Positive political Theory:
an introduction
General information
Credits: 6 (40 hours) for both EPS curricula (EPA&PPP); 3 (20
hours) for Ph.D Students in Political Studies (Political Science)
Period: 22th September - 1st December (no classroom 24th Sept)
Instructor: Francesco Zucchini ([email protected] )
Office hours: Tuesday 15.30-17.30, room 308, third floor, Dpt.
Studi Sociali e Politici
1
Course: aims, structure, assessment
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The course is an introduction to the study of politics from a
rational choice perspective.
Students are introduced both to the analytical tools of the
approach and to the results most relevant to the political
science. We will focus on the institutional effects of decisionmaking processes and on the nature of political actors in the
democracies.
All students are expected to do all the reading for each class
session and may be called upon at any time to provide
summary statements of it.
Evaluation of all students is based upon the regular
participation in the classroom activities (30%) and a final
written exam.
Evaluation of Ph.Students is also based upon individual
presentations (30%).
2
Topics (in yellow also for Ph.D students)
Positive political
Theory: An
introduction
Lecture 1: Epistemological foundation of the
Rational Choice approach
Francesco Zucchini
4
What the rational choice is not
“NON RATIONAL CHOICE THEORIES
Theories with non rational actors:
Theories without actors:
•System analysis
•Structuralism
•Functionalism (Parsons)
•Relative deprivation theory
•Imitation instinct (Tarde)
•False consciouness (Engels)
•Inconscient pulsions (Freud)
•Habitus (Bourdieu)
What the rational choice is
Weak Requirements of Rationality:
1) Impossibility of contradictory beliefs or
preferences
2) Impossibility of intransitive preferences
3) Conformity to the axioms of probability
calculus
Weak requirements of Rationality
1) Impossibility of contradictory beliefs or
preferences:
if an actor holds contradictory beliefs she cannot
reason
if an actor hold contradictory preferences she can
choose any option
Important: contradiction refers to beliefs or
preferences at a given moment in time.
Weak requirements of Rationality
2) Impossibility of intransitive preferences:
if an actor prefers alternative a over b and b over c ,
she must prefer a over c .
One can create a “money pump” from a person with
intransitive preferences.
Person Z has the following preference ordering:
a>b>c>a ; she holds a. I can persuade her to
exchange a for c provided she pays 1$; then I can
persuade her to exchange c for b for 1$ more;
again I can persuade her to pay 1$ to exchange b
for a. She holds a as at the beginning but she
is $3 poorer
Weak requirements of Rationality
3) Conformity to the axioms of probability
calculus
A1 No probability is less than zero. P(i)>=0
A2 Probability of a sure event is one
A3 If i and j are two mutually exclusive events, then
P (i or j)= P(i )+P(j)
A small quantity of formalization...
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A choice between different alternatives
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Each alternative can be put on a nominal, ordinal o
cardinal scale
The choice produces a result
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S = (s1, s2, … si)
R = (r1, r2, … ri)
An actor chooses as a function of a preference
ordering relation among the results. Such ordering is
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complete
transitive
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Utility
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A ( mostly) continuous preference ordering
assigns a position to each result
We can assign a number to such ordering
called utility
A result r can be characterized by these
features (x,y,z) to which an utility value u =
f(x,y,z) corresponds
Rational action maximizes the utility function
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Single-peak utility functions
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One dimension (the real line)
Actor with ideal point A, outcome x
A
Linear utility function:
+
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U = - |x – A|
U
Quadratic utility function: +

x
A
x
U = - (x – A)2
U
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Expected utility
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There could be unknown factors that could come in
between a choice of action and a result
.. as a function of different states of the world M =
(m1, m2, … mi)
Choice under uncertainty is based associating
subjective probabilities to each state of the world,
choosing a lottery of results L = (r1,p1;r2,p2; … ri,pi)
We have then an expected utility function
EU = u(r1)p1+u(r2)p2+ … u(ri)pi
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Strong Requirements of Rationality
1) Conformity to the prescriptions of game
theory
2) Probabilities approximate objective
frequencies in equilibrium
3) Beliefs approximate reality in equilibrium
Strong Requirements of Rationality
1) Conformity to the prescriptions of game
theory: digression..
