Sunanda Roy

On removing Condorcet effects from pairwise election tallies

Condorcet profiles are responsible for a number of well known preference aggregation paradoxes. It is reasonable to claim that these profiles represent complete ties between the candidates and should therefore be excluded to determine election outcomes. Established profile decomposition techniques are of limited usefulness in extracting and removing Condorcet effects, because of the computational complexity involved, even if complete knowledge of voters’ rankings of all candidates are available. The paper discusses an easily implementable method of removing or reducing Condorcet effects from pairwise scores. Pairwise scores (and not complete knowledge of voters’ rankings) are often the only available data based on which an election winner has to be determined.

Rationalizability, adaptive dynamics, and the correspondence principle in games with strategic substitutes

New insights into the theory of games with strategic substitutes (GSS) are developed. These games possess extremal serially undominated strategies that provide bounds on predicted behavior and on limiting behavior of adaptive dynamics, similar to games with strategic complements (GSC). In parameterized GSS, monotone equilibrium selections are dynamically stable under natural conditions, as in parameterized GSC. Dominance solvability in GSS is not equivalent to uniqueness of Nash equilibrium, but is equivalent to uniqueness of simply rationalizable strategies. Convergence of best response dynamics in GSS is equivalent to global convergence of adaptive dynamics, is equivalent to dominance solvability, and implies uniqueness of equilibrium, all in contrast to GSC. In particular, Cournot stability is equivalent to dominance solvability in GSS. The results shed light on predicted behavior, learning, global stability, uniqueness of equilibrium, and dynamic stability of monotone comparative statics in GSS. Several examples are provided.

Comparative Statics with Never Increasing Correspondences

This paper studies models where the correspondences (or functions) under consideration are never increasing (or weakly decreasing) in endogenous variables, and weakly increasing in exogenous parameters. Such models include games of strategic substitutes, and include cases where additionally, some variables may be strategic complements. It is shown that the equilibrium set in such models is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. For a given parameter value, a pair of distinct equilibria are never comparable. Moreover, generalizing an existing result, it is shown that when a parameter increases, no new equilibrium is smaller than any old equilibrium. (In particular, in n-player games with real-valued action spaces, symmetric equilibria increase with the parameter.) Furthermore, when functions under consideration are weakly decreasing in endogenous variables, a sufficient condition is presented that guarantees existence of increasing equilibria (symmetric or asymmetric) at a new parameter value. This condition is applied to two classes of examples.