expected utility

8/7/2019 expected utility

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Decision Making Under

Uncertainty


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Formally, a lottery is a random variable L whose outcomes WI, W2, .... Wn areeconomically meaningful events that occur, respectively, with knownprobabilities P1, P2, Pn

The outcomes may be anything that individual decision maker may value,though we will often assume, without loss of generality, that the outcomes Wirefer to different levels of wealth.

To simplify the 'exposition, we will take the set of possible outcomes WI,W2, ...,Wnas given and fixed and indicate a lottery L simply by the probabilities withwhich these outcomes occur.

Specifically, we write


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..to indicate a lottery in which agent realizes wealth Wi with probability Pi fori = 1,2, ... , n.

Example: Suppose that obtaining an MSfrom OSUwill result in a $100thousand annual salary with probability 0.6 and a $50 thousand annual salarywith probability 0.4; further suppose thatnot obtaining an MSwill result in a$50 thousand annual salary with probability 0.5 and a $10 thousand annual

salary with probability 0.5. With outcomes consistingof annual salaries of

$100. $50, and $10 thousand, the former (MS) lottery may be denoted

L1 = (0.6,0.4,0.0)

and the latter (non MS) lottery may be denoted

L2= (0.0,0.5,0.5).


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Acompound lotteryis a lottery whose possible outcomes are simple lotteries. IfL1 and L2are lotteries and pai 1 and pai2 are a pair of probabilities, we write

to denote the compound lottery in which lottery

is awarded with probability pai1


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and lottery

is awarded with probability 2. We assume agents view compound lotteries as theequivalent simple lotteries implied by iterating probabilities.

Thus, the compound lottery

is equivalent to the simple lottery

Where

Pi= 1p1i+ 2p2i i=1,2,n


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Example 1 The outcome of obtaining an MSdegree from Purdue Universityis alsorandom. With probability 0.4,a Purdue MSwill be equivalent to an OSUMS; withprobability 0.6,a Purdue MSwill equivalent to no MSat all. Thus, continuing thepreceding example, pursuing an MS from Purdue is the compound lottery

L = 0.4L1 + 0.6L2,

which is equivalent to the simple lotteryL = 0.4(0.6,0.4,0.0) + 0.6(0.0, 0.5, 0.5) = (0.24,0.46,0.30)

That is, obtaining an MSdegree from Purdue will result in a $100 thousand annualsalary with probability 0.24 .. a $50 thousand annual salary with probability 0.46, and a

$10 thousand annual salary with probability 0.30


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As in static consumer theory, we will assume that an agent has a well-definedpreference ordering over the set of all lotteries, and write LI L2 to indicatethat the agent prefers lottery L1 to lottery L2 or is indifferent between the two.

If L1 L2 and L2 L1, we write L1~L2 to indicate the agent is indifferentbetween the two lotteries

We will make certain assumptions regarding an agent's preferences overlotteries. In particular, if L1, L2 and L are lotteries, we will assume:

Completeness: EitherL1 L2, L2 L1 or both

Transitivity: IfL1 L and L L2,then L1 L2

Continuity: If L1 L L2 then for some probability L= L1 +(1-)L2

Independence: IfL1 ~ L2,then, for any probability L1+ (1- )L ~ L2 +(1- )L


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The independence axiom states that if the agent is Indifferent betweenlotteries L1 and L2. then he is indifferent between any two compound lotteries

in which L1 is replaced by L2, and vice versa.

The independence axiom is a controversial assumption, but, together withthe other three assumptions, yields a very strong and practical result

In particular. if all four axioms hold, there exists a real-valued function Udefined on the set of all outcomes w such that if L1 = (P11, P12,... ,P1n)andL2= (P21, P22... ,P2n)are lotteries, then

We call the function u a von Neuman-Morgenstern utility of wealth function


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The existence of a von Neuman-Morgenstern utility of wealth function is a veryconvenient result. It allows us to compare lotteries by performing relatively

simple computations.

