Experimental Design
Five subjects were recruited for each experimental session. The amendment rule remained constant within a given experimental session but differed between sessions (four closed rule and four open rule sessions).3
The first session in each treatment had 10 elections. As it was immediately obvious that play was still evolving by election 10, we extended all subsequent sessions to 15 elections.
To minimize the possibility of repeated play game effects, at the start of each election each of the five “legislators” was randomly assigned a new subject number, which was known only to that legislator and changed across elections (but not between rounds of a given election). The number of elections (15) was announced at the start of each session.
Procedures were as follows. First, all subjects filled out a proposal form for allocating the $25.00. After these forms were collected, a roll of a five-sided die determined the standing proposal. This proposal, along with the subject number of the proposer, was posted on the blackboard so that, prior to the voting, subjects saw the amount of money allocated to themselves and to every other voter.
In closed rule sessions each subject immediately voted on the proposed allocation. If a simple majority accepted the proposal, the payoff was implemented and the election ended. If the proposal was rejected, the process repeated itself after applying the discount rate of 0.8 to the total benefits. (Discounting was done by the experimenters, with the total amount of money to be allocated in the next round posted on the blackboard.) Voting results were posted on the blackboard underneath each proposal. These consisted of the total number of votes for and against the proposal but not the votes of individual electors. The blackboard contained information for each of the last several elections.
In open rule sessions, after a proposal was selected, each legislator completed a form either seconding or amending the proposal. If amending a proposal voters were required to propose an alternative distribution of the benefits provided. Although “seconding” forms were collected from everyone (to preserve the identity of the proposer), a roll of a four-sided die determined which legislator, other than the proposer, would be recognized. If the proposal was seconded, an election was held following the same procedures as in the closed rule sessions. If the proposal was amended, the amendment was posted on the blackboard along with the original proposal, and a runoff election was held. The winner of the runoff election became the standing proposal in the next round of the election. Benefits following a runoff election were subject to discounting, so that following a runoff the dollar amounts of the standing proposal were multiplied by 0.8, with the total amount to be allocated posted as well. Election in open rule sessions continued in this way until a proposal was seconded and approved by a simple majority.
Subjects were recruited through announcements in undergraduate classes and advertisements in student newspapers at the University of Pittsburgh and Carnegie–Mellon University. This resulted in recruiting a broad cross section of graduate and undergraduate students from both campuses. At the end of each experimental session four elections were randomly selected, with subjects paid the sum of their earnings in these four elections. Subjects also received a participation fee of $5.
In each session an additional subject was recruited to roll the dice to help assure subjects that the outcomes were indeed randomly determined. This subject received afixed fee of $15. Practice elections were held first to familiarize subjects with the procedures and accounting rules. All experimental sessions were conducted using pencil and paper. Copies of the instructions are posted on the Web site http://www.econ.ohio-state.edu/kagel/fklinstructions.pdf.
Results
We report results in terms of a series of conclusions. This is followed by the data supporting the conclusions reached.
Conclusion 1
Once voters gain some experience with the procedures, all proposals are accepted without delay under the closed amendment rule, and are accepted with delays under the open amendment rule, as the theory predicts. However, delays are less frequent than predicted under the open rule.
Figure 1 reports the frequency at which proposals were accepted in round 1. This occurs in 96.4% (53/55) of the closed rule elections, close to the 100% the theory predicts, the two rejections both occurring early in a session. The open rule, on the other hand, exhibits failures to accept in round 1 throughout, with the failure rate averaging 25.5% (14/55) over all elections and 13.3% (2/15) over the last five elections. These differences between open and closed rules are significant at the 1% level (Z = 3.245) over all elections and are significant over the last five elections as well (Z = 1.464, p < .10). 4
These tests use election as the unit of observation. The remaining tests reported use average amount offered by subject as the unit of observation. One-tailed tests are reported when the theory predicts a difference between treatments; two-tailed tests when there are no predicted differences.
Nevertheless, the failure rate under the open amendment rule was much lower than the predicted 50%. As will be shown, the proximate cause for this is that supermajorities were the norm in these elections.
