Subjective Expected Utility Theory with Costly Actions - Economics ...

a¤ect the consequences associated with some states do not arise exclusively in
the context of principal-agent relationships. Indeed, such situations are common
in economic analysis and often arise even when the decision maker acts in
isolation. For instance, Robinson Crusoe may have to choose between building
his hut on a ‡ood plane or on high grounds and thereby, at a cost, a¤ect the
consequences of ‡ooding. Robinson Crusoe’s choice is not adequately captured
by traditional subjective expected utility theory because in that theory (e.g.,
Savage (1954)) the choice set includes all acts (that is, all assignments of consequences
to states) and, more importantly, acts do not enter the preferences
directly. This framework is inadequate to describe the choice facing Crusoe.
Building his hut on high grounds involves a larger expense of time and e¤ort.
Moreover, because the time and e¤ort to be spent must be determined in advance,
they cannot vary with the state of nature (e.g., they cannot depend on
the amount of precipitation). Put di¤erently, to …t into Savage’s framework,
the time and e¤ort spent building the hut must be incorporated into the description
of the consequences and the set of acts must include all assignments
of consequences to states. The problem is that the decision of where to locate
the hut must be made prior to the realization of the state yet acts, such as f
described below, must be contemplated:
States Flood No Flood
f
Hutlostina‡ood.
Little time and e¤ort spent
building hut on ‡ood plan
Hut stands.
Signi…cant amount of time and e¤ort spent
building hut on high ground
Put in more general terms, let X be the set of outcomes and denote by A the
set of actions (e.g., the hut damaged by a ‡ood is a consequence and time and
e¤ort spent is an action). If consequences are de…ned as action-outcome pairs
then Savage’s set of acts is given by (A £ X) S ; where S denotes the set of states
of nature. But, as the above example suggests, this set contains acts that are
incredible. To suppose that the decision maker is able to express meaningful
preferences as regards such acts creates a conceptual di¢culty with the theory. 2
This di¢culty is averted if, for instance, the only cost of actions is the
…nancial cost and the consequences represent …nancial rewards then actions are
indeed labels of the corresponding acts. To see this, let c (a) denote the …nancial
cost of implementing the action a and denote by x (s; a) the state-contingent
action-dependent monetary payo¤, then the state-contingent payo¤ function
xa (¢) =x (¢;a)¡c(a) is a Savage-type act and a is its label. In this example the
fact that the cost of action and the reward are additive is essential. However,
if the action involves spending time (e.g., while seeking employment) and the
state-contingent payo¤ is monetary (e.g., wages or unemployment insurance
bene…ts depending on the eventual state of employment) then, not being perfect
substitutes, actions must be separated from acts.
2 I am indebted to Steve Matthews for suggesting this discussion of Savage’s theory.
2