In particular, we need only compute the "expected utility" provided by the twolotteries. The agent always prefers the lottery that provides the greatestexpected utility


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Example 2: Suppose an agent is presented with two lotteries

LI = (0.2,0.4,0.1, 0.2, 0.1)

A

ndL2 = (0.1, 0.5, 0.2, 0.1, 0.1)

over five possible wealth levels w= 10,20, 30, 40,and 50.

. Further suppose that the agent's preferences satisfy the von Neuman-Morgenstern axioms and that he ascribes utilities u=1.0,1.4, 1.7,2.0,and2.2 to the five wealth levels, respectively. The agent's expectedutility with lottery 1 is:

EU1 =0.2 1.0 +0.4 1.4 +0.1 1.7 +0.2 2.0 +0.1 2.2 =1.55

and with lottery2 isEU2=0.1 .1.0 +0.5 1.4 +0.2 1.7 +0.1 2.0 +0.1 2.2 =1.56.

Thus, the agent prefers lottery2 over lottery 1.


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Suppose for the sake of argument, that an agent faces the prospect of twopossible Wealth outcomes, w1 and w2, with probabilities P1 and P2,respectively.

Denote the agent's expected utility bv u bar = P1U(w1)+ P2u(w2)

Agents expected wealth by w bar= PI w1 + P2w2.

We consider three distinct possibilities for the curvature ofu: u is convex, u is

linear, and u is concave


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Figure 1: Convex Utility of Wealth function; ubar>uwbar, risk loving

Prefers risky prospects over receiving wealth with certainty


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Figure 2.: Linear utility of wealth function ; ubar=uwbar, risk neutral


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Figure 3; Concave utility of wealth function; uwbar>ubar; risk averse

Prefers expected wealth with certainty over risky prospects


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Jensen's inequality generalises the risk aversion result, above result to moregeneral risky prospects with a continuum of outcomes:

If u is concave and defined for all possible values of a continuous randomvariable wThenU(E(w)Eu(w)

Example: Demand for Insurance

Suppose a risk-averse agent has a current wealth level W, but faces apossible loss L with probability p.

Also suppose that the agent may purchase any amount of coverage Kagainst the loss at a premium rate

That is, the agent can pay a premium K ( for a contract that pays her anindemnity K in the event of a loss, but nothing otherwise.

How much coverage should the agent purchase?


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With insurance, the agent ends up with wealth W - L + K - K with probability pand wealth W- K with probability 1- p.

The agent achieves her most preferred position by purchasing coverage K thatmaximizes her expected utility of wealth

Eu(w)= g(K)= (1 - p) u(W-yK)+ pu (W- L + K-yK)

This is achieved by setting derivative of expected utility with respect to K to 0:

g'(K)= (-y)(1- p) u'(W- yK)+ (1 - y) pu (W- L + K- yK)= 0

y/p. u'(W- y K)= 1- y/1-p. u (W-L+K- yK)(Since the agent is risk-averse, u is concave, and so is expected utility as afunction of the coverage level K; thus, the first-order condition is both necessaryand sufficient for a maximum.)


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Now suppose that insurance is actuarially fair. That is, the expectedindemnity pKequals total premiums yK,so that p = y. Then,

u'(W- yK)= u'(W-L + K- yK).

w-K=w-L+K-KAnd thusK=L

That is, if the insurance is actuarially fair, a risk-averse agent will purchasefull coverage, completely eliminating any uncertainty.


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It can also be shown that if insurance is not actuarially fair,

that is, the insurer sets a premium, that exceeds the loss probability p in order

to make a profit,then a risk-averse agent will purchase less than full coverage and retain someof the risk.

Measuring Risk A version

One way to measure an agent's aversion to a specific risk is by the amount ofmoney the agent would be willing to pay to eliminate it completely.

Suppose an agent faces uncertain wealth w. The certainty equivalent level ofwealth w* is the level of wealth the agent would accept with certainty in

exchange for his uncertain wealth. That is,

Eu(w)= u(w*)


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We know by Jensen's inequality that if the agent is risk-averse,

u(Ew bar)> Eu(w bar)= u(w*)

which implies

E w bar> w*

We define the risk premium to be the difference between the certaintyequivalent wealth and the expected uncertain wealth

= E w bar-w*

The risk premium measures the agent's willingness to pay to eliminate allrisk. It is positive if the agent is risk-averse.