Conclusion 2
Play begins with supermajorities under both closed and open amendment rules. With experience, play converges toward minimal winning coalitions under the closed amendment rule, but supermajorities remain the norm under the open rule.
Figure 2 provides Kernel density estimates of the sum of the two lowest offers in round 1 for the first and last five elections.5
All computations and estimations are performed using Stata 7. Kernel density estimates are obtained using the kdensity command and random-effects probits are estimated using the xtprobit command.
Distributions are remarkably similar in the first five elections, with a single prominent peak near the $10 level in both cases representing a near-egalitarian distribution of shares across voters. However, clear differences between treatments have emerged by the last ve elections. Under the closed rule there is now a prominent peak near the $0 level, as the theory predicts, and a much smaller peak around the $10 level. In contrast, under the open rule there is a single prominent peak spanning the 6–$10 range. As a result the average sum of the two lowest offers over the last five elections is $3.29 under the closed rule versus $7.06 under the open rule (
p < .05, Mann–Whitney test).
These different outcomes in the last five elections are supported by substantially different strategies for allocating the two lowest shares. These can be classified into the following three simple categories. (1) In the double zero (DZ) strategy, the sum of the two lowest offers is less than or equal to $2.00 (e.g., an offer of $9, $7.50, $7.50, $0, $1). Clearly, the $1 share in this example is similar to a $0 share, since voters receiving such shares always rejected them; see Figure 5. This strategy is employed in 65.3% (49/75) of the last five elections under the closed rule versus 4.0% (3/75) under the open rule. Further, a breakdown of offers within the DZ category shows that 51% (25/49) of them gave exactly two zeros to two voters, with 96% (47/49) of them giving at least one zero to one voter and $1 or less to another voter. (2) In the single zero (SZ) strategy, one voter is allocated $1 or less. This occurred almost exclusively under the open rule: 44.0% (33/75) of all open rule offers in the last five elections versus 4.0% (3/75) of closed rule offers. (Of the SZ offers under the open rule, 66.7% (22/33) provided one zero to one player.) (3) In equal split (ES) strategy, all voters are given at least $4 in round 1. This occurred 29.3% (22/75) of the time under the closed rule, compared to 40.0% (30/75) under the open rule, in the last five elections (54.6% [12/22] of these offering perfectly equal splits under the closed rule; 23.3% [7/30] under the open rule).
Conclusion 3
Proposers' take starts out nearly the same under both amendment rules. However, proposers' take grows substantially under the closed amendment rule, averaging between $9 and $10 in the last five elections. This growth under the closed rule results from the increased frequency of minimal winning coalitions. In contrast, there is an immediate decline in proposers' take under the open rule, with allocations quickly converging on the $6–$7 level and staying there.
Figure 3 shows the share that proposers allocated to themselves in round 1. Notice that both the open and the closed rules start with virtually the same share for the proposer in elections 1 and 2. There is substantial and continuous growth in the average share proposers give themselves under the closed rule. But there is an immediate decline in average proposers' share under the open rule, with shares stabilizing at the $6–$7 level after the second election. The null hypothesis that proposers allocate the same amount to themselves under open and closed rules is soundly rejected in the last five elections (p < .01, Mann–Whitney test).
The growth in the average share proposers take for themselves under the closed rule results directly from the increased frequency with which they follow the DZ strategy, which grows more or less continuously, averaging 11.1% (4/36) in the first two elections and 73.3% (22/30) in the last two elections. In contrast, the share proposers take for themselves, conditional on playing the DZ strategy, remains essentially the same, around $10, from election five on (see Figure 4). This results in the steady growth in the share proposers take for themselves, reported in Figure 3.
Conclusion 4
Proposers receive a uniformly larger share of the benefits than coalition members under both closed and open amendment rules so that we can reject a null hypothesis of an equal proportionate distribution between proposers and other coalition members under both amendment rules. Nevertheless, proposers take well below the SSPE prediction in both cases.
Looking at accepted offers, under the closed rule the proposer takes on average $8.30 for him- or herself, while the next highest average share is $6.62. Under the open rule the average take by the proposer is $6.21, while the next highest average share is $5.47. In both cases, using a sign test (Snedecor and Cochran 1980), the null hypothesis that the median of the differences between the proposer's share and the share offered to anyone else is zero can be rejected at the 1% level.