 Uncertainty between choices and outcomes
could also result from the (unknown)
decisions taken by other rational actors
 Game theory studies the strategic
interdependence between actors, how one
actor’s utility is also function of other actors’
decisions, how actors choose best strategies,
and the resulting equilibrium outcomes
15
Principles of game theory
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Players have preferences and utility functions
Game is represented by a sequence of moves
(actors’ – or Nature – choices)
How information is distributed is key
Strategy is a complete action plan, based on the
anticipation of other actors’ decisions
A combination of strategies determines an outcome
This outcome determines a payoff to each player,
and a level of utility (the payoff is an argument of the
player’s utility function)
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Principles of game theory (2)
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Games in the extensive form are represented
by a decision tree
which illustrates the possible conditional
strategic options
The distribution of information:
complete/incomplete (game structure),
perfect/imperfect (actors’ types), common
knowledge
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Principles of game theory (3)
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Solutions is by backward induction, by
identifying the subgame perfect equilibria
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Nash equilibrium: the profile of the best
responses, conditional on the anticipation of
other actors’ best responses
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A Nash equilibrium is stable: no-one
unilaterally changes strategy
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Strong Requirements of Rationality
2) Subjective probabilities approximate
objective frequencies in equilibrium.
Every “player” makes the best use of his
previous probability assessments and the new
information that he gets from the environment.
Beliefs are updated according to Bayes’s rule.
Strong Requirements of Rationality
Bayesian updating of beliefs
• P(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it does
not take into account any information about B.
• P(A|B) is the conditional probability of A, given B. It is also called the posterior probability
because it is derived from or depends upon the specified value of B.
• P(B|A) is the conditional probability of B given A.
• P(B) is the prior or marginal probability of B
Strong Requirements of Rationality
3) Beliefs should approximate reality
Beliefs and behavior not only have to be
consistent but also have to correspond with
the real world at equilibrium
Rational Choice:
only a normative theory ?
Usual criticism to the Rational Choice theory:
In the real world people are incapable of making all the required calculations and
computations. Rational choice is not “realistic”
Usual answer (M.Friedman): people behave as if they were rational:
“In so far as a theory can be said to have “assumptions” at all, and in so far as their “realism” can be
judged independently of the validity of predictions, the relation between the significance of a theory and
the “realism” of its “assumptions” is almost the opposite of that suggested by the view under criticism.
Truly important and significant hypotheses will be found to have “assumptions” that are wildly inaccurate
descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic
the assumptions (in this sense). The reason is simple. A hypothesis is important if it “explains” much by
little, that is, if it abstracts the common and crucial elements from the mass of complex and detailed
circumstances surrounding the phenomena to be explained and permits valid predictions on the basis of
them alone. To be important, therefore, a hypothesis must be descriptively false in its assumptions;
it takes account of, and accounts for, none of the many other attendant circumstances, since its
very success shows them to be irrelevant for the phenomena to be explained.
As if argument claims that the rationality assumption, regardless of its accuracy,
is a way to model human behaviour
Rationality as model argument
(look at Fiorina article)
Rational Choice:
only a normative theory ?
Tsebelis counter argument to “rationality as model argument” :
1)“the assumptions of a theory are, in a trivial sense, also conclusions
of the theory . A scientist who is willing to make the “wildly inaccurate”
assumptions Friedman wants him to make admits that “wildly inaccurate”
behaviour can be generated as a conclusion of his theory”.
2) Rationality refers to a subset of human behavior. Rational choice
cannot explain every phenomenon. Rational choice is a better approach
to situations in which the actors’ identity and goals are established and
the rules of interaction are precise and known to the interacting agents.
Political games structure the situation as well ; the study of political actors
under the assumption of rationality is a legitimate approximation of
realistic situations, motives, calculations and behavior.
