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The Market Value of Weighted Votes as An Alternative Approach to Voting Power: With an Application to the EU Council of Ministers Scott L. Feld, Department of Sociology, Purdue University Bernard Grofman, Department of Political Science, University of California, Irvine Bgrofman@uci.edu Leonard Ray, Department of Political Science, Louisiana State University December 9, 2004 Earlier versions of this paper were presented at the Annual Meeting of the Public Choice Society, Nashville, Tennessee, March 21-23, 2003, the Workshop on Voting Power Analysis, London School of Economics, August 9-11, 2002, the Japanese-American Conference on Mathematical Sociology, Vancouver, July, 2002. We are indebted to helpful comments from James Endersby, Takeshi Minakuchi, and William Zwicker. ABSTRACT We offer an alternative and, we believe, empirically more realistic approach to measuring power in weighted voting games. Rather than power being a function of the number of coalitions in which an actor may find herself to be decisive or pivotal, we look to the bribe-worthiness (market value) of each actor in a situation where some external actor is seeking to put together a winning (or blocking) coalition. For small numbers of actors, we show that the intuitions we get about how much a given member is “worth” in a given weighted voting game can give us very different values than we get from either the Banzhaf or the Shapley-Shubik approach. We then explicitly compare bribe-worthiness values and Banzhaf power scores for each of the EU weighted voting schemes in use in the Council of Ministers from 1958 to 1995. 2 Groups with differentially powerful actors have been extensively studied, both theoretically and empirically. In particular, there exists a large game-theoretic literature on the topic of weighted voting, the focus of this paper. Weighted voting games are quite important empirically are found in many settings, from County legislatures in New York (Grofman and Scarrow, 1981), to special water districts throughout the U.S (Riker, 1983), to the U.S. Electoral College (Brams, 1968; Owen, 1975); to the UN Security Council (Brams, 1975), to the International Monetary Fund (Leech , 2002b); to the European Union Council of Ministers (Felsenthal and Machover, 1997, 2001b; Holler and Widgren, 1999; Hosli, 1993, 1995, 1996; König and Brauninger; Leech, 2002c; Nurmi and Meskanen, 1999; Raunio and Wiberg, 1998; Sutter, 2000a, b; Widgren, 1994), to stockholder meetings throughout the world (Glazer, Glazer and Grofman, 1984; Leech, 1988). In general, when it comes to voting within a committee or legislative setting, not all actors are equal. Some may possess many more votes than others may; some may even possess a veto.1 A key question in this literature is how to assess the "power" of the various unequally weighted actors in a committee or legislature.2 A fundamental and somewhat counterintuitive result of all the game-theoretic models of power is that an actor's power need not be directly proportional to the share of votes that s/he controls. Usually power score calculations yield the result that smaller actors have less power than their weight (and may sometimes even be dummies, i.e., have an assigned power score of zero), while larger actors have more power than their weight would suggest. The game-theoretic literature on committee power is large and rapidly growing and we will not try to review it in depth.3 Among the key topics are axiomatic approaches to power, new power scores, and various paradoxes involving power indices (e.g., Felsenthal and Machover, 1995; Dubey and Shapley, 1979; Brink, 2001; Saari and Sieberg, 2001; Napel and Widgrén, 2000); empirical and theoretical comparisons across approaches (Owen, 1995b; Berg, 1999; Leech, 2002a; Widgren, 2001); and considerations of the relationship between power and equity in representation (Nurmi, 1982; Laurelle and Widgren, 1998).4 The usual approaches to measuring power in weighted voting games involve a priori calculations about the likelihood that any given member will be decisive (Banzhaf Score) or pivotal (Shapley-Shubik Value).5 Over the past several decades, however, there have been a variety of attempts to develop more realistic models of weighted voting power, e.g., by directly positing a probability distribution over expected coalitional patterns, or by predicting coalitional alignments based on measurements of the common interests of a given set of actors, or by embedding actors in a multidimensional issue space. There has also been a very recent extended debate about the usefulness of power indices in understanding power relationships within the Council of Ministers, the major legislative body of the European Union, and between the Council and other organs of governance in the EU, pitting George Tsebelis and George Garrett and some others (Garret and Tsebelis; 1999; Tsebelis and Garrett, 1997; see also Tsebelis and Garrett, 2000; Tsebelis et al., 2001), who largely dismiss the relevance of power index calculations, against advocates of the usefulness of insights derived from power index calculations (see e.g., Felsenthal and Machover, 2001a; Holler and Widgren, 1999; Lars and Berg, 1999). The work we present here is intended as a contribution to that debate, but it also broader in scope. We argue that, in some contexts, both the traditional power indices, and even attempts to 2 modify them to make them "more realistic" by replacing assumptions of equiprobability of coalitions/permutations with calculations based on expected (ideological or other) proximity between actors, or to embed them in larger political settings, will mischaracterize completely the nature of power relationships. Inspired by the literature on rent-seeking, to calculate the power of differentially weighted actors, we ask how much would it be worth to bribe each actor within any particular winning (or perhaps, blocking)6 coalition to vote a particular way on a piece of legislation so as to attain a desired outcome (or block an undesired one). Our calculations focus on the notion of a minimal winning coalition (or minimal blocking