Savage's subjective expected utility model

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,^[1] based on previous work by Ramsey,^[2] von Neumann ^[3] and de Finetti.^[4]

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.

Savage's framework posits the following primitives to represent an agent's choice under uncertainty:^[1]

• A set of states of the world ${\displaystyle \Omega }${\displaystyle \Omega }, of which only one ${\displaystyle \omega \in \Omega }${\displaystyle \omega \in \Omega } is true. The agent does not know the true ${\displaystyle \omega }${\displaystyle \omega }, so ${\displaystyle \Omega }${\displaystyle \Omega } represents something about which the agent is uncertain.

• A set of consequences ${\displaystyle X}${\displaystyle X}: consequences are the objects from which the agent derives utility.

• A set of acts ${\displaystyle F}${\displaystyle F}: acts are functions ${\displaystyle f:\Omega \rightarrow X}${\displaystyle f:\Omega \rightarrow X} which map unknown states of the world ${\displaystyle \omega \in \Omega }${\displaystyle \omega \in \Omega } to tangible consequences ${\displaystyle x\in X}${\displaystyle x\in X}.

• A preference relation ${\displaystyle \succsim }${\displaystyle \succsim } over acts in ${\displaystyle F}${\displaystyle F}: we write ${\displaystyle f\succsim g}${\displaystyle f\succsim g} to represent the scenario where, when only able to choose between ${\displaystyle f,g\in F}${\displaystyle f,g\in F}, the agent (weakly) prefers to choose act ${\displaystyle f}${\displaystyle f}. The strict preference ${\displaystyle f\succ g}${\displaystyle f\succ g} means that ${\displaystyle f\succsim g}${\displaystyle f\succsim g} but it does not hold that ${\displaystyle g\succsim f}${\displaystyle g\succsim f}.

The model thus deals with conditions over the primitives ${\displaystyle (\Omega ,X,F,\succsim )}${\displaystyle (\Omega ,X,F,\succsim )}—in particular, over preferences ${\displaystyle \succsim }${\displaystyle \succsim }—such that one can represent the agent's preferences via expected-utility with respect to some subjective probability over the states ${\displaystyle \Omega }${\displaystyle \Omega }: i.e., there exists a subjective probability distribution ${\displaystyle p\in \Delta (\Omega )}${\displaystyle p\in \Delta (\Omega )} and a utility function ${\displaystyle u:X\rightarrow \mathbb {R} }${\displaystyle u:X\rightarrow \mathbb {R} } such that

${\displaystyle f\succsim g\iff \mathop {\mathbb {E} } _{\omega \sim p}[u(f(\omega ))]\geq \mathop {\mathbb {E} } _{\omega \sim p}[u(g(\omega ))],}${\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 ${\displaystyle \mathop {\mathbb {E} } _{\omega \sim p}[u(f(\omega ))]:=\int _{\Omega }u(f(\omega )){\text{d}}p(\omega )}${\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 ${\displaystyle f\in F}${\displaystyle f\in F} as if he considered only 1) his subjective probability of each state ${\displaystyle \omega \in \Omega }${\displaystyle \omega \in \Omega } and 2) the utility he derives from consequence ${\displaystyle f(\omega )}${\displaystyle f(\omega )} given at each state.

Savage posits the following axioms regarding ${\displaystyle \succsim }${\displaystyle \succsim }:^{[1] [5]}

• P1 (Preference relation) : the relation ${\displaystyle \succsim }${\displaystyle \succsim } is complete (for all ${\displaystyle f,g\in F}${\displaystyle f,g\in F}, it's true that ${\displaystyle f\succsim g}${\displaystyle f\succsim g} or ${\displaystyle g\succsim f}${\displaystyle g\succsim f}) and transitive.

• P2 (Sure-thing Principle)^{[nb 1]} : for any acts ${\displaystyle f,g\in F}${\displaystyle f,g\in F}, let ${\displaystyle f_{E}g}${\displaystyle f_{E}g} be the act that gives consequence ${\displaystyle f(\omega )}${\displaystyle f(\omega )} if ${\displaystyle \omega \in E}${\displaystyle \omega \in E} and ${\displaystyle g(\omega )}${\displaystyle g(\omega )} if ${\displaystyle \omega \notin E}${\displaystyle \omega \notin E}. Then for any event ${\displaystyle E\subset \Omega }${\displaystyle E\subset \Omega } and any acts ${\displaystyle f,g,h,h'\in F}${\displaystyle f,g,h,h'\in F}, the following holds:

${\displaystyle f_{E}h\succsim g_{E}h\implies f_{E}h'\succsim g_{E}h'.}${\displaystyle f_{E}h\succsim g_{E}h\implies f_{E}h'\succsim g_{E}h'.}

In words: if you prefer act ${\displaystyle f}${\displaystyle f} to act ${\displaystyle g}${\displaystyle g} whether the event ${\displaystyle E}${\displaystyle E} happens or not, then it does not matter the consequence when ${\displaystyle E}${\displaystyle E} does not happen.

An event ${\displaystyle E\subset \Omega }${\displaystyle E\subset \Omega } is nonnull if the agent has preferences over consequences when ${\displaystyle E}${\displaystyle E} happens: i.e., there exist ${\displaystyle f,g,h\in F}${\displaystyle f,g,h\in F} such that ${\displaystyle f_{E}h\succ g_{E}h}${\displaystyle f_{E}h\succ g_{E}h}.

