Chapter 92

ENDOGENOUS PROPERTIES OF EQUILIBRIUM ANDDISEQUILIBRIUM IN SPATIAL COMMITTEE GAMES

RICK K. WILSON

Rice University

Unlike a market in which the combination of individual actions yields Pareto optimaloutcomes, a political setting generally involves collective choices over Pareto subopti-mal reallocations. Whereas the combination of private interests in a market leads to theefficient allocation of resources, in politics coalitions decide how to reallocate resources,oftentimes taking from some to give to others. Political decisions are often made under aprocess in which everyone agrees to abide by a choice made by some (e.g., a majority).Models of social choice point out that such processes are hardly neutral and need notreflect the general will of those participating in the decision (Arrow, 1963; Plott, 1967;McKelvey, 1976; Riker, 1980; Austen-Smith and Banks, 1998).

Within many political institutions, collective choices are sensitive to the endogenouspreferences of actors. One such archetypal institution is the spatial committee game. Init there are a set of actors, each of whom hold equivalent powers. An initial status quois arbitrarily imposed on the committee and any actor is free to propose an amendmentto the status quo. Once an amendment is brought to the floor it is voted on under simplemajority rule. If a majority prefers the amendment, it replaces the status quo, otherwisethe status quo remains unchanged. The process of amending the current status quo con-tinues until a motion to adjourn passes with a simple majority. At that point the currentstatus quo is declared the final outcome for the committee process. In this sense theprocess is an idealized version of democratic practice in which any proposal can beconsidered, each proposal is compared under a binary choice procedure to the currentstatus quo, and the final outcome is that which is preferred by the majority.

1. Theoretical Background

The collective choice setting discussed here relies on standard assumptions tied to spa-tial modeling and some of that notation is developed here. Let N = {1, 2, . . . , n}be the n-membered set of decision makers charged with selecting a single alterna-tive, x, from a convex policy space X ⊂ Rm. Each member i ∈ N has a strictlyquasi-concave binary preference relation over all x ∈ X (such that ui(•) : Rm → R1).Taking a strong set of preference relations, for any pair of alternatives, x, xo ∈ X, ifui(x) > ui(x

o), then an individual’s preference ordering is represented as xPixo. In the

case that ui(x) = ui(xo), actors are indifferent among alternatives, or xIix

o. While anactor’s utility function can take on many forms, in the experiments presented here an

Handbook of Experimental Economics Results, Volume 1Copyright © 2008 Elsevier B.V. All rights reservedDOI: 10.1016/S1574-0722(07)00092-3

Ch. 92: Endogenous Properties of Equilibrium and Disequilibrium in Spatial Committee Games 873

individual’s utility declines as a function of distance away from her ideal point. Indif-ference contours can be represented as circles around that ideal point. For the ith actorwith an ideal point located at xi and for xi, xo ∈ X the set of alternatives preferred toxo by actor i is defined as Pi(x

o) = {x ∈ X|‖xi − x‖ < ‖xi − xo‖}.For simple majority rule games the set of winning coalitions is given as S =

{S1, S2, . . . , Sk} where Sj ∈ S if and only if |Sj | > n/2 (if n is odd) or |Sj | > (n+1)/2if n is even. In a spatial committee setting, alternatives can only be implemented by amajority coalition. In the same fashion that the set of alternatives preferred by an in-dividual to some xo was defined, a set of alternatives preferred by the coalition Sj toxo can be defined. Keeping with common usage the existing policy is referred to asthe status quo (xo). The set of alternatives preferred by a winning coalition is madeup of the intersection of those alternatives preferred by members of the coalition tothe status quo, or P(xo) = ⋂

i∈SjPi(x

o). Borrowing liberally from Shepsle and Wein-gast (1984) the set of all socially preferred alternatives is called the win set of xo orW(xo)−⋃

Sj ∈S PSj(xo). The standard finding in social choice is that W(xo) �= ∅. That

is, for any status quo, there exists at least one alternative (and usually many) that defeatsit under simple majority rule. It is only when the preferences of actors satisfy very spe-cific conditions (e.g., pairwise symmetry) that W(xo) �= ∅ and therefore an equilibriumexists.