coalition). i.e., one in which the removal of any member changes the coalition from winning to losing (or from losing to winning) We assume that all minimal winning (or blocking) coalitions are equally valuable to the briber, who is concerned only with winning (or blocking). Thus, because of competitive forces, we would expect that the sum of the bribes paid to the members of any given minimal winning (or blocking) coalition should be identical regardless of which coalition is being bribed. We then seek to find a fixed value for each actor in the game such the expected payoff (bribe) to each actor will be proportional to that value, and such that the sum of the values of the members of each winning coalition is the same. When we cannot find such a unique set of values, we look, instead, to find the closest approximation thereto.7 In the next section, after more formally presenting our model, we begin with a simple example of what is called a homogeneous game (see e.g., Taylor and Zwicker, 1997: 51). For the context where we may think of legislators as being approached by a single briber, we argue that 3 the values derived from our bribeworthiness calculations are much closer to our intuitive ideas about how powerful actors ought to be then are results based on the more usual power index approaches. However, we then go on to present an informal proof sketch that suggests that, as long as there is a substantial degree of (reasonably continuous) variation in the distribution of weights within a committee, as committee size grows larger, the difference between bribeworthiness as we calculate it, and more traditional power scores will tend to shrink. Thus, our approach and the various forms of power score approaches, although logically completely distinct, tend to converge in their results as committee size increases. We illustrate the plausibility of this claim empirically by explicitly comparing bribe-worthiness values and Banzhaf power scores for each of the EU weighted voting schemes in use in the Council of Ministers from 1958 to 2003 -- a period during which EU membership ranged from six (in 1958) to fifteen (in 1995)8 -- and find general support for our expectation that Banzhaf scores, weights, and bribe-worthiness values all tend to be very similar. Finally we consider how to expand our model to deal with transaction costs. What is arguably the most important empirical implication of the work on power scores is that power is not linear in weight, with small actors generally having less power than their weight and large actors having more power. We argue that the larger the transaction costs, the greater will be the discrepancy between an actor’s weight and her market value, with some actors even being of no value when the costs of negotiating with them exceeds the value of their expected contribution to the building of a (minimal) winning coalition. Thus, the key empirical phenomenon that actors with little weight (e.g., small stockholders) have close to zero power; while larger voting blocs 4 often have power dramatically disproportionate to their vote shares can be accounted for within our market share framework once we allow for transaction costs.9 The approach we offer is very similar in spirit to work from the game theoretic literature on bargaining games such as the Aumann-Maschler bargaining set, and to closely related work on “near-core” solutions such as the nucleolus and the kernel (Aumann and Maschler, 1964; Maschler, Peleg and Shapley, 1979; Schmeidler, 1969; see also McKelvey, Ordeshook and Winer, 1978; Sudhölter, 2001). But it is even more similar to the work of Young (1978) and that of Taylor and Zwicker (1997; see also Taylor and Zwicker, 1993) on bribery-based models -work with which we were not familiar with when we developed our own model. 5 II. Market Value Standard Power Score Approaches As noted above, the most common approach is to measure power is in terms of power indices such as the Shapley-Shubik value and the Banzhaf index (Shapley and Shubik, 1954; Banzhaf, 1965). The Banzhaf-Coleman approach (arguably first presented by Penrose, 1946; see also Coleman, 1971) bases its power scores on the likelihood that a particular actor will be decisive, i.e., that the removal of that actor from the coalition will change a winning coalition to a losing one or a losing coalition to a winning one, assuming that all combinations of voter preferences are equally likely.10 The Shapley-Shubik approach bases its power scores on the likelihood that a particular actor will be pivotal in putting a coalition over the threshold to get a winning majority, based upon the assumption that coalitions form in random order.11 Consider a simple situation in which actor A has three votes, and four other actors each have one vote apiece. Under simple majority rule, i.e., with what is called a quota of 4,12 in this example, the largest actor is decisive whenever the four less powerful actors are split evenly two to two or are split three to one or one to three. Looking to the Banzhaf index, of the thirty-two possible combinations of pro-con preferences among those five voters, 12 are split 2/2, another 8 6 are split 3/1, and still another 8 are split 1/3. So actor A is decisive in 28 combinations. Actor B is only decisive when the other actors are split all against the big actor or the big actor against all of them; this occurs in only 4 of the 32 combinations. By symmetry, the situations of C, and D and E are identical to that of B. So, the power scores are proportional to 28, 4 ,4 ,4 ,4; normalized to sum to one, they are approximately (.64, .09,.09,.09,.09). Turning now to calculations of Shapley-Shubik values, we see that, in the example above, there are 120 permutations of the 5 actors. A is pivotal in 72 of them. Each of the other actors are pivotal in 12. So, the normalized Shapley-Shubik power scores are (.60, .10,.10,.10,.10). Note that, although these Shapley-Shubik values are very similar to the normalized Banzhaf scores, they are not identical to them.13 Both Banzhaf and Shapley-Shubik allow for an actor to be decisive /pivotal in winning coalitions that are more than minimal winning. Consider