Collective Decision-Making in Organizations

ww.uni.magdeburg.de

Collective Decision-Making in Organizations

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Collective Decision-Making in Organizations

Dr. Roland Kirstein, Professorship of Business Economics

Summer Term 2007

Organization

• Tue 9.15-10.45, 13.15-14.45

• 2L+2T (2 CP)

• Tutorial (Dirk Matzner) starts April 24.

• Homepage: http://www.ww.uni-magdeburg.de/bizecon => teaching, summer 07

• Downloads (slides, articles), new information

• Link to the internet forum “Game Theory”

• My office hour: Wed, 11-12, Bldg. 22/Room D-003 (better send an email!)

• Email: cdmo@rolandkirstein.de

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Motivation

Economics deals with

=> decision-making under scarcity, optimal allocation of scarce resources

=> individual, interactive, and collective decisions

=> markets, games, organizations/society

=> economic theory understands fairly well individual and interactive behavior,

but is it guaranteed that organizations also act rationally

=> influence of the institutional framework on collective decisions

=> what institutional arrangements may foster optimal collective decisions

Current debate about Corporate Governance, auditing, organization structures...

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Social and individual incentives

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Example

(Ordeshook, p. 54 f., variation of the Pliny case)

A CEO of a small listed company allegedly has committed severe managerial errors. The

supervisory board (consisting of three members: 1, 2, 3) has three options:

x) do nothing (presumption of innocence)

y) “promote” him to a powerless, but higher paid position

z) fire and sue him.

Three cases:

• If the board members agree unanimously on one option, this is the optimal

collective decision.

• If two board members agree, then they should decide.

• But what is the collective decision if each board member has a different

favorite

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Example: Three cases

Unanimosity Two members agree Divergence

scenario 1 Member scenario 2 Member scenario 3 Member

Ranking 1 2 3 Ranking 1 2 3 Ranking 1 2 3

1st y y y 1st z z x 1st y z x

2nd z z z 2nd y x y 2nd z y y

3rd x x x 3rd x y z 3rd x x z

In this example, all three

board members are

convinced that the CEO

is guilty, but they are

reluctant to mete out the

maximum punishment.

In this example, two board

members prefer the maximum

punishment, while the third

would rather let the CEO

unpunished.

Now member 1 would prefer

at least a lenient

punishment => total

disagreement with regard to

the best option.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Example: different voting rules

1. Idea: Majority voting.

Problems:

ties (in small committees),

prevailing option might be supported only by small number of voters

(even in large committees, depending on number of proposals)

=> making no decision (“stalled”) is, however, a decision for the status quo (x).

2. Application of the American civil/criminal procedure rule (for jury trials):

First the jury decides “guilty or not”; if yes, then the judge determines the sanction.

3. Binary / pairwise voting (as in the consumer choice theory)

=> Both procedures would guarantee that the final outcome is supported by a majority,

=> but they do not guarantee consistency!

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Example: American procedure

scenario 3 Member

Ranking 1 2 3

1 st y z x

2 nd z x y

3rd x y z

First the three members decide whether the CEO is guilty or not => 2 yes, 1 no

Then, they decide whether to apply y or z => y

even though two members would have preferred x over y

=> a direct vote between x and y would result in a different outcome!

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Example: pairwise voting => the voting paradox

scenario 3 Member

Ranking 1 2 3

1st y z x

2nd z x y

3rd x y z

Now the outcome depends on the agenda:

• First x against y => x, then the winner against z => z

• First y against z => y, then the winner against x => x

• First z against x => z, then the winner against y => y

Any collective outcome can be implemented by choice of the agenda.

“Transitivity” is not guaranteed by pairwise (binary) voting

=> collective preferences can be irrational.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Collective choice

(or public choice, social choice) analyzes non-market allocations

=> not many decision situations within an organizations can be interpreted as a market

=> economic theory of the firm hardly addresses actual decision procedures

=> economic theory of politics is focused on parliaments/governments

markets

(auctions, trade)

spontaneous

allocation of resources

games

(teams, monitoring)

“constitutions”

(votes, elections)

organized

The (constitutional) choice of the collective decision procedure and of the agenda

(the rules of the intra-organizational game) can be as important as any substantial

argument for the determination of the collective outcome.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Goals of this course

• understand “political decisions” in organizations

• analyze elections, voting in committees, lobbying, power

• analyze spontaneous coordination (teams, tournaments)

• making use of quantitative methods (game theory)

Positive analysis: derive outcomes.

Normative analysis: evaluate institutional arrangements and derive recommendations.

=> Obviously, a normative result rests on a positive analysis (but not vice versa)

=> “Chinese wall” between normative goals and positive outcomes.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Contents of the Course

• Basic Concepts

o positive and normative analysis

o rationality and methodological individualism

• Votes and Elections

o Deeper look into the voting paradox: Condorcet winner

o Generalization: Arrow paradox

o Condorcet Jury Theorem

• Teams and Tournaments

o public goods and free-riding

o rent-seeking

• Hierarchies, Monitoring, Bureaucracy

• Power

• Rule-governed Behavior, Corporate Culture.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Textbooks (find some articles on the homepage)

Ordeshook, P., Game Theory and Political Theory. Cambridge 1986 (reprint 2003)

• ch. 1+2 on basic concepts (or any other Microeconomics textbook)

• ch. 4+6 on voting and elections

• ch. 5 on public goods and interest groups

Hargreaves Heap, S. et.al.: The Theory of Choice. Oxford, Cambridge/MA 1992.

• part 3 on collective choice

Holt, C.A.: Markets, Games & Strategic Behavior. Boston et.al. 2007.

• ch. 14+16 on public goods

• ch. 17 on rent seeking

• ch. 18 on voting

Schelling, T. (1960) The Strategy of Conflict

Williamson, O (1985) The Economic Institutions of Capitalism. Macmillan, NY.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Section 1: Basics (overview)

1. Individual choice under (budget) constraint:

=> preference relation

=> rationality axioms (in particular: transitivity, completeness)

2. Markets (often) work efficiently; if not, some regulation my fix it

=> Why not leave all allocation decision to markets

=> Why create organizations

3. Methodological Individualism: The behavior of organizations is to be explained by the

aggregate behavior of its members.

4. (Under which circumstances) can organizations be seen as rational actors

Two main obstacles:

• Games with Pareto-suboptimal equilibria (e.g., the prisoners’ dilemma)

• Aggregation of individual preferences to a collective decision.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Individual choice

Assumptions:

• Set of options (e.g., commodity bundles) X

• An individual is able to make a sequence of binary comparisons

=> preference ordering (binary relation) P : X 2 →X

e.g., P(x,y)=x means: Individual prefers x over y.

• Observed choices allow us to derive “revealed” preferences!

• If a preference ordering is transitive, reflexive, antisymmetric, complete, then it

can be represented numerically (“utility”) U : X → IR with U(x)≥U(y) P(x,y)=x

In the latter case, choosing a best option from X can also be interpreted as maximizing U.

The “representability axioms” are also termed “rationality” axioms;

hence, rationality (in economics) means nothing else but

• having preferences which obey the representability axioms,

• maximizing utility (under a budget constraint)

=> Demand function x1(p 1 ,p 2 ,B), substitution/income effect, inverse demand p 1 (x 1 )

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Markets

Where demand and supply “meet”

Positive analysis: market equilibrium (intersection of demand and supply curve)

Supply: how many units will profit maximizing firms produce at a given price

Demand: how many units will utility maximizing households buy at a given price

Normative analysis: market equilibrium maximizes net welfare, i.e., the vertical

distance between demand (marginal benefit) and supply (marginal cost).

