8/12/2019 Expected Value of Perfect Information

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Expected Value of Perfect Information

Figure 1 shows the decision tree we developed in the previous lecture. Having completed

the drawing we are now read! to make our decision when the phone rings.

Figure 1" #ecision $ree

It is our %oss telling us that he has someone on the other line who wants to sell us somemarket research data. He wants to know if !ou are interested

Q: How much does it cost&

A: '())).

D: $hat*s dirt cheap for market research data. +e would t!picall! expect to pa! '1()))to ',)))) for even a simple surve!.

Q: +h! is it so cheap&

A: $he data comes from a surve! conducted for another compan! who paid full price forthe work. $he market research compan! is tr!ing to make a little extra mone! %! re-selling the data.

Q: +hat does the data tell us&

A: It will onl! tell us whether the market is favora%le or unfavora%le.

Q: an we use that t!pe of information&

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A: :es.

D: If !ou look at the tree in Figure 1 then %oth high growth and moderate growth would%e considered favora%le. 7ow growth would %e considered unfavora%le.

Q: Is it worth '()))&

A: +e can use the Expected Value of Perfect Information 3EVPI5 to come up with a;uick answer to that 3remem%er < !our %oss is on the phone with !ou5

Q: +hat does =perfect information> mean&

A: It means that whatever it tells !ou is correct.

Q: an !ou actuall! %u! perfect information&

A: ?o not legall! %ut let me set up a h!pothetical situation"

D: $he following is not intended to insult an!one*s religious %eliefs. It is supposed to %ea little %it funn! and to make a point.

:ou are sitting in !our office feet up on the desk and thinking a%out the decision treetr!ing to make up !our mind when a celestial %eing appears in !our office. It is angelin full regalia" wings ro%es halo and harp. $he angel announces that s@he is raisingmone! for a vacation to #isne! +orld %! selling information a%out the future. $heinformation was gleaned %! eavesdropping around the throne of Aod up in heaven.t this point the angel pulls out a stone ta%let which is kept turned awa! from !ou.

Q: +h! does the angel keep the ta%let turned awa! from us&

A: @he doesn*t want !ou to see the information %efore !ou %u! it.

D: $hat*s a funn! thing a%out %u!ing information. $hink of %u!ing a car" do !ou send aletter to a dealership listing the features !ou want on !our new car have them selectone for !ou and the! drop it off when !ou send them a check& Pro%a%l! not most ofus go look at the car %efore we %u! it. If !ou rent an apartment do !ou Cust sign thecontract or do !ou go look at the apartment first 3actuall! there have %een times inm! life where I have %een desperate enough to Cust signed the contract %ut not %!choice5& If !ou %u! even a loaf of %read do !ou turn it over to make sure it isn*t allgreen on the %ottom& Df course !ou do. +henever we %u! something we look at itfirst. :ou don*t do that with information.

Q: +h! do we %u! information sight-unseen&

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A: If the! let !ou see the information first !ou might not pa! them 3though I*m not sureI*d recommend doing this to a celestial %eing5.

D: emem%er that the next time !ou hear someone railing awa! at our Intelligenceagencies. $he I and friends spend their entire lives dealing with a product the!

must purchase %efore seeing 3imagine running the purchasing department for !ourcompan! like that5. $he! naturall! spend a lot of their time tr!ing to verif!information the! have alread! purchased and that*s %efore the! can %egin to anal!eit. It must %e frustrating to sa! the least.

Q: o what is the angel*s information worth to us&

A: $hat depends on what the information sa!s.

D: If we could see the information then we would know what to do and we could Cudgewhat it is worth %! how much we gain 3or how %ig a loss we avoid5. Gnfortunatel!

we won*t know what it sa!s until after we hand over a check.

Q: How do we know how much to pa!&

A: $hat*s a different ;uestion and one we can answer via the EVPI calculation.

Q: How do !ou calculate EVPI&

A: alculate a weighted average of the %est pa!offs for each state-of-nature.

D: +e don*t know exactlywhat the stone ta%let will sa! %ut we can narrow down thethings it might sa! %! considering what possi%le futures exist for our compan!. +ecan get a list of possi%le futures %! looking at the decision tree.

Q: Is that all the futures we have to consider&

A: 7et me answer that ;uestion with another ;uestion"

Q: +hat is the assumption of exhaustiveness as it applies to states-of-nature&

A: $he assumption of exhaustiveness is that we have a complete list of the states-of-nature.