a game with four voters, A, B, C and D, with weights 1, 2, 3 and 4, respectively, in a game with a quota of 6 votes . Looking at Banzhaf scores, we see that for the vector (1, 0, 1, 1), i.e., for the situation where A, C and D are voting yes and B is voting no, Banzhaf will “count” both C and D as decisive voters since their change of vote would change the outcome. But, because B is not decisive in this situation, {A, C, D} is not a minimal winning coalition. Turning to Shapley-Shubik scores for this same game, we see that, if the permutation is ACDB, then D is “counted” as pivotal, while if the permutation is ADCB, then C is “counted” as pivotal, despite the fact that the coalition {A, C, D} is not minimal winning. Other related approaches, however, constrain in some fashion the set of coalitions within which an actor’s decisiveness or pivotality is to be assessed, e.g., by positing 7 that only minimal winning coalitions will form (Deegan and Packel, 1976; cf. Fishburn and Brams, 1996).14 The Banzhaf-Coleman and Shapley-Shubik and related approaches are often described as a priori power indexes. These types of power indices have been criticized for being acontextual, in that they ignore any information about the relationships among the actors or the interest they have in common which might affect the likelihoods that they would (choose to) be the same side of an issue.15 Recent work has attempted to build power indices that take into account expected coalitional propensities, such as work by Guillermo Owen and colleagues on games with a priori unions (Owen, 1982; Carreras and Owen, 1988); games with permission structures (e.g., Gillies, Owen and van den Brink, 1992) and games on graphs (Grofman and Owen, 1982; Owen, 1996). When voter preferences can be thought of as embedded in an n-dimensional issue space, Owen’s interest in more realistic approaches to voting power has led to the Shapley-Owen index for spatially embedded voting games (Owen and Shapley, 1989), first applied in Grofman, Owen, Noviello and Glazer (1987: see also Feld and Grofman, 1990).16 For the case where each separating median line/median hyperplane is equally likely to be chosen, the Shapley-Owen index for the ith voter measures the proportion of all yes-no votes on which that voter will be pivotal.17 While we recognize the utility of such approaches, we take a different tack by presenting a methodology that can be viewed as an alternative to any use of power indices, a priori or not. The basic difference between our approach and that of the power index approach is that our approach is what we will call clone independent. Virtually all of the usual power score 8 approaches can be characterized in terms of an urn model, i.e., a probability distribution across coalitions. In these approaches actors estimate their value based on the likelihood that they will be part of a coalition in which they find themselves decisive (or pivotal). Imagine coalitions being drawn at random from some urn which contains a particular set of (perhaps ordered) coalitions, perhaps with some coalitions appearing more than once, and where we calculate the proportions of such draws in which each actor is pivotal or decisive to determine that actor’s score. Exactly what coalitions we regard as feasible, i.e., which coalitions we put in the urn, and whether we look at ordered or non-ordered coalitions, will determine the power scores. Clearly, by modifying our assumptions about the set of feasible coalitions (e.g., restricting them in some a priori way, such as limiting ourselves to connected coalitions,18 or minimal winning coalitions, etc.) we can change the power scores of the actors -- and generate “new” power indices. Now imagine that we add to that urn a duplicate of some coalition. If the power scores calculations change as a result of the addition of that duplicated coalition, then we will call the power index clone dependent.19 All of the standard power scores approaches of which we are aware are clone dependent. The problem with clone dependence, as we see it, is that the value of an actor is assumed to be a function of how likely it is that a coalition in which that actor is decisive or pivotal will be found in the urn. Yet, if we are looking to solve a set of simultaneous equations to find some set of parameters for each of the variables consistent with the conditions implied by the equational constraints, duplicating one (or more) of these equations does not change the solution. We argue that power score calculations should often be more like solving a set of simultaneous 9 equations involving the values of different winning coalitions to search for a set of scores that are consistent with all these equations, than like drawing from an urn. If actors (both briber and potential bribees) can derive common expectations about what each actor should be worth by examining and comparing the roles each actor plays in each of the set of the set of feasible (minimal) winning coalitions, then that expectation should become the value of the offered (and accepted) bribe. A closely related idea is stated by Maschler (1992) when he asserts that the AumannMaschler bargaining model and related approaches (the kernel and the nucleolus) assume that a given coalition has been formed, and then ask how the members of that coalition will bargain over the spoils that coalition can gain.20 Here, however, rather than assuming bargaining among the members of an already formed coalition, we posit an external entrepreneur making bids to individuals with the aim of assembling a winning coalition, and seeking to minimize the total amount of bribes s/he pays out by paying potential members of some winning coalition no more than s/he is “worth” -- in a setting where the structure of the weighted game is such that one can usually replace that member with some one or more others to achieve a different coalition that will also have a majority sufficient for winning.21 How to model power is, we believe, very much dependent upon the specific assumptions we make about the nature of the bargaining game being played. There is thus, in our view, no a priori right answer to which