1980 Vote Trading_An Experimental Study

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Vote Trading: An Experimental StudyAuthor(s): Richard D. McKelvey and Peter C. OrdeshookReviewed work(s):Source: Public Choice, Vol. 35, No. 2 (1980), pp. 151-184Published by: Springer

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Vote trading:An experimental study

RICHARDD. McKELVEY andPETERC. ORDESHOOK*California nst. of Technologyand Carnegie-MellonUniversity

Abstract

This essayreportsthe resultsof ninety 3-personand5-personbargainingxperimentsusingseveralalternativevote trading cenarios.Theseexperimentsaredesigned o test:

(1) RikerandBrams'controversial ypothesisthat vote tradingcanyield inferiorout-comes as againstthe alternativehypothesis that vote trading nduces 'market-like'

efficiencyin votingbodies;(2) the relativeadequacyof severalgametheoreticsolution

conceptsfor vote tradinggameswithout a Condorcetwinner(core);and, (3) the ade-

quacyof the core itself.

First, on the basis of eighteen experimentswith binding commitments,we findsome support or Riker and Brams'hypothesis:in nine trials,subjectsfirst tradeto aParetodominatedoutcome.In five of these trials,however,these outcomes are even-

tuallydisplacedby Pareto efficient ones. Withoutbindingcommitments,however,wefind little supportfor the 'paradoxof vote trading'hypothesis.Specifically,whilesixof seventrials of a 3-persongameyield 'equitable'outcomes,seventeen rialsof several5-persongameswithout bindingcommitmentsstronglysupportthe competitivesolu-tion as a cooperative amesolutionconcept,andsuggest hat the V-setandM, bargain-ing set areeither redundantor useless. Seventrialseach of six 5-persongameswith acondorcet winner(core), however,suggestthat usual static solution conceptsmay be

inadequateor treatinggameswith any interestingdegreeof strategic omplexity.

The theoreticaland empiricalliteratureson vote trading(or logrolling,asthe terms are frequently interchanged)s largeandgrowing.Yet, when vote

tradingopportunitiesexist, little is knowngenerallyabout how committees

actually operate under alternativerules and patterns of individualprefer-ence. This essay seeks to fill this gapby reportingon ninety 3- and 5-personbargainingxperimentsusingseveralvote tradingscenarios.

The empirical vote-tradingliterature exists almost exclusively in thedomainof legislative scholarshipand focuseson such questionsas: To what

extent does vote tradingoccur in the U.S. Congress,what are the tactics* This researchs supportedby aNationalScienceFoundationgrant.Wealsowishto

acknowledgehe assistanceof RodGretleinand MarkWiner.

PublicChoice 35 (1980) 151-184.All rightsreserved.

Copyrightq@1980MartinusNijhoffPublishersby, TheHague/Boston/London.







152 R.D.McKelveyndP.C.Ordeshook

legislatorsuse to vote trade, and how does the substance of legislationinduce or impede vote trading?The theoretical literature s dominatedbythree concerns:the relationshipbetween vote tradingopportunitiesandthe

paradoxof voting (a relationship hat is now well-understood),he issue ofwhether vote tradinginduces market-likeefficiency in voting bodies as

opposed to prisoners'dilemmatype inefficiency, and the relevanceof co-

operativegame-theoretical olutionconcepts.Principallybecause of methodologicalbarriers,these literaturesrarely

addresseach other. The resultsreportedin this essay,however,arerelevantto both. First, they test Riker and Brams'(1973) controversialhypothesisthat vote tradingcan yield Pareto inferior (inefficient) outcomes in legis-latures as against the alternativehypothesis that vote trading induces

'market-like'action

and, thus,is an aid in

obtainingefficient

legislativeoutcomes (cf. Tullock,1970; Haefele, 1970; andWilson,1969). Second,theresults of the experiments reportedhere test the principalgame-theoreticsolution concepts that are used to solve majoritygameswithout cores andwithout fixed agenda - the competitive solution, the Von Neumann-

Morgenstern olution (V-set), and the bargaining et. Third, we use these

experimentsto show how the bargaining ontext - specifically,the contextof open face-to-face negotiation (as in a 'democratic'legislativesubcom-

mittee) as opposed to a decentralized context but with enforceablevotecommitments (as might occur in a full legislaturewith norms enforcingpromises)- affect ourconclusions aboutthe first two issues addressedhere.But finally, and of concernto both the empiricalandtheoretical iteratures,we raise a disquietingnote about the fundamental solution concept of

cooperativegame theory - the core (and, hence, a disquietingnote aboutall derivativesolution concepts). Specifically, these experiments suggestthat a vote tradinggame'sstructurecan fundamentallyalter the processof

negotiationswhich, in turn, affects the likelihood that a Condorcetwinnerundermajorityruleis chosen.

Section 1 reviews the two theoretical issues that are central to this

essay: the extent to which vote tradinginducesefficiency or inefficiencyin legislativeoutcomes and the adequacyof existent gametheoretic solu-tion theory. Section 2 describes the two bargainingcontexts of our

experiments,while Section 3 details the specific voting games to whichthese contexts are applied. This section also describesthe predictions ofthe competitive solution in each game. Section 4 briefly reviews our

experimentalprocedures.Section 5 discusses the experimentaloutcomesin two parts: (1) an evaluationof the Riker and Brams''paradoxof vote

trading'hypothesis;and (2) an assessment of the relativeperformanceofthe three solution

hypotheses for gameswithout cores-

the competitivesolution, the V-set and the MI bargaininget. Section 6 presentsourresultswith respectto the core, andSection 7 summarizes urconclusionsand the

general mplicationsof the experiments.







Votetrading: nexperimentaltudy 153

Theoretical ssues

Perhapsthe most controversialhypothesis about vote tradingis the one

offered by Rikerand Brams- that suchtradingcan leadto inefficient out-comes in legislatures.Put simply,thishypothesisis at odds with the counter

argument that unrestrictedvote tradinginduces market-likeefficiency in

legislativebodies. Accordingto this argument, implemajorityrule,appliedto issues individually,fails to take account of preferenceintensity, whilevote trading across issues permits legislators to registertheir intensities

by tradingvotes on issues about which they care little for votes on issuesthat concernthem greatly.

The simple example in Table 1, which describes a 3-member egislature

confronted with six bills illustratesthis argument or efficiency. Assumingthat the payoff to each legislatorof a defeated bill is zero andthat payoffsare additiveacrossbills (i.e., the bills areunrelated),then, if simplemajorityrule is applied to each bill individually,all six bills fail and the payoff toeach legislatoris zero.' Legislator1, however, should be willing to tradea vote on bill C or F for a vote on D or E, legislator2 shouldbe willingtotrade a vote on A or D for avote on B or C,while legislator3 should tradeavote on B or E for one on A or F. If all threelegislators rade,the outcomePPPPPPprevails legislator1 passesbills D and E, 2 passesbills B and C,and 3 passes bills A and F. As a resultof trading,then, the net payoff toeach legislatoris two (e.g., -3 -3 -2 +6 +6 -2 = 2), which is Paretooptimal and Pareto preferred to the outcome that results from sincere

voting.Rikerand Brams,nevertheless,constructa simplevariantof this example

in which, if each pair of legislatorstrade a vote on the bill that concernsthem least for a vote on a bill that concernsthem most, a Pareto inferioroutcome results. Furthermore,the legislatorsare in a prisoners'dilemmawith respectto such trades- no legislatorpossessesany incentive to refuse

unilaterally o engage n vote trading.

Table 2 reproducesthis example (we normalizepayoffs here so that thepayoff to eachlegislatorof failinga bill is zero). Rikerand Bramsnow arguethat legislator 1 will trade away votes on bills C and F to block D and Efrom passing; egislator2 will tradeaway votes on A and D to block B and

C; and legislator3 will trade away votes on B and E to block A and F.The outcome that results from these bilateral tradesis all bills failingwiththe correspondingutility outcome (0, 0, 0). This outcome, however, isPareto dominatedby passingall bills and the correspondingpayoff vector

(2, 2,2).

That all bills fail is not a result that cooperative game theory predicts.Regardlessof whether one considers the core, V-set or competitive solu-

tion, the predictedoutcomes areParetooptimal.For the examplein Table2the core is empty, but the competitivesolutioncorrespondso the following







154 R.D. McKelveyandP.C. Ordeshook

Table 1.

Payoff if a bill is passed

Legislator A B C D E F

1 -3 -3 -2 6 6 -22 -2 6 6 -2 -3 -33 6 -2 -3 -3 -2 6

Table2. GameGI

Payoff if a bill is passedLegislator A B C D E F

1 3 3 2 -4 -4 22 2 -4 -4 2 3 33 -4 2 3 3 2 -4

coalitions and outcomes.

Coalition Outcome Payoffs

(1,2} PFFFFP (5,5,-8){1,3)} FPPFFF (5, -8, 5){2, 3} FFFPPF (-8,5, 5)

and in this instance,the set of these threepayoffs is the game'smainsimpleV-set,while these three payoffs, each being offered by the coalitionreceiv-

ing positivepayoffs corresponds o theM1 bargaininget.The reasonfor a divergencebetween these predictionsand the outcome

hypothesized by Riker and Brams is straightforward:Riker and Bramsassume that vote trading is a decentralized process, uncoordinatedbymajoritycoalitions,whereas solution theory assumes that if majoritycoali-tions can realizegains that are unobtainable under uncoordinatedaction,suchcoalitions form.

