10_aa - 10 Anscombe and Aumann Expected Utility Let be a...

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10 Anscombe and Aumann Expected Utility Let Ω be a finite set of m states of the world, with generic elements s, t Ω. Let X be a finite set of n consequences with a generic element x X . It will often be useful to identify a state s with its index and enumerate Ω = { 1 , 2 , . . . , m } . Let H = (Δ X ) Ω denote the space of all functions from Ω to Δ X ; such functions are also called (Anscombe–Aumann) acts. So a generic element h H = (Δ X ) Ω is a function h : Ω Δ X from Ω to Δ X , hence assigns each state s Ω a lottery h ( s ) Δ X . For ease of notation, we let h s = h ( s ) Δ X , so h s ( x ) = [ h ( s )]( x ) [0 , 1]. One way to think of h s ( x ) is as the probability of the consequence x conditional on the state s , i.e. h s ( x ) represents Pr( x | s ), given the act h . For any act h X ) Ω , state s Ω, and lottery π Δ X , define the new act ( h - s , π ) : Ω Δ X by ( h - s , π ) = ( h 1 , . . . , h s - 1 , π, h s +1 , . . . , h m ). So [( h - s , π )]( t ) = ( π if t = s h ( t ) if t 6 = s ; or, in our subscript notation, ( h - s , π ) s = π and ( h - s , π ) t = h t for all states t 6 = s . In words ( h - s , π ) replaces h s , the lottery that the act h assigns to state s , with the lottery π ; the other lotteries are kept the same. In a slight abuse of notation, for any lottery π Δ X , we let π also denote the constant act f : Ω Δ X such that f ( s ) = π for all s Ω. H is a convex subset of the space of functions from Ω to R n , H = (Δ X ) Ω ( R n ) Ω , where if f, g : Ω R n are functions from Ω to R n , then the function αf + βg : Ω R n is defined by the [ αf + βg ]( s ) = αf ( s ) + βg ( s ). The classic interpretation of H is as follows. We can consider elements of Δ X as bets on an objective roulette, where the probabilities of outcomes are physically determined. Consider each of the world as the event that a specific horse, named s , wins a race in a field of horses Ω, where different people can have different assessments of each horse’s strength. Our aim is to identify the decision maker’s personal assessment of the probability that horse s will win the race. To do so, we allow the payout on horse s to be another lottery that depends on the outcome of a roulette spin. So, we first let the horses run and then we spin the roulette. The payoff on the roulette depends on which horse wins. There are at least three natural mathematical representations of the space H : H = (Δ X ) Ω . This is the original mathematical interpretation. For example, suppose Ω = { s 1 , s 2 , s 3 } and X = { x 1 , x 2 , x 3 } . Then a particular h : Ω Δ X would be the following: h ( s 1 ) = (0 . 3 , 0 . 2 , 0 . 5) h ( s 2 ) = (0 . 4 , 0 . 6 , 0) h ( s 3 ) = (0 , 1 , 0) 56
If π = (0 . 5 , 0 . 4 , 0 . 1), then ( h - s 2 , π ) would be: [( h - s 2 , π )] ( s 1 ) = (0 . 3 , 0 . 2 , 0 . 5) [( h - s 2 , π )] ( s 2 ) = (0 . 5 , 0 . 4 , 0 . 1) [( h - s 2 , π )] ( s 3 ) = (0 , 1 , 0 , 0) Another way to represent H is as a set of compound lotteries, where the subjective first stage lottery is over which state s Ω obtains, and the objective second stage lottery, which is conditional on s , is over which x X finally obtains. These compound lotteries can be written as probability trees. For example, h and can be denoted: s 1 s 2 s 3 .

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