Choose decision with max expected utility Derive the utility function using simple CE questions 2.

15 Steps in using utility functions 1.Utility Functions w Utility: Relative value (worth) of each payoff to the decision maker w Utility Theory: Transform payoffs into utility scale (0 1) w Utility & Coherence: Expected utility criterion EU(a j ) ranking of options is consistent with DM certainty equivalents EU(a j ) 9 CE for complex problems w Given: p( )a 1 Excellent0.110,000 Good0.35,000 Average0.31,000 Poor0.2 400 Terrible0.1 3,000 w Evaluating CE(a 1 ) is extremely difficult w Utility theory is used for complex problems.8 CE & Coherence Counter-Example w Example shows that an incoherent criterion makes DM a perpetual money-making machine w For coherence:y = y* w Any evaluation criterion must be subjected to this coherence test w Can we use only CE criterion for all decision problems? No, only for simple 2-outcome problems.w Next, since y* = CE(a j ), DM will not mind switching from a j to y*

7 CE & Coherence Counter-Example w Given: Option a j has CE(a j ) = y*, but Under criterion C, score C(a j ) C(y*) w We can find y such that: C(a j ) = C(y),y > y* w Decision maker (DM) will pay to replace y* by y w Next, since C(a j ) = C(y), DM will not mind switching from y to a j.

6 Certainty Equivalents & Coherence w The concept of CE provides a coherent approach for evaluating (ranking) decisions w A valid criterion must recommend ranking consistent with the CE options w A coherent criterion must provide the same score for an uncertain option and its certainty equivalent.

5 Making Decisions based on CE w Given 4 choices depending on flipping a fair coin w p( )a 1 a 2 a 3 a 4 Heads0.5550700400300 Tails0.50 100100150 CE200150230220 w a 1 is chosen under minimax regret w a 2 is chosen under max EV (but highest risk) w a 3 is chosen under max certainty equivalent (combines expected value with risk) w a 4 is chosen under maximin.

$Z for certain (X Z Y) w The value of Z that makes the 2 options equal to decision maker is the: Certainty equivalent of a 1 : CE(a 1 ) w Stochastic problems can be transformed to deterministic equivalent w Criterion: select option a j with maximum CE(a j )

4 Determining & Using CE In general, given 2 choices: w a 1.

Certain payoff: $X w If X = 0, you will prefer a 1 w If X = 500, you will prefer a 2 w Certainty Equivalent: Value of X (0 X 500) that makes a 1 = a 2 Uncertain payoff: Heads: +$500, Tails: 0 w a 2.

3 Certainty Equivalents Definition w A certain payoff which has the same value to the decision maker as an uncertain payoff w a 1.

$200 for certain: price for selling a 1 w p( )a 1 a 2 Heads0.5500200 Tails0.50200 w Your sale price of a 1 is its certainty equivalence Flip a fair coin Heads: you get $500, Tails: you get nothing w a 2. Certainty Equivalents w Suppose you had 2 choices: w a 1.

1 CHAPTER 3 Certainty Equivalents from Utility Theory.

Power utility certainty equivalent