positive political theory: an introduction general information

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Positive political Theory: an introductionGeneral information

Credits: 9 (60 hours)Period: 8th January - 20th MarchInstructor: Francesco Zucchini (francesco.zucchini@unimi.it )Office hours: Monday 17-19.30, room 308, third floor, Dpt. Studi Sociali e Politici

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Course: aims, structure, assessment The course is an introduction to the study of politics from a

rational choice perspective. The course is an introduction to the study of politics from a

rational choice perspective.In the first two modules we will focus on the institutional effects of decision-making processes and on the nature of political actors in the democratic political systems. In the last module we will focus on the origin of the state, on the democratization process and on the collective action problems.

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 students is based upon the regular and active participation in the classroom activities (20%), a presentation (30%) and a final written exam (50%).

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Positive political Theory: An introduction

Lecture 1: Epistemological foundation of the Rational Choice approachFrancesco Zucchini

What the rational choice is not

Theories without actors:

• System analysis • Structuralism• Functionalism (Parsons)

Theories with non rational actors:

• Relative deprivation theory• Imitation instinct (Tarde)• False consciouness (Engels)• Inconscient pulsions (Freud)• Habitus (Bourdieu)

“NON RATIONAL CHOICE THEORIES

What the rational choice isWeak 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 Rationality2) 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 Rationality3) Conformity to the axioms of probability

calculusA1 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)

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A small quantity of formalization... A choice between different alternatives

S = (s1, s2, … si) Each alternative can be put on a nominal, ordinal o

cardinal scale The choice produces a result

R = (r1, r2, … ri) An actor chooses as a function of a preference

ordering relation among the results. Such ordering is complete transitive

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Utility

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

One dimension (the real line) Actor with ideal point A, outcome x Linear utility function:

U = - |x – A|

Quadratic utility function: U = - (x – A)2

U

A x

U

A x

+

+

-

-

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Expected utility

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

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

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Strong Requirements of Rationality1) 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

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Principles of game theory

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)

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)

Solutions is by backward induction, by identifying the subgame perfect equilibria

Nash equilibrium: the profile of the best responses, conditional on the anticipation of other actors’ best responses

A Nash equilibrium is stable: no-one unilaterally changes strategy

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.

• P(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it doesnot take into account any information about B.

• P(A|B) is the conditional probability of A, given B. It is also called the posterior probabilitybecause 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

Bayesian updating of beliefs

Strong Requirements of Rationality

Bayesian updating of beliefs. ExampleSuppose someone told you they had a nice conversation with someone on the train. Not

knowing anything else about this conversation, the probability that they were speaking to a woman is 50%. Now suppose they also told you that this person had long hair. It is now more likely they were speaking to a woman, since most long-haired people are women. How likely ?Bayes' theorem can be used to calculate the probability that the person is a woman. W = event that the conversation was held with a woman, and L = event that the conversation was held with a long-haired person.It can be assumed that women constitute half the population for this example. So, not knowing anything else, the probability that  W occurs is P (W) = 0.5 Suppose it is also known that 75% of women have long hair, which we denote asP (L | W) = 0.75 (read: the probability of event  given event  is 0.75).Likewise, suppose it is known that 30% of men have long hair, orP (L | M) = 0.3where  M is the complementary event of W, i.e., the event that the conversation was held with a man (assuming that every human is either a man or a woman).

Bayesian updating of beliefs. Example

Our goal is to calculate the probability that the conversation was held with a woman, given the fact that the person had long hair, or, in our notationP (W | L)Using the formula for Bayes' theorem, we have:

where we have used the law of total probability. The numeric answer can be obtained by substituting the above values into this formula. This yieldsi.e., the probability that the conversation was held with a woman, given that the person had long hair, is about 71%.

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) Salience of issues and information2) Learning3) Heterogeneity of individuals4) Natural Selection5) 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 institutionsIn 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 analysisOptimal 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

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Positive political Theory: An introduction

Lecture 2: Basic tools of analytical politics

Francesco Zucchini

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Spatial representation

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

Continuous utility functions in 1 dimensionUtility

Dimension xxi

Spatial representation

..and in 2 Dimensions

Iso-utility curves or indifference curves

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Spatial representation

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

I

X

YP

Z

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 Preferences x Institutions = Outcomes

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 Committee/assembly of N members K = p N minimum number of members to approve a committee’s

decision

In Simple Majority Rule (SMR) K > (1/2)N

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 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.

If no alternative has an empty winset then we have cycling majorities

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

Condorcet Paradox

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

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.

Whoever control the agenda can completely control the outcome

Imagine a legislative voting in two steps. If Leg 1 is the agenda setter..

ranking Leg.1Leg.2Leg.3

1° z y x

2° x z y

3° y x z

x y

x

z

z

Whoever control the agenda can completely control the outcome If Leg 2 is the agenda setter..

ranking Leg.1Leg.2Leg.3

1° z y x

2° x z y

3° y x z

z x

z

y

y

Whoever control the agenda can completely control the outcome

If Leg 3 is the agenda setter.

ranking Leg.1Leg.2Leg.3

1° z y x

2° x z y

3° y x z

z y

y

x

x

Probability of Cyclical MajorityNumber 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 0.251

6 0.202 0.315

Limit 1.000 1.000 1.000 1.000 1.000 1.000

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Median voter theorem A committee chooses by SMR among alternatives

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

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

y z x

1°

2°

3°

Utility

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

x y z

1°

2°

3°

Utility

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

x z

1°

2°

3°

Utility

ranking Leg.1Leg.2 Leg.3

1° z y x2° x z y3° y x z

y

2 peaks

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

y z

1°

2°

3°

Utility

ranking Leg.1Leg.2 Leg.3

1° z y x2° x z y

3° y x z

x

2 peaks

In 2 or more dimensions a unique equilibrium is not guaranteed

ranking Leg.1 Leg.2 Leg.3

1° z z x

2° x y z

3° y x y

ranking Leg.1 Leg.2 Leg.3

1° x z y

2° y y z

3° z x x

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)…

z 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.

Sen’s Value-Restrictions Theorem

ranking Leg.1 Leg.2 Leg.3

1° x x z

2° y z y

3° z y x

x X is not middling for any voter and it is the winning alternative

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).

Sen’s Value-Restrictions Theorem

ranking Leg.1 Leg.2 Leg.3

1° x x z

2° y z y

3° z y x

x is not middling for any voter and it is the winning alternative y x z

Utility

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Electoral competition and median voter theorem

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Theorems 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

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Cycling majorities

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

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

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° y 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)