2. Experimental Design

The experimental design uses a computer-controlled setting in which 5 subjects arebrought together as a committee to make a decision. Each subject is assigned a differentideal point in a two-dimensional alternative space. That space is 300 by 300 units andhas 90,000 unique points. Payoffs to subjects, for any point, are a non-linear decreasingfunction of distance from their ideal point. Sample indifference contours are providedfor a subject and by moving a pointer in the alternative space the computer providesan exact dollar value for any point in the space. The locations of other subject’s idealpoints are common knowledge, while other’s payoffs are not.

Subjects are able to place a motion on the floor at any time. No motion is broughtto a vote unless “seconded” by another. At that time the status quo and the amendmentare highlighted and subjects notified that a vote is forthcoming. After a 20 second delaysubjects are transferred to a voting screen in which the status quo and its value areposted against the same information for the amendment. Subjects are then asked to votefor one or the other and their vote is cast privately. Once voting ends the results arerevealed. If the amendment gains at least three of five votes, it becomes the new statusquo, otherwise the status quo is retained. The committee process then continues with allprevious proposals on the floor and subjects are invited to make additional proposals.The committee ends its deliberations when a motion to adjourn is brought to the floorand a majority votes to end the experiment. At that point subjects earn the value of the

874 R.K. Wilson

current status quo (for detailed discussion of this procedure, see Haney, Herzberg, andWilson, 1992 or Fiorina and Plott, 1978).

3. Endogenous Preferences

The experiment reported here manipulates only the distribution of subject ideal points.Three manipulations are used: the Core, the Star and the Skew Star. Each of these ma-nipulations has unique characteristics (see Plott, 1967 for a more general discussion).The Core is a unique equilibrium located at the ideal point of member 5 and it has theproperty that it defeats any other alternative in a pairwise contest. In the language usedabove, if x5 is member 5’s ideal point, then W(x5) �= ∅. By contrast the Star distributionhas no unique equilibrium prediction. Indeed, any point in the alternative space can beselected via some agenda. In the language used above, for any xo, W(xo) �= ∅. Likewisethere exists no unique equilibrium under the Skew Star distribution. However, in this set-ting there is a natural simple majority coalition of members 2, 3 and 4. These three actorsare located relatively close to one another and at some distance from members 1 and 5.

3.1. The Core

A large number of experiments have been run with the Core as a prediction (seeMcKelvey and Ordeshook, 1990 for a survey). Those results demonstrate two things.First, outcomes converge on the Core. Second, those outcomes seldom end up at theunique point prediction of the Core. The results reported here are consistent with alarge number of experiments run by others. Figure 1 plots the outcomes from sevenexperimental trials using the Core preference configuration. Several things are worthnoting on the figure. The ideal points for subjects are given by red squares and mem-ber 5’s position lies at the Core (it is almost obscured by outcomes). The initial statusquo is denoted by “Initial Status Quo” and its location is given as a small circle. Amend-ments passed by a majority at some point in the experimental trials are plotted in green.Finally, the outcome from each trial is plotted in blue.

The first thing to note from the figure is that no outcome falls at the equilibrium –member 5’s ideal point. As a point prediction the Core fails. However, taking the Coreas the prediction assumes that it is included in the agenda. It is important to rememberthat subjects had 90,000 distinct alternatives to pick from, on average put 27 on the floorand, on average, voted on only 2.9 different alternatives. Moreover, the point prediction,(120, 125) was bound to fail because it was never proposed in any of the experimentaltrials. However, in five of seven trials the final outcome is the Condorcet winner. It coulddefeat all other alternatives on the floor when subjects adjourned the experiment. Thata Condorcet winner existed is due to the structure of preferences. In the remaining twocases subjects at x5 voted against their self interest to move the status quo away fromtheir ideal point. Although they subsequently amended the agenda to move back to theCore, the trial ended before they reached the Condorcet winner.

