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Utility Theory Utility Theory

Utility Theory - PowerPoint Presentation

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Utility Theory - PPT Presentation

John Lee Department of Political Science Florida State University Utility The idea that we can assign value to an action and then choose amongst a set of possible actions based on their value My grandma could offer me 50 or 100 I choose the offer that maximizes my utility In this case ID: 587836

utility lottery 100 expected lottery utility expected 100 1000 probability play equals ticket winning action choose equal benefits participating

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Slide1

Utility Theory

John Lee

Department of Political Science

Florida State UniversitySlide2

Utility

The idea that we can assign value to an action and then choose amongst a set of possible actions based on their value.

My grandma could offer me $50 or $100. I choose the offer that maximizes my utility. In this case, I choose $100.Slide3

Utility

Previous example/slide based on action with

certain

outcome. In other words, I know that if I accept $50 I get $50.

What if I am uncertain?

Grandma could say that if I choose $50 option I get $50 with a probability (

p

) of 4/10 (.4) and $0 with a probability of 6/10.

Alternatively, she says if I choose the $100 option I get $100 with a probability of 3/10 and $0 with a probability of 7/10.

What should I do?Slide4

Expected Utility

Determining utility when an action results in a set of

possible

outcomes.

For example, for each offer from grandma I might get $50/$100, or I might get $0.

Since I might not get $50 if I choose the $50 dollar option, what is my expected utility of this action?Slide5

Expected Utility

Expected Utility

– The utility one expects to receive in an uncertain situation.

Some Notation

p

(a) = probability of event a

EU

i

(x) = Expected Utility of action “x” for individual “

i

”.

U

i

(x) = Utility of action “x” for individual “

i

”.Slide6

Expected Utilities

For any given p(a) it must meet some criterion (probability rules):

p

(a) must be greater than or equal to zero.

p

(a) must be less than or equal to one.

The probability of all outcomes must sum to one.

So if a & b are possible outcomes then the following must be true.

p(a) + p(b) = 1Slide7

Grandma Example

EU

me

($50) = p(gives money)*50 + p(withholds money)*0

EU

me

($50) = (4/10)*50 + (6/10)*0

EU

me

($50) = (4/10)*50

EU

me

($50) = 20

EU

me

($100) = p(gives money)*100 + p(withholds money)*0

EU

me

($100) = (3/10)*100 + (7/10)*0

EU

me

($100) = (3/10)*100

EU

me

($100) = 30

EU

me

($100) >

EU

me

($50)

30 > 20Slide8

Lottery Example

Say you buy a lottery ticket. There is a 1/1000 chance that you win the lottery. If you win you get $1000. Finally, the lottery costs $2.

What is your expected utility?

EU

i

(L) = p(W)*B

W

+ p(L)*B

L

– c

Where B

W

equals the benefits of winning and B

L

equals the benefits of losing. If you lose you zero. “c” represents the cost. Ergo, the following equation represents the Expected Utility of this lottery.

EU

i

(L) = p(W)*B

W

– cSlide9

Lottery Example

Since we know that p(W) equals 1/1000, BW equals 1000, and c equals 2 we can now compute the expected utility of participating in the lottery.

EU

i

(L) = p(W)*B

W

– c

EU

i

(L) = (1/1000)*1000 – 2

EU

i

(L) = 1000/1000 – 2

EU

i

(L) = 1 – 2

EU

i

(L) = -1

Should you do it?Slide10

Lottery Example

In this case you have two possible actions, you can (1) play the lottery or (2) not play the lottery.

If you play the lottery we determined your expected utility equals $-1.

If you do not play the lottery you lose or gain nothing, so your utility is $0.

EU(lottery) < EU(~lottery)

-1 < 0

Clearly you should not play the lottery.Slide11

Lottery Example (2)

Should you buy a lottery ticket? Well, you need to know three things before answering this question.

What are the probabilities of winning/losing?

What are the benefits of winning?

What are the costs of participating in the lottery?

Once we know these we can compute the expected utility of buying a lottery ticket and compare it to our expected utility of not buying a lottery ticket.Slide12

Lottery Example (2)

Should you buy a lottery ticket?

EU

i

(L) = p(W)*B

W

+ p(L)*B

L

– c

Where B

W

is equal to the benefits of winning (let’s say $1000) and B

L

is equal to the benefits of losing ($0). Also, let’s say the lottery costs $2.

But let’s say we DO NOT know the probability of winning the lottery

.

EU

i

(L) = p(W)*1000 - 2Slide13

Lottery Example (2)

Now that we know the expected utility of participating in the lottery, what is the expected utility of not participating in the lottery? Well you get nothing if you don’t buy a ticket so we are left with the following utility.

EU

i

(~L) = 0Slide14

Lottery Example (2)

So how do we figure out if we should play the lottery? Much the case of simple Utilities we compare our expected utilities and see which one is greater.

EU

i

(L) >

EU

i

(~L)

p(W)*1000 – 2 > 0

p(W)*1000 > 2

p(W) > 2/1000

p(W) > 1/500

The probability that we win must be greater than or equal to 1/500 for us to rationally play this lottery.

Since the probability of winning a lottery is never greater than 1/500 we know that we should never play the lottery.