Directional and Static Equilibrium in Social Decision Processes

Author: Matthews, Steven A.

Year: 1978

Degree: Dissertation (Ph.D.)

Advisor: Plott, Charles R.

Committee Member: Unknown, Unknown

Option: Social Science

DOI: 10.7907/078W-7B13

Abstract

This thesis proposes a model of social decision processes that is applicable to situations in which social change must be incremental. In the limit, only the direction and not the speed of a shift in the status quo can be decided at each point in time. Individual preferences over directions are induced myopically via the inner product of direction (unit) vectors with the gradients of utility functions. Then the direction of shift at each instant is taken to be an equilibrium of a game that has directional outcomes.

Both two-person non-cooperative games in which two candidates adopt directional strategies to maximize their shares of cast votes, and n-person simple games of which absolute majority rule is a special case, are studied. Directional equilibria for the former and directional cores for the latter are characterized. Results include the following: (1) directions "pointing" towards point equilibria are directional equilibria; (2) a mobile candidate will diverge from a rigid, extremist opponent; (3) a status quo x simultaneously approaches each winning coalition's preferred-to-x set if and only if it shifts in an undominated direction; (4) given Euclidean preferences, a status quo that shifts in undominated directions will converge to the point core or to the set of points with empty directional cores; (5) an empty directional core at a point implies local cycling occurs in a neighborhood of the point; (6) stringent pairwise symmetry conditions must be satisfied by utility gradients at a point that has a nonempty directional core in a majority rule game; and (7) undominated directions exist at boundary points of a global cycling set and "point back into" the cycling set. Results (6) and (7) indicate that for majority games in spaces of dimension greater than three, directional cores are usually empty and global cycling sets are usually the entire space.

The disseration appendix is a self-contained paper in its own right. In a behaviorally-intuitive fashion, it establishes pairwise symmetry conditions for a point contained in the interior or boundary of a convex feasible set to be quasi-undominated in an anonymous simple game.

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