Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 1

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1 Session 2: Expected Utility In our discussion of betting from Session 1, we required the bookie to accept (as fair) the combination of two gambles, when each gamble, on its own, is judged fair. That is, the bookie posted fair odds for betting on events and, subject only to a concern about the magnitude of the stakes involved in the bets, we required the bookie to accept arbitrary combinations of fair bets. But, is that always your attitude towards a combinations of two fair bets? For example, might you have a different view about an even money bet on a coin landing heads at $40 stake, compared with an even money $2 bet on the same event. Coin lands heads Coin lands tails Even-money $20 bet on heads win $20 lose $20 Even-money $1 bet on heads win $1 lose $1 Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 1

2 The von Neumann Morgenstern theory of cardinal utility (1944) The goal is a theory of qualitative preference over options that admits a quantitative representation, one that captures a notion of strength of preference. That is, the goal is a theory of cardinal (rather than merely ordinal) preference. A simple lottery L is probability distribution over a finite set of rewards Rewards = {r 1,..., r n }. Write a lottery L as a sequence < p 1, p 2,..., p n > where p j 0 and j p j = 1 (j = 1,..., n). The quantity p j is the chance of winning reward r j. Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 2

3 Matrix of m-many lotteries on the set of n-many rewards. r r 1 2 j n r r L 1 p p p p j 1n L 2 p 21 p 22 p 2j p 2n L m p m1 p m2 p mj p mn Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 3

4 Von Neumann - Morgenstern theory introduces one operation for the combination of two lotteries into a third lottery. The convex combination of two lotteries is denoted by. Fix a quantity x, 0 x 1. xl 1 (1-x)L 2 = L 3 where p 3j = xp 1j + (1-x)p 2j (j = 1,..., n). You may think of as involving a compound chance where, first a coin (biased x for "heads") is flipped and, if it lands heads then lottery L 1 is run and if it lands tails then lottery L 2 is run. Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 4

5 Von Neumann - Morgenstern preference axioms. The theory is given by 3 axioms for a binary preference relation over lotteries. Axiom-1 Preference is a weak order: is reflexive L L L is transitive [L 1 L 2 L 3 ] if L 1 & L 2 L 3 then L 1 L 3 and lotteries are comparable under [L 1 L 2 ] either L 1 or L 2 L 1. Define the asymmetric and symmetric parts of in the usual way: L 1 if L 1 but not L 2 L 1 L 1 if L 1 and L 2 L 1 Axiom-2 Independence with a common lottery doesn't affect preference [L 1 L 2 L 3, 0 < x 1], L 1 if and only if xl 1 (1-x)L 3 xl 2 (1-x)L 3. Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 5

6 Axiom-3 (Archimedes) This is a technical condition to allow the use of real numbers to provide magnitudes for cardinal utilities. If L 1 L 3, then 0 < x, y < 1, xl 1 (1-x)L 3 yl 1 (1-y)L 3. Or then 0 < z < 1 zl 1 (1-z)L 3. Von Neumann - Morgenstern Theorem These three axioms are necessary and sufficient for the existence of a unique cardinal utility function U( ) on rewards, U(r j ) = u j (j = 1,.., n) such that: L 1 if and only if j p 1j u j j p 2j u j This is the Expected Utility property of cardinal utilities. Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 6

7 U is unique up to a positive linear transformation. That is, for a > 0 and b an arbitrary real number, the utility U', defined by U' = au + b is equivalent to U. Note: Strength of preference is captured by the fact that the quantity [U(L 1 ) - U(L 2 )] [U(L 3 ) - U(L 4 )] is invariant over equivalent cardinal utility functions. That is, [U'(L 1 ) - U'(L 2 )] [U'(L 3 ) - U'(L 4 )] is the same quantity whenever U' = au + b. Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 7

8 A geometric presentation of von Neumann-Morgenstern Cardinal Utility theory. Assume that r 1 r 2 r 3. Relative to a lottery L, note the region of lotteries that must be strictly preferred to L and the region of lotteries that must be strictly dispreferred to L. Stochastic dominance: Moving probability from less to more preferred rewards. r 2 L r 3 r 1 Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 8

9 Line of indifference between r 2 and L 2 = u 2 r 3 (1- u 2 )r 1. Note: If r 2 then r 2 xr 2 (1- x)l 2 r 2 r 3 L 2 r 1 Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 9

10 Parallel lines of indifference Note: Reasoning by similar triangles shows that indifference lines are parallel. r 2 r 3 L 2 r 1 Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 10

11 Choice set #1: The Allais Problem Option 1: Receive $3.5M (million dollars) for sure. Option 2: Receive $0 with chance.01; receive $3.5M with chance.90, and receive $5M with chance.09. Choice set #2: Option 3: Receive $0 with chance.90 and $3.5M with chance.10. Option 4: Receive $0 with chance.91 and $5M with chance.09. Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 11

12 If I ve picked the dollar amounts right, you have an inclination to pick Option 1 rather than Option 2, and Option 4 rather than Option 3. However, these choices cannot reflect rational (strict) preferences using a von Neumann-Morgenstern utility for money. To see this, imagine that these options are generated with a fair 100-slot roulette wheel where you win the dollar amounts, below, according to which slot results on the spin of the wheel: #1-90 #91-99 #100 Option 1 $3.5M $3.5M $3.5M Option 2 $3.5M $5 M $0 Option 3 $0 $3.5M $3.5M Option 4 $0 $5 M $0 Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 12

13 Define these three lotteries on the 3 rewards {$0, $3.5M, $5M}: Lottery L A by probs {0, 1., 0} for the 3 rewards {$0, $3.5M, $5M} Lottery L B by probs {.1, 0,.9} for the 3 rewards {$0, $3.5M, $5M} Lottery L C by probs {1., 0, 0} for the 3 rewards {$0, $3.5M, $5M}. Then, Option 1 =.9L A.1L A Option 2 =.9L A.1L B And Option 3 =.9L C.1L A Option 4 =.9L C.1L B So, Option 1 Option 2 if and only if Option 3 Option 4. Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 13

14 Class Problems Session 2. These 3 problems deal with the framework of lotteries over 3 rewards, given on slide Regarding slide 9, argue that the coefficient u 2 in the lottery L 2 = u 2 r 3 (1- u 2 )r 1 serves as the utility U(r 2 ) for reward r 2. What are the associated utilities for r 1 and r 3? 2.2 Graph the Allais Problem with the simplex for lotteries involving 3 rewards. 2.3 Using the simplex, formulate modifications of the von Neumann-Morgenstern Theory that will accommodate Allais-styled choices? Notes for Session 2, Expected Utility Theory, Summer School 2009 T.Seidenfeld 14