Rational Choice
Sociology
Lecture 2:
The Explication of the Concept of Rational
Action under Certainty and under Risk in the
Rational Choice Theory
Rational Choice under
Certainty

Differently from psychological theories of behavior RCT is formal, or
logico-mathematical theory. As in many other logico-mathematical
theories (e.g. geometry), the concept of rational action is defined in
axiomatic way:
X is rational action if (and only if) X satisfies (or corresponds, does not
violates) axioms
(1), (2), (3)... (n).
Like statements of logical-mathematical theories, those of RCT are apriorical,
i.e. not DIRECTLY related to empirical experience. If the observed
behavior does not corresponds to axioms od RCT, ir means that it is
irrational, not that RCT is false. Insofar as the behaviour of people
sometimes (how often – open question) corresponds to RCT, RCT can
work or be applied as empirical theory. When observed behaviour violates
axioms of RCT, it serves as normative theory: as a background for
identification of irrationalites in human behaviour (if you are not able to
say what is rational to do in some situation, how you can distinguish
“rational” from “irrational” behaviour)?
Rational Choice under
Certainty
Why accept axioms?
 Self-evident statements (if you understand them, you “see” that
they are true); they express our preanalytical intuitions (insights)
about rationality
 Sufficient to derive (prove) theorems
Preferences which satisfy (or correspond to) axioms are consistent.
Rational behavior (behavior driven by consistent axioms) is
consistent behavior. So the axioms are similar to rules of formal
logic which describe conditions of consistent (not selfcontradictory) thinking. If somebody violates rules of logic, this
does not refute formal logic, but means that that violator says selfcontradictory, absurd things
Some theoreticians call RCT “logic of choice”; Vilfredo Pareto
(famous Italian economist and sociologist) called rational action
“logical action”
Rational Choice under
Certainty:
Axioms
Axioms of the RCT for choice under certainty describe formal
properties of the preferences of an actor who can fully and truly
foresee all outcomes of each alternative action. In such situation
the choice among actions is equivalent to the choice between
outcomes.
(1) Reflexivity: xi ~xi
Due to this equivalence “xi” can be read both “action xi” and “the
outcome of an action xi”.
“~” means “as good as”; “indifferent”; “>” means “better”.
Reflexivity means the same as 1=1, A=A; i.e. “the value of xi= the
value of xi”
(2) Completeness: for each xi and xj from the feasible set, either xi
≥xj, or xi ≤xj, or xi ~ xj.
xi ≥ xj means “xi is as good as or better than xj”; xi ≤ xj means “xi is as
good or worser than xj”.
(3) Transitivity: for each xi, xj, xk, if xi ≥ xj, and xj ≥ xk, then xi ≥ xk
Rational Choice under
Certainty:
Definition
If preferences of an actor satisfy axioms (1)-(3), then for each xi from feasible
choice the utility function U(x) is defined which assigns for each xi an utility
index ui.
If preferences satisfy only axioms (1)-(3), then utility indexes are ordinal
numbers (or utility is a variable measured at ordinal level): they mean simply
the place of an action (or its outcome) from the worst to the best in the order
of actor’s preferences. No mathematical operations with them make sense.
To have consistent preferences means simply to able to arrange all
alternatives of choice from the worst to the best. To choose rationally to
choose the alternative with greatest or maximal utility index; i.e. to maximize
utility.
“Maximize the utility” means “choose the best”, or “optimize”. Utility in formal
sense does not mean “pleasure”, “money” etc.
The definition simply says simply that to “choose the best” means to have
consistent preferences and act according to them or because of them
Rational behavior = behavior caused or driven by consistent (ordered in the way
described by axioms) preferences
Some say : rational behaviour is simply “consistent behavior” (caused or as if
caused by consistent preferences)
Application of RCT in
Economics:
axiom of continuity
When RCT is applied in (neoclassical) Economics, the definition of
rational choice is supplemented by axiom
(4) of Continuity
There are two equivalent formulations:
(4a): preferences are continuous, if for every x, y, z such that xi >yi >
zi there exists combination of x and z (say, {xk,zk}) such that actor
is indifferent between yi and {xk zk}):
(4b): preferences are continuous, if for every xi and yi, such that xi>yi,
there exists the quantity of y yk such that yk>yi and yk>xi
i.e. preferences are continuous, if by increasing the quantity of the
worser alternative one can reverse the order of preferences
Why continuity axiom is
important for Economics?