Adam Smith (1776): “invisible hand”

Two-sided competition, “creative destruction” (Schumpeter, Hayek)

Markets may even work efficiently if ideal conditions are not met

(for an overview: Kirstein/Schmidtchen 2003 – read it!)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Coase (1937): Transaction cost

If markets are so wonderful, why don’t we use them everywhere

Using markets is not costless: transaction costs are the cost of

• finding a partner for a market interaction

• specifying the contract terms

• enforcing them according to the contract.

Other problems with market allocation

• technological externalities (and transaction costs, see Coase 1960)

• asymmetric information (e.g., uninformed customers, informed sellers)

• natural monopoly / market power (entrance/exit cost): economies of scale

• specific investments

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Organizations

Institutional means to overcome the problems with markets:

• transaction costs vanish if partners do not have to make/enforce contracts

• externalities are internalized if the parties merge

• specific investments are protected from hold-up

• vertical integration alleviates some problems of economies of scale

(double-marginalization)

However, new cost types may arise: organizations require

• specification of the organizational contract (constitution)

• coordination / decision procedures

• safeguards against cheating of members (monitoring, incentives)

• an agreement regarding the distribution of returns, costs, profits.

=> a trade-off may exist between transaction cost and organization cost (Coase 1937).

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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$, €

OrgC+TAC

OrgC

TAC

0 i*

1

degree of

integration i

Coase (1937): internal solution for the choice between market/organization => hybrid.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Organizations as Contracts

Buchanan (AER 1975): Economics rather deals with contracts than with choice.

Williamson (1985): four dimensions of contractual relations

• Frequency: one-shot or repeated interaction

• Opportunism: do people pursue their interest even with “cunning”

• Specific investments: do the contributions of a party have a positive value elswhere

• Rationality: do people maximize their utility, or do they only intend to do so (but are

“bounded”, Simon 1957)

=> spot-markets, long-term contracts, organization

Multi-shot interactions: repeated games, shadow of the future

Specific investments: Fundamental transformation of an anonymous market relation

into a bilateral monopoly => problem of “hold-up”.

(Without specific investments, a bilateral relation does not invite for hold-up)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Some constellations of behavioral assumptions

Rationality Opportunism Specificity Adequate contractual form

perfect yes yes “Planning” (PA theory)

bounded no yes Promise

bounded yes no Competition

bounded yes yes “Governance structure”

=> bounded rationality, opportunism, and factor specificity make it inevitable to create a

governance structure, i.e., a long-term contract in which agree upon procedures to

solve problems SHOULD they arise.

Contingent contract = agreement upon the solution of all possible problems, e.g., the

distribution of revenues and costs among the partners.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Choice of contract scheme

Governance structure, safeguards, long-term contract

=> rules of the intra-organizational game.

But can a first best solution be reached at all

=> Incentive and information problems

=> How to aggregate individual decisions/preferences into a collective action

Two main principles of aggregation:

• spontaneously (equilibria of non-cooperative games)

• organized (voting, elections, hierarchies, power)

Game theory serves as a tool to analyze both of these principles.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Non-cooperative games

Rules of the game consist of four elements

• Set of players

• A strategy set for each player (strategy = plan for the whole game)

• A utility function over “outcomes” = strategy combinations

• The information structure (are the moves of other players observable)

A solution of a non-cooperative game is a strategy combination of which no one has an

incentive to deviate (given that all other player do not deviate) => Nash equilibrium.

A NE is “subgame perfect” if it consists of optimal decisions in each part of the game.

It is “perfect Bayesian” if the beliefs are formed according to Bayes rule.

Cooperative game theory assigns utility to coalitions, not strategy combinations.

Axiomatic game theory (bargaining) starts with desirable properties of a solution.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Section 2: Voting and elections

Voting: the game among the voters (committee members, the people...)

Elections: the game among the candidates who want to be elected

Overview:

• Voting paradox and the Condorcet winner

• Arrows dictatorship / impossibility theorem

• Condorcet Jury theorem and further paradoxes

• Legislative bargaining, logrolling

• Median voter theorem

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Recall the voting paradox

The paradox is due to Marquis de Condorcet (1743-1794)

scenario 3 Member

Ranking 1 2 3

1st y z x

2nd z x y

3rd x y z

Any collective outcome can be implemented by choice of the agenda.

Transitivity (of social preferences) is not guaranteed by pairwise voting.

=> binary voting does not necessarily lead to intransitive social preferences;

this will only be avoided if all preferences are “single-peaked”.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Condorcet voting paradox

Unanimosity Two members agree Divergence

scenario 1 Member scenario 3 Member scenario 4 Member

Ranking 1 2 3 Ranking 1 2 3 Ranking 1 2 3

1st y y y 1st y z x 1st z z x

2nd z z z 2nd z x y 2nd x y z

3rd x x x 3rd x y z 3rd y x y

All three members have

identical preferences,

which can be

rearranged such that

they are single peaked

(e.g.: x-y-z would do it)

In this example, the options

cannot be rearranged such

that all preferences are

single-peaked

=> CVP occurs

In scenario 4, the options can

be arranged s.t. preferences

are single- peaked (x-z-y)

=> no CVP (binary voting

leads to collective preferences

which are transitive and

independent of the agenda).

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Condorcet winner

(Ordeshook p. 76)

Definition: If O is the set of options (possible outcomes),

then x∈O is a Condorcet winner if, for any other element y∈O,

the number of members who strictly prefer x to y

exceeds the number of members who strictly prefer y to x.

=> no option defeats a Condorcet winner in majority voting.

Theorem: Under general assumptions (convex preferences, majority, n-persons),

the social preferences are cyclic if no Condorcet winner exists.

(Proof in Ordeshook, p. 77-80, but this is not going to be a topic for the finals)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Application of the CVP: The “killer amendment”

(Ordeshook p. 66f)

Assume that a committee has to decide among two options, x and y and, in this situation,

x would win (e.g., x is a legislative initiative, while y is the status quo).

Opponents of x may introduce a third option z that prevails over x but loses to y.

Assume that z consist of an amended version of x (“the killer amendment”)

Parliamentary rules require that an amendment is first voted against the original bill.

=> final outcome is y.

=> no Condorcet winner (x loses to z, z loses to y, y loses to x)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Arrow’s impossibility (or: dictatorship) theorem

Assumptions

• X = set of available options {x,y,z...}

• N = set of individuals {1,2,3...}

• Each individual has a rational preference order, denoted R i , over (pairs out of) X.

• Rationality means, in particular, complete and transitive.

• Individual decision problem: (R i ,X)

• Individual Decision Function of individual i is denoted D i : (R i ,X) → x∈X and

assigns an option (outcome, decision) to each decision problem.

• Social preference profile R=(R 1 , R 2 ,...)

• A Social Decision Function D: (R,X) → x∈X assigns a social outcome to each

combination of a preference profile R and an option set X.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Arrow’s axioms:

D(R,X) should obey

• collective rationality (CR)

• universality (UN)

• Pareto-principle (PP)

independency of irrelevant alternatives (II)

• non-dictatorship (ND)

(and who would object to this)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Collective rationality: D is transitive and complete.

Universality: D is defined for all possible combinations of preference orders.

Pareto-principle: If, according to R, all members of N prefer x over y,

then D(R,{x,y})=x (Note that the Pareto-principle says little about disagreement cases)

Irrelevance: For a given preference profile, the social decision between two options x,y

does not depend on the availability of a third option z.

