4 research outputs found
Impossibility theorems involving weakenings of expansion consistency and resoluteness in voting
A fundamental principle of individual rational choice is Sen's
axiom, also known as expansion consistency, stating that any alternative chosen
from each of two menus must be chosen from the union of the menus. Expansion
consistency can also be formulated in the setting of social choice. In voting
theory, it states that any candidate chosen from two fields of candidates must
be chosen from the combined field of candidates. An important special case of
the axiom is binary expansion consistency, which states that any candidate
chosen from an initial field of candidates and chosen in a head-to-head match
with a new candidate must also be chosen when the new candidate is added to the
field, thereby ruling out spoiler effects. In this paper, we study the tension
between this weakening of expansion consistency and weakenings of resoluteness,
an axiom demanding the choice of a single candidate in any election. As is well
known, resoluteness is inconsistent with basic fairness conditions on social
choice, namely anonymity and neutrality. Here we prove that even significant
weakenings of resoluteness, which are consistent with anonymity and neutrality,
are inconsistent with binary expansion consistency. The proofs make use of SAT
solving, with the correctness of a SAT encoding formally verified in the Lean
Theorem Prover, as well as a strategy for generalizing impossibility theorems
obtained for special types of voting methods (namely majoritarian and pairwise
voting methods) to impossibility theorems for arbitrary voting methods. This
proof strategy may be of independent interest for its potential applicability
to other impossibility theorems in social choice.Comment: Forthcoming in Mathematical Analyses of Decisions, Voting, and Games,
eds. M. A. Jones, D. McCune, and J. Wilson, Contemporary Mathematics,
American Mathematical Society, 202
Essays on the Computation of Economic Equilibria and Its Applications.
The computation of economic equilibria is a central
problem in algorithmic game theory. In this dissertation, we
investigate the existence of economic equilibria in several
markets and games, the complexity of computing economic
equilibria, and its application to rankings.
It is well known that a competitive economy always has an
equilibrium under mild conditions. In this dissertation, we study
the complexity of computing competitive equilibria. We show that
given a competitive economy that fully respects all the conditions
of Arrow-Debreu's existence theorem, it is PPAD-hard to compute an
approximate competitive equilibrium. Furthermore, it is still
PPAD-Complete to compute an approximate equilibrium for economies
with additively separable piecewise linear concave utility
functions.
Degeneracy is an important concept in game theory. We study the
complexity of deciding degeneracy in games. We show that it is
NP-Complete to decide whether a bimatrix game is degenerate.
With the advent of the Internet, an agent can easily have access
to multiple accounts. In this dissertation we study the path
auction game, which is a model for QoS routing, supply chain
management, and so on, with multiple edge ownership. We show that
the condition of multiple edge ownership eliminates the
possibility of reasonable solution concepts, such as a
strategyproof or false-name-proof mechanism or Pareto efficient
Nash equilibria.
The stationary distribution (an equilibrium point) of a Markov
chain is widely used for ranking purposes. One of the most
important applications is PageRank, part of the ranking algorithm
of Google. By making use of perturbation theories of Markov
chains, we show the optimal manipulation strategies of a Web
spammer against PageRank under a few natural constraints. Finally,
we make a connection between the ranking vector of PageRank or the
Invariant method and the equilibrium of a Cobb-Douglas market.
Furthermore, we propose the CES ranking method based on the
Constant Elasticity of Substitution (CES) utility functions.Ph.D.Computer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64821/1/duye_1.pd
PageRank as a Weak Tournament Solution
We observe that ranking systems—a theoretical framework for web page ranking and collaborative filtering introduced by Altman and Tennenholtz—and tournament solutions—a well-studied area of social choice theory—are strongly related. This relationship permits a mutual transfer of axioms and solution concepts. As a first step, we formally analyze a tournament solution that is based on Google’s PageRank algorithm and study its interrelationships with common tournament solutions. It turns out that the PageRank set is always contained in both the Schwartz set and the uncovered set, but may be disjoint from most other tournament solutions. While PageRank does not satisfy various standard properties from the tournament literature, it can be much more discriminatory than established tournament solutions