Conclusion 5
Voting patterns show a clear bifurcation, with round 1 shares below $4 always rejected under both rules and shares at or above $5 almost always accepted. Regressions show that subjects vote primarily on the basis of their own share of the benefits, with minimal concern for the shares of the least well-off and for their share compared to the proposer's take.
Figure 5 shows votes, by shares offered, in round 1, excluding the votes of proposers themselves. Under the closed rule shares of less than $4 are always voted against (66/66). Votes change rather dramatically for offers of $4–$4.99, with 69.4% (25/36) of these offers accepted and virtually all offers greater than or equal to $5 accepted (113/118). There is a similar pattern under the open rule, with all offers less than $4 rejected (17/17), 91.3% (42/46) of all $4–$4.99 offers accepted, and all offers greater than or equal to $5 accepted (101/101).
Table 2 reports estimates of the following voting equation:
where 1 {·} is an indicator function that takes the value one if the left-hand side of the inequality inside the braces is greater than or equal to zero and the value zero otherwise. Explanatory variables include own share (sit); DZ, an indicator variable taking value one if the DZ strategy is on the floor; and the share the proposer takes (PS). (Since the DZ strategy did not take the floor in the open rule, an indicator variable for SZ is included.) The equation is estimated using a random effects probit, with a one-way subject error component for all rounds. The sign of the coeffcient (presented in Table 2) for own share is positive, large in value relative to the other coefficients, and statistically significant in both treatments. The coefficients for both the DZ and the SZ strategies, and for proposer's share (PS), do not achieve statistical significance. The implication is that subjects are voting primarily out of concern for their own share of the benefits, with limited concern for the shares of the least well-off and for the proposer's share.
6 Limiting the analysis to round 1 votes, PS is negative and statistically significant at the 10% level in the closed rule. (In round 1 of the open rule there exists a perfect fit with all shares greater than 0.16 accepted, so that the normality assumption underlying the probit regression is not satisfied.) The results are robust to alternative specifications such as replacing DZ (or SZ) and PS with a summary measure of inequality such as the effective number of parties (enpv); i.e., whenever PS or DZ is statistically significant, so is enpv, and if neither is significant, neither is enpv. The disaggregated measures of inequality help to pinpoint the precise inequality factors at work.
Conclusion 6
Seconding and amending patterns in the open rule sessions followed the voting pattern reported in Figure 5. Further, when proposing amendments, the two worst-off voters typically receive increased shares.
Any time a voter was allocated a share of less than $4, he or she chose to amend the round 1 proposal (35/35 cases). In contrast, in 45.0% (9/20) of the cases where a voter was allocated $4, he or she voted to second the motion, and offers greater than $4 and less than $5 were almost always seconded (94.3% of all cases; 33/35), as were offers of more than $5 (seconded 98% of the time; 100/102).7
Voting patterns closely follow amendment patterns, with all those who seconded a proposal voting in favor of it and 95.1% of amenders voting against a proposal if it was seconded.
Further, when proposing amendments, in all but one instance, amenders offered the two worst-off subjects improved shares (typically, themselves and one other subject). Subjects were not immediately successful in creating amendments that would win a runoff election: In the first five elections, only 53.3% offered improved shares to at least two other voters beside themselves. However, they learned to create more successful amendments, with improved shares offered to at least two other voters 84.2% of the time in the last five elections.
Discussion
The results of Expt 1 provide clear qualitative support for the BF model, as (1) proposals pass without delay under closed amendment rules, but there are delays under the open rule, as voters left out of the coalition amend the standing proposal, (2) there is a more equal distribution of benefits among coalition members under the open compared to the closed amendment rule, and (3) proposers get a significantly larger share of the benefits than coalition members (and substantially larger shares than the average “legislator”) under both amendment rules.8
Diermeier and Morton (2000) examine allocations under a closed amendment rule in a finitely lived legislature, with members having unequal recognition probabilities and unequal vote shares. They report finding no evidence of proposer power, including one treatment with very equal vote shares (34/99; 33/99, 32/99). We have substantially more observations than they do to identify proposer power, and the experimental designs are very different. This makes reconciling their results with ours problematic without further experimentation.