approach to measuring power is best.22 Nonetheless, in many real world situations, we believe our “market value” approach makes more sense than the usual approaches, and it can be generalized so as to incorporate one key feature of the world, the 10 existence of transaction costs, in a way that does not seem to be true for the more standard approaches to measuring power. Homogeneous Games To understand the nature of the approach we propose, we begin by considering some uncomplicated weighted voting games in which we can precisely calculate what we will henceforth call bribeworthiness, or alternatively, the market value of a vote. Perhaps the most straightforward case occurs when we have what are called homogeneous weights, i.e., a weighted voting game in which all minimal winning coalitions (henceforth abbreviated MWCs) have equal total weights (Zwicker and Taylor, 1997: 51). The five voter example in the previous section, where A is given a weight of 3, and the other four actors a weight of 1, with a quota of four, is a homogeneous game. The minimal winning coalitions (MWCs) are A, and any one of B, C, D and E, or the set {B, C, D, E}. Of course there are an infinite number of possible weights (plus quotas) which give rise to the same MWCs.23 Nonetheless, if we are given a game which gives rise to some set of minimal winning coalitions, it seems reasonable to represent that game in the most parsimonious way possible, and if we can find homogeneous weights to represent a game it seems reasonable to do so.24 For games with homogeneous weights, weights can be regarded as measures of each actor’s market value/bribeworthines. Moreover, even for such seemingly easy to model games, the market values so determined can be quite far from the values determined by indices such as those of 11 Banzhaf or Shapley-Shubik. We begin with an analysis of simple majority games. In such games, neglecting ties, winning coalitions and coalitions with power to block winning outcomes are one and the same. Conceptualizing Bribeworthiness for Weighted Majority Rule Voting Games with Homogeneous Weights Our model assumes that the primary value of votes is their value in exchange. The question we examine is how much each voter’s share of weights is worth to an outsider who would purchase a collective decision. For a majority rule game, purchasing a collective decision requires purchasing a majority vote. Since it is reasonable to assume that the purchaser will not pay any more than necessary, it seems reasonable that he will purchase no more than a minimal winning coalition. Furthermore, since purchasers are willing to pay the same amount for any minimal coalition, if actors are otherwise indifferent among alternatives, the value of all minimal winning coalitions should converge towards having the same total price. For any weighted majority rule voting game, if we could set up a series of linear equations in which the sum of the values of the actors in each minimal winning coalition is equal to the sum of the values in every other minimal winning coalition, and we normalize those values to sum to one, and that set of simultaneous equations has a unique solution, then the values so arrived at would give us a natural way to establish the relative market values (bribeworthiness) of each actor. For homogenous games, we have a nice result. 12 Proposition 1: If a weighted majority rule voting game is homogeneous and decisive,25 then this set of linear equations involving minimal winning coalitions26 has a solution in which the market value (bribeworthiness) of any non-dummy actor is proportional to that actor’s weight.27 For games with homogeneous weights we can thus arrive at a notion of what we shall call fungible power, i.e., a situation in which weight and power are the same. We will illustrate this proposition with the previous example, where A has three votes and each of the other four voters has a single vote, and thus each minimal coalition has exactly four votes. Our approach focuses attention on the fact that the three votes of actor A are directly substitutable for the votes of any three other actors, and that should be reflected in their power scores -- at least if we neglect transaction costs.28 But, then, by symmetry, the actor whose share is three votes should have market power equal to the total of any other three voters. If values are normalized, then the market value scores are (.430, .143, .143, .143, .143). 13 Note that these weights are substantially different from what we got from the two standard power scores: the Banzhaf values of (.64, .09, .09, .09, .09), and the Shapley-Shubik power scores of (.60, .10, .10, .10, .10). Unlike the usual power score approach we do not assume that an actor who is pivotal in many coalitions will be more heavily bribed. Rather, since only one coalition will actually form, we ask what, in that coalition, is a reasonable "bribe" for that actor, given that all actors in the winning coalition will have to be paid "what they are worth.” Note also that the method we have initially used to assessing market value makes no assumptions whatsoever about the likelihoods of particular coalitions, or about the extent of common interests among actors. Indeed, in the calculations we gave, it turns out not to matter how many different coalitions an actor is decisive or pivotal in. Rather, what matters is her imputed market power relative to set of coalitions in which s/he might find herself. 