If D(R,{x;y;z})=x, then D(R,{x;y})=x

Non-dictatorship: No individual exists whose preferences determine the social outcome

even if all other individuals disagree.

Dictatorship: D(R,X)=D i (R i ,X) ∀ R -i

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Dictatorship

...is not defined by malevolence or the usage of coercion.

Even a wise person whom all members choose to follow blindly is a dictator in this

sense.

Moreover: Ownership of a resource (or a control right) without veto power of other

actors would also constitute “dictatorship” with regard to this resource.

Compare this to the definition of “power”, according to Max Weber:

Chance to carry out a plan despite the resistance of others.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Arrow’s theorems

If X and N contain at least three options, respectively, then

a Social Decision Function that satisfies (CR), (UN), (PP), (II) violates (ND)

(dictatorship theorem).

no Social Decision Function exists which satisfies (CR), (UN), (PP), (II), and (ND).

(impossibility theorem).

Proof => tutorial

Intuition of the proof: From (II) it follows that options have to be compared pairwise.

This may lead to intransitivity if more than one person is involved (see the CVP).

Transitivity of a pairwise comparison procedure is only guaranteed if just one rational

individual decides.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Arrow’s theorem, more general:

If any 4 of the 5 axioms hold, then the fifth one is violated

e.g., if a Social Decision Function obeys CR, UN, PP, ND, then preference profiles exist

under which it is intransitive => voting paradox.

The five axioms look rather sensible (even desirable), but Arrow has demonstrated that

they contradict each other.

The mess would become even worse if more axioms were required,

such as

• uniqueness of the social outcome

• strategy-proofness (no individual is able to manipulate the outcome by strategic

voting).

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Lessons from the Arrow theorem

In brief: A group decision is guaranteed to obey the four axioms only if it made by a

single person => strong justification for private property.

Main lesson: be careful with collective actors (“national interests”, “the governement

has decided”, “Corporation A implements a new strategy”) as rational players.

=> collective preferences are perhaps not rational, even if all members are rational.

Group decisions are not the same as individual decisions;

their analysis requires taking into account

individual preferences

• aggregation procedure (e.g.: majority voting, or binary voting)

institutional framework (e.g.: who is allowed to set the agenda)

The topic may become more complicated if players are boundedly rational.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Tullock’s criticism af Arrow’s theorems

(Tullock 1967)

The crucial axiom is the irrelevance axiom (II);

if it is discarded, then the impossibility vanishes.

=> Many voting systems obey CR, UN, PP, ND.

=> Hence, Arrow’s theorem is irrelevant at least for collective decisions in very large

groups (democracy).

However, for (smaller) organizations, the problem exists:

No social decision function (i.e., aggregation of individual preferences to a collective

decision) exists which makes sure that collective preferences are rational (obey all five

axioms), except for “dictatorship” (e.g., ownership, delegation to one manager...).

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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A Generalization of Arrow

(McKelvey; see Ordeshook p. 77)

Arrow and CVP rest on the assumption that the option set is discrete (three or more

options)

Does the paradox also occur in continuous option sets

=> “spatial” preferences with two players/two dimensions: Edgeworth box.

=> we are interested in three or more players and two or more dimensions

Three steps:

1. Preferences in option space

2. Pareto-optimality /-improvement for two or three players.

3. Impact on agendas

Result: under fairly general assumptions, the collective decision situation is never

agenda proof

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


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Indifference circles and contract curves for three players/two dimensions

Three heterogeneous players

have circular indifference

curves in a two-dimensional

option space =>

contract curves form triangle.

y’’ is Pareto-optimal for 1 and

2 (not for 3).

Compared to y’, y*** is

Pareto-superior for 1, 2, 3

(yet no Pareto-optimum).

No option is a Paretooptimum

for all three players.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


38

An agenda that leads to a Pareto-inferior

outcome

Start in y0, and draw a line that is orthogonal

to the CC between 1 and 2, choose a point

which is closer to this CC, e.g., y1

=> this point prevails over y0.

Repeat this procedure for 2 and 3 => y2,

repeat this procedure for 1 and 3 => y3,

and so on.

Binary voting yields y6>y5>y4>y3>y2>y1>y0

but y6 (just as any yi, i>0) is clearly Paretoinferior

to y0

=> y0 would prevail over y6 in a direct vote

=> cyclic collective decision

=> no Condorcet winner.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


39

McKelvey-Theorem: If

• three or more players have convex preferences over two (or more) dimensions,

• the decision rule is pairwise majority voting

• a Condorcet point does not exist,

then social preferences are cyclic

(i.e., for any two points x,y a finite agenda exists that leads from x to y and back to x).

=> looking at Arrow/CVP and McKelvey simultaneously, any institution’s collective

decisions can be manipulated by an elite that

• chooses the aggregation procedure

• sets the agenda (if social choice function is pairwise voting)

even if the voting members are fully rational and dictatorship is ruled out.

Can the voters “strike back”, i.e, also manipulate the outcome => strategic voting

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


40

Further problems with voting

1. Condorcet’s voting paradox rests on three crucial assumptions: Voters

• are non-experts

• do not really DELIBERATE about their decision-problem => Discursive dilemma

• are “sincere” (decide according to their true preferences, hence non-strategically).

=> voters just vote, for whatever reason, without thinking too much, myopic.

We will look at committee members who

• vote strategically: agenda abuse,

• have to justify their decisions: discursive dilemma,

• are experts: Condorcet Jury Theorem.

2. Do voting rules obey much weaker “reasonable” requirements, such as monotonicity:

the success probability of a proposal is non-decreasing in the number of votes.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


41

Example for violation of monotonicity:

Election of the Turkish president, May 2007

Turkish parliament consists (stylized) of two groups, ruling AKP with about 60%, and

the opposition with 40% of the seats.

AKP seeks to have its candidate elected, which requires two conditions:

1. more than 2/3 of the members of parliament have to take part in the election

2. the candidate needs the absolute majority.

Abstention (better: boycott) is feasible => three options (yes, no, boycott)

• members of the AKP: yes > boycott > no;

• members of opposition: no > boycott > yes.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


42

Turkish president

If all members vote for their most preferred outcome, then the AKP candidate prevails.

If just some members of the opposition boycott, then AKP prevails.

If complete opposition boycotts, then the candidate of AKP receives 100 percent of the

the submitted votes, but he loses.

=> by choosing their second best option, the minority can secure the status quo.

=> voting rule based on participation quota violates is not “decisive”.

Two reasons:

• strategic misrepresentation of preferences

• double condition violates monotonicity requirement (some more votes for “no” may

lead to a victory of the AKP candidate, while a boycott secures the status quo)

=> Opponent’s dilemma: explicitly vote for no, or choose abstention

=> Implemented by the quota rule, alternative rule may avoid this dilemma

=> Support quota instead of participation quota.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


43

(2/3,0,1/3)

yes

(1,0,0)

Double condition with participation quota

(1/3,0,2/3)

yes

boycott

(0,0,1)

SQ

(1/2,1/2,0)

no

(0,1/3,2/3)

(0,2/3,1/3)

no

(0,1,0)

An n-dimensional „simplex“ visualizes all

possible decision combinations of the

members if exactly (n+1) options are

given, n>1.

Simplex for two options: line

Simplex for three options: triangle

E.g., the outer borders represent

switches between two options (while the

respective third option receives no

votes).