• P3 (Monotonicity in consequences): let ${\displaystyle f\equiv x}${\displaystyle f\equiv x} and ${\displaystyle g\equiv y}${\displaystyle g\equiv y} be constant acts. Then ${\displaystyle f\succsim g}${\displaystyle f\succsim g} if and only if ${\displaystyle f_{E}h\succsim g_{E}h}${\displaystyle f_{E}h\succsim g_{E}h} for all nonnull events ${\displaystyle E}${\displaystyle E}.

• P4 (Independence of beliefs from tastes): for all events ${\displaystyle E,E'\subset \Omega }${\displaystyle E,E'\subset \Omega } and constant acts ${\displaystyle f\equiv x}${\displaystyle f\equiv x}, ${\displaystyle g\equiv y}${\displaystyle g\equiv y}, ${\displaystyle f'\equiv x'}${\displaystyle f'\equiv x'}, ${\displaystyle g'\equiv y'}${\displaystyle g'\equiv y'} such that ${\displaystyle f\succ g}${\displaystyle f\succ g} and ${\displaystyle f'\succ g'}${\displaystyle f'\succ g'}, it holds that

${\displaystyle f_{E}g\succsim f_{E'}g\iff f'_{E}g'\succsim f'_{E'}g'}${\displaystyle f_{E}g\succsim f_{E'}g\iff f'_{E}g'\succsim f'_{E'}g'}.

• P5 (Non-triviality): there exist acts ${\displaystyle f,f'\in F}${\displaystyle f,f'\in F} such that ${\displaystyle f\succ f'}${\displaystyle f\succ f'}.

• P6 (Continuity in events): For all acts ${\displaystyle f,g,h\in F}${\displaystyle f,g,h\in F} such that ${\displaystyle f\succ g}${\displaystyle f\succ g}, there is a finite partition ${\displaystyle (E_{i})_{i=1}^{n}}${\displaystyle (E_{i})_{i=1}^{n}} of ${\displaystyle \Omega }${\displaystyle \Omega } such that ${\displaystyle f\succ g_{E_{i}}h}${\displaystyle f\succ g_{E_{i}}h} and ${\displaystyle h_{E_{i}}f\succ g}${\displaystyle h_{E_{i}}f\succ g} for all ${\displaystyle i\leq n}${\displaystyle i\leq n}.

The final axiom is more technical, and of importance only when ${\displaystyle X}${\displaystyle X} is infinite. For any ${\displaystyle E\subset \Omega }${\displaystyle E\subset \Omega }, let ${\displaystyle \succsim _{E}}${\displaystyle \succsim _{E}} be the restriction of ${\displaystyle \succsim }${\displaystyle \succsim } to ${\displaystyle E}${\displaystyle E}. For any act ${\displaystyle f\in F}${\displaystyle f\in F} and state ${\displaystyle \omega \in \Omega }${\displaystyle \omega \in \Omega }, let ${\displaystyle f_{\omega }\equiv f(\omega )}${\displaystyle f_{\omega }\equiv f(\omega )} be the constant act with value ${\displaystyle f(\omega )}${\displaystyle f(\omega )}.

• P7: For all acts ${\displaystyle f,g,\in F}${\displaystyle f,g,\in F} and events ${\displaystyle E\subset \Omega }${\displaystyle E\subset \Omega }, we have

${\displaystyle f\succsim _{E}g_{\omega }{\text{ }}\forall \omega \in E\implies f\succsim _{E}g}${\displaystyle f\succsim _{E}g_{\omega }{\text{ }}\forall \omega \in E\implies f\succsim _{E}g},

${\displaystyle f_{\omega }\succsim _{E}g{\text{ }}\forall \omega \in E\implies f\succsim _{E}g}${\displaystyle f_{\omega }\succsim _{E}g{\text{ }}\forall \omega \in E\implies f\succsim _{E}g}.

Theorem: Given an environment ${\displaystyle (\Omega ,X,F,\succsim )}${\displaystyle (\Omega ,X,F,\succsim )} as defined above with ${\displaystyle X}${\displaystyle X} finite, the following are equivalent:

1) ${\displaystyle \succsim }${\displaystyle \succsim } satisfies axioms P1-P6.

2) there exists a non-atomic, finitely additive probability measure ${\displaystyle p\in \Delta (\Omega )}${\displaystyle p\in \Delta (\Omega )} defined on ${\displaystyle 2^{\Omega }}${\displaystyle 2^{\Omega }} and a nonconstant function ${\displaystyle u:X\rightarrow \mathbb {R} }${\displaystyle u:X\rightarrow \mathbb {R} } such that, for all ${\displaystyle f,g\in F}${\displaystyle f,g\in F},

${\displaystyle f\succsim g\iff \mathop {\mathbb {E} } _{\omega \sim p}[u(f(\omega ))]\geq \mathop {\mathbb {E} } _{\omega \sim p}[u(g(\omega ))].}${\displaystyle f\succsim g\iff \mathop {\mathbb {E} } _{\omega \sim p}[u(f(\omega ))]\geq \mathop {\mathbb {E} } _{\omega \sim p}[u(g(\omega ))].}

For infinite ${\displaystyle X}${\displaystyle X}, one needs axiom P7. This inclusion makes P3 redundant.^[8] Furthermore, in both cases, the probability measure ${\displaystyle p}${\displaystyle p} is unique and the function ${\displaystyle u}${\displaystyle u} is unique up to positive linear transformations.^{[1] [6]}

• Anscombe-Aumann subjective expected utility model
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• Expected utility
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