14 Unfortunately, there are sets of weights for which there is no functionally equivalent homogeneous set of weights.29 For example, consider a weighted voting game in which A, B, C, D, E, F have, respectively, weights 6, 5, 4, 3, 2, 1, totaling 21, with a vote quota of 11 votes. It can be readily determined that these weights are not homogeneous. ACD is a minimal winning coalition with 13 votes, and AB is another MWC, but it has 11 votes. Since all MWCs do not have the same total votes, the set of weights is not homogeneous. With further analysis, it is also apparent that there does not exist any homogeneous set of weights that is functionally equivalent to these weights in terms of giving rise to the same set of minimal winning coalitions. To see this, we simply observe that if all minimal winning coalitions in the game above would have the same totals, then i.a., the weight of ACE would have to equal that of ADE, and therefore the weight of C would have to equal that of D. On the other hand, BCE is a minimal winning coalition, but BDE is not-- so C and D can never have the same weights under this decision rule. This is sufficient to show that there can be no homogeneous set of weights for this game. We can generalize this intuition as follows. Proposition 2: Market values are uniquely defined only for games that can be represented as ones with homogeneous weights.30 We can illustrate this proposition using the weighted majority rule voting game example discussed immediately above. The minimum winning coalitions are {A, B}, {A, C, D}, {A, C, E}, {A, C, F}, {B, C, D}, (A, D, E}, {B, D, E, F}. If we equate the values of the members of 15 each of these minimal winning coalitions to one another, we get the set of equations below: A + B = A + C + D = A + C + E = A + C + F = B + C + D = A + D + E = B + D + E + F. Normalizing, we also require: A + B + C + D + E = 1. However, it is easy to see that there is no consistent solution to this set of equations.31 The fact that most games cannot be represented as game with homogeneous weights might seem to be a major limitation of our approach. But we will show how we can extend our notion of fungible power scores beyond the case of games with homogeneous weights for the case of games where a set of constraining equations does not have a solution, by considering the best approximation to homogeneous weights. Conceptualizing Bribeworthiness for Weighted Majority Rule Voting Games without Homogeneous Weights by Calculating Approximate Market Values We previously suggested that all minimal winning coalitions should have the same total value, because an outsider would not pay more for any one minimal winning coalition than for any other. But we would argue that, even if there is no way to make the values exactly equal, then there will still be a tendency towards making the values of the minimal winning coalitions as similar as possible. Specifically, we suggest that the set of values that minimizes the range of values (i.e. highest minus lowest) will determine what we might call the approximate fungible power scores, or approximate market values. Determining precise bounds for the set of weights with minimal discrepancies is a complex linear programming problem, but we have been able to estimate these power scores by using 16 successive approximations.32 This process seems to work well. Starting with an arbitrary set of weights, we can generally find a new set of weights that reduce the difference in weights of the largest and smallest minimum winning coalition; but Abetter@ weights become more elusive as we seek to improve further.33 Let us again consider the system of equations above, for a game with weights of 6, 5, 4, 3, 2, and 1, respectively, with a majority quota of 11, now with the aim of generating a "plausible" set of weights. <<Table 1 about here>> While both our approximate market values and the Banzhaf scores recognize the equivalence of the actors initially given weights of 2 and 3, there are big differences in the relative weights of some of the other actors. Specifically, our value for the actor with smallest weights is more than twice her Banzhaf score, and our value for the actor with the fourth-highest weighting is much less than her Banzhaf score. As a consequence, the ratio between the fourth actor and the first is 2:1 for our approach, compared with 5:1 for the Banzhaf scores.34 The ideas discussed above can be extended to weighted voting games with quotas higher than simple majority. Approximately Fungible Power in Qualified Majority Rule in Non-Homogeneous Weighted Voting Games Note that our discussion above applies to Amajority rule;@ i.e., where either a coalition is 17 winning or its complement is winning. These ideas can readily be extended to Aqualified majority rule@ (i.e. involving higher than majority quotas). A qualified majority rule is homogeneous when all minimal winning coalitions have the same values as one another and all minimal blocking coalitions have the same value as one another. Such sets of weights, if they exist, are unique up to a scalar multiplier, just as for majority rule. For homogeneous qualified majority rule weighted voting games it seems straightforward to use the weights as market values. When a qualified majority rule game does not possess a representation in terms of homogeneous weights then, just as with majority rule games, there is a discrepancy between the largest and smallest MWC, but there is also a discrepancy between the smallest and largest minimum blocking coalition (MBC). We would now suggest that, for qualified majority rule games, approximately fungible power or approximate market share values be defined as the weights that minimize the larger of these two discrepancies. Market-Based Power in the Limit Although, we can readily find hypothetical (or even actual) examples in which the best estimates of market share values give us minimal winning coalitions of substantially unequal weight, in practice, often the “optimal” weights/approximate market share values generated by our computer program will produce small differences between the weight share of the largest and the smallest MWC, even when the size of the weighted voting body is relatively small.. Moreover, for very large voting bodies, in general, we expect the differences between the weight share of the largest and the smallest MWC to be very small, indeed. 