Outcomes, Probabilities, and Objective Lotteries

Outcomes, Probabilities, and Objective Lotteries
The earliest formal representation of uncertainty came from founders of modern probability theory such as Pascal and Fermat. In this approach, the uncertainty attached to any event is represented by a numerical probability p between 0 and 1. Because probability theory derived from the study of games of chance that involved virtually identical
repeated events, such probabilities were held to be intrinsic properties
of the events in the sense that an object’s mass is an intrinsic property of the object. These probabilities could either be calculated from the principles of combinatorics, for an event such as being dealt a royal flush, or measured by repeated observation, for an event like a bent coin landing heads up.
For an individual making a decision under objective uncertainty, the objects of choice are objective lotteries of the form P = (x1, p1;...; xm, pm), which yield outcome xi with objective probability pi, where p1 + … + pm = 1. The theory of choice under uncertainty treats lotteries in a manner almost identical to the way it treats commodity bundles under certainty. That is, each individual’s preferences over such lotteries can be represented by a real-valued preference function
∗ ∗∗
V(⋅), in the sense that for any pair of lotteries P* =( x1 ∗ , p ;...; , p
1 xm∗ m∗)
and P = (x1, p1;...; xm, pm), the individual prefers P* over P if and only if V(P*) = V (, p ∗;...; x ∗ m∗, p ∗ m∗) exceeds V(P) = V(x1, p1;...; xm, pm), and is
∗
x11
indifferent between the two lotteries if and only if V(P*) =
∗ ∗∗
( x1, p1 ∗;...; xm∗, pm∗)exactly equals V(P) = V(x1, p1;...; xm, pm).1