If preferences are continuous, then utility indexes are cardinal numbers
(=utility is variable, measurable at least at interval level). So, mathematical
operations like addition or subtraction are possible
One can express or measure the utility of one outcome by the utilities of
other outcomes, i.e. to find out what is their price (some say economics are
about things items that have price; only exchangeable things have prices;
price is ratio of exchange ; for things to have prices money is not necessary;
there are relative prices and monetary prices)
If preferences doesn’t satisfy axiom (4), this means that they are
discontinuous or lexicographic: in the feasible set, there are “priceless”
goods such that an actor will not trade or barter for no matter how much
large amount of other goods (including money).
Do you have such “priceless goods”? Are there things that you will never do no
matter how much other valuable goods will be proposed in exchange?
If yes, your preferences are lexicographic!
See for more:
Norkus Z. Max Weber ir racionalus pasirinkimas. V.: Margi raštai, 2003, sk. 8 (pp.
255-268); sk. 15.3.1-15.3.2 (pp. 396-405) or
Norkus Z. Max Weber und Rational Choice. Marburg: Metropolis Verlag, 2001 S.
288-302; 444-456.
Application of RCT in (neoclassical) Economics:
substantive assumptions
Besides the axiom of continuity, application of RCT in (neoclassical) economics
involves supplementing of 4 formal axioms by some substantive axioms, that
transform RCT from purely formal logic of choice into empirical theory
(1)
Self-interest: actors are indifferent to the consequences of their choices for the
welfare of other people (externalities).
Assumption of self-interest is not a part of the definition of rational bevaviour in RCT.
RCT doesn’t say rational behaviour=self-interested behaviour or self-interested
consistent behavior. Pure or “thin” (See next slide) RCT says just that rational
behaviour is consistent, i.e. driven by reflexive, complete, transitive and maybe
continuous preferences but says nothing about their content. An altruist, if rational,
also maximizes utility. How often people choose self-interestedly, and how often
(negatively and positively) altruistically, is an empirical question.
(2)
Decreasing marginal utility (in the theory of consumer’s choice)
(3)
Decreasing marginal productivity (in the theory of producer’s choice, where the
concept of production function is central concept)
(4)
Insatiability of consumer wants
(5)
Scarcity of resources
(6)
Specification what “utility” means. E.g., in the model of the behaviour of the producer
in the competitive market: profit maximization.
Central problem in (micro)economics: which allocation of scarce resources among
alternative uses is optimal (maximizes the utility of an actor).
In (rational) consumer choice theory: given the budget constraints and consumer wants
(preferences), which allocation of the budget maximizes consumer’s satisfaction?
In (rational) producer choice theory: under given prices for production factors and produced
goods, capital and technology, what output to produce, with which inputs to produce
and how much to produce to maximize the profit?
“Thin” and “thick” concepts of rationality
In the literature on the rational choice, one can find distinction between “thin” and “thick”
concepts of rationality.
 Thin concept of rationality is formal, or logico-mathematical; it is defined only by
conditions of consistency of the preferences and probabilistic beliefs (in the case of the
choice under risk);
 Thick concept of rationality contains, in addition, more or less substantive assumptions
about the content of the preferences
 Depending on character of these assumptions there are several ways to get
“thick”concept of rationality (e.g. in neoclassical economics)
 Generally, applying RCT to explain empirical behavior, some assumptions about the
content of preferences usually are made (=“thickening” of thin rationality).
 Importantly, the questions, which preferences are good, which bad doesn’t belong to
RCT. It is the question of topics of ethics (is it good to maximize profit? Consumer
satisfaction? To maximize votes (for politicians)? etc.) Should people care about others?
Or behave in strictly self-interested way?
RCT doesn’t discuss, what our ultimate ends should be; only what is to be done given
some specific set of consistent preferences.
 However, maybe one can choose not only according preferences, but the preferences
themselves? Possibly yes, but the metapreferences should be assumed etc. See more:
Norkus Z. "Apie plonąjį praktinį racionalumą ir jo pastorinimus". In: Problemos. 1998. Nr.
54. P. 39-53 (Corrected version of the paper should be used!); Norkus Z. Max Weber ir
racionalus pasirinkimas. V.: Margi raštai, 2003, pp. 185-191; 196-201, 228-230
 Maybe there are some “last preferences” common for all people? Which ones? On these
difficult topics
See Norkus Z. “Pirmenybių endogenizacijos problema racionalaus pasirinkimo teorijoje”
Seminarai 2001. Atviros visuomenės kolegija. V.: Strofa, 2002 pp.29-37
Becker G. S. Accounting for Tastes. Cambridge, Mass.: Harvard UP, 1996. Ch.1 “Preferences
and values“, Ch. 2 "De Gustibus Non Est Disputandum", p. 1 -49.