5 arguments
Five arguments in defense of the Rational
Choice Approach (Tsebelis)
1)
2)
3)
4)
5)
Salience of issues and information
Learning
Heterogeneity of individuals
Natural Selection
Statistics
Five arguments in defense of the Rational
Choice Approach (Tsebelis)
3) Heterogeneity of individuals: equilibria with some
sophisticated agents (read fully rational) will tend
toward equilibria where all agents are sophisticated in
the cases of “congestion effects” , that is where each
agent is worse off the higher the number of other
agents who make the same choice as he. An
equilibrium with a small number of sophisticated
agents is practically indistinguishable from an
equilibrium where all agents are sophisticated
Five arguments in defense of the Rational
Choice Approach (Tsebelis)
3) Statistics: rationality is a small but systematic component
of any individual , and all other influences are
distributed at random. The systematic component has
a magnitude x and the random element is normally
distributed with variance s. Each individual of
population will execute a decision in the interval [x(2s), x+(2s)] 95 percent of the time. However in a
sample of a million individuals the average individual
will make a decision in the interval [x-(2s/1000),
x+(2s/1000)] 95 percent of the time
Rational choice:
a theory for the institutions
In the rational choice approach individual action is
assumed to be an optimal adaptation to an institutional
environment, and the interaction among individuals is
assumed to be an optimal response to each other. The
prevailing institutions (the rules of the game) determine
the behavior of the actors, which in turn produces
political or social outcomes.
Rational choice is unconcerned with individuals or
actors per se and focuses its attention on political and
social institutions
Advantages of the Rational choice Approach
• Theoretical clarity and parsimony
Ad hoc explanations are eliminated
• Equilibrium analysis
Optimal behavior is discovered, it is easy to formulate
hypothesis and to eliminate alternative explanations.
• Deductive reasoning
In RC we deal with tautology. If a model does not work , as
the model is still correct, you have to change the
assumption (usually the structure of the
game..).Therefore also the “wrong” models are useful for
the cumulation of the knowledge.
• Interchangeability of individuals
Positive political
Theory: An
introduction
Lecture 2: Basic tools of analytical politics
Francesco Zucchini
29
Spatial representation
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In case of more than one dimension, we have
iso-utility curves (indifference curves)
Utility diminishes as we move away from the
ideal point
The shape of the iso-utility curve varies as a
function of the salience of the dimensions
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Spatial representation
Utility
Continuous utility functions in 1 dimension
xi
Dimension x
..and in 2 Dimensions
Iso-utility curves or
indifference curves
Spatial representation
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In case of more than one dimension, we have
iso-utility curves (indifference curves)
Utility diminishes as we move away from the
ideal point
The shape of the iso-utility curve varies as a
function of the salience of the dimensions
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Indifference curve
X
I
P
Y
Player I prefers a point
which is inside the
indifference curve (such
as P) to one outside
(such as Z), and is
indifferent between two
points on the same
curve (like X and Y)
Z
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A basic equation in positive political
theory
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Preferences x Institutions = Outcomes
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Comparative statics (i.e. propositions) that
form the basis to testable hypotheses can be
derived as follows:
As preferences change, outcomes change
As institutions change, outcomes change
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A typical institution: a voting rule
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Committee/assembly of N members
K = p N minimum number of members to approve a committee’s
decision
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In Simple Majority Rule (SMR) K > (1/2)N
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Of course, there are several exceptions to SMR
 Filibuster in the U.S. Senate: debate must end with a motion of
cloture approved by 3/5 (60 over 100) of senators
 UE Council of Ministers: qualified majority (255 votes out of 345,
73.9 %)
 Bicameralism
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A proposition: the voting paradox
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If a majority prefers some alternatives to x, these set
of alternatives is called winset of x, W(x); if an
alternative x has an empty winset , W(x)=Ø, then x
is an equilibrium, namely is a majority position that
cannot be defeated.
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If no alternative has an empty winset then we have
cycling majorities
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SMR cannot guarantee a majority position – a
Condorcet winner which can beat any other
alternative in pairwise comparisons. In other terms
SMR cannot guarantee that there is an alternative x
whose W(x)=Ø
37
Condorcet Paradox
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Imagine 3 legislators
with the following
preference’s orders
Alternatives can be
chosen by majority rule
Whoever control the
agenda can completely
control the outcome
ranking
Leg.1 Leg.2
Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
1,2 choose z against x but..
ranking
Leg.1 Leg.2
Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
2,3 choose y against z but
again..
ranking
Leg.1 Leg.2
Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
1,3 choose x against y..
z defeats x that defeats y that defeats z.
ranking
Leg.1 Leg.2
Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
Whoever control the agenda can
completely control the outcome
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Imagine a legislative voting in two steps. If
Leg 1 is the agenda setter..
y
x
x
z
z
ranking
Leg.1 Leg.2Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
Whoever control the agenda can
completely control the outcome
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If Leg 2 is the agenda setter..
x
z
z
y
y
ranking
Leg.1 Leg.2Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
Whoever control the agenda can
completely control the outcome
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If Leg 3 is the agenda setter.
