Utility Theory Decision Theory

$1 vs. ($0 or $3) If you had a choice between Getting $1 Flipping a coin, heads you get $3, tails $0 Which would you do and why? Expected returns 1) (100%)($1) = $1 2) (50%)($3) + (50%)($0) = $1.50

Risk neutral thinking Since $1 and $3 isn’t a lot of money, you might pick the choice that would return the highest expected return. If tails came up, and you get $0, you might not care too much and at least you feel you put yourself in the position to get the best expected return.

$1M vs. ($0 or $3M) If you had a choice between Getting $1M Flipping a coin, heads you get $3M, tails $0 Which would you do and why? Expected returns 1) (100%)($1M) = $1M 2) (50%)($3M) + (50%)($0) = $1,500,000

Risk Averse thinking Since $1M is a lot of money, you might think this would be a great thing to happen. Having $3 would be better but having $0 would be much worse that $1M. Note: if you feel that $1M is not a lot of money, change Million to Billion and start over.

Quantify this problem (Utility value) Let’s say on a scale of 0 to 100, you feel that: $0 is a U=0 $1M is a U=94 $3M is a U=96 Choice 1) E[U] = 94 Choice 2) E[U] = (50%)0 + (50%)96 = 48 94 is a lot better than 48

$1 vs. ($0 or $3) If you had a choice between Getting $2 Flipping a coin, heads you get $3, tails $0 Suppose you really need $3 Which would you do and why? Expected returns 1) (100%)($2) = $2 2) (50%)($3) + (50%)($0) = $1.50

Quantify this problem (Utility value) Let’s say on a scale of 0 to 100, you feel that: $0 is a U=0 $2 is a U=5 $3 is a U=90 Choice 1) E[U] = 5 Choice 2) E[U] = (50%)0 + (50%)90 = 45 45 is a lot better than 5

Payout = Utility (Risk Neutral) As the payout grows the utility grows linear with it Payout Utility $10 10 $20 20 $30 30 $40 40 $50 50

Low Payout = High Utility (Risk Averse) Lower payouts have relatively high utility, then it tapers off with high values Payout Utility $1M 94 $2M 95 $3M 96 $4M 96.5 $5M 96.8

Low Payout = Very low Utility (Risk Seeking) Lower payouts have relatively very low utility, then values rise dramatically with small increases in payouts Payout Utility $1 $2 1 $3 3 $4 8 $5 15

Risk (neutral, averse, seeking)

Von Neumann-Morgenstern utility theorem In decision theory, the von Neumann- Morgenstern utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. This function is known as the von Neumann- Morgenstern utility function. The theorem is the basis for expected utility theory

Axioms 1) Completeness 2) Transitivity 3) Continuity 4) Independence L < M, M < L or L = M 2) Transitivity L < M, M < N then L < N 3) Continuity L < M, M < N then there is a p (0,1) where pL + (1 – p)N = M 4) Independence If L < M, then for any N, there is a p (0,1] where pL + (1 – p)N < pM + (1-p)N

Buying insurance? You pay a premium of $700/year to insure you car against damage/liability. How does the insurance company make a profit? profit = $700 – Expected accident payouts insurance employ people to calculate expected, then set the premium to that number plus profit They why would you buy insurance? You are risk averse and the premium has a low utility and the payout has a high utility.

Why did Google pay $1.15B for Waze? Facebook, Google and Apple bid to buy Waze Google won with a bid of 1.15B but Google already had Google Maps.

Marginal Utility Marginal Utility is the difference from the previous total Quantity Total Utility Marginal Utility 1 120 2 210 90 3 270 60 4 300 30 5 6 -30 7 240 -60