It is surprising hat Riker and Bramsdiscount the effects of majoritycoalitions, especiallywith the unconvincingargument hat such coalitionsare

necessarilyunstable'.Much of Riker'searlier

experimentalresearch,or

example, arguesthat the cooperativesolution notion of the V-setdefines

stability,even rationality,in these n-personsituations.Thisexample,never-

theless, raises an important substantivequestion. It seems reasonable to







Vote trading:An experimental tudy 155

hypothesize that Pareto inferior outcomescan prevail f the communicationand coordinationnecessaryto form winningcoalitions is rendereddifficult

by the bargaining ontext - as in decentralizedegislatureswith specialized

authority and expertise. The question their analysis raises, then, is: Dopeople necessarilytrade to inefficient outcomes or do cooperativecoalition

processes emerge, wheneverprofitable, even in systems with unrestrictedand enforceablebilateralvote tradingopportunities?The second questionthat their analysisraises,albeit indirectly, is: To the extent that an unre-stricted coalitionprocessprevails,whichcoalitionsform and what outcomes

prevail?This second question also addressesthe theoreticalliteraturethat links

vote trading opportunitieswith the paradoxof voting. The existence of a

paradoxof an

intransitive yclein the social

orderingunder

majorityrule

- is oftentimes taken to mean that social outcomesare unstable andunpre-dictable. But, such situations are simply examples of majorityrule gameswithout cores - without legislativeproposalsthat cannot be defeated byother proposals in a majority vote. The competitive solution, the V-set,and the bargainingset, however, are intended explicitly to treat gameswithout cores and, hence, our second questioncan be restated as a queryabout the relativeadequacy of these solution concepts, their applicabilityto the study of legislativeprocesses,and the extent to whichthe outcomesof majority rule situations without a Condorcet winner are nevertheless

predictable.2

Two experimentalbargaining ontexts

The realworld suggestsa variety of contexts for legislativevote trading.Weconsider two in the experimentsreported here - one that is designedtobest fit the preconditionsfor the applicabilityof the core and the com-

petitive solution, and another that seems best suited to induce the kinds

of inefficient tradeshypothesizedby Riker andBrams.

Context 1

The players (subjects) are confronted with a list of 'bills' and they mustdecide which bills to passand which to fail (where they can passall, none,or any subset). Discussion is unrestricted,but when some majority can

agree on the disposition of one or more of the bills, that dispositionbecomes the social choice. Discussion continues until all bills are disposedof

(passedor

failed).








Con ext2

Each playeris initially given one vote on each bill in the form of a ballot.Trades of these ballots can be discussed without restriction and trades

themselves can be made freely. Once a ballot is traded,however, it cannotbe recalledwithout the consent of the playerpossessing t, while all playersknow the holdings of all other players.At some prespecifiedtime or byunanimousagreementto terminatethe tradingperiod, all playerscast theballots in theirpossessionasthey choose, beginningwith bill A, then B, andso on. The socialchoice is the result of thisballoting.

Clearly,Context 1 is best suited for testing the adequacyof cooperativegame theoretic solution concepts and it parallels he proceduresused else-

where in spatial games to assess the core and the competitive solution(cf. Berl et al., 1976; Laingand Olmstead,1978). If it can be said to corre-

spond to anything in a real legislativeenvironment,it corresponds o thedeliberationsof small committees or conference committees debatingthe

pros and cons of amendments or additions to existing legislation. Thevote trades that Context 2 permitsmight seem unusualin any legislativesituationsince it entails tradingsome physicalandpotentiallyunretrievable

commodity - ballots. These ballots, however, are intended to representbinding commitments - contracts - that legislativenorms presumably

enforce.That these two contexts are strategicallydistinct from the view of theplayers is clear. In Context 1, promisescan be made and readily brokenin the event of a better proposal. In Context 2, on the other hand, deci-sions and commitments can be rendered irreversible.The question weaddress when conducting experimentsusing Context 1, then, concerns the

adequacy of n-person game theoretic solution concepts. The questionweaddressusing Context 2 is whether the Pareto dominated outcomes of aprisoners' dilemma prevail or whether the players learn to avoid suchoutcomes.

Vote tradinggames

The three player example discussedby Riker andBrams which we repro-duce in Table 2 and which we call G1 for future reference- is one gameconsideredin the seriesof experimentswe reporthere. In this section, wedescribe five additionalgames that we use in our experiments,but whichall entail five players.

Table 3 describesa game that seems even more likely to yield prisoners'dilemmatype tradesunderContext 2 than does Riker and Brams'originalexample. In this game, each player is concerned principallywith passinga singlebill whereas f all bills pass, all playersareworseoff than if all bills







Vote trading:An experimentalstudy 157

failed. That is, if each player successfully trades to secure two additionalvotes on the bill that concerns him most, all bills pass and each player'spayoff is -2 (i.e., 10 - 2 - 4 - 4 - 4 - 2 = -2); but if no tradingoccurs

and everyone votes sincerely all bills fail and each player receives apayoff of zero.

Table3. GameG2

Payoffs if bills arepassedPlayer A B C D E

1 10 -2 -4 -4 -2

2 -2 10 -2 -4 -43 -4 -2 10 -2 -44 -4 -4 -2 10 -25 -2 -4 -4 -2 10

This game is interesting also from the standpoint of testing solution

theory. That it possesses an empty core is readily confirmed. Unless we

resortto lotteries,moreover,its V-setis also empty. The competitive solu-

tion,K,on the other

hand,exists but it not

necessarilyobvious.

Specifically,K does not predict simply that any three playerswill coalesceto passtheir

bills and fail the remaining wo. Rather,the competitivesolution consistsof

the followingset of five coalitionsand outcomes.


{1,2, 3) PPPFF (4, 6, 4, -10, - 10)

2, 3,4)} FPPPF (-10, 4, 6, 4, - 10)

{3, 4, 5} FFPPP (-10, -10, 4, 6, 4)

{4, 5, 1 PFFPP (4, - 10, - 10, 4, 6)

{5, 1, 2} PPFFP (6, 4, - 1, - 1, 4)

Hence, K predicts that five of the ten minimal winning coalitions will

not form. For example, if the coalition {1, 3, 5} attempts to form with

the proposalPFPFP and utility outcome (4, -8, 2, -8, 4), it is not viable

againstthe proposal of {2, 3, 4} - the pivot, player3, strictlypreferswhat

{2, 3, 4} offersto what {1, 3, 5} offers.3Thus,whileanyof the tenper-mutationsof three passesand two fails is possibleundermyopic vote trad-

ing, only five of these permutationsis predictedby cooperative solution

theory.Another, but perhapsmore subtle version of this gameis the one por-

trayed in Table 4, which is obtainedfromG2 by permuting he playersand

by multiplyingthe payoffs on bills B and C by - 1.








Table4. GameG3

Payoffs if bills arepassed

Player A B C D E

1 -4 2 -10 -2 -42 -2 -10 2 -4 -43 -4 4 2 10 -24 -2 4 4 -2 105 10 2 4 -4 -2

Again,this game,underContext 2, presentsmuch the sameincentivesforeach playerto secure first the ballots of a singlebill, which, unlessblockedby secondary trades or coalitions, leads to the Paretoinferioroutcome ofpassingbillsA, D andE and the payoff vector (- 10, -10, 4, 6, 4). That thisoutcome is Pareto dominated can be seen by consideringthe alternativeoutcome FPPFF, whichyields the payoff vector (-8, -8, 6, 8, 6).

As with gameG2, the V-set s empty if lotteriesarenot permittedwhere-as the competitivesolution exists but is distinctly less symmetricthan thesolutions to gamesG

1and G2. Here,K conists of the

followingfive coali-

tions andoutcomes:


{1, 2, 3} FFFPF (-2, -4, 10, -2, -4){1,3, 4} FPFPP (-4,-18, 12, 12,-4){1, 2, 5} PFFFF (-4, -2, -4, -2, 10){3, 4, 5} PPPPP (-18,-18, 10, 10, 10){2, 4, 5} PFPFP (- 18, -4, -4, 12, 12)

The coalition predictionsof K in gamesG1, G2 and G3 are not trivial,but gamesthat yield even more restrictivepredictionsarepossible.Princip-ally, then, to test K further,gameG4 in Table 5 is also consideredin ourexperimentalsequence.

The competitivesolution for this gameis,


{1,2, 4} FPF (-1,7,- 1, 6,-5){1,3, 5} FFP (- 1,- 1,5, -4,2)

{2, 3, 4} PPP (-7, 7, 5, 6, -4)

In G4, then, K predicts that only three of the ten minimalwinningcoalitions form. In this instance,the payoff predictionsof K also constitute








Table5. GameG4

Payoffs if bills arepassed

Player A B C

1 -5 -1 -12 1 7 -13 1 -1 54 4 6 -45 -1 -5 2

Table6. GameG5

Payoffsif bills arepassedPlayer A B C D E

1 10 -2 -5 4 -52 -2 10 -5 -5 43 4 -8 -5 3 -84 -8 4 3 -5 -8

5 -5 -5 4 -10 4

a V-set, but this gameis nevertheless nterestingbecause the coalition pre-dictionsof K are the most restrictiveof any gameconsidered.

GamesG1 through G4 are all similar n that they all possess an emptycore - no disposition of bills is undominatedunder majority rule by allother admissabledispositions. Because of the core's intuitive plausibility

and the importancethe social choice literatureattaches to the notion of aCondorcetwinner,however,it seems reasonable o examinechoice in gameswith nonempty cores as a yardstick for judgingthe performanceof othersolution concepts. Table 6, then, describesan additional vote tradinggamein which, excludinglotteries, the sincerevoting outcome of failingall bills

(FF FFF) is a core.Game G5 is interestingnot simply becauseit possessesa core, but also

becausethere seemsampleopportunityfor outcomes other than the core to

prevail.For example, even underbargainingContext 1 it seems reasonable

in both gamesfor players 1 and 2 to agree,formallyor informally,to tradevotes on bills A and B - their most preferredand least costly bills. Withthat single agreement, all players can vote sincerely otherwise and passbills A and B. Hence, while G5 possesses a core, the myopic vote trading








agreements nvisionedby RikerandBramsmight proveirresistibleand resultin otheroutcomes.

A potential difficulty with this game is that the core might also be

deemed 'equitable'and, hence, be chosen for reasonsother thanits domin-ance property.To insureagainstthis possibility,we consider an additionalcore game,G6, that is derived from G5 by multiplyingthe payoffs of each

player on bills C and E by -1 (see Table7). For the game n Table 7, then,the core corresponds o the outcome FFPFP.