Ch. 92: Endogenous Properties of Equilibrium and Disequilibrium in Spatial Committee Games 875

Figure 1. Outcomes under an experiment with a Core.

A more important feature of this agenda process is that outcomes are convergent.Plotted in Figure 1 are the successful amending steps to the agenda. These points aremore broadly distributed in the alternative space than the outcomes. What is not readilyapparent is the manner in which an agenda converged toward the Core. However, thiscan be shown by noting that in five of the seven trials every successful amendment wascloser to the Core than its predecessor. Moreover, the agenda process rapidly converged.While, on average, it took under 10 minutes of floor discussion to adjourn the exper-iment, it took very little time to reach the final outcome for the trial. On average thefinal outcome was selected within 41.6 seconds of beginning the trial. This meant thatthe final outcome was quickly selected, was chosen from a handful of proposals on thefloor, and was reached via a short agenda.

3.2. Star Preferences

By comparison under the Star preference manipulation there is no endogenous equi-librium. The outcomes from 18 trials under this configuration are plotted in Figure 2.While many of these outcomes fall in the central portion of the alternative space, they

876 R.K. Wilson

Figure 2. Outcomes under an experiment with symmetrically distributed preferences and no equilibrium.

are located in a very different area of the alternative space than outcomes under the Coreconfiguration and they are more widely dispersed.

Several things are worth noting in these trials. First, given that there is no equilibrium,outcomes do not converge on any particular point. Instead the agenda wanders acrossthe alternative space. This is especially pronounced in 6 of 12 trials with long agendas.In those 6 trials voting cycles were observed in the sense that an alternative A wasdefeated by an alternative B which was defeated by an alternative C which was thendefeated by alternative A. While predicted by theory (Arrow, 1961), voting cycles arealmost impossible to observe, yet here they occurred at least half of the time.

Second, because outcomes fail to converge, there are far more proposals placed onthe floor, far more votes taken, and far more changes to the status quo than under theCore preference trials. For instance in these trials subjects averaged 62.3 proposals onthe floor, called just under 21 amendment votes on average, and changed the status quomore often (6.7 times versus 2.9 times, on average). Some sense of this can be gainedby comparing the dispersion of successful amendments (noted in green) in Figures 1and 2. Under the Star configuration the agenda was much more likely to wander in thealternative space. All of this proposing, voting, and amending activity is a hallmarkof disequilibrium behavior predicted by standard models of collective choice. There is

Ch. 92: Endogenous Properties of Equilibrium and Disequilibrium in Spatial Committee Games 877

Figure 3. Outcomes with a skewed distribution of preferences and no equilibrium.

substantial instability in choice, the dynamics of the agenda path appear chaotic and theoutcomes are shifted to a different part of the alternative space than outcomes under theCore.

3.3. Skew Star Preferences

The Skew Star preference configuration, like the Star configuration, has no majorityrule equilibrium. It is introduced to show that outcomes are sensitive to the endoge-nous preferences of actors, independent of any equilibrium prediction. Figure 3 plotsthe outcomes from nine trials using this preference configuration. Those outcomes areclustered around the coalition {2, 3, 4}, even though no equilibrium prediction exists.This clustering represents two remarkable features about this voting process. First, out-comes are clearly dependent on the distribution of preferences. Second, the presence ofa specific majority rule coalition, defined by its proximity, exerts enormous influenceover the decision process. In this design no subject is granted the right to control theagenda. However these outcomes reflect the ease with which a majority coalition canimpose outcomes – especially when its members share common interests.

878 R.K. Wilson

Even though outcomes are clustered, this did not mean that subjects had an easy timesettling on them. The agenda process in these trials resembled that in the Star manipula-tion. First, under the Skew Star configuration, subjects took a good deal of time to reachan outcome. On average they spent about 12.5 minutes in proposal making – less thanunder the Star but greater than under the Core configuration. Second, subjects in theSkew Star manipulation cast almost as many amendment votes as their counterparts inthe Star manipulation (although they were less successful in amending the status quo).

