Introducing Action Costs EUIS0l=62-5=57 SO EU2|S0=74-25=49 Euso=maxEulso], Eu2 So 57 Al A2 5 S1 S2 S3 S4 0.2 0.70.20.1 .8 100 50 70 MLA 2011, Tsinghua University
Introducing Action Costs s0 s1 s2 s3 A1 0.2 0.7 0.1 100 50 70 A2 s4 0.2 0.8 80 • EU1[S0] = 62 – 5 = 57 • EU2[S0] = 74 – 25 = 49 • EU[S0] = max{EU1[S0], EU2[S0]} = 57 -5 -25 11 MLA 2011, Tsinghua University
Theoretical justification of meu ◆[ Theorem]: Assume we have an agent whose preferences over lotteries satisfy the axioms (A1)-(A6). Then there exists a function U: O-R, such that for any Pair of lotteries T, T', we have that t < 'iff U(r)<U(T') here we definerecursively) the expected utility of any lottery as 1 丌k:k )=∑aU(x) =1 MLA 2011, Tsinghua University
Theoretical Justification of MEU [Theorem]: Assume we have an agent whose preferences over lotteries satisfy the axioms (A1)-(A6). Then there exists a function , such that for any pair of lotteries , we have that iff , where we define (recursively) the expected utility of any lottery as 12 MLA 2011, Tsinghua University
Postulates of Preferences/Rationality ◆(A1) Orderability (x1x丌2)∨(1<丌2)V(丌1~丌2) ◆(A2) Transitivity (丌1>丌2)∧(丌2>丌3)→(丌1>丌3 ◆(A3) Continuity 1 2 a∈(0,1),丌2~[丌1:a:;73(1-a) ◆(A4) Monotonicity (x1>丌2)∧(a>B)→{x1:a;2:(1-a)>[r1:;丌2:(1-6) ◆(A5) Substitutability (丌1~丌2)∧a∈(0,1)→{m1:a;丌3:(1-a)~[x2:a:;a(1-a) ◆(A6) Decomposability ;x2:B;73:(1- MLA 2011, Tsinghua University 1:c;2:(1-a)6;丌3:(1-a)(1-6)
Postulates of Preferences/Rationality (A1) Orderability (A2) Transitivity (A3) Continuity (A4) Monotonicity (A5) Substitutability (A6) Decomposability 13 MLA 2011, Tsinghua University
Utility Function ◆ Existence proved ◆ How about its form? ◆ a utility function a assigns numeric values to possible outcomes a most obvious is monetary gain /loss e What's the utility of money a is it linear of money? MLA 2011, Tsinghua University
Utility Function Existence proved How about its form? A utility function ❑ assigns numeric values to possible outcomes ❑ most obvious is monetary gain/loss What’s the utility of money? ❑ is it linear of money? 14 MLA 2011, Tsinghua University