کنترل برنامه کاری به عنوان یک بازی گفت و گوی ارزان: نظریه و آزمایش با امتیاز انبار شدنی

The paper studies a committee voting sequentially on a known series of binary proposals. Each member is granted, in addition to a vote for each proposal, a single extra bonus vote – a streamlined version of Storable Votes. When the order of the agenda is exogenous, a sufficient condition guarantees the existence of welfare gains, relative to simple majority voting. But is efficiency compromised if a chair controls the order of the agenda? The agenda becomes cheap talk and can be used to transmit information about the chairʼs priorities. The game has multiple equilibria, differing in the precision of the information transmitted, but the welfare impact is minor, and the comparison to simple majority voting is unchanged. In laboratory experiments, subjects have difficulty identifying the informative strategies, but payoffs are effectively identical to theoretical predictions. The bonus vote matters; the chairʼs control of the agenda does not.

Consider a group of voters faced with a series of binary decisions, each of which can either pass or fail. Decisions are taken according to the majority of votes cast, but suppose that, in addition to a regular vote for each decision, each voter is endowed with some “bonus votes” that can be spent freely over the different decisions. This is the idea behind Storable Votes, a simple voting scheme designed to elicit and reward voters’ intensity of preferences. By inducing a voter to cast more votes on decisions he considers relatively more important, storable votes typically increase ex ante welfare, relative to simple majority voting. One concern that previous work has not addressed is the potential for agenda manipulation. Because storable votes allow voters to modify their “weight” in decision-making by cumulating or reducing the number of votes they cast, control of the agenda could be particularly important. The goal of this paper is to take a first step towards the study of storable votes with endogenous agenda. Because agenda-setting problems are difficult, and the storable votes game quite complex, the paper exploits two simplifications. First, the voting mechanism is streamlined: in addition to their regular votes, agents are endowed with a single indivisible bonus vote to be cast freely over any of the proposals. The analysis shows that when the agenda is fixed, this simplified mechanism can be sufficient to achieve welfare gains over majority voting. More precisely, the paper identifies a simple sufficient condition guaranteeing higher expected utility relative to majority voting. Welfare gains always obtain if the value of the bonus vote, relative to regular votes, is chosen correctly (in practice, is not too large), and either the number of voters is even or large enough, or the differences in intensity of preferences across proposals are important enough. Suppose now that one of the voters assumes the role of committee chair and is granted some agenda power. The second simplification of the paper is to limit such power to the order with which decisions are brought to a vote. At the start of the game, the chair decides and announces the order of the agenda. Would the chair choose the order so as to exhaust other voters’ bonus votes before presenting his own favorite proposal? Could he then carry it through the strength of his bonus vote, even with a narrow support and an efficiency loss? And would the efficiency loss be magnified in equilibrium, as the other committee members adjust their own strategies, possibly saving their bonus votes and failing to register their intensity of preferences? In this framework the agenda’s order acquires the character of a cheap talk message: the chair can use it to transmit information about his priorities. In line with the results of the cheap talk literature, the game has multiple equilibria that differ systematically in the precision of the information conveyed. A babbling equilibrium exists, where no information is conveyed, and the game remains identical to the exogenous agenda case. But informative equilibria also exist where, on the basis of their position in the agenda, voters are able to identify decisions over which the chair is sure not to cast his bonus vote. When only one such decision is identified, the information is very imprecise; when two are identified, more information is conveyed in equilibrium, and so on up to the fully informative equilibrium where all but one of the decisions are known not to be targets of the chair’s bonus vote, and hence the one decision over which the chair casts the bonus vote is identified precisely. The ability of the chair to transmit information, and varying degrees of information, through the order of the agenda alone may seem a bit “too subtle” to have practical implications, but we can interpret the cheap talk message as a norm: it is plausible to think of committees where the chair is known, for example, never to put on the table his highest priority decision first, or second, or last.In equilibrium, when information is transmitted the chair effectively commits to casting his bonus vote on a subset of decisions only. The commitment is valuable because other voters refrain from competing with the chair’s vote, and the chair sees his probability of being pivotal increase on the decisions he cares most about. As a result, the chair’s expected utility is higher: the power to set the order of the agenda is valuable. As for the other voters, the impact of the information on their expected utility is less clear-cut: by avoiding competition with the chair, in equilibrium they face higher competition from other non-chair voters on the remaining decisions. The end result is ambiguous, as is the expected aggregate welfare effect. As for the quantitative importance of these effects, precise numerical results for all parameterizations I have studied yield very small magnitudes. Briefly then: granting control of the agenda’s order to a chair need not change the game at all, if