The Politics of Lineland Last time: She blinded me with Science Why and how should we study legislatures and legislators? Today Introduction to spatial voting theory

“Nothing shocks me. I'm a scientist.” --Indiana Jones political science need be no less scientific than the natural sciences –uncover and explain empirical regularities, using explanations as general as possible; test competing explanations against one another using public and replicable procedures; rigorous peer review social phenomena are not specially complex –cosmology, ecology, biology all deal with highly complex subjects

There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact. – Mark Twain Politics is a hard subject to study systematically, nonetheless –politics makes winners and losers; scientific knowledge about politics thus could affect who wins and who loses this means that our subject matter may respond to being studying (analogous to Heisenberg’s uncertainty principle) and that we have to be extra careful about data collection and interpretation

The dinosaurs became extinct because they didn't have a space program. – Larry Niven An introduction to the spatial theory of voting and elections political scientists seek to develop general, parsimonious explanations of political phenomena –formal logic offers a convenient way for saying a lot with a little and for building complex structures from simple ones in replicable ways –we will talk about some simple, formal tools, use the tools to develop models, then argue by analogy from formal model to real world

Rationality and Preferences We are going to assume without proof that the actors we study are rational –they can rank-order from best to worst alternatives over which they get to choose (or the outcomes associated with different courses of action they may undertake) –they choose courses of action based on their rankings of the alternatives, preferring actions leading to higher-ranked outcomes to actions leading to lower-ranked ones

Rationality and prefs, cont. “rational” individual prefs thus are complete and transitive But politics is about collective action/choice, not merely individual action/choice. How should individuals combine to produce group outcomes? –Can my action make a difference to the group outcome (i.e., do others need to care about what I will do)? –Will others’ actions make a difference to my welfare?

Interdependency and choice If my choices affect your welfare and your affect mine, we are in the world of game theory –basic solution concept to strategic interaction is the Nash equilibrium: A combination of strategies, one for each player in the game, such that no player can improve her outcome through a unilateral change in strategy –a strategy is a completely contingent plan of action –strategies can be “pure” or “mixed”, where a mixed strategy is a lottery over a set of pure strategies every strategic-form game with finite “pure” strategy sets has at least one Nash equilibrium in pure or mixed strategies

A simple model of elections Suppose that we can characterize voters’ preferences in degrees of “liberalness” and “conservativeness”, left to right –no voter who prefers Ted Kennedy to Diane Feinstein should also prefer Trent Lott to Diane Feinstein –this is a “single-peaked preference” condition With single-peaked prefs for voters and only two candidates competing by offering ideological platforms the “platform game” between the candidates has an unique, “centripetal” equilibrium –neither candidate can do better than to offer the median voter’s favorite ideological platform

A simple model of legislation Legislatures typically use “binary agendas” –consider policy change by evaluating alternatives two at a time, eliminating the loser from further consideration If legislators’ induced preferences are single- peaked over an ideological dimension, then the uniquely unbeatable policy alternative will be the median legislator’s most preferred policy and the winner in every pairing will be the alternative the median legislator likes better

The median voter theorem If choosers have single-peaked preferences over a set of feasible alternatives that can be ordered on a single dimension (e.g., smallest to biggest) and they choose via a binary agenda process with no abstention, then the median voter’s most preferred alternative on the agenda will be the winner. The alternative closest to the median chooser’s “ideal point” (in-principle most favored alternative) will also be a Condorcet winner (can beat any other alternative in pairwise vote). In 2-candidate elections, adopting a Condorcet-winner platform would be part of a Nash equilibrium In legislating, the procedural objective may often be to introduce/block the introduction of the median voter’s favorite policy

A spatial model X5X5 X1X1 X2X2 X3X3 X4X4 q X 3 is the favorite point of voter 3; q is the status quo ante (policy or incumbent’s platform), and points on the X line inside the half-circle centered on X 3 going through q are points that 3 likes better than q, i.e., 3’s “preferred-to set of q”, P 3 (q). Voter 3 is the median voter in this space, so only things that she likes better than q can win by majority-rule vote if preferences are single-peaked. We predict that in any binary choice, the alternative closest to X 3 will win

Extending the one-dimensional model of group choice Median voter theorem tells us that under certain conditions, group choice tends toward “central” outcomes Agenda control models –If the median dude’s favorite choice can’t get on the agenda, then he takes his favorite thing that DOES get on the agenda

Some terms Binary relation Rationality –Transitivity, completeness, efficiency Single-peaked preferences Nash equilibrium Status quo ante/reversion point Preferred-to set Indifference curve/set Win set