Any move parallel to a border leaves the

respective third component unchanged.

An move which is orthogonal to a border

line leaves the ratio between two

components unchanged.

Social outcome: if boycott receives less

than 1/3, then the candidate is elected if

he receives more than 50% of the

submitted votes => yes.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


44

(1/3,0,2/3)

(1/2,0,1/2)

yes

(1,0,0)

Double condition with support quota

boycott

(0,0,1)

(1/3,1/3,1/3)

(1/2,1/2,0)

no

(0,1,0)

Under a support quota, a proposal (the

AKP candidate) prevails if he receives

the absolute majority of the submitted

votes and this majority represents at

least a share q of all members.

The figure shows three examples of

such a quota q:

q=0 (dot in the upper corner)

q=1/3 (upper diagonal line)

q=0.5 (lower diagonal line)

The diagonal lines are parallel to the

right border => moving on them leaves

the first component unaffected.

=> „yes“ in the area to the southwest of

the respective line, „no“ elsewhere

=> support quota rule is „decisive“

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


45

Participation quota vs. support quota

boycott

boycott

SQ

no

no

AKP

AKP

yes

no

yes

no

With participation quota, voters who

switch from „boycott“ to „no“ (arrow

parallel to right border) might help

the AKP candidate prevail.

If a proposal requires the support

of a certain quota of all voters,

whether they participate or not, this

dilemma is avoided.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


46

Strategic voting and agendas

Recall the CVP (no ordering of the three options exists such that all members have

single-peaked preferences):

scenario 3 Member

Ranking 1 2 3

1st y z x

2nd z x y

3rd x y z

Now consider the following decision rule:

• x is the status quo

in the first voting round, the challenger (y or z) is determined

Naive/myopic approach: y prevails in 1 st round, then x prevails (and 3 is happy).

=> What happens if members vote strategically What would YOU do as 1 or 2

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


47

Strategic voting in scenario 3 / challenger rule

(Holt, section 18.5)

y

z

first round: challenger

x y x z

second round: final decision

3. 1. 3. 2.

Player 1‘s preferences

By strategically voting for z (instead of y) in the first round, player 1 could turn the

collective decision from x (his worst outcome) to z (his second best outcome).

Experimental results: x in first rounds, then z observed

=> subjects start with sincere voting, but learn strategic voting

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


48

Case study on strategic voting:

Federal funds for state schools

(Ordeshook p. 83)

Status quo (a0): no federal funding

1956 federal aid bill (a1) aimed at support for school construction

Powell amendment (a2): federal funds only “for schools open to all children without

regard to race”, in conformity to a recent US supreme court decision.

Analysis (of former congressional votes) shows: a2 > a1, a1 > a0, a0 > a2.

=> Without Powell amendment, a1>a0 (simple bill passes)

With amendment: congressional procedural rules require voting on the amendment first,

then the winner runs against status quo => a0 prevails.

Allegedly, 97 Republican representatives, who favored segregation (at that time) have

voted for a2 against their preferences, in order to “kill” a1 altogether.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


49

Gibbard/Satterthwaite

(Ordeshook p. 86)

A social decision rule D(R,X) is called

• “resolute” if it yields a unique outcome (special case of “decisive”);

• “manipulable” if some member exists who can benefit from misrepresenting his true

preferences.

Formally: Let the preference profiles R and R’ differ with respect to player i’s

preferences (R i , not R i ’, represents this member’s true preferences, i.e., his choice if he

was the dictator). If member i prefers D(R’,X) over D(R,X), then D is manipulable.

Gibbard (1973)/Satterthwaite (1975)-Theorem: If the option space contains more than

two elements and if D is non-manipulable and resolute, then D is dictatorship.

=> Generalized by Schwartz (1982) to the case of non-resolute decision-functions.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


50

Lessons of the GS-Theorem

(Ordeshook p. 88)

• for the positive analysis of institutions: there is no reason to assume that voters

vote honestly ( = reveal their preferences).

• for the normative (design) analysis: the above insight has to be kept in mind when

comparing two institutional alternatives because, if an institution is altered, this may

influence its members’ decisions.

It is irrelvant to ask: given the voters’ choices, what would have been the outcome if

another institution had been applied => ceteris paribus can be misleading.

Politics: under an alternative election rule, voters’ decisions (and campaign strategies)

might have been different.

Business: an alternative voting rule for general assemblies / bilateral committees after

mergers / team decisions may alter the shareholders’/managers’/team members’ behavior

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


51

Vote trading / logrolling

If you vote for my proposal (even though you dislike it),

I will vote for yours (even though I hate it).

Organized form of strategic misrepresentation,

• very common in parliaments (intergroup negotiations, coalitions, within groups)

• also well known in the academic world (citation circles)

Problem: majority voting (one person, one vote) does not reflect intensity of wishes,

while Pareto-efficiency requires to take into account the marginal rates of substitution.

=> logrolling may serve as a remedy, but it may come with additional inefficiency if the

parties of the barter can externalize (part of) the cost of their programs.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


52

Example for logrolling

The entries indicate the utility gain/loss of a voter if a proposal passes

(utility of status quo is normalized to zero).

Example 1: proposal

Example 2: proposal

Voters A B A B

i -2 -2 -20 -20

ii 5 -2 5 -2

iii -2 5 -2 5

welfare +1 +1 -17 -17

In both examples

• just one voter attaches high utility gain to one proposal;

• majority voting leads to rejection of both proposals;

• voters ii and iii can let their favorites prevail as a bundle by logrolling.

In Expl.1 this is welfare enhancing (and the coalition between ii and iii is stable);

in expl. 2 it is welfare decreasing (but the coalition is unstable, as i can bribe ii or iii).

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


53

Logrolling and Transitivity

(Mueller ch. 5 J)

Definition: Let X={A;B;(AB);SQ}. A social choice function D(X,R) puts society into a

logrolling situation if

• D({A;SQ},R)=SQ,

• D({B;SQ},R)=SQ,

• D({(AB);SQ},R)=(AB);

i.e., the pair (AB) defeats the status quo SQ which defeats both A and B.

Theorem: The existence of a logrolling situation implies intransitive social preferences;

the existence of a transitive social preference ordering implies absence of logrolling.

(Bernholz 1973)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


54

Alternatives to majority voting

(Mueller, ch. 7)

Majority rule: choose the option that is ranked first by more than 50% of voters

Majority rule with runoff election (French President): if no candidate reaches more

than 50%, a second election is held between the two best candidates.

Plurality rule: choose the option that is ranked first by the largest number of voters.

Condorcet criterion: choose the option which defeats all others in pairwise votes.

Hare system: each voter indicates the option he ranks highest; remove the option which

is ranked highest by the fewest voters. Repeat for remaining options.

Coombs system: Each voter indicates the option he ranks lowest. Remove the option

which has been chosen by most voters. Repeat until one options remains.

Borda count: Option set X contains n elements; each voter may submit a ranking of

m≤n options; the arbiter assigns to each option m points for each first place, m-1 points

for each second place, and so on. The option with the highest number of points prevails.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


55

Application of Borda count: European Song Contest ("Allemagne – deux points").

Problem: Violation if irrelevance axiom is possible, see example in Ordeshook, p. 69f.

(inverted-order paradox, winner-turns-loser paradox).

Variations:

• Each voter submits ranking; if no option receives absolute majority, the last option

will be cancelled; the votes this option has received will be distributed according to

the second preferences.

• Voters may distribute weights (e.g., 100 points) over the m options.