Table7. GameG6

Payoffsif bills arepassed

PlayerA B

CD E

1 10 -2 5 4 52 -2 10 5 -5 -43 4 -8 5 3 84 -8 4 -3 -5 85 -5 -5 -4 -10 -4

Experimentalprocedures

The subjectsin the experimentswe reporthereare drawnfromtwo sources- 152 undergraduates t Carnegie-MellonUniversity,and 42 MS studentsfrom C-MU'sGraduate School of IndustrialAdministrationand School ofUrban and Public Affairs. Thesetwo pools arekept distinct,and while most

subjects play in more than one game, subjects are scheduled so as tominimize the extent to which they play with each other more than once.4

Further, subjectswith no experience in these experimentsare not mixed

with subjects who had played in at least one earliergame.No additionalcontrols,however,areplacedon scheduling.The instructions for subjects are presentedin the appendix.Briefly, all

players sit at an 8 foot circular table to discuss the disposition of bills

openly; secret negotiations are not permitted. Further, while preferencesover bills are induced by money, no mention of money or schemes fordividingtheir winningsarepermittedduringthe bargaining.Thus, negotia-tions can occur only over the disposition of the several bills confrontingthe subjects. Each subject is told, nevertheless,the bills with which each

playerassociatives a

positivevs. a

negative payoff,and

subjectscan com-

municatewhich bills or combinationsthey most preferto pass and which

they most prefer to fail. Hence, communication about ordinal but notcardinal nformation s permitted.








To keep subjects from inferringthe exact magnitudesof other players'payoffs, the payoff schedulesin Tables2 through7 aremultipliedby j, 1,or 2, where the multiple for each player is chosen randomlybeforehand.

Hence, in any one of the 5-person games, for example, there are 35 = 243possible sets of payoff schedules,and in each experimentone of these isdrawnrandomly,with replacement.In addition,each subjectis assigneda

positive or negativenumber,a, that is revealedafter the terminationof the

experimentand added or subtractedto his or her final score. Since subjectsare told that, if their total score is less than a dollar, they will be paidonedollar for their participation, his number seeks to insure strictpreferenceover outcomes that promisenegativerewards i.e., if a is sufficientlygreat,it can cancel out negativerewardsand, hence, a person shouldnot be in-

differentbetween two proposalsthat promise negativepayoffs). Also, withrespectto outcomes with relatively largerewards, he threatof a negativeais intended to minimize satisficing.The comments of the subjectsduringand after the experiments suggestthat the inclusion of the unknowna pro-duces the intended effect. In fact, it is frequentlythe case that some playersuse a's existence to argue hat they shouldget 'more'- in fear of a negativea - when others suspect that they are already doing relativelywell in a

potential agreement. Overall, the final payoffs to subjects are nontrivial,rangingfrom a low of $ 1 to, in one instance,a maximum of $ 30, with

approximately$ 5.50 beingthe averagepayoff persubjectper experiment.Bargainingunder Context 1 proceedsin two stages:A 15-minutediscus-

sion stage followed by a decision stage without time limit. During thediscussionstage, players are free to discuss the dispositionof bills as theychoose, but no formally binding decisions are permitted. In the decision

stage, discussionmay continue, but when some majoritycan agreeon the

disposition of one, some or all of the bills, the membersof that majoritycan implementthat dispositionby each of them signingan agreementcard,noting that disposition.If threeormore cardsagreecompletely,a final deci-sion is taken on the bills in question. If three or morecardsdisagreen any

respect, no decisionis takenon any of the bills.Thisprocesscontinues untilall decisions are disposedof. Note that subjectsare not required o decideon allbills simultaneously,but can do so if they so choose.

Bargainingunder Context 2 proceeds in three stages: A three-minute

study period in which no tradescan be discussed;a tradingperiodthat lastsfor 15 minutes or until there is unanimousagreementto terminate;and, a

ballotingperiod. Duringthe trading period, discussion s open, as in Con-text 1, except that now the players possessballots on each bill that can betraded freely 'like baseballcards'.During the ballotingperiod, the ballots

on each bill are voted, first on bill A, then on bill B, and so on, at whichpoint the experiment erminates.One variant of bargainingunder Context 2 is also considered,but used

sparingly. Specifically, after the balloting is concluded,subjectsare asked








to vote on whether to approveor reconsider he resultantdispositionof thebills. If a majorityvotes to approve,the experimentterminates;otherwise,all ballots are redistributedwith each player again receivingone vote on

each bill and the tradingperiodbeginsanew. Underthis variantof Context2, then, the playersareafforded the opportunityto correct,if necessary,forthe Pareto nferioroutcomes Riker andBramshypothesize.

Experimental esults

Our discussionof experimentalresults is best divided into three parts: anevaluation of the Riker-Brams ypothesis, an evaluation of solution theoryfor

gameswithout

cores, and,in a

separatesection,an

analysisof

votinggameswith a Condorcetwinner(a core). The reasons for this ordershouldbecomeclearshortly.

Efficiencyvs. inefficiencyin vote trading

The experimental outcomes of particularrelevance to the question ofwhether vote trading fosters inefficiency or efficiency in legislativeout-comes are those from games G1, G2 and G3, usingbargainingContext 2.Table 8, which describes the relevant sixteen experimental outcomes,appears, at first glance, to provide only modest support for Riker andBrams'hypothesis.5The outcomes of only four experiments 7, 8, 13 and15 - correspond dentically to what their analysisof individuallymyopic,decentralized radingpredicts.And, more to the point, 8 of 12 experimentswith 'inexperienced'subjects do not arriveat the Paretoinferioroutcomewhile none of the six experimentswith playerswho havepriorexperienceas

subjects (althoughnot necessarily using Context 2) arriveat such an out-come.

These results,however,disguisesubstantialsupport for the vote trading

process their analysisassumes.First, in five additionalexperiments- 3, 5,9, 10 and 12 - the initial sequence of trades leads to the appropriatePareto dominatedoutcome (failing everything n 3 and5, passingeverythingin 9, 10 and 12). But, upon reachingthese outcomes, the subjectsrealizethe consequences of unrestrictedtrading, whereuponin experiments3, 5,10 and 12, an 'equitable'outcome is explicitly sought and chosen while in

experiment9, the coalition {1, 2, 3 forms, takes advantageof its oppor-tunity to redistribute he ballots,andpassesonly the bills its membersmost

prefer- an outcome in the competitivesolution,K.We

can go evenfurther,however, n findingsupportfor theprocesscom-ponent of their hypothesis, for oftentimes a Pareto inferior outcome isavoided more, we suspect, by 'accident' than by sophisticated strategicplanning.Considerexperiment11 in which players2, 3, and 4 arriveat an








outcome in K. It is tempting to view this outcome as a consequenceof

sophisticated strategy, yet a review of the players' discussionssuggestsotherwise.Specifically,the actual tradingsequence in this experimentis as

follows:

Ballotholdingsof players1 2 3 4 5

Initialholding ABCDEABCDE ABCDEABCDEABCDE2 & 3 tradeB & C ABCDE ABBDE ACCDEABCDE ABCDE2 &4 tradeB &D ABCDE ABBBE ACCDEADCDE ABCDE3 &4 tradeC &D ABCDEABBBE ACCCEADDDE ABCDE3 & 5 tradeA & E ABCDEABBBE ACCCAADDDE EBCDE

1 &2 tradeA & E ABCDA EBBBE ACCCAADDDE EBCDE

At this point no additional radesareworthwhile,and,inparticular,players1and 5 cannot offer the other players anythingof value for their votes on Aor E. The trades between players 3 and 5 and 1 and 2 insures this, andwhile advantageousn the short run for players 1 and 5, it is disadvantage-ous in the end. Thus, in this trial, it appearsthat, while players trade

myopically as Riker and Bramspredict, the particularsequence of tradesavoids an inefficient outcome accidentally.

Essentiallysimilar

processes explainthe

remaining experimentsdes-

cribedin Table 8. For example, the sequence of trading n experimentno.16 - in whichan outcomeinK alsoprevails is as follows:

Ballotholdingsof players1 2 3 4 5

Initialholdings ABCDE ABCDE ABCDEABCDEABCDE4 & 5 trade on A &E ABCDE ABCDE ABCDE BCDEEAABCD1 & 3 tradeon C &D ABCCE ABCDE ABDDE BCDEEAABCD2 & 5 tradeon B & A ABCCE BBCDEABDDE BCDEEAAACD

2 & 4 tradeon E & B ABCCE BBBCD ABDDE CDEEEAAACD2 &4 trade on D & C ABCCE BBBDD ABDDE CCEEEAAACD

Again, now, players 2, 4, and 5 are positioned to close out players 1and 3. But, as in experimentno. 11, an outcome predicted by cooperativesolution theory prevailsnot because of any apparentcoalitionalstrategy,but rather,becauseseemingly approrpiate rades, coupled with a particularsequence of bilateralagreements,result in something other than a Paretoinferior outcome. Experiment 4 exhibits an equivalentpattern, differing

only in the sequence of trades. In experiments 1, 14, and 17 a majorityeventually find themselves able to close out one player from one advan-

tageous trade. In experiments4 and 6, on the other hand, two inexperi-enced playersnaively trade to an outcome that benefits the thirdplayer.