voters coordinate on the non-informative equilibrium; if the game does change, it is through the transmission of information, with a positive effect on the chair’s expected utility, and an ambiguous effect on all other voters. Even then the best guess is that the magnitudes are very small. The welfare comparisons to simple majority voting derived under exogenous agenda carry over with little change when the chair controls the order of proposals. But would voters be able to identify these rather subtle equilibrium strategies? And if they do not, could the chair’s power in fact lead to sensible declines in efficiency? The second part of the paper reports the results of a laboratory experiment where subjects are confronted exactly with the voting game described by the model. Not surprisingly, they have clear difficulties identifying the possible informative role of the agenda order, a hard task, and one made still more difficult by the multiple equilibria. Non-informative equilibrium strategies best explain the observed behaviors, and even then the strategic mistakes are higher than in the simple scenario with exogenous agenda. And yet, and this is the interesting part of the results, realized experimental payoffs track very closely the theoretical predictions of the non-informative equilibrium. In fact, because theoretical predictions are quantitatively very similar across equilibria, experimental payoffs are almost indistinguishable from equilibrium payoffs for any of the equilibria of the experimental parameterization. Even when subjects deviate from equilibrium strategies, they disproportionately cast their bonus vote on their highest intensity proposal, a simple, intuitive course of action sufficient to approximate very closely the equilibrium welfare properties of the voting rule. The robustness of storable votes to strategic mistakes has been a recurrent finding in experimental studies and a strong argument in their favor. The paper’s conclusion is that both in theory and in the experiments, the bonus vote does matter; the chair’s control of the agenda does not, at least in the limited form studied here. Stronger conclusions will need to wait, but the paper ends with some discussion of why the insight provided by this simplest agenda game is likely to carry over to more general scenarios. This study is part of a larger research project studying the theoretical and experimental properties of storable votes. First proposed in Casella (2005) and studied experimentally in Casella et al. (2006, 2008), storable votes resemble Cumulative Voting, a voting system used in US corporate boards and local jurisdictions and allowing voters to spread freely across multiple candidates a given total budget of votes. Cumulative voting is designed to increase the representation of minorities (Guinier, 1994; Issacharoff et al., 2002), and has been shown to do so effectively, theoretically (Cox, 1990), empirically (Pildes and Donoghue, 1995; Bowler et al., 2003) and experimentally (Gerber et al., 1998). It applies to single multi-candidate elections, and thus is a tool of representative democracy; storable votes instead are designed for series of binary decisions and apply to committee decision-making and direct democracy. Like cumulative voting, storable votes make it possible for a numerical minority to win occasionally, but in the case of storable votes the representation of minority interests need not be a goal in itself; it is the result of giving weight to voters’ intensity of preferences, and thus it is typically efficient, in addition to being fair. Proposition 2 in this paper, defining a sufficient condition for the welfare superiority of storable votes to simple majority rule with an arbitrary number of voters, complements existing results. Previous work allowed for multiple bonus votes, but outside some examples, equilibrium welfare results were obtained only when the voters are either 2 (Casella, 2005), or a very large number (Casella and Gelman, 2008). From a theoretical point of view, storable votes are related to the mechanism designed by Jackson and Sonnenschein (2007), and to the almost identical voting scheme proposed independently by Hortala-Vallve (2006).The literature on agenda games typically studies the introduction of different alternatives when a single choice must be made among several (see for example McKelvey’s 1976 classic paper, Ordeshook and Palfrey, 1988, for an early discussion of incomplete information, and, for more recent contributions, Dutta et al., 2004, and the works discussed there). Here the problem is different because all proposals are unrelated; each of them either passes or fails, independently of the others. The possible importance of the agenda comes solely from the existence of the bonus vote. The agenda game studied in this paper recalls results and intuitions from the literature on cheap talk games, originating from Crawford and Sobel (1982). In particular, through the agenda order the chair can communicate the ranking of his intensities over the different proposals – a multidimensional cheap talk problem that has been shown to allow typically more communication than the onedimensional case (Austen-Smith, 1993; Battaglini, 2002; Levy and Razin, 2007). The closest comparison is to comparative cheap talk (Chakraborty and Harbaugh, 2007), but the specific model is quite different: the message comes through an action that although not costly constrains the remainder of the game, there are multiple agents, the game is sequential, and, as the analysis will show, the strategic interaction between the players depends non-trivially on the number of voters and proposals, and on the value of the bonus vote. Yet, the multiple equilibria distinguished by the coarseness of the information transmitted are clearly reminiscent of this literature’s results. Similarly, the experiments recall experimental analyses of cheap talk games (for example, Palfrey and Rosenthal, 1991; Sopher and Zapater, 1993; see Crawford’s 1998 survey and discussion) but both the design and