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.

Primitives and problem

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

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

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

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

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

The model thus deals with conditions over the primitives ${\displaystyle (\Omega ,X,F,\succsim )}$ —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 ${\displaystyle \Omega }$ : i.e., there exists a subjective probability distribution ${\displaystyle p\in \Delta (\Omega )}$  and a utility function ${\displaystyle u:X\rightarrow \mathbb {R} }$  such that

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

where ${\displaystyle \mathop {\mathbb {E} } _{\omega \sim p}[u(f(\omega ))]:=\int _{\Omega }u(f(\omega )){\text{d}}p(\omega )}$ .

The idea of the problem is to find conditions under which the agent can be thought of choosing among acts ${\displaystyle f\in F}$  as if he considered only 1) his subjective probability of each state ${\displaystyle \omega \in \Omega }$  and 2) the utility he derives from consequence ${\displaystyle f(\omega )}$  given at each state.

Axioms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Savage's representation theorem

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

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

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

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

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

See also

• Anscombe-Aumann subjective expected utility model
• von Neumann-Morgenstern utility theorem

Notes

1. Referring to axiom P2 as the sure-thing principle is the most common usage of the term,^[6] but Savage originally referred to the concept as P2 in conjunction with P3 and P7,^[1] and some authors refer to it just as P7.^[7]

References

1. Savage, Leonard J. (1954). The Foundations of Statistics. New York: John Wiley & Sons.
2. Ramsey, Frank (1931). "Chapter 4: Truth and Probability". In Braithwaite, R. B. (ed.). The Foundations of Mathematics and Other Logical Essays. London: Kegan Paul, Trench, Trubner, & Co.
3. von Neumann, John; Morgenstern, Oskar (1944). Theory of Games and Economic Behavior. Princeton University Press. ISBN 978-0691130613.
4. de Finetti, Bruno (1937). "La prévision : ses lois logiques, ses sources subjectives". Annales de l'Institut Henri Poincaré. 7 (1): 1–68.
5. Abdellaoui, Mohammed; Wakker, Peter (2020). "Savage for dummies and experts". Journal of Economic Theory. 186 (C). doi:10.1016/j.jet.2020.104991. hdl:1765/123833.
6. Gilboa, Itzhak (2009). Theory of Decision under Uncertainty. New York: Cambridge University Press. ISBN 978-0521741231.
7. Kreps, David (1988). Notes on the Theory of Choice. Westview Press. ISBN 978-0813375533.