However, recall Arrow theorem – no social decision function obeys all five axioms.

=> what are criteria for choosing a decision rule

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


56

Criteria for choosing voting rules

(May 1952, Young 1974, Mueller ch. 7.D)

A social decision rule D(X,R) should be

• decisive: D picks a winner for each R;

• monotone: if another voter chooses an option, this does not decrease its probability

to prevail (which would rule out strategic/insincere voting).

• neutral: D is impartial to the options/candidates (or do their names matter);

• “canceling”: D cancels out one voter’s statement “I prefer x to y” if another voter

states “I prefer y to x” (impartiality with respect to voters) – only the number of

voters who support an option are relevant for the outcome, not their identity;

• faithful: if D is applied to a society of only one individual, it picks his favorite;

• consistent: assume that society N (with preference profile R) consists of two

disjoint subsets N1, N2 (with preference profile R i ), i∈{1;2}. Application of a

voting rule yields outcomes D i =D(X,R i ), which are not necessarily unique (rather a

set). If D 1 ∩ D 2 ≠ ∅, then the voting rule is called consistent if D(X,R) ∈ D 1 ∩ D 2 .

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


57

E.g., majority voting

• satisfies neutrality, cancellation, faithfulness,

• but not decisiveness.

• It only satisfies consistency if a Condorcet winner exists (i.e., no voting cycles).

The Borda count is vulnerable for strategic voting, if the voters who consider this know

the preferences of the other voters.

Related topics (which I will not cover in this course):

• Rules to assign parliament/committee seats to parties (Hare-Niemeyer, d’Hondt).

• Split votes (first vote, second vote) in German federal elections.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


58

Voting: what did we learn

Condorcet: if there is no C. winner, then pairwise majority voting is cyclic

=> outcome depends on agenda

=> McKelvey for continuous choice sets

Arrow: if social decision function guarantees four desirable axioms, it is dictatorship

Gibbard/Satterthwaite and Schwartz: If a social decision rule is non-manipulable and

resolute, it is dictatorship.

=> justification of property rights

=> comparison of institutions always requires positive analysis of players’ behavior

Logrolling: can increase efficiency (stable coalition), or decrease (instable).

Alternative voting rules (like Borda count): may satisfy some desirable properties, but

always have at least one drawback

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


59

Juries (or experts’ committes)

So far, we have learned that voting

• does not guarantee social rationality

• and can be manipulated,

regardless of the aggregation rule.

Is it, thus, better to leave social decisions to experts => “Jury theorems”

Problems:

• How should jury decide if members disagree => majority

• Does the jury have to justify its decision => discursive dilemma possible

• When to employ just one expert, a small jury, a larger jury => decision quality

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


60

Competent juries/committees (1):

The “discursive dilemma”

Consider a board of three voters which have to decide an issue, e.g., the termination of a

CEO’s contract for weak performance of the firm => only two options, no CVP!

Termination is justified if, and only if two requirements are satisfied:

• the CEO has violated a contractual obligation (violation);

• and this act has caused the weak performance (causation).

=> this is what the board members have to determine

=> and what may case a paradox if board decides with majority.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


61

Discursive dilemma

Member violation causation termination

1 T T T

2 T F F

3 F T F

Majority T T T

Collective decision is agenda dependent:

a) If board members individually decide whether termination is justified, and then

vote, two of them would reject termination.

b) If board members first vote on whether the criteria are satisfied (which implies the

collective decision on termination without further vote), the outcome is

termination.

Deliberation (reasoning about more than one necessary condition to justify a decision)

may lead to a voting paradox even if the number of options is (too) small.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


62

Competent juries/committees (2):

The Condorcet Jury Theorem

Voting paradox, Arrow => uninformed (and sincere) voting

Condordcet Jury Theorem: decision of experts (still sincere voting).

Original version: A jury

• makes a binary decision

• with absolute majority,

• consists of k=2h+1, h∈IN, homogeneous members,

• each of whom decide correctly (and sincerely) with a probability q ∈ [0,1].

Amendments:

• two independent error probabilities

• correlated choices (influence), more than two options, other majority decision rules,

strategic voting, jury members with different competency parameters.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


63

Original Condorcet Jury Theorem

Denote as Q(q,k) the probability that a jury of k members (who decide correctly with q)

takes a correct majority decision.

General formula for Q(q,k):

Q(

q,

k)

=

k


j=

h+

1

k


q

j

j

(1 − q)

k − j

; j ≤ k

Recall that

k


=

j

k!

j!(

k −

j)!

E.g.: k=3 => 8 cases how the three members can cast their votes; in one case all three of

them are correct (probability q 3 ), in 3 cases, two of them are correct while one of them

votes wrongly (probability (1-q)q 2 ) => Q(q,3)=q 3 +3(1-q)q 2 =3q 2 -2q 3 .

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


64

Original Condorcet Jury Theorem

The theorem makes three claims for h>1: If q∈(0.5,1) then

1. Q(q,k) > q

2. Q(q,k+2) > Q(q,k)

3. lim Q(q,k) = 1 for k → ∞

Corresponding claims: If

q∈(0,0.5) Q(q,k)1 q 2 -1.5q+0.5>0

(q-0.5)(q-1)>0 => the roots of the left-hand side are q=0.5 and q=1; for any value of

q with q>0.5 and q


65

=> for 0.5


66

Interpretation of the (original) Condorcet theorem

A decision of a jury which consists of homogeneous members (who decide with absolute

majority) is “better” than the decision of each single member if the individual probability

of making the correct decision is greater than 0.5

Justification for having a democracy instead of a “philosopher king”:

=> A majority of half-wits can make better decisions than one brilliant expert.

=> formally: Q(q,k)>q E despite 1>q E >q>0.5 is possible.

Wikipedia, Google, ebay perform well when aggregating information.

Galton’s example: the bull guess (Surowiecki, Wisdom of the Crowds).

Criterion for “better” is the probability of making a correct (collective) decision

=> If the “correct” decision is the one that provides higher welfare, then expected

welfare is also as the quality criterion.

=> Group decision-making can be welfare-superior (compare to Arrow’s theorem).

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


67

Modified Condorcet Jury Theorem

(Kirstein/v. Wangenheim 2007)

Original CJT addresses binary decision situations, but models only one error probability.

Binary decision situation: X={A;B}

Either A is better (with probability ) or B is better (1-)

Gain from carrying out A if it is better: G>0

Loss from carrying out A if B is better: L


68

If the parameters (, G, L) describe the decision situation completely,

society should carry out A if G+(1- )L > 0 -(1- )L/G < 1

The left hand side is positive and will subsequently be denoted as T.

Society has to decide between three different modes how to pick one option:

• blindly pick A (if T1);

• ask one single expert (=person who is less likely to commit an error);

• ask a group of such (homogeneous) experts which decides with majority.

Research questions:

• When is it better to ask a group of homogeneous experts rather than just one

• When is it better to ask a group of semi-experts rather than a highly competent one

The “ask the auditory” vs. the“phone a friend” lifeline

in “Who Wants to be a Millionaire”

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


69

A model of an expert

(“phone-a-friend”)

An expert can decide the question (should we carry out A or B) better than by pure

chance. A simple model of experts draws on conditional probabilities:

r = prob (expert chooses A | A is better)

w = prob (expert chooses A | B is better)

Then:

• r > w => positive detection skill

• r = 1,w = 0 => perfect detection skill

• r = w => zero detection skill

• 1 > r > w > 0 => imperfect, but positive detection skill

All possible r-w-combinations can be represented in a “unit box”.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


70

Single expert

State

Option A is better B is better

A G L

B 0 0

prior 1-

Expert’s decision

for A r w

for B 1-r 1-w

with 1>r>w>0.

