Exercise on TossingCoin EMV EVPI EOL Answer

8/12/2019 Exercise on TossingCoin EMV EVPI EOL Answer

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Answer to

An Exercise on EMV, EVPI, and EOLBSNS2120 Jinchang Wang

Name _________________________

Suppose you are playing a game by guessing the result of flipping a coin. The possiblerewars or payoffs are gi!en as below"

0.# 0.# $%ow e&pecte !alue'

(aning )*ea+ (aning )Tail+ ,-

/uessing )*ea+ 100 0 20

/uessing )Tail+ 30 1#0 4#

5olumn -a& 100 1#0 12#

6ou may play this game for many times.

$1' 7f ta8e it as a ecision ma8ing problem9 then$a' the ecision alternati!es are"

/uessing )*ea+ an /uessing )Tail+.

$b' the states of nature are"(aning )*ea+ an (aning )Tail+.

$c' the probabilities of states of nature are"#0: for each of the two states of nature.

$2' 7s this a ecision ma8ing uner uncertainty or ecision ma8ing uner ris8;This is a problem of ecision ma8ing uner ris8.

$4' 5alculate e&pecte payoff $,-' for each ecision alternati!e.,- for /uessing )*ea+ < 100=0.# > $0'=0.# < #040 < 20 $ollars',- for /uessing )Tail+ < $30'=0.# >1#0=0.# < ?0>@# < 4# $ollars'

$?' What is your strategy of )guessing+ when playing this game;/uessing )Tail+ for e!ery game.

$#' What is the !alue of -a& ,- in this game; What oes it mean in this game;

-a& ,- < 4#.7t means that you woul get 4# per game on a!erage if you guess )Tail+ all the timewhich is the best strategy.

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Suppose that someone offers perfect information $A7' about the result of a flipping with# charge per flip. o you ta8e that offer;

$@' 7f A7 says )it lans on *ea+9 then you woul )guess+ _*ea__9 an ha!e rewar of

_100____.

$3' 7f A7 says )it lans on Tail+9 then you woul )guess+ _Tail____9 an ha!e rewar of

_1#0___.

$C' The probability for A7 to say )it lans on *ea+ is _0.#_____.

The probability for A7 to say )it lans on Tail+ is __0.#_____.

$10' 5alculate the e&pecte !alue if you ta8e the offer an follow the A7.100=0.# > 1#0 =0.# < #0 > @# < 12# $ollars'

$11' What we calculate in item $10' is calle _____________.a. ,wDoA7 b. ,wA7 c. ,A7

$12' *ow much of rewar on a!erage o you e&pect to win per game if you play on yourselfwithout using any aitional information;

4# $-a& ,- as in item $#''

$14' *ow much more woul the perfect information bring to you on a!erage for a flip9 compareto playing on yourself without using any aitional information;

12# 4# < C0

$1?' What is the !alue of ,A79 e&pecte !alue of perfect information;C0 $because the !alue of aitional information is eEual to the aitional benefit it can

bring to the ecision ma8er9 which is gi!en in item $14''

$1#' Fre you better off or worse off if you pay # to purchase the perfect information; Why;Better off. 7t is because you can get aitional C0 on top of 4# by paying Gust #.

$1' $a' What is your )net win+ per game on a!erage if you pay # to purchase the perfectinformation;

,wA7 #< 12# #< 0Hor,wDoA7 > net aitional benefit ue to aitional information

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< ,wDoA7 > $,A7 #'< 4# > $C0 #'< 4# > 2#< 0

$b' What is your )net win+ per game on a!erage if you play on yourself without using theaitional information;,wDoA7 < 4# $as in item $12''

$1@' /enerate the opportunity loss table $regret table' for this problem"

0.# 0.# $%ow e&pecte !alue'

(aning )*ea+ (aning )Tail+ ,I(

/uessing )*ea+ 0 210 10#

/uessing )Tail+ 130 0 C0

$13' 5alculate e&pecte opportunity loss $,I(' for each ecision alternati!e.,I( for /uessing )*ea+ < 0=0.# > 210=0.# < 10#H,I( for /uessing )Tail+ < 130=0.# > 0=0.# < C0.

$1C' Base on the ,I(s9 the best alternati!e is _/uessing )Tail+__ with ,I( < _C0_.

Now lets look at the relations among EMV, EOL, and EVPI.

$20' The alternati!e with -a& ,- is _ /uessing )Tail+___.

$21' The alternati!e with -in ,I( is __ /uessing )Tail+___.

$22' 7f you )guess *ea+9 then the e&pecte payoff $,-' is _20__9 an the e&pecte

opportunity loss $,I(' is _10#__9 an $,->,I('10# < 12#___.

$24' 7f you )guess Tail+9 then the e&pecte payoff $,-' is _4#___9 an the e&pecte

opportunity loss $,I(' is _C0__9 an $,->,I('C0 < 12#_____.

$2?' alue of -in ,I( < _C0_$7tem 1C'__.

$2#' alue of ,A7 < _C0_$7tem $1?'__.

$2' Fs we calculate abo!e9 ,wA7 < _12# $7tem $10' or $11''___.

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$2@' Summary:(a !he alternati"e with Max EMV is alwa#s the alternati"e with Min EOL.

($ EMV%EOL& EVwPI 'or e"er# alternati"e.

(c Max EMV & EVwoPI.(d Min EOL & EVPI.

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Exercise on TossingCoin EMV EVPI EOL Answer