## Essays in collective decision making

##### Author

Derya, Ayşe Mutlu

##### Advisor

Kerimov, Azer

##### Date

2014-10##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

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Show full item record##### Abstract

Four different problems in collective decision making are studied, all of which
are either formulated directly in a game-theoretical context or are concerned with
neighboring research areas.
The rst two problems fall into the realm of cooperative game theory. In the
first one, a decomposition of transferable utility games is introduced. Based on
that decomposition, the structure of the set of all transferable utility games is
analyzed. Using the decomposition and the notion of minimal balanced collections,
a set of necessary and sufficient conditions for a transferable utility game
to have a singleton core is given. Then, core selective allocation rules that, when
confronted with a change in total cost, not only distribute the initial cost in the
same manner as before, but also treat the remainder in a consistent way are studied.
Core selective rules which own a particular kind of additivity that turns out
to be relevant in this context are also characterized.
In the second problem, different notions of merge proofness for allocation rules
pertaining to transferable utility games are introduced. Relations between these
merge proofness notions are studied, and some impossibility as well as possibility
results for allocation rules are established, which are also extended to allocation
correspondences.
The third problem deals with networks. A characterization of the Myerson
value with two axioms is provided. The first axiom considers a situation where
there is a change in the value function at a network g along with all networks
containing g. At such a situation, the axiom requires that this change is to be
divided equally between all the players in g who are not isolated. The second
axiom requires that if the value function assigns zero to each network, then each player gets zero payo at each network. Modifying the rst axiom, along a
characterization of the Myerson value, a characterization of the position value
is also provided.
Finally, the fourth problem is concerned with social choice theory which deals
with collective decision making in a society. A characterization of the Borda
rule for a given set of alternatives with a variable number of voters is studied
on the domain of weak preferences, where indi erences between alternatives are
allowed at agents' preferences. A new property, which we refer to as degree
equality, is introduced. A social choice rule satis es degree equality if and only
if, for any two pro les of two nite sets of voters, equality between the sums of
the degrees of every alternative under these two pro les implies that the same
alternatives get chosen at both of them. The Borda rule is characterized by the
conjunction of faithfulness, reinforcement, and degree equality on the domain of
weak preferences.

##### Keywords

Game theoryCooperative game theory

Transferable utility games

Core

Allocation rules

Allocation correspondences

Additivity

Proportionality

Merge proofness

Shapley value

Networks

Myerson value

Position value

Social choice theory

Borda rule