Rational choice under risk: the concept of
prospect
 Choice under certainty (perfect foreknowledge or prediction)
happens, but perhaps rather rarely. Therefore, many theorists
doesn’t consider it very important or interesting, and focus on the
choice under risk. Some of them assert that choice under certainty
may be considered as a limiting case of the choice under risk. So
(they maintain), theory of the RCT under risk is more general, and
if we have satisfactory explication of the more complex concept of
the rational choice under risk, then this explanation will imply as its
part also what it means to choose rationally under certainty
 Choosing under risk, an actor is not able foresee truly and fully the
consequences of alternative actions. There is no equivalence
between the choice among the actions and the choice among the
outcomes. Depending on the circumstances (not known for an
actor in advance), an action can lead to different outcomes.
However, choosing under risk, an actor is able to estimate relative
probabilities of outcomes for each action. So she chooses among
prospects that are associated with action. Her preferences have
as their objects the prospects. Prospect is the set of probable
outcomes of an action (something like lottery)
Choice under risk: the concept of prospect
(an example)

Jonas is businessman and has an urgent need to come from Vilnius to Svetlovsk
(somewhere in Russia). There are two possibilities to travel: by train and by plane. If he
takes train he will arrive to Svetlovsk after 7 hours (no matter which weather) . If he
takes the plane he will be Svetlovsk after 2 hours if good weather over Svetlovsk, but it
will take 16 hours if because of bad weather the plane will not be able to land in due time
in Svetlovsk. The probability of good weather over Svetlovsk believed by John is p=0,8;
that of bad 1-p=0,2. Considerations of the the price of the ticket and travel comfort do
not matter.
There are two ways to model choice situation. The first is decision table or
matrix. It consists of 4 submatrixes:
1) Outcome matrix, including two prospects train {r11, r12}, plane (r21, r22)
Good
weather
Bad
weather
To take train
r11= 7 hours
r12= 7 hours
To take plane
r21= 2 hours
r22= 16 hours
Choice under risk: probability and utility
matrixes
Good weather Bad weather
Train
p11=0,8
p12=0,2
Plane
p21=0,8
p22=0,2
Train
Plane
Good weather
Bad weather
u11=-7
u21=-2
u12=-7
u22=-16
Choice under risk: expected utility matrix –
derived from utility matrix by weighting utility
with probability
Good
weather
Bad
weather
Take train
eu11= -7×0,8=
-5,6
eu12= -7×0,2=
-1,4
eu (train) =
eu11 + eu11=
-7
Take plane
eu21= -2×0,8=
-1,6
eu22=-16 ×0,2=
-3,2
eu (plane)=
eu21+ eu22 =
-4,8
Choice under risk: combined
decision matrix
Good weather
Bad weather
Take train
r11= 7 hours
p11=0,8
u11=-7
eu11= -5,6
r12= 7 hours
p12=0,2
u12=-7
eu12= -1,4
eu (train) =
eu11 + eu11=
-7
Take plane
r21= 2 hours
p21=0,8
u21=-2
eu21=-1,6
r22= 16 hours
p22=0,2
u22=-16
eu22=-3,2
eu (plane)=
eu21+ eu22 =
-4,8
Modeling rational choice under risk: decision
tree method (generally, more commendable)
Chance node
r11 ;
p11=0,8; u11=-7; eu11=-5,6
train
eu (train)=-7
r12; p12=0,2; u12=-7; eu12=-1,4
Actor
(decision
node)
Chance
node
plane
r21; p21=0,8; u21=-2; eu21=-1,6
eu(plane)
=-4,8
r22; p22=0,2; u22=-16; eu22=-3,2
The concept of the
rational choice under risk

To behave rationally under risk is to maximize expected utility; to choose the
action with the prospect that has the maximal possible expected utility.

This is so-called Bayesian rule (not to conflate with Bayesian theorem which
is the rule of the rational learning from experience; see Norkus Z. „Tikėjimas:
racionalaus pasirinkimo teorijos perspektyva“, kn. Tikėjimo prieigos. Sud. N.
Putinaitė. V.:Aidai, pp. 92-140 or Elster J. Explaining Social Behavior. More
Nuts and Bolts for the Social Sciences. Cambridge: Cambridge UP, 2007, Ch.
7, pp. 124-144, Ch.11, pp. 202-206.

However, strictly speaking, what was maximized by Jonas in this example,
was not the expected utility (eu) , but expected value (ev); expected
utility=expected value if actors attitudes to risk are neutral. Normally this is
not the case. Besides, we got the u-values making simplifying assumption
that Jonas cares only about the travelling time. Again this is not the case.
Also, p values were simply assumed. Can one measure the u values and p
values of an actor, instead of assuming or postulating them in more or less
arbitrary way?
This the subject of next lecture