Society now has to decide whether to employ the expert;

if not: whether to carry out “blindly” A (if T1)

Interpretation: Let B be the status quo, then A is an innovation/deviation.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


71

Case 1: T>1

It is better to employ the expert than to blindly carry out B if

if r G + w (1- ) L > 0 r/w > T

=> unit box!

Case 2: T G+(1- )L (1-r)/(1-w) > T

=> unit box

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


72

More than one expert => committee or jury

(“ask-the-audience” with two options and majority rule)

If the decision about A or B is assigned to a jury which

• consists of an odd number (k=2h-1, h>1)

• of homogeneous experts (i.e., all have the same r-w-values)

• who decide with absolute majority (no abstention),

then Q(r,k) is the probability of correctly deciding for A,

while Q(w,k) is the probability of wrongly deciding for A.

• 1 > Q(r,k) > Q(w,k) > 0: positive, but imperfect jury detection skill;

• Q(r,k)=1, Q(w,k)=0: perfect jury;

• Q(r,k)=Q(w,k): blind jury.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


73

Research goals

What is a criterion for decision quality of juries / single experts

Original CJT: the probability of making a correct decision, q.

=> Recall that Q(q,k)>q for 0.5


74

Expected welfare as a quality criterion

Expected welfare

• of a single expert’s decision: r G + w (1- ) L

• of a jury decision: Q(r,k) G + Q(w,k) (1- ) L

A jury decision is welfare superior if, and only if,

Q(r,k) G + Q(w,k) (1- ) L > r G + w (1- ) L

[Q(r,k)-r] G + [Q(w,k)-w] (1- ) L > 0

[Q(r,k)-r] > [Q(w,k)-w] T

The algebraic treatment becomes very tedious

=> simulations (provides a hunch of the shape and properties of the boundaries)

=> do the tedious proofs ;-)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


75

Simulation result for k=11, T=1 (r on the vertical, w on the horizontal axis;

grey areas: jury is welfare superior)

r>0.5, but single expert is better

(contradicts CJT).

w > 0.5, but the jury is better

(contradicts original CJT).

1> r > 0.5 > w => jury is better

(in line original CJT).

this is white ;-(

see next slide

r model would even allows us to compare single expert to a jury of semi-experts.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


76

k=11; T < 1 T=1 T > 1

Figure 3 from Kirstein/v.Wangenheim (2007)

=> With r>0.5>w, the jury is always superior (except for r=1, w=0) => in line with CJT.

=> If each juror has r>w, then only the upper triangle is relevant.

=> Possible cases: r>0.5, but jury is worse; w contradicts CJT.

=> Again: a group of half-experts might be better than a single competent expert.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


77

Power in the EU (or: why Poland is right and wrong)

Decision procedure of the EU council (assembly of governments):

A) Treaty of Nice assigns votes to members states (e.g., Germany/Italy 29,

Spain/Poland 27, ...Malta 3); three stage procedure: a proposal is acknowledged if

• qualified majority of votes (at least 258 out of 345 votes) if this represents

• a majority (absolute or even 2/3) of the 27 members states

• and 62 % of the EU population (if a member state invokes this criterion).

B) So called “Constitutional”: two stage procedure (“double majority”)

• 55% majority of the member states (i.e., 15 out of 27 states; no voting weights)

• if they represent at least 65% of the EU population.

=> “Constitution” abandons the arbitrary voting weights of the Nice treaty,

=> but fails to implement “equal influence” of each EU citizen (to the contrary).

=> Power index analysis shows that Poland’s “square root” comes closer to that.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


78

Power index analysis

• Introduced by Penrose and Banzhaf (1946), alternative concept by Shapley.

• Attempts to measure ex ante power (in front of a veil of ignorance).

• Research question: if members of a committee are uncertain about their future

positions (yes or no) on a multitude of issues,

• what influence (ability to change the outcome) is provided to them by the voting

procedure

Core concept: a member is in a pivotal position if the social outcome depend on how

this member casts his vote; power = share of pivotal constellations.

=> allows for a comparison of different voting procedures (e.g. with respect to majority

quorum, assigned voting weights, size of the committee...)

=> if a social outcome has a certain value to a player (e.g., normalized to one), then the

voting power is a direct measure of the utility effect of a voting procedure.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


79

Example: committee of three members

=> 8 possible configurations of preferences

=> first example: equal voting weight

A B C Y N outcome pivotal

1 y y y 3 0 y -

2 y y n 2 1 y AB

3 y n y 2 1 y AC

4 y n n 1 2 n BC

5 n y y 2 1 y BC

6 n y n 1 2 n AC

7 n n y 1 2 n AB

8 n n n 0 3 n -

=> each player is pivotal in four cases (out of eight)

=> identical voting power (symmetric case)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


80

Example 2: unequal voting weights

=> let A have three votes

A B C Y N outcome pivotal

1 3y y y 5 0 y A

2 3y y n 4 1 y A

3 3y n y 4 1 y A

4 3y n n 3 2 y A

5 3n y y 2 3 n A

6 3n y n 1 4 n A

7 3n n y 1 4 n A

8 3n n n 0 5 n A

=> only player A is pivotal (formally: in eight cases out of eight)

=> A has total voting power.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


81

Definition of voting power

I={1;...n} Set of members of the committee

I-i = I\{i} Set of all members except for member i.

n Number of members

w i Voting weight of member i∈I

i∈I w i =2h+1, h∈IN => absolute majority is h+1

v i ∈{1;0} Vote of member i; v i =1 if i votes y, v i =0 if i votes n.

i∈I w i v i Number of weighted votes for y

=> 2 n is the number of possible preference constellations

=> Social outcome is y if i∈I w i v i ≥ h+1, otherwise n.

=> Voter k∈I is pivotal if

• i∈I w i v i ≥ h+1 > i∈I-k w i v i (with k: y, without k: n)

• i∈I w i v i < h+1 ≤ i∈I-k w i v i (with k: n, without k: y)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


82

Voting power of member k

• absolute: number of constellations in which k is pivotal, divided by the number of

possible preference constellations 2 n (denote this ratio as P k ).

• relative: p k =P k / i∈I P i

• total: p k =1

• zero: p k =0

increases in w k , decreases in n and in the quorum (here: q=h+1) from which on a

decision is valid (unless some member has total /k has zero voting power,

think of n=2, w 1 =51% and w 2 =49%.).

Assume that the EU wants each citizen be represented with equal voting power in the

EU council.

Should the EU council members (=representatives), hence, be assigned voting weights

that directly reflect (i.e., that are linear in) the population size of their member states

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


83

Penrose 1946 Journal of the Royal Statistical Society 109(1), 53-57

see Leech, D. 2002: Designing the Voting System..., Public Choice 113, 437-464.

If a committee consists of delegates of member states, and each committee member

receives voting weights according to the population size of the state he represents,

then his voting power is proportional to the square of the population size.

Example: if a member state has two times the population of another state, it would have

four times the voting power of the other state.

Hence: To give each citizen the same ex ante influence (in the Penrose/Banzhaf sense)

on the outcomes of the EU council, the voting weights of his member state delegate

should be proportional to the square root of the population size.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


84

Relative power indices under different

procedures (for EU25): the “constitution”,

the Nice treaty, and the square root rule.