Further, there is convergence toward minimal winning coalitions under the closed amendment rule, as the SSPE predicts.
There are, however, major quantitative deviations from the SSPE. These are twofold. First, for our parameter values, under the open rule the SSPE predicts a minimal winning coalition. This did not occur. Rather, play converged on supermajority coalitions. As a result, round 1 proposals were seconded and passed at a frequency higher than predicted in open rule sessions. Second, within the closed rule sessions, there was a much more equal distribution of shares among coalition members than the theory predicts, even conditioning on DZ allocations.9
McKelvey (1991) also finds a more equal distribution of benefits between coalition partners than the BF model predicts under closed amendment rule procedures. In his experiment there are three voters choosing between three or four predetermined allocations, resulting in a mixed strategy equilibrium.
These quantitative deviations from the SSPE are entirely consistent with deviations from subgame perfect equilibrium outcomes observed in bilateral bargaining game experiments. The main insight from this literature is that responders in bilateral bargaining games have some minimal threshold for the share of the pie that they are willing to accept, which is typically well above the subgame perfect equilibrium prediction, and that they consistently reject offers below this threshold. For example, in the ultimatum game responders consistently reject offers below 30%–40% of the pie, even thought the subgame perfect equilibrium prediction calls for accepting any allocation short of zero.10
In the ultimatum game a proposer has a fixed sum of money to allocate between him- or herself and the responder. If the proposed allocation is accepted, it is binding. If it is rejected, both players receive nothing. The unique subgame perfect equilibrium prediction for this game (under the assumption that bargainers seek only to maximize own income) is that the proposers will receive all the money (or almost all of it if payoffs are discrete). These experiments typically employ stakes similar to those used here. However, the results extend to much higher-stakes experiments as well (see, e.g., Slonim and Roth 1999).
In anticipation of such responses proposers offer a substantial share of the pie, with median returns to responders of about 40% of the pie (see
Roth 1995 for a review of this literature).
11 Dictator game experiments, in which player 2 has no opportunity to reject player 1's offer, result in a wholesale reduction in offers to player 2 (Forsythe et al. 1994), ruling out altruism to explain these results.
Similar results occur in our game: Responders typically reject offers below $5, and proposers, anticipating this, typically fail to ask for anything approaching the SSPE. As a result, under the open amendment rule it actually pays (in an expected value sense) to play the ES and the SZ strategies as opposed to the DZ strategy, conditional on what proposers were actually asking for: The average return for proposers under the ES and SZ strategies was $6.14 and $6.12, respectively. This compares to an expected return from the DZ strategy of $5.03 using the average share that DZ proposers asked for under the open rule ($8.48) or $6.03 using the average share that DZ proposers asked for in the last five elections under the closed rule ($10.37).12
These counterfactual computations assume that (i) an amendment, once it is proposed, always beats the proposal on the floor, (ii) any proposal, once it is seconded, will be passed, (iii) a DZ amendment always completely excludes the previous proposer (which is what amenders did), and (iv) only a DZ amendment can beat a DZ proposal. This last assumption is the most restrictive, but it helps make the computations manageable and is based on the assumption that the amender must provide at least one member of the original coalition a superior payoff to win his or her vote.
Thus, conditional on what proposers felt comfortable asking for, there was little chance for the DZ strategy to emerge naturally under the open rule.
13 For the the open rule it is also probably much more salient that there is a 50% chance of one's proposal failing under the DZ strategy and getting zero, compared to a 25% chance under the SZ strategy and 0% chance under the ES strategy.
These more equal shares between coalition members (relative to the SSPE predicted outcome) are also consistent with the deviations from subgame perfection reported in bilateral bargaining game experiments. However, minimal winning coalitions still tend to emerge under closed amendment rules as proposers learn to give zero and have impunity from the two voters excluded from the coalition. Further, coalition members have few compulsions about voting against such proposals as long as their share of the pie is sufficiently large. This last result is information about tastes for “fairness,” which obviously cannot be discovered from bilateral bargaining game experiments.14
This lack of concern for players receiving minimal or zero shares, provided own share is large enough, has been reported for three-person ultimatum games (Güth and van Damme 1998).