Table

8.Outcomes

from

games

G1,

G2

and

G3

using

bargaining

Context

2(paper

ballots)

Game

Experiment

Outcome

Actual

payoffs**

'Riker-Brams'

'Riker-Brams'

number

predicted

predicted

outcome

payoffs

G1

1*t

FFFPFF

(-8,

2,6)

FFFFFF

(0,

0,0)

G1

2t

FPPPFP

(1.5,-6,

8)

FFFFFF

(0,

0,0)

G1

3*t

PPPPPP

(1,4,

4)

FFFFFF

(0,

0,0)

G1

4t

FPFPFF

(2,

-2,

10)

FFFFFF

(0,

0,0)

G1

5

PPPPPP

(3.5,

6.5,

4)

FFFFFF

(0,

0,0)

G1

6t

FPFPPF

(-2.5,

2,7)

FFFFFF

(0,

0,0)

G2

7t

PPPPP

(-1,-2,-4,-2,-2)

PPPPP

(-1,-2,-4,-2,-2)

G2

8*t

PPPPP

(-1,

-2,-2,-4,-1)

PPPPP

(-1,-2,-2,-4,-1)

G2

9*

PPPFF

(4,

3,8,-20,

-10)

PPPPP

(-2,-1,

-4,

-4,

-2)

G2

10

FFFFF

(0,

0,0,

,)

PPPPP

(-2,

-2,

-2,

-2,

-1)

G2

ll*t

FPPPF

(-5,

4,6,4,-5)

PPPPP

(-1,-2,

-2,

-2,

-1)

G2

12

FFFFF

(0,

0,,

0,)

PPPPP

(-2,-

1,-2,-

1,-2)

G3

13*t

PFFPP

(-5,

-20,

4,6,2)

PFFPP

(-5,

-20,

4,6,2)

G3

14t

PFPPP

(16,

5,6,-4,

-20)

PFFPP

(-10,

-5,

4,3,6)

G3

15t

PFFPP

(-5,

-10,

8,12,4)

PFFPP

(-5,

-10,

8,12,

4)

G3

16

PFPFP

(-18,

-4,

-2,

12,

24)

PFFPP

(-10,

-10,

2,6,8)

G3

17*t

PPPPP

(-9,

18,

20,

14,20)

PFFPP

(-5,

-10,

8,6,8)

G3

18

PFFFP

(-8,

-12,

-12,4,4)

PFFPP

(-

10,

-20,

8,3,2)

*

Denotes

games

inwhich

players

are

permitted

toreconsider

outcome

and

redistribute

votes.

**

Since

actual

payoff

schedules

given

toplayers

are

randomized

by

multiples

of1/2,

1,and

2,the

numbers

reported

here

differ

from

the

numbers

in

Tables

2-4.

The

payoffs

reported

there

are

the

'normalized'

payoffs

corresponding

to

the

payoff

schedule

which

would

have

resulted

if

the

multiple

were

1.

t

Denotes

games

consisting

ofplayers

who

are

playing

for

the

first

time

inany

experiment.








In both of these experiments,one playeris involved in the first threetrades,leaving, to the surpriseof the other two subjects, no additional tradesworthwhile.

The question remains,then, as to whatwe can conclude about vote trad-ing generallyand about Riker and Brams'analysis in particular.The first

apparent conclusion is that Riker and Brams incompletely describe theseveral alternative vote trading sequences, and that not all of these se-

quences lead to Pareto dominated outcomes. Specifically, whether byaccident, miscalculation,or subtle strategic planning, a Pareto dominatedoutcome is neverreachedat any time in 9 of 18 experiments.In six of thesenine games, tradingyields an outcome that, while not necessarilypredictedby cooperative solution theory, benefits a majority at the expense of a

minorityand this

majoritydeterminesnot to trade further.The

remainingsequencescorrespondto inexperiencedsubjectsmakingunderstandable,fnot optimal, tradesthat benefit only one player,but with no opportunityto reverse he result.

Our second conclusion, however, is that simplevote tradingcanproducePareto dominated outcomes. In the remaining nine experiments usingContext 2, an outcomeconsistentwith RikerandBrams'analysis s reachedat some point in the tradingprocess.That is, in one-halfof the 18 experi-ments reported in Table 8, vote tradingleadsfirst to a Pareto dominatedoutcome.

But, our third conclusion is that not all subjects agree that such out-comes must necessarilyprevail,for, as we note earlier, n five of those nine

experiments,subjectsdevisea means for reversinghe consequencesof theirtrades.Themeansareinterestingandinformative:

Experiment3The subjectsof this 3-persongameare not required o vote on whetherto re-considerand redistribute he ballots. In perhaps he most convolutedagree-ment observed,aftertrading o FFFFFF andvotingto reconsider, hey agree

unanimouslyto trade back to where, under sincerevoting, all bills failed,as this Paretodominatedoutcome is regardedas the 'fair'security point foreach player. They then vote, nevertheless, to pass all bills. During the

balloting, one player thinks aloud about taking advantageof the votingsequence, but the other two players threaten that if he attempts this

deception, they will vote againto reconsiderandsubsequentlyclose him out

completely.

Experiment5The

subjectsof this

3-person gamearenot

requiredto vote on whetherto

reconsider and redistribute the ballots. However, after trading to thePareto dominated outcome FFFFFF, they redistributethe ballots them-selves(i.e., effectively cancelledall trades),unanimouslyagreenot to trade,








andthen vote sincerelyto passallbills.

Experiment9

The subjectsof this 5-personexperimentfirst tradeto the outcomePPPPF,which closes out player 5. However,player5 (and we are not sure how orwhy) talks two others into votingto reconsider,whereupon hey then tradeto the Pareto dominated outcome of passingeverything.Once againtheyvote to reconsider and redistribute the ballots. But, on this third try, theplayersbegin discussingcoalition possibilities,refusingto tradeat all, with{ (1, , 3} eventually decidingto impose the outcome by tradingamongstthemselveson billsA, B, andCandfailingD andE without trades.

Experiments10 and 12The subjects of these two five-persongames are not requiredto vote onwhether to reconsider,but, after tradingto the ParetodominatedoutcomePPPPP, hey agreeunanimouslyto rotate the ballots in their possession byone player.Voting sincerely,now, allbills fail.

These five experimentssupportthe objectionsof someto RikerandBrams'analysis - that if Pareto dominated outcomes prevail,playersand presum-ablylegislatorswill devisemechanisms or avoidingsuch outcomes.

Of course, in actual legislative environments, the mechanisms for

enforcingvote trades are not the sameas in our laboratorysetting. Hence,we should not predict that legislatorswill avoid Paretoinefficiency in thesame ingenuous fashion as our subjects. The precedingdiscussionshows,however, that tradingneed not cease at Pareto inferioroutcomes. To theextent that subjects can devise a means of formingthe coalition of thewhole to resolve the 'paradoxof vote trading',we shouldexpect that legis-lators are capable also of devisingmechanisms that similarlyresolve the'paradox'.

Testingsolution theoryin gameswithoutcores

If Riker and Brams'analysisis restatedto hypothesizethat the outcomesfrom vote trading in decentralizedlegislatureswith binding agreementsdiffer sharplyfrom the outcomes predicted by cooperative gametheoreticsolution notions, the results reported in Table 8 confirmthat hypothesis.In only four experimentsthere - 9, 11, 16 and 17 - are the outcomesconsistentwith the predictionsof the competitivesolution,andin only oneexperiment, 9, do we observe explicitly the coalitional bargainingthat

cooperativetheoryassumes.

But,in

defendingtheir

hypothesis,Riker and

Brams go further by arguingthat even coalitional processses should notdisruptthe 'paradoxof vote trading'.Specifically,they assert hat








Since here s no preferredetof trades n which o basea stable oalition, ostablecoalitions possible.Theparadoxf votetradingannot, herefore,e simply olvedby wavingt awaywitha coalition.Rather,t is inherentn thenature f thelegisla-tive

process nd,givenan

appropriateistributionf tastes ndexternal

osts,cannot

be avoidedpp.1245-6).

Consider,however, the 24 experiments reportedin Table 9, which we con-ductwith bargainingContext 1.

First, the outcomes of all but one experiment,23, of the twenty-fourexperimentsreportedin Table 9 are Pareto optimal- which includes seventrials of the game Riker and Bramsuse to illustratethe 'paradoxof vote

trading',G1. Clearly,bargaining ontext matters.While the inefficiency of

vote tradinghypothesis possessessomevalidityin decentralized ommitteesthat permit binding vote trades, it is obviously false for committees or

legislaturesthat operateunder fuller information or where agreementscanbe withdrawn n the face of a better deal.

This point is perhapsbest made by recalling that, under Context 1,playersare permittedto dispose of bills one at a time, in pairs,and so on,in any order if they so choose. But, in only one experimentthat Table 9

reports, 25, are the bills processed with other than a majority agreementon all bills simultaneously. Certainly,in numerousexperiments,proposalsto dispose of one or two bills are made - typically under the guise of

simplifying the groups'task - but such proposalsfall before a universalargument: the players are unwilling to relinquishthe opportunity to usetheir votes on those bills for purposesof bargaining.Subjects almost im-

mediately recognize the value of voting other than sincerely to securethe votes of others on other bills,but only if such agreementscover all bills

simultaneously.The inescapableconclusion, then, is that the instability of vote trading

scenarios has nothing whatsoever to do with the ability of coalitions toresolve Riker and Brams''paradox'.This then, raises our second question:

If an unrestrictedcoalition process can prevailto avert the paradox,whichcoalitions form and what outcomes prevail?Specifically, does instabilityrender outcomes unpredictable, or are any of the cooperative solution

concepts of gametheory useful?The seven 3-person experimentscan hardlybe describedas a confirma-

tion of solution theory as only one experimentyields an outcome pre-dicted by K, V, and M1, experiment20. But, a clearpattern emergesto

explain five of the remainingsix games.6 In each case, the players agreeto search for a fair or equitable outcome after realizingthat two players

can be made better off only at the expense of a third (the outcome ofexperiment 23 is a miscalculationby all three playerswho do not noticethat passingall bills makes everyonebetter off). When we turn to the five-

player games, on the other hand, fairness is almost never mentioned.