y
z
y
x
x
ranking
Leg.1 Leg.2Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
Probability of Cyclical Majority
Number of Voters (n)
N.Alternatives
(m)
3
5
7
9
11
limit
3
0.056
0.069
0.075
0.078
0.080
0.088
4
0.111
0.139
0.150
0.156
0.160
0.176
5
0.160
0.200
0.215
6
0.202
Limit
1.000 1.000 1.000 1.000 1.000 1.000
0.251
0.315
Median voter theorem
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A committee chooses by SMR among alternatives
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Single-peak Euclidean utility functions

Winset of x W(x): set of alternatives that beat x in a committee
that decides by SMR

Median voter theorem (Black): If the member of a committee G
have single-peaked utility functions on a single dimension, the
winset of the ideal point of the median voter is empty. W(xmed)=Ø
46
When the alternatives can be disposed on only one dimension namely when the utility
curves of each member are single peaked then there is a Condorcet winner: the
median voter
Utility
ranking Leg.1 Leg.2 Leg.3
1°
z
z
1°
x
2°
2°
x
y
z
3°
y
x
y
3°
y
z
x
When the alternatives can be disposed on only one dimension namely when the utility
curves of each member are single peaked then there is a Condorcet winner: the
Utility
median voter
ranking Leg.1 Leg.2 Leg.3
1°
x
z
1°
y
2°
2°
y
y
z
3°
z
x
x
3°
x
y
z
When there is a Condorcet paradox (no winner) then the alternatives cannot be
disposed on only one dimension namely the utility curves of each member are not
single peaked
2 peaks
Utility
ranking
Leg.1Leg.2 Leg.3
1°
z
y
1°
x
2°
x
z
y
3°
y
x
z
2°
3°
x
y
z
When there is a Condorcet paradox (no winner) then the alternatives cannot be
disposed on only one dimension namely the utility curves of each “legislator” are not
ever single peaked
2 peaks
Utility
ranking
Leg.1Leg.2 Leg.3
1°
z
y
1°
x
2°
x
z
y
3°
y
x
z
2°
3°
y
x
z
In 2 or more dimensions a unique equilibrium is not guaranteed
Electoral competition and median voter
theorem
51
Theorems
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Chaos Theorem (McKelvey): In a multi-dimensional
space, there are no points with a empty winset or no
Condocet winners, if we apply SMR (with one
exception, see below). There will be chaos and the
agenda setter (i.e. which controls the order of
voting) can determine the final outcome
Plot Theorem: In a multi-dimensional space, if
actors’ ideal points are distributed radially and
symmetrically with respect to x*, then the winset of
x* is empty
Change of rules, institutions (bicameralism,
dimension-by-dimension voting) can produce a
stable equilibrium
52
Cycling majorities
53
Plott’s Theorem
Plott’s Theorem
Instability, majority rule and multidimensional space
How institutions can affect the stability (and the nature) of the decisions ? Example
with bicameralism
Imagine 6 legislators in one chamber and the following profiles of preferences.
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
x
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z
2,3,5,6 prefer x to z but..
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
y
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z
1,4,5,6 prefer w to x, but..
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
y
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z
all prefer y to w, but..
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
y
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z
1,2,3,4 prefer z to y, ….CYCLE!
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
y
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z
Imagine that the same legislators are grouped in two
chambers in the following way (red chamber 1,2,3 and blue
chamber 4,5,6) and that the final alternative must win a
majority in both chambers.
2, 3, and 5, 6 prefer x to z
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
y
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z
However now w cannot be preferred to x as in the Red
Chamber only 1 prefers w to x. …once approved against z ,
x cannot be defeated any longer
What happen if we start the process with y ?
All legislators prefer y to w..
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
y
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z
However now z cannot be chosen against y as in the Blue
Chamber only 4 prefers z to y. …once approved against
w , y cannot be defeated any longer.
We have two stable equilibria: x and y. The final outcome
will depend on the initial status quo (SQ)
1) If x (y) is the SQ then the final outcome will be x (y)
2) If z (w) is the SQ then the final outcome will be x (y)
ranking Leg.1
Leg.2
Leg.3
Leg.4
Leg.5
Leg.6
1°
z
x
x
z
y
y
2°
y
z
z
y
w
w
3°
w
y
y
w
x
x
4°
x
w
w
x
z
z