One question that Expt 1 leaves open is why proposer shares do not converge closer to the SSPE in the closed rule sessions. Looking at voting records it is clear that (i) zero shares to noncoalition members do not significantly jeopardize the chances for a proposal passing and (ii) based on votes for $5 shares, the expected payoff (in round 1) for a $15, $5, $5 proposal is $13.38. The latter is reasonably close to the SSPE and is considerably higher than the mean proposer share of $10.37 for DZ proposals in the last five elections. Yet there are virtually no proposals of $15, $5, $5 in the last ve elections.15
These proposals are reasonably secure as well, with an 87% probability of passing. In contrast, the expected (round 1) payoff to a proposer under the SSPE is $7.76, as $4 shares are much more likely to be rejected.
Given the large adjustments in behavior under the closed rule, it may well be that, with more elections, play would move closer to the SSPE. Further, it is clear that it is a lot easier, and less risky, for a proposer to recognize that he or she can give zero shares to a minority of voters under the closed rule than to fine-tune shares within a minimal winning coalition. Experiment 2 explores these possibilities.
EXPERIMENT 2
Experimental Design
Two basic changes were made in the procedures for Expt 2 in an effort to see if play would converge closer to the SSPE under the closed rule. First, the number of elections was increased from 15 to 25 to give play more time to evolve. Second, we attempted to speed up the learning/adjustment process as follows: Each experimental session employed six subjects, five student “legislators,” as before, and a sixth legislator, an economics graduate student, who, it was announced, “has been instructed to make proposals and to vote according to a computer algorithm.” The computer algorithm called for maximizing the proposer's share from the following two alternatives: (i) a proposal of $10, $8, $7, $0, $0, with the proposer's amount listed first16
Alternative (i) was chosen from the more extreme, but common set of values observed in Expt 1, in order not to give the “computer's” strategy away and to avoid possible demand induced effects. Postexperiment questioning of subjects indicated that they were unable to identify the computer's proposals successfully. The introduction of the “computer's” strategy formally eliminates the SSPE. However, given that behavior is subject to a strong trial-and-error adjustment process, we have introduced changes that should push behavior closer to the SSPE, which is what Expt 2 is designed to achieve.
or (ii) take the highest proposal passed to date, add $2 to the proposer's share, and give equal shares to two coalition members and zero shares to the two noncoalition members.
17 If the computer's proposal was rejected once, it was repeated. If it was rejected twice, the proposer added $1 (instead of $2) to the highest proposal passed to date.
As a voter, the graduate student approved any proposal that gave her a share at least as large as the SSPE (0.16) and voted against any smaller share. The nature of the computer's strategy, and who the graduate student was, were
not announced.
18 The graduate student was paid a flat fee for helping, was a classmate of the experimenters who would not stand out from other participants, and was willing and able to help out. No one inquired regarding who was following the computer algorithm, what they were paid, or what the algorithm was.
With the “computer” in the mix we stood a good chance of having proposals reach the floor that came closer to the SSPE than any proposal in Expt 1. The fate of such proposals, whether they were accepted or rejected and, if accepted, whether other voters would make similar or more extreme proposals, should provide insight into the likelihood of play ever converging, on its own, to the shares predicted within minimal winning coalitions under the SSPE.
Payments were made for six elections, selected at random at the end of the session. As before an additional subject was recruited to act as dice roller for all random decisions. Three sessions were conducted using inexperienced subjects drawn from the same subject population as for Expt 1.
Results
The main results from Expt 2 can be summarized as follows.
- After the first five elections, 85% (51/60) of all proposals were passed in round 1 in Expt 2, compared to 100% (35/35) in Expt 1. These rejections were largely the result of “computer”-generated proposals at (or close to) the SSPE (three such cases with DZ allocations and $4 or $4.50 allocations to coalition members) and a number of cases (four) where subjects' own proposals gave more to the proposer than had typically been observed in Expt 1.