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Figure 4: Certainty Equivalent Income

Consider Figure 4, which illustrates the case of an agent who faces an

uncertain wealth prospect


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In this figure, the wealth level w* provides the same expected utility as hisuncertain wealth prospect.

Thus, the agent is indifferent between receiving w* with certainty and facing hisuncertain wealth prospect. As such, w* is the agent's certainty-equivalentwealth and

= w bar w*

is his risk premium.


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Example: Mr. Smith is a salesman who works on commission.

In a good year, he earns $80,000 and, in a bad year, he earns $60,000.

Good years and bad years are equally probable, so that Smith's expectedannual income is $70,000.

Smith's employer offers him the opportunity to continue in his current position,but at a fixed salary that is to be negotiated.

As he enters the negotiations with his employer, Smith decides he wouldaccept $66,000, but not a penny less.

It follows that Smith's certainty-equivalent income is $66,000, implying that heassesses the cost of risk associated with working on commission at $ 4,000 =$70,000-$66,000.


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Example: An investor with utility of wealth function u(w) = w and initial wealth

100 is offered a risky asset with two equally likely payoffs, -10 and 10. The riskpremium placed on the risky asset by the investor satisfies

Eu(w bar)= u(Ew bar- )

that is,0.5 90+ 0.5 110 = 100 -


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which can easily be solved with a calculator for= 0.2506


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Two widely used measures of risk aversion are, respectively, theArrow-Pratt measures of absolute and relative risk aversion:

A(w)= -u"(w)/ u (w)

R(w)= -wu"(w)/ u(w)

The two measures are independent of the utility function used to represent

agent preferences;

the latter, but not the former, is also independent of the units used to measurewealth.

The sign of the absolute and relative risk aversion measures indicate theagent's attitude toward risk. Since we assume that u'(w) > 0, both measures willbe positive if the agent is risk-averse


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both measures will be zero if the agent is risk-neutral; and both measures willbe negative if the agent is risk-loving

Both measures of risk aversion are intimately related to the risk premium.

Suppose an agent with current wealth w is presented with a small pure risk. That is, he faces an uncertain wealth

Wbar=w+

where is a zero-mean random variable with standard deviation andcoefficient of variation v= /wrelative to current wealth.

Then an approximate expression for the risk premium ,the amount ofmoney the agent is willing to pay to entirely eliminate the pure risk ,is

A(w) sq.

That is, the risk premium is directly proportional to the agent's absolute riskaversion and to the absolute magnitude of the risk, as measured by its

variance.


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Similarly

That is, the risk premium, as a proportion of current wealth, is directlyproportional to the agent's relative risk aversion and the relativemagnitude of the risk, as measured by the square of the coefficient ofvariation.

It is generally presumed that the wealthier an agent is, the less he iswilling to pay to eliminate a given risk.

In other words, for a given risk, the risk premium should decline withwealth. It can be shown that, for any risk, the risk premium decreaseswith wealth

if, and only if, the coefficient of risk aversion is globally decreasing inwealth, that is, if and only if, A'(w)< 0 for all w.

An agent whose utility of wealth function satisfies this condition is said to

exhibit decreasing absolute risk aversionor DARA preferences


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Example : Suppose an agent faces terminal wealth levels of4 and6 with equalprobability, so that his wealth has an expectation of5, a variance of1, and acoefficient of variation1/5. Further suppose that the agent possesses a utility ofwealth function u(w)= -w-2. Then the agent's certainty-equivalent income w' ischaracterized by:

-w*-2=-.5.4-2-.5.6-2=-.02257

So that w*=4.707

And his risk premium is

=w bar-w*=5-4.707=.293

UseArrow-Pratt approximation and compute pai?


