States, Events, Outcomes, and Acts

States, Events, Outcomes, and Acts
From a mathematical perspective, the representation of uncertainty by means of additive, numerical probabilities allows us to apply the tremendous body of analytical results of modern probability theory (e.g., Feller 1968, 1971; Billingsley 1986). But from a modeling perspective,
the assumption that uncertainty comes prepackaged with well-defined, measurable “objective” probabilities is unrealistic. Outside
of the gambling hall, most economic decisions and transactions involving uncertainty—investment decisions, search decisions, insurance
contracts, financial instruments—are defined in terms of uncertain
events rather than numerical probabilities.
This approach to representing uncertainty and uncertain pros-pects—formalized by Savage (1954) and now known as the subjective approach—involves the following basic constructs:
χ = {..., x , ...} an arbitrary space of outcomes or consequences.
26 Machina
S = {..., s , ...}
E = {..., E , ...} ƒ(⋅) = [x1 on E1;...; x
m
A = {...,ƒ(⋅),...} W(⋅) and

a space of mutually exclusive and exhaustive states of nature, representing all possible alternative unfoldings of the world. an algebra of events, each a subset of S.
on Em] subjective act yielding outcome xi in event Ei, for some partition {E1,...,Em} of S (or equivalently, yielding outcome ƒ(s) in state s). the set of all such subjective acts. the individual’s preference function and corresponding preference relation over A.
A wonderful example of the use of this framework to represent an uncertain decision was provided by Savage (1954, pp. 13–15): Say you are making omelets and have already broken five of your six eggs into a mixing bowl. The decision you must make is: Do you break the sixth egg? The uncertainty arises from the fact that this sixth egg has been around for some time and might be rotten. You can either break this egg into the bowl with the other eggs, break it into a separate saucer to inspect it, or throw it away unbroken. Savage represents this problem in terms of states, acts, and outcomes by means of the following table:
State
Act
Egg is good
Egg is rotten
Break into bowl
Six-egg omelet
No omelet, and five good
eggs destroyed
Break into saucer
Six-egg omelet, and
Five-egg omelet, and
a saucer to wash
a saucer to wash
Throw away
Five-egg omelet, and
Five-egg omelet
one good egg destroyed