- The growth in the proportion of DZ proposals closely matches the pattern reported in Expt 1, with no significant differences between the two experiments in any consecutive set of three elections.19
Mann–Whitney tests where the unit of observation is subject value data for three consecutive elections (i.e., elections 1–3 and 4–6).
- The amount the proposer takes grows steadily in Expt 2, with virtually the same growth pattern as in Expt 1.20
Outside of elections 4–6, when proposers ask for a higher share in Expt 2, there are no significant differences between the two in any consecutive set of three elections.
- Voting patterns are quite similar to those reported in Expt 1, with all shares less than $4 rejected in both cases and almost all shares of $5 or more accepted (95.8% [113/118] in Expt 1 vs 93.6% [117/125] in Expt 2). Estimates from the voting equation (1) for Expt 2 reported in the last column in Table 2 show that own share is the dominant factor influencing voting for or against a proposal (accounting for 92% of the variance explained in the data), with proposers' share having a negative, and statistically significant, impact on the probability of voting for a proposal (p < .10).21
In contrast, looking at round 1 votes, DZ is negative and statistically significant (p < .10) and PS is negative but becomes statistically insignificant. These two variables are highly colinear. These results are computed excluding one outlier subject. Including that subject affects the result by leaving only own share and the constant as statistically significant regressors.
This holds for round 1 data and is robust to all the specifications employed.
Conclusion 7
Experiment 2 shows somewhat closer conformity to the SSPE than Expt 1 in that proposers take larger shares and there is increased frequency of DZ (and strict DZ) allocations and virtual elimination of ES allocations. However, proposed allocations are still substantially more equal than the SSPE predicts, and in the three elections closest to the SSPE allocation, coalition members all rejected the small shares provided.
Discussion
The rejections of $4–$4.50 shares in Expt 2 associated with the SSPE (or near-SSPE) allocations imply that play has little if any chance of converging to the SSPE on its own, as these proposals have essentially no chance of receiving a majority vote. As such it is highly unlikely that they would be proposed often enough to get voters to accept such small shares (see, e.g., Roth and Erev 1995). This still leaves the possibility of convergence close to the SSPE, e.g., a $15, $5, $5 allocation. This proposal was offered three times by the computer and passed in all cases. Further, based on votes for $5 shares observed in Expt 2, this proposal has an expected return to the proposer (in round 1) of $14.63, with about a 98% chance of passing. But here too we observed no spontaneous proposals of this sort so that it remains an open question if even more experience would result in such proposals emerging.
What is left unexplained from Expt 2 is the relative role of more elections versus the “computer” proposals in pushing play closer to the SSPE. Over the first 15 elections there are minimal differences in the frequency of DZ play and in the shares proposers take for themselves between the two experiments, so the existence of the computer proposals appears to have had no effect on these dimensions. Where the computer proposals may have had an effect is in the near-complete elimination of ES (equal split) proposals by election 13 in Expt 2, compared to a sizable remanent (26.7%) in Expt 1.23
One question raised by Expt 2 is that perhaps subjects' awareness of the computer's presence fundamentally altered their behavior. Although we have no direct evidence on this point, Winter and Zamir (1999) report results from an ultimatum game experiment in which a significant proportion of the subject population was played by computers making relatively unequal offers as well as accepting such offers. They report minimal differences between treatments where subjects were told of the computers and their strategy versus when they were not told. The minimal differences in proposer behavior over the first 15 elections in Expts 1 and 2, and the similarities in voting patterns between the two experiments, suggest no systematic effect on behavior here as well.
What, if any role, the near-complete elimination of ES proposals played in the adjustments over the last 10 elections in Expt 2 is unknown. But it may have sped up these adjustments.
CONCLUSIONS
The question underlying our experiment is whether the amendment process—open versus closed amendment rules—can, of and by itself, have the impact on legislative outcomes that the literature suggests? That is, does a closed amendment rule favor minimum winning coalitions with proposers obtaining greater benefits than under an open amendment rule? Do open amendment rules facilitate a more widespread and egalitarian distribution of benefits and lead to greater delays in the legislative bargaining process?