the goals are different. As opposed to allowing subjects to exchange messages and testing whether the messages are truthful and whether they affect the subsequent play of the game, the experiments described in this paper investigate whether subjects are aware that the order of the agenda can be used as a message, and have as goal the evaluation of the welfare properties of the voting scheme, and their robustness to strategic mistakes. The paper was motivated by concerns about the chair’s potential ability to manipulate the game; the theory suggests that the chair may indeed benefit, although probably little, but the experiments see no evidence of a bias in favor of the chair. From this perspective, the result recalls experimental tests of the Baron and Ferejohn (1989) legislative bargaining game, where again experimental subjects regularly come short of achieving the proposer’s advantage hypothesized by the theory (see McKelvey’s 1991 original paper, and the discussion and references in Fréchette et al., 2005). Finally, the topic invites comparison with the body of work studying information transmission and deliberation in committees before a decision is taken by voting (starting with Austen-Smith, 1990; see also Coughlan, 2000, Austen-Smith and Feddersen, 2006, and Gerardi and Yariv, 2007). But again the focus is different: the model of this paper is a pure private value model, where individuals differ in their preferences, as opposed to a common value model where individuals differ in their information.1 There is no deliberation, but a single agent, the chair, is allowed to send a non-costly signal. The next section describes the model; equilibrium strategies and efficiency are studied in Section 3 when the agenda is exogenous, and in Section 4 when the chair controls the order of the agenda; Section 5 describes the experimental design and the experiment’s results, and Section 6 concludes. Most proofs and detailed calculations are in Appendix A; experimental instructions for one of the treatments are reproduced in Appendix B.

Storable votes are designed to elicit and reward each voter’s relative intensity of preferences, and thus increase the efficiency of decision-making and the representation of the minority. One important concern, however, is whether the possibility to shift votes across decisions makes storable votes particularly sensitive to agenda manipulation. This paper studies a scenario where the set of decisions to be voted upon is fixed, but the chair controls the order with which they are called to a vote. The game is then transformed into a cheap talk game, where the chair can use the order of the agenda to transmit information about his priorities. In line with the results of the cheap talk literature, several equilibria are possible, distinguished by the precision with which the information is conveyed. When information is transmitted, it discourages the other voters from competing with the chair, and increases the chair’s expected payoff. The effect on the other voters’ utility and aggregate welfare is ambiguous, and in all parameterizations studied quantitatively very small. In a laboratory experiment, subjects have clear difficulties in identifying the informative equilibria: even when the individual chairs’ ordering strategy is consistent, the informational content of the chair’s strategy is not recognized by the other group members. Nevertheless, because bonus votes are disproportionately cast on high intensity proposals, the experimental payoffs are remarkably close to the theoretical predictions, a robustness result already observed in past storable votes experiments with exogenous agendas. In this paper, the main theoretical prediction is that the chair’s control of the order of the agenda should have a negligible effect on the welfare properties of storable votes. And this is indeed the final conclusion of the experiment: the bonus vote does matter; the chair’s control of the agenda does not. It is true, however, that the agenda power granted to the chair is minimal. Even restricting the chair to control the order of the proposals, suppose the order was announced sequentially, after observing how many votes have been cast, as opposed to being chosen once and for all at the beginning of the game. Would results change? Some equilibria of the game studied in this paper remain equilibria here: the babbling equilibrium, where no information is conveyed, the fully informative equilibrium where the chair always calls his highest intensity proposal first, and the partially informative equilibrium where the chair never calls his highest intensity proposal first. In general the other equilibria break down because the uncertainty over the agenda’s order prevents non-chair voters from comparing their intensities over decisions over which the chair’s bonus vote is expected, or not expected; other equilibria may well appear. But the game remains essentially a game of information transmission, an interesting but, in this setting, very subtle role for agenda control. At least in the equilibria that carry over from this paper, the differences in expected outcomes with respect to an exogenous agenda remain minor. How about the more interesting case where the chair controls not only the order but also the content of the agenda? A plausible model would describe voters facing a set of possible decisions, only a subset of which would be voted upon, with the chair choosing sequentially the next decision to be brought to the table. I leave this problem for future work, but some preliminary thoughts suggest that the insights derived from this paper may carry over. The difficult problem facing voters who are not chair is the uncertainty over which decisions will be presented in the future. But this uncertainty is present even when the sequential agenda is exogenous – it is a different problem from protecting oneself from possible manipulations by the chair. Where agenda power matters is once again in granting the chair the possibility to transmit information about his priorities.