Abstract
We conducted a sensitivity analysis of the results of weighted voting experiments by varying two features of the experimental protocol by Montero et al. (Soc Choice Welf 30(1):69–87, 2008): (1) the way in which the roles of subjects are reassigned in each round [random role (RR) vs. fixed role (FR)] and (2) the number of proposals that subjects can simultaneously approve [multiple approval (MA) vs. single approval (SA)]. It was observed that the differences in these protocols had impacts on the relative frequencies of minimum winning coalitions (MWCs) as well as how negotiations proceed. 3-player MWCs were more frequently observed, negotiations were much longer, subjects made less mistakes, and proposal-objection dynamics were more frequently observed, under the protocol with FR and SA than under the protocol with RR and MA.
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Notes
Felsenthal and Machover (1998), for example, noted this point in their study of the Council of Ministers in the European Economic Community.
Fréchette et al. (2005), Drouvelis et al. (2010), and Kagel et al. (2010) conducted their experiments in non-cooperative game environments based on variants of a legislative bargaining model by Baron and Ferejohn (1989). The analysis by Fréchette et al. (2005) was based on the demand bargaining model by Morelli (1999).
The formation of MWCs is consistent with the “size principle” proposed by Riker (1962), but the main purpose of the experiment conducted by Montero et al. (2008) was to test the “paradox of new members” i.e., adding a new member to a committee can increase the voting powers of some existing members, while reducing the powers of others. Aleskerov et al. (2009) tested the effect of asymmetry regarding favorable coalition partners among committee members on their coalition formation and resource allocation.
Packel and Deegan (1980) generalized the DP index by giving a system of axioms for the weighted DP index.
In Esposito et al. (2012), the number of votes of some, but not all, subjects varied when subjects were faced with the same game, because their focus was on learning about the relationship between voting weights and voting powers.
Montero et al. (2008) and Aleskerov et al. (2009) noted the potential (positive or negative) effects of RR on how subjects learned to play the games, compared to FR, but they did not investigated experimentally how the difference between RR and FR affects subjects’ behavior.
Table 1 The four treatments (protocols) examined In a game, e.g., \([4; 3, 2, 2]\), if the players were ordered from left to right as players 1, 2, and 3 in the proposal-input table every round, most of the observed winning coalitions might be either \(\{1, 2\}\) or \(\{2, 3\}\),even though \(\{1, 2\}\) and \(\{1,3\}\) are essentially the same. Aleskerov et al. (2009) showed that randomizing the order of players’ appearance in the proposal-input table every round excludes this type of effect from being observed in experiments.
Fig. 1 
Screen shot for player 2. His or her proposal-input table is on the left-hand side. A proposal is made by entering four integers into the empty cells in the table and pressing the propose button (in red). The integer in a cell represents the points allocated by the proposer to the player that corresponds to the cell. The order of players’ appearance in the table is randomized. (Color figure online)
Montero et al. (2008) set the random time limit to be between 300 and 600 s and reported that there was no group that could not reach an agreement within 300 s. Aleskerov et al. (2009) set a fixed time limit of 300 s and reported that there were several groups that failed to reach an agreement before the time limit, but their results were otherwise almost the same as those obtained by Montero et al. (2008).
The sentences appearing in the screens and instructions were translated into Japanese for sessions in Tsukuba and into French for those in Montpellier.
“Appendix 2” shows the test results for (1) normality of sample (Shapiro–Wilk test), (2) homogeneity of variance of samples (Levene test), and (3) normality of residuals (Shapiro–Wilk test) conducted before the two-way ANOVA.
The p values are 0.7906 and 0.2089 for the factors FR and SA, respectively, as shown in Fig. 7.
The p values are 0.3961 and 0.0269 for the factors FR and SA, respectively, as shown in Fig. 8.
Fig. 8 
Distribution of the average number of proposals (left) and approvals (right) in the last 5 rounds under the four protocols. RR–MA (solid gray), FR–MA (solid black), RR–SA (dashed gray), FR–SA (dashed black). The results of a two-way ANOVA with mixed effects are also shown. Primary factors are role assignment and approval scheme. A possible interaction between them are taken into account. The adjustment for stratification factors is made with the country (Japan \(=\) 1 or Japan \(=\) 0 for France). A boldfaced value indicates the rejection of the null hypothesis at the 10 % significance level. No data transformation is carried out. The degrees of freedom for the numerators and denominators are 1 and 27, respectively. (Color figure online)
The p values are 0.3812 and 0.7136 for the factors FR and SA, respectively, as shown in Fig. 8.