Results from Expt 1 show that the amendment rules have the impact the literature suggests, as proposals under a closed amendment rule are more likely to pass immediately than under an open rule, and the share of benefits the proposer takes is substantially greater under the closed rule. Further, under both amendment rules proposers obtain a larger share of the benefits than do others' allocated benefits, resulting in decisive rejection of a model where benefits are proportional to relative vote shares within the winning coalition (Gamson 1961).24
Proposer power is a central implication of distributive models of the legislative bargaining process, such as the Baron–Ferejohn model, that underlies our experimental design.
There are, however, important quantitative differences between our results and the Baron–Ferejohn model underlying the experiment. For our parameter values the theory predicts minimal winning coalitions under open amendment rules. Instead we observe supermajorities which, at least as a proximate cause, can be attributed to proposers' reluctance to ask for a large enough share that it pays to brave the 50% probability that their proposals will be amended. Instead proposers achieve higher expected returns from the more egalitarian distributions offered. In closed rule elections we observe a more egalitarian distribution of benefits among coalition members than the SSPE predicts. However, under the closed rule proposers have impunity from voting behavior of noncoalition members, and the additional resources that such impunity provides can be used both to increase the proposer's share and to provide greater shares for coalition members. The latter increases the chances of the proposal being accepted. As a result proposers receive a much larger share in the closed rule elections, as the Baron–Ferejohn model predicts, with the institutional forces postulated as being responsible for these differences playing a major role. Only the details of how these institutional forces play themselves out differ from the theory's characterization under the SSPE refinement.
Experiment 2 was designed to understand better the reasons why the distribution of benefits within winning coalitions was consistently more egalitarian than predicted under the closed amendment rule. To do this we increased the number of elections and introduced a “computer” player designed eventually to propose the SSPE. There is continued learning/adjustments in proposals in the extra elections, with behavior at the end of the experiment closer to the SSPE (i.e., increased frequency of minimal winning coalitions and larger benefits to proposers) than in Expt 1. However, the results also indicate that there are fundamental barriers to achieving the SSPE, as coalition members consistently reject the small share of benefits the theory predicts.
There are obvious connections between our results and the large experimental literature on shrinking-pie bilateral bargaining games (including the ultimatum game; Roth 1995 surveys the experimental literature). In the latter, play consistently deviates from the subgame perfect equilibrium in favor of a more equal distribution of benefits between bargainers. This in turn has led to the development of a literature designed to explain these deviations in terms of arguments other than own income in agents' utility function, something commonly referred to as “fairness” considerations (see, e.g., Bolton and Ockenfels 2000, Charness and Rabin 2000, and Fehr and Schmidt 1999, to cite a few of the more prominent attempts to organize the experimental data).25
There is also a learning literature designed to explain these outcomes (see, e.g., Roth and Erev 1995).
Fairness considerations appear to play an important role in our game as well, with subjects concerned primarily about receiving their “fair” share of any given allocation. One prominent focal point for the minimum acceptable share in our game would be 1/n, or $5 in round 1 (Bolton and Ockenfels 2000). In contrast, calculating the continuation value of the game as required under the SSPE is no doubt beyond the abilities of most of our subjects. Indeed, subjects appear to rely on this minimal “fair” allocation, as shares much below $5 are routinely rejected in round 1, while shares at or above $5 are usually accepted under both closed and open amendment rules. As such, under the closed rule subjects frequently reject offers at or slightly above the continuation value ($4), while under the open rule subjects always accept $5 offers even though these are below the continuation value ($6).26
The latter cannot be explained by the large number of equal share proposals offered since, if anything, this should result in a higher continuation value. Similarly, risk aversion cannot explain these differences, as under the open amendment rule voters act as if they are risk-averse, accepting smaller shares than the continuation value, and under the closed amendment rule they act as if they are risk-loving, often rejecting shares at the continuation value.
