility curves coincide at the points a and b in this graph only for convenience of exposition.)

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The value of a gamble

Suppose that your utility function for total wealth $x is u(x) = x1/2 For simplicity, we won’t bother here to normalize the utilities between 0 and 1

Consider a gamble that pays $5000 with probability 0.5 and -$5000 with probability 0.5 (i.e., you have to pay $5000 if you lose)

The following table shows that your risk premium depends on your wealth:

Thus with this utility function, the more money you already have, the less money you would require in compensation for accepting the gamble That is, your risk premium is decreasing with your wealth.

Initial Wealth Exp. Utility Certain Equiv Risk Premium

x U = .5(x+5000)1/2

+ .5(x-5000)1/2

CE = U2 EV – CE

= x – CE

$ 5000 50 $ 2500 $2500

$15000 120.7 $14571 $ 429

$25000 157.3 $24747 $ 253

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The value of a gamble

The fact that the risk premium can vary with initial wealth means that we can’t consider the value of a gamble in isolation from our total wealth (this is often called framing).

It may be inconvenient if we have to constantly have to refer to our current level of wealth in order to judge the value of a gamble

There are certain utility functions, however, for which the value of a gamble can be judged apart from total wealth

For example, consider the utility function u(x) = 1 – exp(-x/10000). The following table lists the risk premiums for each:

Initial Wealth Exp. Utility Certain Equiv Risk Premium

x U = .5(1 – e-(x+5000)/10000)

+ .5(1 – e- (x-5000)/ 10000)

-10000 ln(1 – U) EMV – CE

= x – CE

$ 5000 .316 $ 3799 $1201

$15000 .748 $13799 $1201

$25000 .907 $23799 $1201

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Utility functions with the delta property

Utility functions for which the risk premium of a gamble does not depend on the initial level of wealth are said to have the delta property

It can be shown that all such utility functions are either of the form (up to a linear scaling)

u(x) = x, oru(x) = 1 – exp(-cx),

for some c > 0 The parameter c is known as the risk

aversion constant: the higher the value of c, the more risk-averse one is (see graph at right)

Risk-neutral utility functions have the delta property

Is this a desirable property of a utility function?

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1.2

0 5000 10000 15000 20000 25000 30000

Total Wealth

Uti

lity

c = 1/5000

c = 1/10000

c = 1/20000

more risk-averse

less risk-averse

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Example: the jellybean game

Suppose there are 50% red jellybeans in a jar, and your utility function for total wealth $x is u(x) = 1 – exp(-x/10). You win $10 if you draw a red jellybean and $0 otherwise

The fair value of this game is $5 – if you were risk-neutral, you would pay up to this amount to play the game

We now compute the certain equivalent of the game. Because the utility function has the delta property, you can ignore total wealth – just assume x = 0.

Expected utility is then U = .5(1 – e-1) + .5(1 – e0) = .316 The certain equivalent is then u-1(.316) = -10•ln(1 - .316) = 3.80 You would pay up to $3.80 to play the game (regardless of your

current wealth) Using utilities that have the delta property is convenient. You can

use such a utility function if you assume that your aversion to risk is constant, regardless of your current wealth

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More on risk aversion

The linear (risk-neutral) and exponential utility functions we have discussed so far are convenient functions to use Both of these functions presume that your level of risk aversion is constant,

regardless of your level of wealth This is what it means to have the delta-property Another measure of the risk-aversion associated with a utility function u(•) at a level

of wealth x is the value r(x) = –u’’(x) / u’(x). For linear utility functions, r(x) = 0. For u(x) = a – bexp(-kx), r(x) = k

Constant risk-aversion is not always be a reasonable assumption – as you grow richer, you may become more tolerant of risk – your risk-aversion should decrease

Other commonly used utility functions are of the form u(x) = xk, for 0 < k < 1 or u(x)=ln(x) The risk-aversion associated with these utility functions decreases as wealth

grows For example for u(x) = ln(x), r(x) = 1/x.

Ultimately, for any major decision, you will have to think hard about how much you value the outcomes – you don’t want to assume a utility function merely because it has a convenient form

Measuring risk aversion

We say that utility curve u is more risk averse than utility curve v if for all outcomes x and y and every lottery (x,p,y) the certain-equivalent of u is less than certain-equivalent of v.

Theorem (Pratt)

This is true if and only if –u’’(x)/u’(x) > -v’’(x)/v’(x) for all outcomes v.

Proof The quantity –u’’(x)/u’(x) is called the Arrow-Pratt

measure of absolute risk aversion Example u(x) = x , v(x) = x2 for x >0.

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Expected value of perfect information under utility

If your utility function has the delta property, then the difference in the certain equivalents of these decisions represents the amount that you should be willing to pay to acquire information If you are risk-neutral, then this is simply the difference in the expected

monetary value of the two decisions If your utility function does not have the delta property, then computing

how much you should be willing to pay to acquire information is more difficult. The following approach will work. Suppose you must pay x for perfect information. If you pay x, then each endpoint on the PI decision tree is reduced by x. Then vary x so that the utility of the perfect information decision tree equals

that of the perfect information decision tree.

Buying vs. selling

Suppose there is a lottery such that with probability .5 you get 1000 and with probability .5 you lose 200.

Would you pay the same price to buy it as you would to sell it?

Yes, if you use expected value Maybe, if you are risk averse

Yes if your utility has the delta property No, otherwise e.g. u(x)=sqrt(500+x)

What if you are risk seeking?

Summary so far Utility provides a way of valuing outcomes that takes your

risk attitude into account. To use it in the context of decision analysis replace all

endpoints by their utilities and do backward induction as in the expected value case.

Utility functions differ between decision makers. You can assess a decision maker’s utility function by

asking him/her to evaluate gambles The certainty equivalent gives the monetary value of a

decision problem. For risk averse decision makers, the utility function is

concave and the risk premium is positive. But behavioral studies suggest that utility theory might not

describe how people really behave. We will not cover assessing multi-attribute utilities or

utilities for health outcomes.