18 . Below we provide an informal proof sketch suggesting that, as long as there is a substantial degree of (reasonably continuous) variation in the distribution of weights within a committee, as committee size grows larger, the difference between estimated bribe-worthiness as we calculate it and the original weights. Proposition 3: As long as there is a substantial degree of (reasonably continuous) variation in the distribution of weights within a committee, as committee size grows larger, bribe-worthiness will tend to approximate the committee weights. Informal proof sketch: If the committee weights gave rise to a homogeneous game, we have already shown that bribe-worthiness and weight would be identical. In that situation, there exists a unique set of normalized weights to represent the game. In general, games will be nonhomogeneous, and thus there will not be a unique set of weights to represent the game. However, if the range of variation in committee weights is large but the distribution of weight options is also “smooth,” then we expect that all minimal winning coalitions (or MBCs, when those are relevant) will be “roughly” the same size. Thus, we may approximate the best-fitting set of bribe-worthiness scores with the weights themselves. In general, we expect that. for a fixed distribution, the larger the number of elements whose (appropriately normalized) sum we examine, the more likely it is that the largest sum and the smallest sum will be very close to one another, since the standard deviation of the sampling distribution decreases with the square root of n. Indeed, for large enough sample sizes (i.e., committee sizes), this discrepancy should tend 19 to some small number, which can be taken to be a measure of the degree of “continuity” in the underlying distribution of feasible weights.35 Later in the paper we present empirical results for the EU Council of Ministers about the discrepancies between the largest and smallest MWCs and MBCs to provide unequivocal evidence that problems caused for our approximation approach to market values by the existence of non-homogeneous MWCs may not be that serious in at least some real world settings. First, however, for the case of large scale games, we turn to a powerful connection between our concept of bribeworthiness and more standard approaches. Relationship Between Market-Based Power and Banzhaf (or Shapley-Shubik) Values in the Limit The central claimed justification for making use of power scores is that the weights themselves are not good indicators of a voter’s bargaining power. Yet, when we define voter bargaining power in “bribeworthiness” terms, the arguments above suggest that, for homogeneous games, power will actually be directly proportional to voter weight, and even for small non-homogenous games, power will often be approximately proportional to weight; while for large non-homogeneous games with many players, if the weight distribution is “smooth,” power will converge to be proportional to the voter weight. But we say even more than this. We can show that despite the quite disparate underpinnings of our approach and the standard power score approaches, in the limit, the two will tend to give the same answer. In other words, for 20 large-scale games, under reasonable assumptions, approximate market share values and Banzhaf scores and Shapley-Shubik values will all tend to converge toward the same answer. Proposition 4: As long as there is a substantial degree of (reasonably continuous) variation in the distribution of weights within a committee, as committee size grows larger, Banzhaf scores (and Shapley-Shubik Values) will tend to approximate the committee weights. Informal proof sketch: The same sort of argument given for Proposition 3 can be developed here. And it is already known from the work of Lindner (Lindner, 2004; Lindner and Machover, 2004) that, as we increase committee size toward infinity, Banzhaf scores and Shapley-Shubik Values will tend to approximate the committee weights if we have many votes with the same weights, such that, as we increase committee size, the number of players of any given weight concomitantly increases.36 Indeed, while the above proposition holds only in the limit, the empirical results we give in the next section for the European Council of Ministers suggest, that even for the relatively limited number of actors in the various historical Common Marker and EU weighted voting games, there is considerable concordance of weights and power scores – especially once we specify the appropriate equivalent set of weights in each case that minimizes the discrepancy between largest and smallest minimum winning coalition. In this context we would also note that 21 recent work of Gelman, Katz and Bafumi (2004: 662) finds that weights in the U.S. Electoral College majority rule weighted voting game in 2004 are very close to the a priori ShapleyShubik values for that game, while Owen (1975) found a similar result for the 1970 Electoral College. 22 III. Empirical Comparisons for Weights in the EU Council of Ministers, 1958-1995 Analysis of the actual weights for the countries in the EC indicates that the power scores, whether calculated by Banzhaf or by the market share approach in this paper, tend to be very highly correlated with the actual weights assigned to the actors, and that the fit increases over time, as the number of actors increases. We show in Table 2, for the period 1958-1995, the actual EU weights, the Banzhaf scores, and our estimate of the best-fitting market values, i.e., the weight assignments that bring the game closest to homogeneity. For each of the five years, the correlations among these three indices are shown in Table 3. <<Table 2 about here>> <<Table 3 about here>> We can show why the relationship between bribe-worthiness and weights is roughly linear by showing how close we are to situations with homogeneous weights.37 We show the differences between the sizes of minimum and maximum MWCs (minimal winning coalitions) and sizes of minimum and maximum MBCs (minimal blocking coalitions) in Table 4 for the actual weights. 