Table

9.Outcomes

from

games

Gl-G4

using

bargaining

Context

1

Game

Experiment

Coalition*

Outcome

Actual

payoffs

Iscoalition

andpayoff

fair

number

inK

G1

19

1,2,

3

PPPPPP

(1,1,2)

No

G1

20t

1,2,3

PPPPPP

(4,

1,4)

No

G1

21

2,3

FFFPPF

(-8,

10,

5)

Yes

G1

22

1,2,

3

PPPPPP

(4,

2,4)

No

G1

23

1,2,

3

PFPFPF

(2,

1,2)

No

G1

24

1,2,

3

PPPPPP

(1,4,

2)

No

GI

25t

sequential

PPPPPP

(2,

1,2)

No

G2

26t

1,4,

5

PFFPP

(2,-10,-10,2,6)

Yes

G2

27

1,4,5

PFFPP

(2,

-10,

-10,

8,3)

Yes

G2

28

2,3,4

FPPPF

(-10,

4,6,4,-

10)

Yes

G3

29t

1,3,4

FPFPP

(-4,-9,

24,12,-2)

Yes

G3

30t

3,4,5

PPPPP

(-18,-18,5,14,5)

Yes

G3

31

3,4,5

PPPPP

(-9,

-9,

5,7,20)

Yes

G3

32

3,4,

5

PPPPP

(-9,

-9,

10,14,

5)

Yes

G3

33

2,4,5

PFPFP

(-18,

-8,

-4,

24,

12)

Yes

G3

34

1,3,4

FPFPP

(-4,-

18,

6,12,

-8)

Yes

G3

35

1,2,5

PFFFF

(-2,-1,-2,-1,

10)

Yes

G4

36t

1,3,5

FFP

(-1,-2,5,-4,

11)

Yes

G4

37t

1,3,5

FFP

(-1,

-1,10,-8,4)

Yes

G4

38

2,3,4

PPP

(-9,

2,10,

34)

Yes

G4

39

1,2,

4

FPF

(-2,

2,-

2,3,-5)

Yes

G4

40

2,3,4

PPP

(-9,

4,5,3,-2)

Yes

G4

41

2,3,4

PPP

(-9,

8,10,

6,-5)

Yes

G4

42

1,2,4

FPF

(-1,4,-4,

12,

-10)

Yes

*

Coalitions

are

identified

by

players

who

sign

agreement

cards.

t

Denotes

first

time

players.








Apparently,two 'losers' and three 'winners'are sufficient to establish the

anonymitythat disposesof suchinterpersonal onsideration.The results of the seventeen5-person experimentsreported n Table9,

moreover,are a bit embarrassing.Wecannot reporta single failure of thecompetitive solution - including the five experiments with first time

subjectsin terms of its coalitionpredictions, ts outcome predictions,andthe conjunctionof both. Only three trials of gameG2 are attempted sinceK there might be too transparent o providea meaningfultest. But not a

single failure is observed also with the less symmetricgames G3 and G4.

Further,of the five coalitions predictedby K for G3, four form at leastonce while for gameG4, in whichK predictsthat seven of the ten minimal

winningcoalitionswill not form,only the threepredictedcoalitionsform.7

Clearly,these experimentaloutcomes impressivelysupportthe competi-tive solution. In fact, we might even hope for a few failuresto rendertheseresultsmore believable.Yet no failurescan be reported.AcroC the three

5-persongames,there area total of 720 possibleoutcomes entailingminimal

winning coalitions (combinations of coalitions and permutationsof pass-fail). Of these, K predictsonly 13, andin sixteenexperiments, en of thosethirteen are observedat least once, while none of the remaining707 out-comesevercorresponds o an experimentaloutcome.

Despitethis impressive upportforK, we shouldalsoexaminethe relative

performanceof two

competingsolution

concepts- the V-set and the M1

bargainingset. With respect to game G1, K, V and M1 are essentiallyequivalentand hence these 3-person gamesprovidelittle supportfor V or

M, over K. Games G2 and G3 are considerablymore interesting.First,without resort to lotteries, neither of these two games possessesa V-set.The success of K in these games, then, appearsto argueagainst V as an

appropriate hypothesis. The M1 bargainingset, on the other hand, is

wholly disjointfrom K. In particular, or gameG2, the followingcoalitionsand outcomes are stableusingthe definitionM1:

Billspassed CoalitionABD {1,2,4)}ACD {1, 3, 4}ACE { 1,3, 5}BCE {2, 3, 5}BDE {2, 4, 5}

An equivalentbargainingset, disjoint fromK, exists for gameG3. Hence,sinceeveryexperimentalutcome s inKforgamesG2 andG3 nooutcome

is inM1.Thus, or thesegames t least,M, mustbedeemeduseless.Finallyfor gameG4, the followingoutcomes and coalitions are stablein M1:








Billspassed Coalition

A {1,3, 4}B {1,2,4}, {1,2,5}, {1,4,5}BC (1,2,3}, {1,3,4}, {2,4,5}ABC {2,3,4), {2,3,5}, {2,4,5)}, {3,4,5}

Lookingat Table 9 againwe see that the outcomesof threeexperiments-

36, 37 and 39 - are not in M1. And, again,since all outcomes arein K, wesee that thepredictionsof M1aremet only if they correspond o thepredic-tionsof K.

We can summarize his sequence of twenty-two experiments,then, with

at least three observations.First, subjectsaregenerallyunwilling o excludea player from a majority agreementin 3-person gamesand instead seek afair outcome. In 5-person games, however, considerations of fairness or

equity rarely arise. This pattern, while certainly warrantingadditional

experimental investigation, seriously questions the adequacy of earlierstudies that seek to test cooperative solution theory in 3-person games.Specifically,in a 3-persongame (but not in a 5-persongame)considerationsof equity or of fairness arise to distort the preferencesthe experimenterseeks to induce - thereby invalidatingthe application and test of the

theory of the abstractgame.Our second conclusion is thatwhen considera-tions of equity do not arise (all of our 5-person games), the competitivesolution performsremarkablywell. The V-set and the bargainingset, onthe other hand, seem either useless or redundantas solution hypotheses.8Third, under bargainingContext 1 - which correspondsmore to cen-tralized, full informationcommittee processes- the vote tradingprocesshypothesizedby Riker andBrams s not observedand,in fact, playersrefuseto process bills sequentiallyand relinquishtheir ability to use their voteson these bills to securemoreadvantageous greements.

Testingthe core

Substantialexperimentalsupport alreadyexists for the core as a solution

concept when it exists (cf. Berl et al., 1976; Fiorina and Plott, 1978), andwe might anticipateequivalentconclusionshereusing bargainingContext 1.The fourteen outcomes we report in Table 10, however, are somewhat

surprisingas only four outcomes correspondto the core (46, 46, 51 and

56).9 And, these results are disconcertingnot only for the core, but alsofor cooperativesolution theory in generalsince the core is but a specialcase of the competitive solution. While in experiments 43, 44, and 54,for example, we might hypothesize that player 5 or 3 is indifferent over








fifty cents, the core dominates the remainingnon-core outcomes byminimumpayoffs of $ 2, $ 2, $ 2, $1, $ 1, $ 1 and$2 - rewarddifferentials

that do not appear trivial.Anotherpossible explanationis that if subjects

have prior experience with any one of gamesG1 throughG4, they mightcome to expect that tradingand deals are essential to reachingany major-ity agreementand never considerthe sincerevoting outcome. But, of thefive experiments entailing experienced subjects, the core fails to prevaileven once. On the contrary, then,prior experienceappears o yield a betterfit with the core'spredictions.

Another possible explanation is that subjects knowlingly or unknow-

ingly deviate from the core so as to maximize their 'total take', or, at

least, deviatefor the benefit of playerswith the largestscalepayoffs. Table

11, then, contrasts the subjects with large as againstsmall scale payoffswith the subjects who agree to dictate the eventualoutcome, and it con-trasts the maximum possible total payout with the actual total payout.Briefly, this table provides little support for the hypothesis that payoffmagnitudes account for the experimental outcomes. First, there is no

apparentrelationshipbetween the eventual coalition and the playersthatare assignedhigh as opposed to low schedules:45% of the playerswith

high payoffs are in the eventual agreementto terminatean experimentwhile 47%of the playerswith low payoffs are in suchagreements.Second,while the

gamesare structured to

yield higher payoffsif the outcome

deviates from the core (owing to the minimum$1.00 payoff), the actual

payoffs on the averageare closer to the core thanto the maximum.

Finally, the core's failuremight be attributedto sloppy experimentaldesign, poor subject motivation, subject ignoranceand confusion, and thelike. However, the impressiveexperimental support for the competitivesolution reported in Table 9 and even the readily interpretabletradingprocessesof the experimentsreportedin Table 8 belies these explanations.We are left, then, with the conclusion that somethingmuch more subtleandimportant s beingrevealed o us in theseexperiments.

We consider briefly two potentially interactivehypotheses. First, wenote from the structureof gamesG5 and G6 that players3, 4 and 5 are

'nearly indifferent' between a great many outcomes that, nevertheless,greatlybenefit or harmanotherplayer. In gameG5, for example, player4assocoates only a one dollar difference (assumingthe scale as shown inTable 6) between the core and the outcomes PPPFFand FPPFP.Similarly,player 5 assocates only a one dollar difference between the core andPFPFF and FPPFF. Finally, player 3 suffers only a one dollarloss, with

respectto the core, fromthe outcomesPPFPFand PFFPP.Equivalent near

indifferences'arise in game G6 as well. Moreover,a review of the payoffschedulesactually employed on the basis of our randomselection proce-dures reveals that in every experiment, player 3, 4, or 5 possess a payoffschedule corresponding o a multiple of 1 or y - that is, a schedule that








Table

10.