The table shows relative power indices, as

the figures in the columns add up to 100.

Hence, you can compare the figures in

each line directly, e.g., Poland:

• with a population of 38 Mio. and 27

votes (almost as much as Germany,

popul. 80 Mio.), the polish power is

about the same as Germany’s under

the Nice treaty (in EU25).

• Under the Constitution, Germany

gains a lot of power, while Poland

loses a bit.

• The square root would endow

Poland with

• about the same power as Nice did,

while Germany still gains a bit.

=> This is why Kirsch proposes SQR and

a quorum of 62% of the EU population.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de

Source: Kirsch, W.: Europa, nachgerechnet;

in ZEIT 2004, Nr. 25 of June 9.


85

Discussion

For a single stage voting procedure, power index is easy (however tedious) to compute.

Voting weights of Nice treaty are completely arbitrary, while Poland proposes a general

principle (no more barter necessary).

=> puzzling that German politicians now accuse Poland of “political barter”.

=> Poland will even lose (compared to Nice treaty) when “square root” is applied.

Recall Arrow and Gibbard/Satterthwaite:

If scientists call the square root rule “just”, this must not be confused with “collectively

rational” or “non-manipulable”. It only refers to the “one citizen, one vote” (better: same

influence) idea.

Voting procedures in Nice treaty and “Constitution” are more complicated (two/three

stages), which has impact on voting power analysis => example.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


86

Example for power index under a two-stage procedure

Consider three firms who send delegates to the supervisory board of a joint venture:

C, F, S; voting weights proportional to their equity shares: w C =48, w F =9, w S =43

C (48) F (9) S (43) social outcome who is pivotal

y y y y -

y y n y CF

y n y y CS

y n n n SF

n y y y SF

n y n n CS

n n y n CF

n n n n -

=> Penrose-indices for majority voting: p C =p F =p S =1/3

=> number of shares (


87

Now introduce a second criterion, e.g., annual returns (market shares). Assume that

• r C =60, r F =5, r S =20 (in Mio. $)

• a proposal has to be backed by more than 50% of votes AND at least 70 Mio. $ in

returns to become the social decision “y”.

C (48/60) F (9/5) S (43/20) social outcome who is pivotal

y y y y CS

y y n n (only 65) S

y n y y CS

y n n n S

n y y n C

n y n n -

n n y n C

n n n n -

=> C and S are pivotal in 4 out of 8 cases, resepctively, while F is never pivotal.

=> relative voting power p C =p S =1/2, p F =0.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


88

Discussion of voting power (continued)

Leech (2002) on Nice, and (2001) on Corporate Governance and Power in UK.

Garrett/Tsebelis, Lane/Berg, Steunenberg et. al. (J. of Theoretical Politics 1999).

Power indices are blind for

• other game structures than simple voting games (such as proposal or veto rights),

exception: “strategic power index” of Steunenberg et. al.;

• heterogeneous valuations of social outcomes,

value of prevailing is normalized to one, no different “intensities”, no efficiency

gains from switching to other constitutions;

• non-uniformly distributed preference configurations,

some implicit coalition may be highly unlikely.

Example for the last problem: I={C;S,F}, w C =48, w F =9, w S =43.

a) Power indices if all (informal) coalitions are possible

b) Power indices if agreement between C and S has to be ruled out

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


Example for power index with heterogeneous preference distribution

Consider again the three firms who send delegates to the supervisory board of a joint

venture; voting weights proportional to their equity shares: w C =48, w F =9, w S =43

C (48) F (9) S (43) social outcome who is pivotal

1 y y y y -

2 y y n y CF

3 y n y y CS

4 y n n n SF

5 n y y y SF

6 n y n n CS

7 n n y n CF

8 n n n n -

and assume that C and S will NEVER agree

=> probability of lines 1, 3, 6, 8 is zero

=> voting power of the three players

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de

89


90

Reduced game

C (48) F (9) S (43) social outcome who is pivotal

2 y y n y CF

4 y n n n SF

5 n y y y SF

7 n n y n CF

Now, F is pivotal in 4 out of 8 cases, C and S in 2, respectively.

=> p F =1/2, p C =p S =1/4 (absolute as well as relative).

Addition of the return criterion (more than 70 out of r C =60, r F =5, r S =20)

=> p C =p S =1/2, p F =0 (but the social decision is “n” in all four cases, as C and S disagree)

Compare this to

initial situation p C =p F =p S =1/3

• added criterion (annual returns) p C =p S =1/2, p F =0

=> who might be interested in introducing which rule

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


91

Voting games

=> interactive behavior between voters

=> programs, proposals taken as given

=> agenda setting, strategic voting

Election games

=> candidates (only) care for being elected (=> probability to prevail)

=> offer programs, proposals

=> voter vote according to their preferences (honestly)

=> zero-sum games

Ordeshook 4.6: elections with a single issue

(one dimensional policy space => single peaked preferences of voters)

=> Median Voter Theorem (MVT)

Ordeshook 4.7: Two-dimensional policy space => trade offs, indifference curves

(but with limited budget, two-dimensional problems can be reduced to one dimension)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


92

Median Voter theorem

Ordeshook, p. 162, see also Congleton (2002)

Strong version of the MVT:

In symmetric two-candidate plurality-rule elections with only one issue (on which voters

have single-peaked preferences), in equilibrium both candidates adopt the median

ideal point.

=> median = same number of voters to the left and to the right.

=> Not necessarily the “middle” of the policy space, or the average policy.

=> Proof: unique strategy combination without incentive to deviate from.

Weak version of the MVT:

In the above setting, the ideal point of the median voter is the implemented policy.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


93

Assumptions of the MVT

• One-dimensional issue space, voters have single-peaked, symmetric preferences.

=> what happens with multi-peaked utility functions

• Two candidates/parties

=> what is the equilibrium with 3 or more parties

• Candidates are only interested in prevailing, not in ideology (=> zero-sum-game).

=> a genuine interest in policy would alter the parties’ yield functions.

• All voters participate and vote according to their preferences regarding the issue.

• Candidates know the voters’ preferences (at least shape and ideal points).

=> otherwise it is more difficult to determine the median voter.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


94

Median voter theorem is based on Hotelling (1929)

• “Hotelling road”: consumers are uniformly distributed over [0,1] (or a circle).

• Transportation cost = harm caused by the distance between supplier and consumer.

• Suppliers max. profit, consumers minimize “full price” (price + transportation cost

if borne by buyers, higher price if transportation cost are borne by suppliers).

=> with zero production cost, consumers buy from the closest competitor.

Hotelling 1: If two oligopolists are located at, e.g., 0 and 1, what prices do they set

Hotelling 2: What locations will two (or more) oligopolists choose

Hotelling 3: Simultaneous choice of location and price

Puer location choice model:

=> normative analysis – which allocation is efficient

=> positive analysis – what location is chosen in equilibrium

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


95

Hotelling – location choice with 2 suppliers

Normative analysis: minimize total transportation cost T(A,B)

Assume that one competitor sits on A, the other on B>A

=> all buyers between 0 and A buy at A

=> total transportation costs in this interval amount to 0.5A*A=0.5A 2

(where 0.5A is the average cost, and A is the number of consumers in [0,A])

=> same analysis for intervals [A,(A+B)/2], [(A+B)/2,B], [B,1]

=> T(A,B) = 0.5A 2 + 0.5[0.5(A+B)-A] 2 + 0.5[B-0.5(A+B)] 2 + 0.5(1-B) 2

= 0.5A 2 + 0.5(1-B) 2 + 0.25(B-A) 2

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Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


96

=> ∂T/∂A = A+0.5(A-B)=0 => 3A=B

=> ∂T/∂B = B-1+0.5(B-A)=0 => 3B=2+A

=> 9A*=2+A*

=> A*=1/4, B*=3/4 are the optimal locations (from a normative point of view)

T(A*,B*)=0.5*1/16+0.5*1/16+0.25*1/4=1/8 is the transportation cost minimum.