23 <<Table 4 about here>> In the last row of Table 4 we show the maximum normalized difference between adjacent weights. We anticipate that this normalized maximum difference will tend toward zero as we increase the number of actors (and the effective total weight). This difference is a measure of the “smoothness” of the weight distribution. Of course, we must be careful in that the calculations we report are using the actual weights, yet we know that these weights are not unique., i.e., we know that, in each year, there exist other sets of weights that will give us the same set of MWCs and MBCs, and it is at least conceivable that for some of these sets of weights we might get quite different results about discrepancies between (normalized) minimum and maximum MWCs and MBCs. However, we can show that, in all equivalent representations of the five actual sets of voting weights in the EU Council of Ministers over the period 1958-1995, we get similar intuitions about how close to homogeneity we are, i.e., all feasible representations of these sets of MWCs and MBCs have very similar weight assignments once we normalize. However, that the market share weights and the actual weights are close does not mean they are identical. In particular, in 1981 the EU game is actually homogeneous, a fact that we do not see from the actual weights, but only when we look for weights that will bring us closer to homogeneity.38 24 IV. Discussion W e have suggested a way of assigning values to each actor in a weighted voting game that is based upon the market value of vote shares for that game. Specifically, when there is a weighted voting rule such that all minimal winning coalitions have the same total, then the weights in that rule are the power scores. Such a set of weights is called a homogeneous set of weights, and such a rule is a homogeneous rule. Any set of weights that is equivalent to a homogeneous set of weights has its power scores determined by those homogeneous weights. In general, very few weighted voting games will have homogeneous rules; however, for nonhomogeneous rules, we have proposed that the approximate market value scores are determined by the set of weights that minimizes the discrepancies between the highest total value of a MWC and the lowest value of a MWC. While those values are generally not strictly unique, we would expect that they vary within narrow bounds, especially as the number of actors grows larger. Indeed, we have conjectured that, for large numbers of actors, and for weights that are roughly speaking “continuous,” i.e., where there are few large “gaps” across the range of weights (judged relative to the size of that range), market power will come closer and closer to the weights, themselves. We have devised a computer program that finds approximations to these market values incrementally, by varying the weights and determining the discrepancies among the total values of the MWCs. When we apply this program to calculate approximate market share values for the 25 various weighted voting rules that have been in use in the European Union Council of Ministers since 1958, we find, as expected, that MWCs vary in weight, but that the discrepancies between the weights of the smallest and largest MWC are not that large relative to the quota.39 Mechanisms for Bribery We can imagine a number of different mechanisms by which bribery of the sort contemplated in this paper might be implemented, but two stand out for their intuitive plausibility. In each, the potential briber has in mind how much a favorable outcome is worth to him/her. In one, the briber offers a total bribe to some particular minimal winning coalition, and expects the members of this coalition to bargain among themselves about how to share this bribe among themselves. In such a situation, actors might assess their marginal value to coalitions in terms of the likelihood that if a given coalition fails to form because they, a pivotal member of that coalition refuse to join it, they could still expect to be a member of some other winning coalition in which they would be pivotal. While this probability can be defined in urn model terms, even in this case, we might think that actors would consider not their probability of being in a coalition in which they are decisive (or pivotal) but rather their value to the coalitions in which they are in, with equivalent (sets of) actors being paid the same bribe. This latter type of collective bargaining agreement story offers one type of rationale that has been offered for the nucleolus and related concepts (Maschler, 1992: 611).40 A second way to think of the coalition formation process makes use of ideas from the literature on sequential coalition formation (See e.g., Brams, 1972, Brams and Garrigo-Pico, 1975; Brams and Riker, 1972; Grofman, 1982; Straffin and Grofman, 1984; Grofman, 1996; Grofman, Straffin and Noviello, 1996.) Here the bribe offerer can be thought of as fishing for bribe-takers, one or more at a 26 time, and throwing them back in if they demand too large a bribe. This “fishing” continues until a winning coalition is reached.41 Robustness of Minimal Winning Coalitions Because our modeling approach is based on the idea of those with a stake in the outcome “bribing” voters to either achieve a particular outcome or to block change from the status quo, there might appear to be no reason for a briber to purchase more than a minimal winning coalition. Nonetheless, for two reasons, we might anticipate that observed voting coalitions would be larger than minimal winning. First, some unbribed voters might vote for the winning outcome even if they get no bribe. Consequently, even though actors are only willing to pay for minimal winning coalitions, the “free” votes might make the actual winning coalitions considerably larger than minimal. Of course, if there are voters who can be counted on to vote as the briber wants without being bribed, and if who they are is known to the briber, then the size of the coalition which must be bribed in order to assembly a minimal winning coalition is reduced, and we can simply do the market value calculations for only the set of remaining voters.42 .Second, however, some processes for forming minimal winning coalitions can actually result in larger than minimal outcomes. Consider, for example, what happens if we apply the sequential “fishing for minimal winning coalitions” process to the game with four voters, A, B, C and D, with weights 1, 2, 3 and 4 , respectively and a quota of 6 -- a game that we have previously considered. If the briber’s sequence of contacts is first A, then C, than D, and each accepts the bribes offered them, and no reneging on the part of either briber