Outcomes

from

games

G5

and

G6

using

bargaining

Context

1

Game

Experiment

Outcome

Coalition

Actual

payoff

Core

Core

number

outcome

payoffs

G5

43

FPPFF

2,4,5

(-7,

2.5,

-26,

7,-.5)

FFFFF

(,0,

0,0,0)

G5

44t

PPFPF

1,2,

3

(12,

6,-.5,

-4.5,

-12)

FFFFF

(0,0,,

0,0,0)

G5

45

FFFFF

1,3,

4

(0,0,

0,0,0)

FFFFF

(,0,

0,0,

,)

G5

46

FFFFF

1,3,

4

(0,0,

0,0,0)

FFFFF

(0,

0,0,

0,0)

G5

47

PPFFF

1,2,3

(8,8,

-8,

-2,

-20)

FFFFF

(0,0,,

0,0,0)

G5

48t

PPFFF

sequential

(16,

16,

-2,

-4,

-20)

FFFFF

(0,

0,0,0,0)

G5

49

FPPFP

2,5,4

(-6,

4.50,

-42,

-2,

3)

FFFFF

(0,0,

0,0,,0)

G6

50

PPPPP

1,2,3

(22,

8,12,

-4,

-56)

FFPFP

(10,

2,13,5,

-16)

G6

51

FFPFP

2,4,5

(10,

.5,26,

5,-16)

FFPFP

(10,

.5,26,5,

-16)

G6

52t

FPPFP

1,2,3,4,5

(4,11,

10,

18,

-6.5)

FFPFP

(5,

1,26,

10,

-4)

G6

53

FPPFP

2,4,5

(4,5.5,

2.5,

18,

-6.5)

FFPFP

(5,

5.5,

6.5,

10,

-4)

G6

54t

PFFFP

sequential

(15,

-6,

6,0,-

18)

FFPFP

(10,

1.6,

5,2.5,

-6)

G6

55t

FPPFP

3,4,5

(8,11,5,

9,-26)

FFPFP

(10,

1,13,5,

-16)

G6

56

FFPFP

2,3,4,5

(20,

.5,

13,

2.5,

-4)

FFPFP

(20,

5,13,2.5,

-4)

*Denotes

first

time

players.







Vote rading: n experimentaltudy 173

Table

11.

Comparison

ofmaximum

and

actual

payoffs

forG5

and

G6

Number

Players

Players

Payoffs

without

a*

Payoffs

with

a

with

high

with

low

schedules

schedules

Coalition

Maximum

Actual

Core

Maximum

Actual

Core

42

3

2,5

2,4,5

$31.00

$12.50

$5.00

$37.00

$21.50

$21.50

43

2

3,4,5

1,2,3

32.00

21.00

5.00

41.50

32.50

19.00

44

3

2,5

1,3,4

32.00

5.00

5.00

41.00

21.50

21.50

45

1,2

3,4,5

1,3,4

35.00

5.00

5.00

49.00

22.00

22.00

46

3,5

4

1,2,3

31.00

19.00

5.00

42.00

26.00

31.50

47

1,2,5

3

sequential

35.00

35.00

5.00

51.00

49.00

35.00

48

3,4

1,2

2,4,5

24.00

8.50

5.00

36.00

26.50

29.00

49

2

3

1,2,3

47.00

44.00

31.00

40.00

36.00

25.00

50

3

2,5

2,4,5

71.00

42.50

42.50

53.00

30.50

29.50

51

3,4,5

1

all

53.00

44.00

43.00

38.00

36.00

34.00

52

4,5

1,2,3

2,4,5

31.00

31.00

23.50

40.00

29.50

22.00

53

-

3,4,5

sequential

36.50

24.00

21.00

28.50

16.00

18.00

54

-

5

3,4,5

47.00

34.00

30.00

35.00

27.00

24.00

55

1,5

2,4

2,3,4,5

74.00

37.50

37.50

57.00

23.00

23.00

Average

1

$41.00

$23.00

$19.00

$42.00

$28.00

$26.00

*Payoffs

are

computed

assuming

a=0nd

ainimum

payoff

ofone

dollar

per

player.








Table

12.

Games

G7,

G8,

G9

andG10

using

bargaining

Context

1

Game

Experiment

Outcome

Coalition

Payoff

Core

Core

number

outcome

payoff

G7

57

FFFFF

3,4,5

(0,

0,0,0,0)

FFFFF

(0,0,00,

0)

G7

58

FFFFF

1,3,

5

(0,

0,

0,,

0)

FFFFF

(0,

0,,0,

0)

G7

59

PPFFP

sequential

(1.5,

12,

-6.5,

-26,

-3.5)

FFFFF

(0,0,

0,0,0)

G7

60

FFFFF

1,2,

3,4,5

(0,0,

0,0,0)

FFFFF

(0,0,

0,,0)

G7

61

FFFFF

3,4,5

(0,

0,0,0,0)

FFFFF

(0,0,

0,0,0)

G7

62

PPFFF

1,3,4

(4,

8,-

10,-5,-

10)

FFFFF

(0,

0,0,,0)

G7

63t

FFFFF

1,2,

4

(0,

0,

0,0

)

FFFFF

(0,

0,0,,0)

G8

64

FFPFP

3,4,5

(5,

.5,

13,

10,-

12)

FFPFP

(5,.5,

13,

10,-12)

G8

65

PFPFP

1,3,5

(40,

-.5,

17,

-1.5,

-5.5)

FFPFP

(20,

.5,

13,

2.5,-3)

G8

66

FFPFP

3,4,5

(10,

2,26,

10,-

12)

FFPFP

(10,

2,26,

10,

-12)

G8

67t

FFFFP

3,4,5

(2.5,-8,

4,8,-6)

FFPFP

(5,2,

6.5,

5,-12)

G8

68t

FFPFP

1,2,

3,5

(5,2,

6.5,

5,-3)

FFPFP

(5,2,

6.5,

5,-3)

G8

69

FFFFP

3,4,5

(5,-4,

4,4,-6)

FFPFP

(10,

1,6.5,

2.5,

-12)

G8

70

FFPFP

1,3,

5

(5,

2,26,

2.5,

-6)

FFPFP

(5,2,

26,

2.5,

-6)

G9

71

FFFFF

1,2,

3,4,5

(0,

0,0,0,O0)

FFFFF

(0,0,

00,,

0)

G9

72

FFFFP

sequential

(-5,8,

-4,

-16,

8)

FFFFF

(,0,

0,0,)

G9

73

FPPFP

2,4,5

(-22,

9,-42,

1,

1.5)

FFFFF

(0,

0,

0,0)

G9

74

FFFFF

1,3,4

(0,

0,0,0,0)

FFFFF

(0,00,0,

,)

G9

75

FFFFF

2,4,5

(0,0,

0,0,0)

FFFFF

(0,0,,

0,0)

G9

76

FFPFP

2,4,5

(-10,-1,

-6.5,-5,16)

FFFFF

(0,0,

0,0,

0)

G9

77t

PFPFF

sequential

(2.5,

-3.5,

-1,-10,

-.5)

FFFFF

(0,0,

0,0,0)

GIO

78

FFPFP

1,3,5

(20,

.5,26,

10,-8)

FFPFP

(20,

.5,

26,

10,

-8)

GIO

79

FFPFP

1,3,

5

(10,

2,13,

10,-4)

FFPFP

(10,

2,13,

10,

-4)

GIO

80

FPFFP

sequential

(-7,

12,

0,12,-4.5)

FFPFP

(10,

2,6.5,

5,-4)

GIO

81

FFPFP

sequential

(20,

1,6.5,

2.5,-4)

FFPFP

(20,

1,6.5,

2.5,-4)

GIO

82

PFPFP

1,3,5

(20,-1,

17,-6,-6.5)

FFPFP

(10,1,

13,10,-4)

GIO

83t

PPPFP

sequential

(4,

18,

4.5,

-.5,

-18)

FFPFP

(5,2,

6.5,

2.5,

-8)

GIO

84

FFFFP

3,4,5

(10,-4,

8,16,-8)

FFPFP

(20,

1,13,

10,-16)

t

Denotes

first

time

players.







Vote rading: nexperimentaltudy 175

Table

13.

Success

rate

ofthe

core* Original

game,

Indifference

Vote

trade

G5,

G6

modification

modification

G7,G8

G9,

G10

All

Experienced

All

Experienced

All

Experienced

All

Experi-

subjects

subjects

subjects

subjects

enced

2*

2

5

4

3

3

10

9

G5,

G7,G9

-

-

-

-

-

-

-

-

7

6

7

6

7

6

21

18

2

2

4

3

3

3

9

8

G6,

G8,G10....

-

-

-

-

-

-

-

-

7

6

7

5

7

6

21

17

4

4

9

7

6

6

-

-

-

-

-

-

14

12

14

11

14

12

*

Numerator

isnumber

ofcore

outcomes,

denominator

isnumber

of

trials.








admits a one dollar to 50 cents indifference across severaloutcomes nearthe core.

These non-core outcomes, however, greatly benefit players 1 or 2 and

threaten at least one remainingplayerwith a largeloss. To the extent thatthese proposals are offered and appear viable, they may cower other

players into less than optimal agreements.The question, then, is: Whyshould these alternativeproposals be made and why might they appearviable?

Our second hypothesis, imprecisely formulated, is that myopic vote

trading of the kind hypothesized by Riker and Brams is occurringto'distort'the bargainingprocess. Specifically,note that in gamesG5 andG6,players 1 and 2 possess a clear and (judgingfrom our experiments)easilyidentified vote trade on bills A and B.

Further,once this trade is

agreedto, players 1 and 2 can 'inform' player 5 that a majoritynow prefersto

pass A and B (players 1, 2 and 3 for A, players 1, 2 and 4 for B). Hence

player 5 might agreeto pass only one of these to fail the other, especiallyif he can secure a simultaneousagreementto fail D. In both games, then,the passageof A or B may appear nevitableto all players,and it is fromthis point (as opposed to the sincere outcome) that they attempt to

improve their positions on the remainingbills. Note also, that bill A is

player 3's most preferredbill, which tends to focus his attention there,while player4 may be unwilling o considerdefeatingbill B.

The preceding discussion greatly simplifies the bargainingprocesseswe observe in these experiments- a process that awaits furtherinvesti-

gation.' Equally impressionistically, hough, we should also add that thevote trade between players 1 and 2, the preferenceof player3 for A, of 4for B, and of 5 for avoidingthe passageof both tendsto fragmentnegotia-tions in that subjectsappearto be less likely to considernegotiatinga com-

plete package.In fact, players 1 and 2 possess a clear incentive to avoidsuch discussionssince, once they are entered, players3 and 4 are likely tosee the advantageof failingboth A andB.