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The agent's utility of wealth function exhibits constant relative riskaversion of3, since

R(w)= -wu"(w) /u'(w) = (-) -6w-3 /2w-3 =3

Thus, from the Arrow-Pratt approximation above, ~ 0.5 5 . 3/25 = 0.300which is clearly a reasonable approximation for the exact figure0.293


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Paradoxes ofExpected Utility

Expected utility theory has been criticized for not adequately explaining allobserved behavior under uncertainty.

For example, many people who gamble also purchase insurance.

This phenomenon is typically explained away by incorporating anentertainment value from gambling

and noting that people will gamble small amounts but will purchase insuranceagainst big potential losses.

Two other, more formal puzzles that are frequently cited in the economicsliterature are known as the Allais and Ellsberg paradoxes


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Allais Paradox

Consider four lotteries that yield winnings of 0, 1 million, and 5 milliondollars. respectively according to the following probabilities:

L1= (0.00.1.00,0.00)L2= (0.01, 0.89, 0.10)

L3 = (0.89,0.11, 0.00)L4 = (0.90, 0.00, 0.10)In actual experiments, most people prefer lottery 1 over lottery 2 andprefer lottery 4 over lottery 3. However, these choices violate theaxioms of expected utility theory.


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To prove this, suppose an agent's preferences satisfy theaxioms of expected utility and that he ascribes utilities uo,u1,and u5to wealth of 0, 1 million, and 5 million dollars,respectively. Then

L1 >L2=> Eu(L1)> Eu(L2)0.00uo + 1.00u1+ O.OOu5 > O.Oluo + 0.89u1 + 0.10u50.89uo + 0.11u1 + 0.00u5 > 0.90uo + O.OOUI + O.10u5=> Eu(L3 ) > Eu(L4)L3 >L4

That is, if an agents preferences satisfy the axioms ofexpected utility and he prefers lottery 1 over lottery2, heshould prefer lottery 3 over lottery 4.


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Ellsberg Paradox

Suppose Hat X has 50 red balls and 50 black balls and Hat Y has 100 red andblack balls, but in unknown proportions.

Given a choice between the two hypothetical gambles

XR= Get $100 if a red ball drawn from hat X, $0 otherwise

YR= Get $100 if a red ball drawn from hat Y, $0 otherwise

most people choose X R

Given a choice between the two hypothetical gambles

X B= Get $100 if a black ball drawn from hat X, $0 otherwise

Yo = Get $100 if a black ball drawn from hat Y, $0 otherwise

most people choose XB


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However, these two choices are inconsistent according to expected utility theory.Let p denote the unknown proportion of red balls I hat Y.ThenXR>YR=>. Eu(XR)> Fu(YR)==:- 0.5u(O)+- 05u(100)> (1 - p)u(0)+- pu(100)=>-0.5u(0) 0.5u(100) > -pu(0)-+ (p - l)u(100)pu(0)+- (1 - p)u(100) > O.5u(O)+- 0.5u(100)

EU(YB):> Eu(XB)YB>XB

Thus, regardless of the value ofp,if an agent's preferences satisfy the axioms ofexpected utility and he chooses XRoverYR,then he should choose YBoverXB.


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How does one explain this paradox? 'Most people are distrustful.

If offered a choice between the first pair of gambles, in which a red ball earns areward, an individual would suspect that the lottery manager has placed veryfew red balls, if any, in hat Y;

that is, he would assume that p is small, making hat X the intelligent choice.

If offered a choice between the second pair of gambles, in which a black ball

earns a reward, an individual would suspect that the lottery manager hasplaced very few black balls, if any, in hat Y


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that is, he would assume that p is large, making hat X the intelligent choice,

In other words, the individuals assessment of the number of black and red balls in hat Yis contingent on how the gambles are presented

If the individual were convinced that the proportion of black and red balls in hat Y wasDetermined by a fair process that was not biased in favour of either black or red balls,then the individual would conclude that all four gambles offer a 50% chance of paying$100, making him/her indifferent between them.

Which would be consistent with expected utility theory

expected utility

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