As noted above, the close proximity of subjects 2, 3 and 4 granted an advantage tothat coalition. The agenda process did not discriminate against the outside players, theysimply had little to offer that was attractive to the coalition. Indeed, the two outsideplayers, 1 and 5, were responsible for almost 82 percent of the amendments that failed.Although they could, and often did, call votes, they were rarely successful in pushingan amendment. Consequently, as Figure 3 makes clear, the distribution of successfulamendments is focused around the simple majority rule coalition of members 2, 3 and 4.

4. Discussion

These experiments demonstrate the extent to which outcomes are responsive to thestructure of preferences. Where a preference-induced equilibrium exists, outcomes con-verge toward that equilibrium. Where a natural majority coalition exists, outcomescluster around it. Where preferences are widely distributed, so too are outcomes.

In a different vein Eavey (1991) suggests that subjects eschew their own private con-cerns and pay attention to relative payoffs among the group. In small groups, withface-to-face interaction, this is an interesting possibility worth further study. Grelak andKoford (1997) re-analyze Eavey’s data and compare them with the Fiorina and Plott(1978) data. Grelak and Koford (1997) do not reach the same conclusion about fairness,but they cannot dismiss the fact that subjects may hold other-regarding preferences andthat such preferences have an impact on spatial committee games. By and large, thefact that outcomes are sensitive to endogenous preferences comes as some relief. Theseresults indicate that there is a clear patterning to outcomes and they are dependent onthe preferences of actors. Rather than throwing up our hands in frustration over the “im-possibility” of understanding basic collective choice processes, these results point to thepossibility of predicting outcomes. Work on the “Uncovered Set” (McKelvey, 1986) andextensions by Austen-Smith and Banks (1998) provide insight into the results reportedhere. Those theoretical constructs aim at capturing the complex relationships betweenpreference and choices.

Acknowledgements

Support by the National Science Foundation (SES 87-21250) is gratefully acknowl-edged as is support by the Workshop in Political Theory and Policy Analysis at IndianaUniversity. Neither organization bears any responsibility for the content of this article.

Ch. 92: Endogenous Properties of Equilibrium and Disequilibrium in Spatial Committee Games 879

References

Arrow, K.J. (1963). “Social Choice and Individual Values”. Yale University Press, New Haven.Austen-Smith, D., Banks, J.S. (1998). “Positive Political Theory I: Collective Preference”. University of

Michigan Press, Ann Arbor.Eavey, C.L. (1991). “Patterns of distribution in spatial games”. Rationality and Society 3, 450–474.Fiorina, M.P., Plott, C.R. (1978). “Committee decisions under majority rule: An experimental study”. Amer-

ican Political Science Review 72, 575–598.Grelak, E., Koford, K. (1997). “A re-examination of the Fiorina–Plott and Eavey voting experiments: How

much do cardinal payoffs influence outcomes?” Journal of Economic Behavior and Organization 32, 571–589.

Haney, P., Herzberg, R., Wilson, R.K. (1992). “Advice and consent: Unitary actors, advisory models andexperimental tests”. Journal of Conflict Resolution 36, 603–633.

McKelvey, R.D. (1976). “Intransitivities in multidimensional voting models and some implications for agendacontrol”. Journal of Economic Theory 12, 472–482.

McKelvey, R.D. (1986). “Covering, dominance, and institution free properties of social choice”. AmericanJournal of Political Science 30, 283–314.

McKelvey, R.D., Ordeshook, P.C. (1990). “A decade of experimental research on spatial models of elec-tions and committees”. In: Enelow, J.M., Hinich, M.J. (Eds.), Advances in the Spatial Theory of Voting.Cambridge University Press, Cambridge.

Plott, C.R. (1967). “A notion of equilibrium and its possibility under majority rule”. American EconomicReview 57, 787–806.

Riker, W.R. (1980). “Implications from the disequilibrium of majority rule for the study of institutions”.American Political Science Review 74, 432–446.

Shepsle, K.A., Weingast, B.R. (1984). “Uncovered sets and sophisticated voting outcomes with implicationsfor agenda institutions”. American Journal of Political Science 28, 49–74.