Thus, a fundamental barrier to achieving the SSPE under the closed rule is that the rule of thumb voters rely on for their minimal “fair” share is greater than the SSPE for coalition partners. By the same token, one explanation for why play comes as close as it does to the SSPE in Experiment 2 is that the rule of thumb underlying votes is reasonably close to the continuation value of the SSPE. As a result, we would predict that factors that reduce the continuation value of the game relative to the 1/
n focal point (for example, increasing the discount rate), or that complicate the game so that a simple focal point no longer exists (such as unequal recognition probabilities or unequal voting shares), will result in greater deviations from the SSPE.
One prominent feature of our results related to the fairness literature in economics is the frequency of the DZ allocations in the closed amendment rule elections and the acceptance of these allocations. Clearly there is a strategic component to these DZ allocations, as, provided that coalition members get their “fair” share, they can be implemented with impunity, while simultaneously increasing the proposer's share. With these DZ allocations, both proposers and coalition members largely ignore the shares of the two worst-off voters, contrary to the conclusions others have reached (see, e.g., Charness and Rabin 2000).
Although experiments provide the investigator with the opportunity to conduct direct qualitative and quantitative tests of theories where, unlike with field data, the institutional assumptions of the model are satisfied by construction, they too suffer from well-known limitations. Central among these is the use of undergraduate students in the role of decision makers, subjects who are unlikely to be as sophisticated as experienced politicians, the target population for applications of the theory. In addition, the amount of money at stake in laboratory experiments is trivial compared to the money at stake in real legislative bargaining situations, so that choices may not be taken as seriously, and subjects act absent the advice of expert staff consultants that real politicians have at their disposal. These issues threaten the external validity, or generalizability, of experimental results. Further, while the best experiments faithfully implement the theory, real political situations are typically much more complicated, with multiple, oftentimes competing, forces at work that are typically abstracted away from in modeling choices. To the extent that the model in question fails to capture certain essential institutional elements of the target situation, the experimental implementation of the model is incapable of uncovering or accounting for these missing institutional forces.
Although there is no guaranteed solution to the first of these limitations (the problem of external validity), to the extent that a given model works well in the laboratory or fails there, it shifts the burden of proof for those who would dispute the results to show that distinctly different forces are likely to be at work outside the laboratory. Further, once one has made such a case, it often points the way to further experiments designed to follow up on the hypothesized explanation. With respect to the second limitation, failure to account for institutional forces at work in the target institutional setting, the solution is to model the factors in question and to bring the revised model into the laboratory to conduct further tests.
In terms of external validity we do note one striking parallel between our results and apparent deviations from the Baron–Ferejohn model identified in field data. The general consensus from studies of coalition governments using cross-country data is that the distribution of ministerial positions between coalition partners is more evenly distributed, relative to the number of votes each coalition member contributes, than the Baron–Ferejohn model implies, as in the latter the prime minister's party (the proposer) should have a disproportionate share of cabinet positions (Warwick and Druckman 2001). While there are many alternative explanations for this deviation from the predicted outcome in field data (e.g., repeated play game elements and the fact that the formateur is likely to be constrained with respect to the ideological positions of coalition partners), the tendency for ex post allocations to be more evenly distributed than predicted is consistent with the results of the present experiment and a host of bilateral bargaining game experiments as well.
Further probes of the external validity of the results reported here could involve employing different subject populations, particularly those with more experience and expertise with budget allocation problems of the sort studied here. For example, one might anticipate that with political science undergraduates, or at least graduate students, minimal winning coalitions would develop more rapidly under closed amendment rule procedures, as this option may be more transparent to them as a consequence of their studies. However, it is not clear that allocations within winning coalitions, or the size of winning coalitions under the open rule, would differ substantially, as there is abundant evidence from bilateral bargaining games in modern industrialized societies that subjects are reluctant to accept any share much below the 1/n threshold (see Roth et al. 1991). Similarly, it would be interesting to see how professional politicians would behave in an experiment of this sort. Here it is not at all clear what would be observed. On the one hand, professionals should be well attuned to the strategic possibilities inherent in the different amendment rules. On the other hand, professionals too are likely to be subject, at least to some extent, to the same behavioral forces that underlie bilateral bargaining games and the multilateral bargaining game reported on here. Answers to these and other questions form the agenda for future research.
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