Aumann and Mascher (1964) defined a solution concept called “bargaining set,” based on “objection” and “counter-objection.” We only employ objection and not counter-objection in our paper.
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This paper is a revised version of our earlier work “A Note on the Experiments of Weighted Voting: Human Mistakes in Cooperative Games” presented at TCER conference 2011. We have greatly benefited from comments and suggestions from Maria Montero and Yoichi Hizen. A part of this research was conducted while Hanaki was visiting the Higher School of Economics (Moscow). We have benefited from discussion with Fuad Aleskerov and Alexis Beliani. The Experimental Economics Lab at Montpellier (LAMETA), in particular, Dimitri Dubois, has kindly hosted us for running this experiment. Financial support from IEF Marie Curie research fellowship n.237633-MMI (Guerci), Japan Economic Research Foundation (Hanaki), MEXT Grants-in-Aid 20330037, 24330078, and 25380222, (Watanabe), and JSPS-CNRS bilateral research project, JSPS-ANR bilateral research project “BECOA” (ANR-11-FRJA-0002) are gratefully acknowledged. The views expressed are purely those of the authors and may not in any circumstances be regarded as stating an official position of the European Commission.
Appendices
Appendix 1: Instructions
The instructions for the four protocols are, for the most part, identical. Thus, in this appendix, we show generic instructions while noting the parts that differ between the protocols. The subjects in a particular protocol are only instructed about the relevant protocol. These instructions were translated into Japanese and French for our experiment in Tsukuba and Montpellier. Both Japanese and French versions of these instructions are available upon request.
INSTRUCTIONS OF THE EXPERIMENT
Welcome! Thank you very much for taking part in our laboratory experiment.
You are a participant in an experiment on group decision making. During the experiment, you, as well as other participants in this room, will be making decisions. The experiment will take about two hours.
RECOMMENDATION
We ask you to comply with these rules and respect the instructions of the experimenter. Any communication with other participants is strictly prohibited. During the experiment, you must not talk, exchange notes, watch other participants’ actions, or use mobile phones. It is important that during the experiment you remain SILENT. If you have any questions, or need assistance of any kind, RAISE YOUR HAND but DO NOT SPEAK. We expect and appreciate your cooperation.
PROTOCOL
There are 20 rounds in this experiment. In each round, you and three other randomly chosen participants will form a group of four people and the four players will decide how to divide 100 points in the manner described later.
Matching
============ ONLY FOR THE FR ============
At the beginning of the experiment, the computer will randomly assign you a player ID number, either 1, 2, 3, or 4. Your player ID number will remain the same throughout the experiment, and will be shown on the computer screen.
At the beginning of each round, the computer will randomly group four participants with different player ID numbers into one group. You will not be able to know which participants are in the same group.
You will repeat the same procedure for 20 rounds. Your ID number does not change from round to round, but the other people in your group change.
============ ONLY FOR THE RR ============
At the beginning of each round, the computer will randomly group four participants into one group. You will not be able to know which participants are in the same group. You will repeat the same procedure for 20 rounds, but your ID number, either 1, 2, 3 or 4, may change from round to round, and the other people in your group will also change. In each round, you will be clearly informed of your player ID for that round.
=========================================
The negotiation
You will be making a decision in a group with three other people on how to divide 100 points among the four of you.
You will not know who the people in your groups are, and the people in your group will change randomly every round.
Each player has a certain number of votes and the information will be shown in the table on the left-hand side of the screen. In the first 10 rounds, the same vote allocation will be used, and a different vote allocation will be used from round 11 to the end.
Any member of the group at any moment during the negotiation may make a public proposal about how to divide the 100 points. To make a proposal, you need to enter four numbers (integers) in the respective boxes on the left-hand side of the screen and press the “propose” button shown in red.
