Optimizing group judgmental accuracy in the presence of interdependencies

Consider a group of people confronted with a dichotomous choice (for example, a yes or no decision). Assume that we can characterize each person by a probability, p _i, of making the ‘better’ of the two choices open to the group, such that we define ‘better’ in terms of some linear ordering of the alternatives. If individual choices are independent, and if the a priori likelihood that either of the two choices is correct is one half, we show that the group decision procedure that maximizes the likelihood that the group will make the better of the two choices open to it is a weighted voting rule that assigns weights, w _i, such that $$w_i \propto \log \frac{{p_i }} {{1 - p_i }}.$$ We then examine the implications for optimal group choice of interdependencies among individual choices.     

If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule

We consider four factors relevant to picking a voting rule to be used to select a single candidate from among a set of choices: (1) avoidance of Condorcet losers, (2) choice of Condorcet winners, (3) resistance to manipulability via strategic voting, (4) simplicity. However, we do not try to evaluate all voting rules that might be used to select a single alternative. Rather, our focus is restricted to a comparison between a rule which, under the name ‘instant runoff,’ has recently been pushed by electoral reformers in the US to replace plurality-based elections, and which has been advocated for use in plural societies as a means of mitigating ethnic conflict; and another similar rule, the ‘Coombs rule.’ In both rules, voters are required to rank order candidates. Using the instant runoff, the candidate with the fewest first place votes is eliminated; while under the Coombs rule, the candidate with the most last place votes is eliminated. The instant runoff is familiar to electoral system specialists under the name ‘alternative vote’ (i.e., the single transferable vote restricted to choice of a single candidate). The Coombs rule has gone virtually unmentioned in the electoral systems literature (see, however, Chamberlin et al., 1984). Rather than considering the properties of these two rules in the abstract, we evaluate them in the politically realistic situations where voters are posited to have (at least on balance) single-peaked preferences over alternatives. Evaluating the two rules under this assumption, we argue that the Coombs rule is directly comparable in that Coombs is always as good as AV with respect to two of our four criteria and it is clearly superior to AV with respect to one of the four criteria, namely criterion (2), and is potentially inferior only with respect to criterion (3). Key to this argument are two new propositions. The first new result shows that, under the posited assumption, for four alternatives or fewer, AV is always as likely or more likely to select the Condorcet winner than plurality. The second new result shows that, under the same assumptions, the Coombs rule will always select the Condorcet winner regardless of the number of alternatives.     

Nash equilibrium strategies in directional models of two- candidate spatial competition

The standard approach to two-party political competition in a multi-dimensional issue space models voters as voting for the alternative that is located closest to their own most preferred location. Another approach to understanding voter choice is based on preferred direction of change with respect to some specified neutral point (e.g., an origin or status quo point). For the two-dimensional Matthews directional model (Matthews, 1979), we provide geometric conditions in terms of the number of medians through the neutral point for there to be a Condorcet (undominated) direction. In this two-dimensional setting, the set of residual locations for which no Condorcet directions exist is identical to the null dual set (Schofield, 1978) and to the heart (Schofield, 1993). In two dimensions, for most locations of the origin/status quo point, a Condorcet direction exists and points toward the yolk, a geometric construct first identified by McKelvey (1986). We also provide some simulation results on the size of the null dual set in two dimensions when the underlying distribution of points is non-symmetric.     

Rethinking the partisan effects of higher turnout: So what's the question?

Controversy persists over the link between turnout and the likelihood of success of Democratic candidates (e.g., DeNardo, 1980, 1986; Zimmer, 1985; Tucker and Vedlitz, 1986; Piven and Cloward, 1988; Texeira, 1992; Radcliff, 1994, 1995; Erikson, 1995a, b). We argue that the authors in this debate have largely been talking past one another because of a failure to distinguish three quite different questions. The first question is: “Are low turnout voters more likely to vote Democratic than high turnout voters?” The second question is: “Should we expect that elections in which turnout is higher are ones in which we can expect Democrats to have done better?” The third question is the counterfactual: “If turnout were to have increased in some given election, would Democrats have done better?” We show the logical independence of the first two questions from one another and from the third, and argue that previous researchers have failed to recognize this logical independence – sometimes thinking they were answering question three when in fact they were answering either question one or question two. Reviewing previous research, we find that the answer to the first question once was YES but, for more recent elections at the presidential level, now appears to be NO, while, for congressional and legislative elections, the answer to the second question appears generally to be NO. However, the third question is essentially unanswerable absent an explicit model of why and how turnout can be expected to increase, and/or analyses of individual level panel data. Thus, the cross-sectional and pooled data analyses of previous research are of almost no value in addressing this third question.

     

Centripetal forces in spatial voting: On the size of the Yolk

The yolk, the smallest circle which intersects all median lines, has been shown to be an important tool in understanding the nature of majority voting in a spatial voting context. The center of the yolk is a natural ‘center’ of the set of voter ideal points. The radius of the yolk can be used to provide bounds on the size of the feasible set of outcomes of sophisticated voting under standard amendment procedure, and on the limits of agenda manipulation and cycling when voting is sincere. We show that under many plausible conditions the yolk can be expected to be small. Thus, majority rule processes in spatial voting games will be far better behaved than has commonly been supposed, and the possible outcomes of agenda manipulations will be generally constrained. This result was first conjectured by Tullock (1967).     

Why representatives are ideologists though voters are not

Though few voters appear to hold consistent ideological views, the roll call votes of congressmen and senators can be well predicted by ideological terms. An explanation for this puzzle is that ideology allows candidates to succinctly explain their views. Because it is difficult to explain detailed positions to voters, a candidate who presents his position in ideological terms may be able to defeat a candidate who supports a set of issue positions that would, in toto, be preferred by a majority of well-informed voters were the voters aware of all the views of that candidate. This effect can be a powerful one. Moreover, ideology may be a source of electoral stability, and a means of providing regularity and structure to elite political debate.Amihai Glazer acknowledges support from the Graduate School of Business at Stanford University. Bernard Grofman received partial support from Grant SES #85-06376, Decision and Management Science Program, National Science Foundation, and NSF Grant BNS #80-01194 to the Center for Advanced Study in the Behavioral Sciences at Stanford University, at which he was a 1985–86 Fellow. We are indebted to Gordon Tullock for his comments.