D: $hat means the tree shows all possi%le futures which further means that the treeshows all the possi%ilities of what might %e written on the stone ta%let.

Q: re the assumptions for a decision tree the same as the assumptions for a pa!offta%le&

A: :es.

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D: :ou might have guessed that when I said 3in an earlier lecture5 that an! pa!off ta%lecan %e drawn as a tree. $he onl! difference is that the assumptions appl! to the nodes3circles and s;uares5 not the tree as a whole. $hus each node is mutuall! exclusiveexhaustive and deterministic.

Q: ould !ou give an example of calculating the EVPI&

A: ure.

D: tart off %! listing the possi%le futures as shown in Figure 2"

High 6oderate 7ow

Figure 2" Possi%le Futures for EVPI alculation

$hen for each state-of-nature look at the tree 3Figure 1 is repeated as Figure ,%elow5 and select the %est pa!off.

Figure ," #ecision $ree 3repeated5

$o help I have shown all the =High> state-of-nature pa!offs in green all the=6oderate> pa!offs in %lue and all the =7ow> pa!offs in red. I have also put thewinner for each state-of-nature in bold. op!ing the winning 3highest5 pa!off foreach state-of-nature into our ta%le we get Figure 4"

High 6oderate 7ow2()) 12)) -()

Figure 4" Possi%le Futures for EVPI alculation

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$he instructions sa! to calculate a weighted average of these =%est> pa!offs so weneed some weights.

Q: +hat is our current %est estimate that the stone ta%let will read =High Arowth&>

A: 4)

D: $his came from the decision tree our pro%a%ilit! of =High Arowth> occurring. $hatis likel! to %e our %est estimate %ecause it is our onl! estimate. $aking the rest of thepro%a%ilities from the tree we get the calculation in Figure ("

High 6oderate 7ow).4)32())5 J ).4)312))5 J ).2)3-()5 K 149)

Figure (" EVPI alculation

?ow we have calculated EVPI %ut we still don*t know how much to pa! the angel.

Q: +ould we pa! the EVPI amount '149)L 3remem%er that the pa!offs are in )))*s5&

A: ?o.

D: +e have to stop and ask ourselves how well we could do without the angel*s help.7ooking up at Figure , again 3I refuse to cop! down that tree a third time5 our %estexpected value is '1,0)L. onsidering that the angel*s information perfect thoughit is is worth onl! the difference %etween those two as shown in Figure /"

EVPI '149)L - EVw@oI - 1,0)L

Purchase Price 8)L Figure /" Purchase Price for Perfect Information

Q: +e would pa! '8)L for the perfect information&

A: :es.

D: It might seem that we are cheating the angel 3perhaps not all that %right an idea givenhis@her connections5 %ecause this information will %e worth a lot more than merel!'8)L. emem%er though that we can work without the information and make somedecision and that we still don*t know what the information will sa!. ince we don*tknow we have to consider all the possi%ilities and that drives down the price.

Q: #oes all this have an!thing to do with our pro%lem&

A: Df course it does.

D: $o get %ack to the pro%lem at hand our %oss is still on the phone wanting to know ifwe are willing to pa! '(L for some decidedl! non-perfect information.

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Q: +ill our %oss reall! wait while we calculate EVPI&

A: :es.

D: If !ou drop out m! sill! stor! know how the calculation works and have the data infront of !ou then this calculation should take less than a minute.

Q: How does EVPI tell us what to pa! for imperfect information&

A: It doesn*t directl!.

D: +hat !ou do with EVPI is find out the cost of the information 3we know that '(L5and a ;uick estimate of the accurac! of the information. o !ou ask !our %oss if themarket research compan! is an! good. @he sa!s =:es we*ve worked with them%efore and the!*ve %een prett! good.> $he onl! ;uestion then is"

Q: re !ou willing to spend '(L for =prett! good> information when !ou know !ouwould pa! '8)L for perfect information&

A: +ell I would.

D: $he EVPI acts as an upper limit for us. :ou will %e amaed at how ;uickl! that value3'8)L5 drops off as the accurac! decreases 3we will calculate a version of that in thenext lecture5. $he onl! virtue of the EVPI is that it is ;uick. Dnce !our %oss asks!ou can perform the calculation and have a %asis for making a decision in less than /)seconds. It is an extremel! rough estimate 3;uick ones usuall! are5 %ut at least it issomethingto %ase !our decision on.

Q: Is there an! other calculation one that perhaps takes longer %ut is more accurate&

A: Funn! !ou should ask. ee the next set of lecture notes on Expected Value ofample Information.

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