Any member of a group can also vote for already submitted proposals. Proposals made by others are shown on the right-hand side of the screen. You can vote for a proposal by pressing an “approve” button shown in red.
============ ONLY FOR MA ============
You can use your votes to support more than one proposal. Each proposal you support will receive all your votes. For each proposal, who are supporting it will be clearly shown.
To submit a new proposal, you need to withdraw your current proposal. You can withdraw your proposal by pressing the “withdraw” button on the left-hand side of your screen at any time during the negotiation. All the members in your group will be informed about the withdrawal of your proposal.
You can also withdraw your votes for another’s proposal by pressing the “withdraw” button shown on the right side-hand of the screen at any time during the negotiation.
============ ONLY FOR SA ============
Please remember, you can only be in favor of at most one proposal, including your submitted proposal, at any given time. You cannot divide your votes up and support multiple proposals. All your votes will be cast for the proposal that you decide to support. For each proposal, who are supporting it will be clearly shown. You can change your approval whenever you want during the negotiation.
You can withdraw your proposal in order to propose a new one or to vote for another’s proposal by pressing the “withdraw” button on the left-hand side of your screen. All the members in your group will be informed about the withdrawal of your proposal.
You can also withdraw your vote for another’s proposal to propose or to vote for a different proposal by pressing the “withdraw” button shown on the right-hand side of the screen.
=============================================
The first proposal that receives the necessary number of votes will be implemented and the negotiation will end. Each of your group members will receive the number of points specified in that proposal.
There is a time limit to the negotiation. The time limit will be between 300 and 420 s. In each round, before the start of the negotiation, the computer will randomly set the time limit, and you will not be informed of the exact time limit. This means that the round could end suddenly at any time between 300 and 420 s after its start. If none of the proposals has received the necessary number of votes within this time limit, then all the members of your group will receive 0 points in this round.
If you have any questions, please raise your hand.
PAYMENT
At the end of the experiment, the computer will randomly select 3 rounds out of the first 10 rounds and 3 rounds out of the second 10 rounds. You will be paid only according to the points you have obtained in these selected rounds, and not according to the points of the whole protocol. The total points you have earned in the selected 6 rounds will be converted to cash at the exchange rate of 1 point \(=\) 14 JPY (13 cents in Euro).
In addition to this, you will be paid 1500 JPY (5 EUR) as a show-up fee. The maximum earning you can make is, therefore, \(1{,}500 + 0.14 \times 6 \times 100\) JPY \(=\) 9,900 JPY (\(5+0.13 \times 6 \times 100 =78\) EUR). The minimum earnings you can make is the show-up fee of 1500 JPY (5 EUR).
PRACTICE ROUND
In order to make you familiar with the interface and mechanism of the experiment, we now do 1 practice round. What you will do in the practice will not affect your final payment. The number of votes given to the four members of your group is not related to what you will see in the real experiment to follow.
IF YOU HAVE ANY QUESTIONS, PLEASE RAISE YOUR HAND.
Appendix 2: Background tests for ANOVA
We adopted two-way ANOVA models with mixed effects to test which factor, role assignment (FR–RR) versus approval scheme (MA–SA), mainly accounts for the difference in observations. There are several assumptions to assure a correct ANOVA modeling. We tested (1) normality of samples, (2) homogeneity of variance of samples, and (3) normality of residuals. For some of the variables, the data transformation (square root followed by arcsine) were used according to the classical procedure (Rao 1960; Winer et al. 1971).
Table 3 shows the results of the Shapiro–Wilk tests for normality of samples. Note that ANOVA is not very sensitive to moderate deviations from the normality of samples. Simulation studies have shown that the false positive rate is not affected too much by violation of the normality assumption (Harwell et al. 1992; Lix et al. 1996). Table 4 shows the results of the Levene tests and the Shapiro–Wilk test for normality of residuals. Please note that we need the null hypotheses to be not rejected for the validity of ANOVA.
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Guerci, E., Hanaki, N., Watanabe, N. et al. A methodological note on a weighted voting experiment. Soc Choice Welf 43, 827–850 (2014). https://doi.org/10.1007/s00355-014-0814-y
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DOI: https://doi.org/10.1007/s00355-014-0814-y