To test both the indifferenceand the vote trade hypothesis separately,we modify games G5 and G6 slightly. First, with respectto indifference,gamesG5 and G6 aremodifiedthus,

Player3: payoff from bill C decreased to -10from -5 or payoff frombill B decreasedto -9

G5 - G7 Player4: payoff from B decreased o 3 from 4

Player5: payoffs from C and E decreased rom 3

to4

G6 G8 equivalent changes except that, on bills C and E,multiplypayoffs by -1








Second, with respectto the vote trade between players 1 and 2 on bills AandB, we form two new games,G9 andG10, thus:

G5 - G9 Player l's payoff from bill B decreased o -12 fromG6- G10 -2

These modifications do not eliminateall potential vote trades or all payoffdifferences ess than two dollars(lest we render he gamestrivial),but theydo move things in the intended direction. The results of twenty-eightexperimentsusingthese modificationsarereported n Table 12.

Table 13 summarizes he comparisonbetween Tables 10 and 12 insofaras the core's success is concernedfor all gamesand for gamesentailingonly

experiencedplayers. Overall,we see some improvement n the core's per-formance,with the indifferencemodificationappearing o have the greaterimpact. The improvement, however, can hardly be described as over-

whelming,for at best we can say that experiencedsubjectschoose the core

approximatelyhalf of the time.We reiteratethat we might not find a success rate for the core of fifty

percentexceptionalwere it not for the hundredpercentsuccess rate of K in

5-persongames.Another clue to the source of this discrepancy,though, isfound by noting that while no seuqnetialballoting occurs in the sixteentrialsof

gamesG2, G3 and G4, a

sequentialdispositionof bills arises

eighttimes (out of 42 trials)in the core games.And,while these eightsequentialdispositions account for but a few core failures(seven of twenty-three),they are perhapsindicative of a strategic ncentivecommon to the severalcore games- an incentivefor players1 and3 or 2 and 4 to disaggregatehedecision - to turn attention from agreementovercomplete packages- to

passbill A orbill B, respectively.Severalsubjects, n fact, volunteeredafter-wardsthat they deliberatelyattemptedto disaggregatehe decisionprocess.And, it is evident that a sequentialdispositioncan yield outcomes otherthan the core.

Conclusions

Aside from the usual disavowal that much additional experimentation,theorizingand analysisremains,we can offer severalconclusionswith confi-dence. First, the vote tradingprocesshypothesized by RikerandBramscanbe induced in an experimental setting. And, while this process need notresultin Pareto dominated outcomes - especiallywith experiencedplayers

- it need not result also in outcomes predictedby cooperativesolutiontheory. Wecannot say, however,whetherthis theory will orwill not prevailultimately with subjects possessingsignificantexperience in such games-

experience that makes them fully awareof the pitfalls of bilateraltrading.








It is evident, nevertheless,that Riker and Brams uncover incentives fordecentralizedlegislative processes to move towards outcomes other thanthose predictedby cooperativesolutiontheory.

In the context of open face-to-facenegotiationswithoutbindingcommit-ments, however, as in centralized committees without significantcosts tocoalition formation, Riker and Brams'hypothesis is less than satisfactory.And, the results,in fact, strictlycontradicttheirassertion hat 'the paradoxof vote tradingcannot . .. be simplysolvedby waving t awaywith a coali-tion'.

Turningto the severalsolution concepts of cooperativen-persongametheory, in our experiments the competitive solution is demonstrablysuperiorto the V-set and the M1 bargaining et. We should emphasizethat

gamesG

1 throughG4 are

designedo insurethe existence of

K,butK need

not exist in all such finite games.Hence,K's strongperformanceheremaybe attributableto its correspondenceto some more general,undiscoveredsolution hypothesis. Nevertheless,while we preferto test severaladditional

games before stating a definitiveconclusion, it is our tentative decision toabandon V and M, altogether as adequate hypotheses about collectiveaction.

Before we can assert that K or some other concept is the best available

hypothesis,however,we must explorefurtherthe problemof why the core,which coincideswith K, meets with only modest successhere. Put simply,we must understandwhy solution theory in general performs better inunstable,asopposedto stable,vote tradinggames.Froma wholly subjectivepoint of view, it appears that in unstable games, the proposals thatconstitute a competitivesolution arisenaturallyas the focus of negotiationswhereas n gameswith a core, this proposaloftentimesis never mentionedatall. This, of course,generatesa seriesof researchquestions.First,as a ques-tion for both theoretical and empiricalresearch,we should ask: Aregameswithout core points nevertheless more determinate(predictable)because

instabilitynaturallyelicits the balancingof proposalsthat K assumes n its

definition? Answeringthis question necessitates a better understandingofthe dynamicsof bargainingand the convergentpropertiesof solution con-

cepts. In particular,can negotiations in stable games get trapped in sub-

cycles that are brokenonly with the fortuitousdiscoveryof the core?We reemphasizethat K and the core are equivalent when the core

exists. Hence another important question is: Can vote trading games be

designedthat also lead negotiations away fromK. That is, since we do not

fully understandwhy the core performspoorly in ourexperiments,we can-not now precludethe possibilitythat the strategic propertiesof gamesG2,G3 and G4

are especiallyadvantageous or solution theory while gamesG5throughG10, for not wholly understood reasons, are especiallydisadvan-

tageousfor that theory.To date, the core andK have been tested principally n the context of








spatialgames- gamesthat presentno incentive or disaggregatingheeventual hoiceinto its component artsor thatprecludeuchdisaggrega-tion. On the basisof the experimentsonsidered ere,however,we con-

clude hatthesecontextsseemtoo devoidof strategic omplexityo revealthe forces hatmightoperaten real,nonexperimentalnstitutions uch aslegislatures forces that invite outcomesrather hanthosepredictedbysomesolutionconceptoperating vermathematicallybstract epresenta-tionsof these nstitutions.

Appendix:Instructions

Generalnstructionsor allsubjects

In this experiment,your task, muchlike a legislative ommittee, s to decidewhichofseveralbills should be passedandwhich shouldbe failed.Eachof you will gainor losea certainamount of money from the passageof eachbillandatthe end of theexperi-ment, the amountyou will be paidfor your participationdependsonwhichbillspassandwhich fail.

To assist you in understandinghis experiment,each of you has been given anidenticalsampleworksheetfor a fictitiousplayer.At the beginningof the experimentyou will each be given a similarsheet, but the numbers here may differfrom this

sample.At the top of the worksheet s a table thatdenoteshow muchmoneythefictitious

playergainsor loses from the passageof eachbill.Thus,the passageof billA is worththreedollars o this player.Eachnumber n this tablerepresents he payoffif abill is

passed.If a bill is defeated,the amountof moneyreceivedby eachplayer or thatbillis zero - thatis, neithera loss nor a gain.

For each of you, though,your final payoff at the terminationof the experimentwill depend on what action is taken acrossall bills. To determineyour payoff at thetermination f thegame,each of you has been assigneda positiveor anegativenumber,which we call a and which appearson the bottom of your worksheets n the spacelabeledac.In the actualexperiment, his will be covered,andwill only be revealed oyou at the terminationof the experiment.When he experiment nds,the payoffsyouhaveaccumulated romthe billsthathavebeen passedwill be addedand entered n the

spacelabeledX. Yourfinal payoff from participationn this experimentwill then be

computedby addingoa,or subtractingt if it is negative,to X. If thistotal is less than

one, you will receive$ 1. If it is greater han 1, you will receive hat amount, n dollars.

Lookingat the sampleworksheet,for example, supposethat onlybillsA, CandDarepassed.Then this decisionis worth 3 dollars rombillA, minus5 dollars rom bill

C, and minus 2 dollars from bill D, or a total of minus4 dollars,so X = -4. Now

supposethat for this player,a = 6: then his final payoff is -4 + 6 or two dollars.Notice then that each of you canwinmore thanyourminimumof a dollareven if the

total value,X, you associatewith the committee'sdecision s zero or negative.On theother hand, even if X is positive,a may be negative,so that any valueof X greaterthan 1 is not a guaranteehatyou will win morethan$ 1.

In the actual worksheetsof this experiment,each of you will not necessarily








associatethe samevalue with a particularbill. Some of you maylose and some of youmay gainif a bill passes,and thus some of you may preferto passa bill while others

preferto fail it. Further,even if two or more of you preferto passa bill, there is no

necessaryequivalenceo

yourdollar reward.That is, one of

youmay

getone dollar

if a bill passeswhile anotherreceivesthree. Finally, the valueof a may differ from

playerto player,so that two playerswith the sameaccumulateddollarvalue need not

ultimatelyreceive he same amountof money.After your worksheetsare handed out to begin the experiment,you will not be

permittedto show yours to anotherplayer.Hence,each of you will know only yourown payoffs,but not the payoffs of any other player.As the sampleworksheet ndic-

ates, though, you will know which bills each player prefersto pass and to fail. In

particular,n the second table on your worksheeta P tellsyou thata playerassociatesa positivepayoffwith the bill in questionwhile an F denotesa negativepayoff.

I emphasizenow that one essentialrule of this experiment s thatat no timeare

you to makeany mentionof money. Further,at no time areyou permitted o discussschemesfor dividingyour winningsat the terminationof play. At the terminationofthe experiment, n fact, eachof you will be paid n the absenceof the otherplayers,sothere s no need to everrevealyour totalwinnings, houldyou decide not to do so.

[STOPFORQUESTIONS]

Context1 instructions

Let me now discuss heway

inwhich theexperiment

willproceed.

Theexperiment

willbe divided nto two phases,a discussionand a decisionphase.At the beginningof the

experiment,you will be given your actualworksheets,and will beginthe discussion

phase,whichwill last for fifteen minutes.Duringthis period,you maydiscussamongyourselveswhat your final decisionmightbe; however,you cannot reachanybindingdecisionsduring his time.

