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Utility Theory

Utility Theory. 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.

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Utility Theory

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  1. Utility Theory John Lee Department of Political Science Florida State University

  2. 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.

  3. 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?

  4. 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?

  5. Expected Utility • Expected Utility – The utility one expects to receive in an uncertain situation. Some Notation p(a) = probability of event a EUi(x) = Expected Utility of action “x” for individual “i”. Ui(x) = Utility of action “x” for individual “i”.

  6. 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) = 1

  7. Grandma Example EUme($50) = p(gives money)*50 + p(withholds money)*0 EUme($50) = (4/10)*50 + (6/10)*0 EUme($50) = (4/10)*50 EUme($50) = 20 EUme($100) = p(gives money)*100 + p(withholds money)*0 EUme($100) = (3/10)*100 + (7/10)*0 EUme($100) = (3/10)*100 EUme($100) = 30 EUme($100) > EUme($50) 30 > 20

  8. 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? • EUi(L) = p(W)*BW + p(L)*BL – c • Where BW equals the benefits of winning and BL 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. • EUi(L) = p(W)*BW – c

  9. 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. • EUi(L) = p(W)*BW – c • EUi(L) = (1/1000)*1000 – 2 • EUi(L) = 1000/1000 – 2 • EUi(L) = 1 – 2 • EUi(L) = -1 • Should you do it?

  10. 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.

  11. 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.

  12. Lottery Example (2) • Should you buy a lottery ticket? • EUi(L) = p(W)*BW + p(L)*BL – c • Where BW is equal to the benefits of winning (let’s say $1000) and BL 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. • EUi(L) = p(W)*1000 - 2

  13. 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. • EUi(~L) = 0

  14. 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. • EUi(L) > EUi(~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.

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