Compare this to T(1,1) = T(0,0) = 1/4.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


97

Positive analysis:

If B is chosen and A replies A incentive to choose greater A.

If A chooses A>B, he yields 1-(B+A)/2=(2-B-A)/2;

derivative with respect to A is negative => incentive to choose smaller A.

If A chooses A=B, he receives 0.5 of the market,

while any deviation makes him worse off.

B’s best reply to A choosing his immediate neighbourhood:

(A-e) if A>0.5, and (A+e) if A the unique equilibrium is A=B=0.5

=> T(1/2,1/2) = 1/4 > 1/8 = T(A*,B*)

=> the equilibrium is inefficient.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


98

Hotelling – location choice with n>2 suppliers

optimal allocation

equilibrium in pure strategies

n=3 A=1/6, B=1/2, C=3/6 none

n=4 A=1/8, B=3/8, C=5/8, D=7/8 A=B=1/4, C=D=3/4

Cox (see Ordeshook 4.11, p. 198): the spatial voting game with n candidates has

• a unique Nash equilibrium if n is even,

• no equilibrium in pure strategies if n is odd.

=> Mixed-strategy equilibrium: see Ordeshook 4.10 (no topic of this course).

=> Computations: Tutorial!

Intuition:

• If n is odd, players have an incentive to move towards the middle where a single

player sits; this player has incentive to “jump” over a new neighbour.

• If n is even, a pair of players sits at the equilibrium points, none of which has an

incentive to deviate.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


99

Median-voter theorem and the Condorcet voting paradox

=> The median ideal point is a Condorcet winner (i.e., prevails in any pairwise voting)

=> the social preference under simple majority rule is transitive

=> and ranks the median ideal point highest.

(D. Black, 1950)

With single-peaked preferences, neither the Arrow theorem nor the Gibbard-

Satterthwaite theorem applies:

Arrow:

=> Collective rationality despite non-dictatorship.

=> Violation of unrestricted domain (restriction to single-peaked preferences).

Gibbard-Satterthwaite:

=> No incentive to misrepresent individual preferences.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


100

Critical evaluation of the median voter theorem

+ In two-party systems (US), sometimes the parties are hard to distinguish.

+ Provides explanation why parties have a tendency towards the center.

- However, often parties show significant differences with respect to certain issues.

- If voters may abstain, the ex-post median is hard to determine.

- Voters do not always have single-peaked preferences (=> min distance to ideal point).

- Long-term loyality to a specific party is excluded.

- Incumbent bonus is excluded.

- Fund-raising is also excluded, but may make an extreme position more lucrative.

- More than two parties change the equilibrium

In the Hotelling model, the equilibrium depends on

• number of competitors

• shape of the transportation/distance cost function

• shape of the Hotelling “road” (line, circle)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


101

Two dimensional issues and the MVT

Often, two issues can be represented in a one-dimensional figure (“simplex”), e.g., if two

expenditure proposals have to be financed out of one budget (as x=B-y).

Examples for independent issues:

• degree of export orientation

• number of new models to be introduced

• recruit outside managers or rely on internal promotions

If a group has to decide upon two of such topics simultaneously, they have a genuine

two-issue problem => two-dimensional spatial voting model with individual ideal points.

If indifference curves are circular, the Euclidian distance is a measure for (dis)utility.

=> Does such a model always have an equilibrium (unique, in pure strategies)

=> Theory: only in special cases.

=> Experimental results with “open” (direct communication) and “closed” rules.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


102

Special case with five voters

and two intersecting contract

curves, the fifth ideal point is

on this intersection.

=> this is the median

(or: central) voter.

Any other combination than

his ideal point can be blocked

by three parties (who are

better off from another

allocation).

Experimental results:

Under both rules, the median

outcome is mainly chosen.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


103

This figure shows the normal

case without a median voter:

The central contract curves

create a pentagon => any

proposal can be blocked by a

coalition of three (moving

towards a different side of the

pentagon).

Even though the pentagon

does not contain theoretical

solutions, under both rules it

attracts (most of) the

outcomes, in particular under

the closed rule.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


104

Leftovers

• Rent-seeking, tournaments

• Teams, externalities, public goods

• Endogenous hierarchies, strategic power index analysis

• Bayesian Monitoring

• Rule-governed behavior, corporate culture.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


105

Wrap up: Collective Decision-Making in Organizations

Major topics:

• Voting (including Condorcet/Arrow/Gibbard-Satterthwaite

• and the Turkish president)

• Logrolling (welfare effect / stability of coalitions)

• Committees/Experts

• Power indices

• Median voter theorem (supply side of a political oligopoly market)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


106

Major insights: voting

• Simple majority voting among many alternatives may lead to a decision which is

supported by only a small minority.

• Absolute majority decisions requires more complex procedures (e.g., a series of

pairwise decisions, or Borda/Hare...).

• Pairwise majority voting may lead to the Condorcet paradox: in the absence of a

Condorcet winner, the social outcome is agenda dependent.

• Arrow: four reasonable requirements can only be satisfied by dictatorship (however,

this is more harmless than it sounds; property rights are a sensible interpretation).

• Gibbard/Satterthwaite: only dictatorship is resolute and non-manipulable.

=> Unless we apply dictatorship, there is always scope for strategic voting.

=> Comparison of institutions/voting systems requires taking this into account

(you cannot just keep constant the behavior of the members of the organization).

• Strategic voting can be prevented by adequate complex rules (e.g., support quota

instead of participation quota).

• Different voting rules violate different desirable properties.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


107

Major insights: committees

• Discursive dilemma

• Condorcet Jury Theorem

• Generalized CJT for expert committees

Major insights: power

• In representative bodies, influence/power is influenced by the voting weights,

• but is not just identical (depending on the one-/multi-stage decision rule) – a small

member can have enormous influence (i.e., can often be pivotal).

• In representative bodies, voting weights proportional to the represented population

would give the citizens of larger states a greater (indirect) influence.

• According to the Penrose index, this power is proportional to the square of the

voting weights.

• Equal (indirect) influence for each citizen, thus, requires assigning voting weights

proportional to the square root of the respective population.

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de


108

Major insights: median voters

• Prerequisites: one issue, symmetric+single-peaked preferences, known to suppliers,

no abstention.

• Weak version: the median voter (not necessarily in the “center”) prevails.

• Strong version: two suppliers will define their offer at the median position.

• Two-issue problems either have to be rescaled into one issue space (budget),

• or do have an equilibrium only in exceptional cases (if one voter is situated on the

intersection of all interior contract curves).

• However, in experiments, suppliers seem to behave as if a unique equilibrium did

exist.

Hints for the final:

• Understand the intuition behind the models/results, be prepared to explain them

verbally and by presenting the figures and tables.

• Define/explain symbols, figures, variable... make clear how you derive your results.

• Check whether your result really addresses the question ;-)

PD Dr. Roland Kirstein, Professorship of Business Economics

Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg

and Center for the Study of Law and Economics, Saarland University, Saarbrücken

http://rolandkirstein.de

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