Positive political Theory: an introduction General information. Credits: 9 (60 hours ) Period: 8 th January  20 th March Instructor: Francesco Zucchini ( [email protected] ) Office hours: Monday 1719.30, room 308, third floor, Dpt. Studi Sociali e Politici. 1.
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Positive political Theory: an introduction
General information
Credits: 9 (60 hours)
Period: 8th January  20th March
Instructor: Francesco Zucchini ([email protected] )
Office hours: Monday 1719.30, room 308, third floor, Dpt. StudiSociali e Politici
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Course: aims, structure, assessment
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Positive political Theory: An introduction
Lecture 1: Epistemological foundation of the Rational Choice approach
Francesco Zucchini
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“NON RATIONAL CHOICE THEORIES
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.
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
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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Utility
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x
A
A
x
U
U
Singlepeak utility functions
+

+

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Expected utility
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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..
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Principles of game theory
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Principles of game theory (2)
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Principles of game theory (3)
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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.
Bayesian updating of beliefs
Positive political Theory: An introduction
Lecture 2: Basic tools of analytical politics
Francesco Zucchini
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Spatial representation
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Spatial representation
Utility
xi
Dimension x
..and in 2 Dimensions
Isoutility curves or indifference curves
Spatial representation
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X
I
Y
P
Z
Indifference curve
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)
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A basic equation in positive political theory
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A typical institution: a voting rule
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A proposition: the voting paradox
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ranking
Leg.1
Leg.2
Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
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
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Leg.3
1°
z
y
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2°
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3°
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z
ranking
Leg.1
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1°
z
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3°
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y
ranking
Leg.1
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1°
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2°
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3°
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x
Median voter theorem
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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
ranking
Leg.1
Leg.2
Leg.3
1°
z
z
x
2°
x
y
z
3°
y
x
y
Utility
1°
2°
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 median voter
ranking
Leg.1
Leg.2
Leg.3
1°
x
z
y
2°
y
y
z
3°
z
x
x
Utility
1°
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 member are not single peaked
ranking
Leg.1
Leg.2
Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
2 peaks
Utility
1°
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
ranking
Leg.1
Leg.2
Leg.3
1°
z
y
x
2°
x
z
y
3°
y
x
z
2 peaks
Utility
1°
2°
3°
y
x
z
In 2 or more dimensions a unique equilibrium is not guaranteed
Preference rankings that allow to dispose the alternatives in one dimension (Single peakedness condition) share one feature: one alternative is never worst among the three for any group member. Therefore we can affirm that for every subset of three alternatives if one is never worst among the three for any voter then majority rule yield a stable outcome ( the median voter most preferred alternative or median ideal point).Such a condition however is sufficient but not necessary to prevent the Condorcet Paradox ( namely the collective intransitivity and the cycling majorities)…
ranking
ranking
Leg.1
Leg.1
Leg.2
Leg.2
Leg.3
Leg.3
z
1°
1°
x
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z
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2°
2°
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3°
3°
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x
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y
SMR yields coherent group preferences ( a stable outcome) if individual preferences are value restricted. In other terms if for every collection of three alternatives under consideration, all members of the voters agree that one of the alternatives in this collection either is not best, not worst, not middling.
ranking
Leg.1
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1°
x
x
z
2°
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x
Sen’s ValueRestrictions Theorem
x
There is no way to dispose the alternatives on only one dimension, namely to have single peaked utility curves for all voters. However there is Condorcet winner (a stable outcome).
ranking
Leg.1
Leg.2
Leg.3
1°
x
x
z
2°
y
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3°
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Sen’s ValueRestrictions Theorem
Utility
x
y
x
z
Electoral competition and median voter theorem
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Theorems
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Cycling majorities
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Plott’s Theorem
Plott’s Theorem
Instability, majority rule and multidimensional space
ranking
Leg.1
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Leg.5
Leg.6
1°
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4°
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Imagine 6 legislators in one chamber and the following profiles of preferences.
ranking
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2,3,5,6 prefer x to z but..
ranking
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1,4,5,6 prefer w to x, but..
ranking
Leg.1
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1°
z
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x
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2°
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3°
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4°
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all prefer y to w, but..
ranking
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1°
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1,2,3,4 prefer z to y, ….CYCLE!
ranking
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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
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1°
z
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4°
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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
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1°
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3°
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4°
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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)
If x (y) is the SQ then the final outcome will be x (y)
If z (w) is the SQ then the final outcome will be x (y)