In decision theory, Savage's subjective expected utility model (also known as Savage's framework, Savage's axioms, or Savage's representation theorem) is a formalization of subjective expected utility (SEU) developed by Leonard J. Savage in his 1954 book The Foundations of Statistics, based on previous work by Ramsey, von Neumann and de Finetti. Savage's model concerns with deriving a subjective probability distribution and a utility function such that an agent's choice under uncertainty can be represented via expected-utility maximization. His contributions to the theory of SEU consist of formalizing a framework under which such problem is well-posed, and deriving conditions for its positive solution.
Primitives and problem Savage's framework posits the following primitives to represent an agent's choice under uncertainty:
A set of states of the world Ω {\displaystyle \Omega } , of which only one ω ∈ Ω {\displaystyle \omega \in \Omega } is true. The agent does not know the true ω {\displaystyle \omega } , so Ω {\displaystyle \Omega } represents something about which the agent is uncertain. A set of consequences X {\displaystyle X} : consequences are the objects from which the agent derives utility. A set of acts F {\displaystyle F} : acts are functions f : Ω → X {\displaystyle f:\Omega \rightarrow X} which map unknown states of the world ω ∈ Ω {\displaystyle \omega \in \Omega } to tangible consequences x ∈ X {\displaystyle x\in X} . A preference relation ≿ {\displaystyle \succsim } over acts in F {\displaystyle F} : we write f ≿ g {\displaystyle f\succsim g} to represent the scenario where, when only able to choose between f , g ∈ F {\displaystyle f,g\in F} , the agent (weakly) prefers to choose act f {\displaystyle f} . The strict preference f ≻ g {\displaystyle f\succ g} means that f ≿ g {\displaystyle f\succsim g} but it does not hold that g ≿ f {\displaystyle g\succsim f} . The model thus deals with conditions over the primitives ( Ω , X , F , ≿ ) {\displaystyle (\Omega ,X,F,\succsim )} —in particular, over preferences ≿ {\displaystyle \succsim } —such that one can represent the agent's preferences via expected-utility with respect to some subjective probability over the states Ω {\displaystyle \Omega } : i.e., there exists a subjective probability distribution p ∈ Δ ( Ω ) {\displaystyle p\in \Delta (\Omega )} and a utility function u : X → R {\displaystyle u:X\rightarrow \mathbb {R} } such that
f ≿ g ⟺ E ω ∼ p [ u ( f ( ω ) ) ] ≥ E ω ∼ p [ u ( g ( ω ) ) ] , {\displaystyle f\succsim g\iff \mathop {\mathbb {E} } _{\omega \sim p}[u(f(\omega ))]\geq \mathop {\mathbb {E} } _{\omega \sim p}[u(g(\omega ))],}
where E ω ∼ p [ u ( f ( ω ) ) ] := ∫ Ω u ( f ( ω ) ) d p ( ω ) {\displaystyle \mathop {\mathbb {E} } _{\omega \sim p}[u(f(\omega ))]:=\int _{\Omega }u(f(\omega )){\text{d}}p(\omega )} . The idea of the problem is to find conditions under which the agent can be thought of choosing among acts f ∈ F {\displaystyle f\in F} as if he considered only 1) his subjective probability of each state ω ∈ Ω {\displaystyle \omega \in \Omega } and 2) the utility he derives from consequence f ( ω ) {\displaystyle f(\omega )} given at each state.
Axioms Savage posits the following axioms regarding ≿ {\displaystyle \succsim } :
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