At the terminationof this 15-minute discussionphase, you will proceed to thedecisionphase,for which thereis not time limit.During hisphase,you arepermittednot only to discussthe dispositionof the bills, but also to makebindingdecisions,bymajorityrule, about them. Specifically,when some majorityof you can agreeon the

dispositionof one, any, or all of the bills, you can formalizethe agreementwith the

white 3 x 5 agreementcards. Each of you in this majorityshouldsignone of yourcards,indicatingon it your player number,and the dispositionof the bills involved

(writeP for pass,F for fail, leavingblankthose bills not involved n the agreementf

any). I will then collect these cards,one from each of you. If a majorityagreecom-

pletely, a decision will have been made on those bills, andI will recordthis on theblackboard. f there is not a majority agreement,no final action on thesebills willbe

assumed,andyou will continueas before.Once a decisionhasbeenmade on any bill,it cannot be alteredsubsequently.The decision phase will proceed,in this manner,until all bills havebeen disposedof. At this point, the experimentwill terminate,and

you will all be paidas described arlier.

Note that it is not necessary or you to be in agreementwith the final decisionforyou to receive some payoff. The payoff each of you receiveswill correspond o the

payoff you get from the final dispositionof the bills, plusyour a value,regardless fhow thatagreements established.








[SAMPLE]

Playerno. Gameno.

2.............. .....(fictitious)

Bills

A B C D E

Payoffifbill is 3 1 -5 -2 -.50

passed

Preferences f otherplayers

Player A B C D E

1 P F P P P

2 P P F F F

3 P P F P P

4 F F F F P

5 F P P P F

Playerno. Gameno. Experiment D Ca X Total

2 (fictitious) 6 +

Name(print) (Signature) Date Received

z i c s n g w l a b u r d . p e t f v h xm ~ z i c s n g w l a y b u ~ k r d p e t q j f o v h x m z i...............

. . . . c ~ s n g w ~ l








Context2 instructions

Letme now discuss heway in which the experimentwillproceed.Theexperimentwillconsist of threephases,calledthe study, trading,andballotingperiods.At the begin-

ning of the experiment,each of you will be given [six/five] paperballots, each ofwhich is worth one vote on a particularbill. Therewill then be a three-minute tudy

period duringwhich you may study your worksheet.Duringthis time, you may notcommunicatewith any of the otherplayers.At the end of the studyperiod,you will

proceedto the tradingperiod.Duringthis period,you maytradeballots, amongyour-selves,in any way you choose.Thus, one playermay agreeto trade his ballot on billA to anotherplayerin exchangefor his ballots on billsC and D. The tradingperiodwill proceedfor 15 minutes,or until there is unanimousagreement o proceedto thenext period.During he entireexperiment, he ballots shouldbe kept on the table infront of you, so that all playerscan observewho holdswhat ballotson a givenbill.

At the endof the tradingperiod,you will thenproceed o the ballotingperiod.During he ballotingperiod,each of you may cast the ballotsyou holdin anywayyou choose. This will be done by a sequentialvoice vote on each bill, first on bill A,then on billB, and so on. For example, f you hold two ballotson a particular ill, youmay vote those two ballots any way you choose on that bill. If you hold no ballotson the bill, you haveno vote in it. A bill will be passed f amajorityof its ballotsarecast for passage. will recordthe resultson the blackboard, t whichpointthe experi-ment will endandyou will be paidasI described arlier.

Note that it is not necessary or you to be in agreementwith the final decisionfor

you to receive some payoff. The payoff each of you receiveswill correspond o the

payoff you getfrom the final

dispositionof the

bills, plusyoura

value, regardlessf

how thatagreements established.

NOTES

Weare not hypothesizinghat separability f billsdescribes ctual egislative itua-tions. This assumption,however,which permitsus to evaluatepackagesof bills

by simply adding he benefits a personassociateswith the dispositionof eachbill,is an experimentalnecessity: it simplifiesthe subjects'tasks and renders the

experiment ractible or them. Consequently,we assumeseparabilityhroughout

thisessay.2. While we include an assessmentof the V-set andM1 bargaining et here, we

should state at the outset that, for reasonsdetailedelsewhere(see McKelvey,OrdeshookandWiner, 1978), we regard hese two 'solution'hypothesessuspectin the presentcontext. In particular,while no generalexistencetheoremsapplyhere for any of the solutionnotions in question,thereis some reasonto conjec-ture that the competitivesolution exists more generally.Second, the V-setin

generalandthe mainsimpleV-set n particular re puremathematicalnventionswithoutbehavioral ationaleand, as such,are simplyad hoc hypotheses.The V-

set, moreover,s unconcernedwith coalitionpredictions,which is a seriousdefi-

ciency in the presentcontext sincethis is equivalento not predicting he specificvote tradesthat occur.The bargaininget, on the otherhand,while not origin-ally intendedto providecoalitionpredictionscan be interpreted o predictthata coalition will not form if it is not associatedwith a stablepayoffconfiguration.








The difficulty with the bargaining et, though, is that it is poorly equippedto

treatgames n whichexternalities represent.3. The formal definition of K is presentedin McKelvey,Ordeshook and Winer

(1978), and elaboratedn

McKelveyandOrdeshook

(1978a). Throughoutthisessay, moreover,we consider only the strong version of K, which eliminates

greaterthan minimalwinningcoalitionsand 'extraneous'coalitions- coalitionsthat can form only if they simultaneously dopt the proposalsof coalitions hatareviable n the strongcompetitivesolution.

4. While pairs of subjects played together a second time in severalexperimentsreportedhere,no effect canbe detected eitherin experimental utcomes or in the

bargaining rocess.It is evident that our experimentswerewidely discussedon

campus(overheard unchtime conversations, or example, included '. .. but Iwould have passed bill B if you'd have agreed to . . .') but the general attitude

expressedby studentswas that this was an enjoyable,competitivesituation and

that, as in parlourgames,everyone should try to do as well for themselvesas

possible.We shouldpoint out, moreover, hat, to our knowledge,no subjecthadreceivedanyformaltrainingn gametheorypriorto his or herparticipation.

5. The experimentnumbers do not correspond o the order n which experimentsarerun,but areaffixedfor convenientreference.

6. The outcome of experiment25, however, s arrivedat sequentiallyand also acci-

dentally in that one player votes incorrectly.Thus, this outcome cannot be

explainedby equityconsiderations.7. Admittedly,there is a bias in our experimentalprocedures,n conjunctionwith

K, for predictingcoalitions of playersthat areadjacentat the bargainingable.

But, not all coalitions predictedby K for games G3 and G4 entail adjacentplayers.The expected numberof adjacentcoalitionspredicted n these games, fall predictedcoalitions are equiprobable, s 7, while adjacentcoalitions in K

actuallyform seven times;moreover,K predictsthat sevenout of ten adjacentcoalitionswill not form, and, in fact, none of these coalitions ever prevails odictatein the outcome.Thus,seatingarrangementso not seem to biasoutcomes.

8. For even strongerexperimentalevidencesupportingK and discreditingM, andV in the non-transferabletility context, seeMcKelveyandOrdeshook 1978b).

9. Table 10 does not report two additionalgamesthat fail to result in core out-comes. In one instance,owingto a scheduling rror,four inexperiencedplayers

werepittedagainstanexperiencedplayer,who took advantage f the situationbymisleading hem as to whether verbalagreementsarebinding.In the secondin-

stance,priorcollusionor sabotages apparentby one playeradmittingafterwardsthat he sought o minimizehispayoff.

10. Presently,we possessapproximately eventyhoursof taperecordedexperimentalprotocols,and, clearly,it will be some time beforewe transcribe heseconversa-tionsandsubject hem to rigorousanalysis.








REFERENCES

Berl,Janet, McKelvey,RichardD., Ordeshook,PeterC., andWiner,Mark. 1976). 'An

ExperimentalTest of the Corein a

SimpleN-Person

CooperativeNonsidepaymentGame.'Journalof ConflictResolution,20: 453-79.

Fiorina, Morris,and Plott, Charles.(1978). 'CommitteeDecisions Under MajorityRule:An Experimental tudy.'AmericanPoliticalScienceReview(forthcoming).

Haefle,EdwinT. (1970). 'Coalitions,MinorityRepresentation, nd Vote TradingProb-abilities.'PublicChoice,8: 75-90.

Isaacs,R. Mark,andPlott,Charles.1978). 'CooperativeGameModelsof the Influenceof the Closed Rule in ThreePersonMajorityRuleCommittees:TheoryandExperi-ments.' In P.C.Ordeshook(Ed.), GameTheoryand Political Science. New York:NewYorkUniversityPress.

Laing,James D., and Olmstead,Scott. (1978). 'Policy Makingby Committees:An

Experimental ndGameTheoreticStudy.' In P.C. Ordeshook Ed.),GameTheoryand PoliticalScience. New York: New YorkUniversityPress.

McKelvey,RichardD., and Ordeshook,Peter C. (1978a). 'CompetitiveCoalition

Theory.'In P.C. Ordeshook Ed.), GameTheoryand PoliticalScience.New York:NewYorkUniversityPress.

McKelvey,RichardD., andOrdeshook,P.C.(1978b). 'AnExperimentalTest of SeveralTheories of CommitteeDecision-MakingUnder MajorityRule.' Carnegie-MellonUniversity.Mimeo.

McKelvey,RichardD., Ordeshook,P.C.,andWiner,M.(1978). 'TheCompetitiveSolu-tion for N-PersonGamesWithoutSidepayments.'AmericanPolitical ScienceRe-

view,72: 599-615.Riker,WilliamH., andBrams,StevenJ. (1973). 'The Paradoxof VoteTrading.'Ameri-

canPoliticalScienceReview,67: 1235-47.

Tullock,Gordon.(1970). 'A SimpleAlgebraicLogrollingModeL'Americanconomic

Review,56: 1105-22.

Wilson,Robert.(1969), 'An AxiomaticModelof Logrolling.'AmericanconomicRe-

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Documents

1980 Vote Trading_An Experimental Study