or bribee is possible, 27 then the winning coalition formed will be of weight 8, and will be non- minimal-winning, since member A can be deleted without affecting the winning status of the coalition.43 Taking Transaction Costs into Account The failure to consider transaction costs is, in our view, one of the chief failings of the literature on power scores and of most of the literature on intergroup bargaining.44 Not all winning coalitions are equal. Some will be much harder to put together than others because they require the concurrence of more actors. It is straightforward to expand our model to deal with transaction costs. The fundamental intuition underlying our extension is quite simple: we posit that the larger the transaction costs, the greater the discrepancy between an actor’s weight and her market value. Indeed, we find that, when transaction costs are set high enough, some actors find themselves as having no value to a briber since the costs of negotiating with them exceeds the value of their contribution to feasible minimal winning coalitions. As we increase transaction costs, it is immediately apparent that, in the market value approach, the value of actors with low weights diminishes and the value of actors with high weights rises. If we make the transaction costs high enough then only a handful of actors will have market value. Once we incorporate transaction costs, we can account for an important empirical phenomenon, the existence of strong incentives for individuals to join together in coalitions, that our model without transaction costs does not allow us to explain. If we do not allow for transaction costs, the market value approach suggests that m voters each with weight k have collectively the same market value as one voter with weight mk. Clearly, there are situations where this is not a realistic assumption. In general, ceteris paribus, we expect that situations with very large number of actors, most of them quite small, will lead to those small actors having little or no estimated market value once we take transaction costs in to 28 account. Thus, like the usual power score approaches, once we incorporate transaction costs, our approach is also able to account for the important empirically observed phenomenon of small stockholders having even less influence than their weights might suggest, and large stockholders holding all or virtually all the power (Zwiebel, 1995). Moreover, because market value is monotonic in weight it does not suffer from some of the paradoxes that other measures of power do, e.g., the possibility that giving additional weight to a voter will reduce that actor’s actual power (Saari and Sieberg, 2001). Conclusions While we regard our results as an important contribution to the ongoing debate about how best to measure power in the context of weighted voting games, we do wish to note some important limitations of what we do. First, we present results for the case where all minimal winning coalitions are taken as feasible. We do not consider, for example, the limitations imposed by spatially embedded issue proximities among actors, e.g., by requiring that only connected (or k-connected) coalitions could form.45 Second, because we are not looking at situations where actors can be thought of as having positions of their own on bills being brought up for a vote, we cannot take into account what we expect to be true in the real world, namely that the bribe you will have to pay an actor to change his vote depends not just on that actor’s weight but also on that actor’s preferences. Ceteris paribus, the further away from the actor’s own ideal point the briber wants the actor to move, the greater will be the necessary bribe.46 Third, and relatedly, we will necessarily be restricted to what Felsenthal and Machover (2001) refer to as p-power, i.e., we will not be investigating how far a given actor can shift the status quo.47 Fourth, we take the weights and 29 quotas as given. We do not seek to explore the nature of the “constitutional” bargaining game that led, say, the members of the EU, to agree to play a particular voting game henceforth. Fifth, we do not address equity issues, e.g., about what weights countries “deserve,” or about whether we should be thinking of power using countries as our units of analysis or using citizens as our units. Sixth, we restrict ourselves to results for weighted voting games in isolation from some broader political context. In particular, in applying our models to the EU Council of Ministers, we implicitly neglect bargaining between the Council and the EU Parliament and the rules defining interactions between the two. Finally, although our approach is inspired by rent-seeking, we have, in this preliminary work, limited ourselves to what should happen if there is only one potential briber. We have not considered the complexities of modeling competition for committee members’ votes among two or more bribers,48 although we suspect that the basic intuitions of our modeling would still go through. Nonetheless, what we hope to have demonstrated is that (1) We can approach power from a bribe-offering and bribe-taking perspective that is much closer to the rent-seeking literature in Public Choice than it is to the relatively apolitical stance of the classic power index approaches. (2) Realistic appreciation of the implications of different approaches requires examination of real data, not hypothetical worst case scenarios. (3) For the kinds of distribution of weights found in many real-world settings, especially as the number of actors increases, the bribe-worthiness approach and more traditional power score approaches will tend to give near identical answers. And, as long as we neglect transaction costs, those answers will tend to reproduce the weights themselves! Thus, a lot of the fuss in the literature over which definition of power is best is it 30 seems, much ado over nothing -- at least for game-situations that do not resemble oceanic games, i.e., games with lots and lots of very small players and only a handful of big ones. 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Peyton. 1978. “Power, Prices and Incomes in Voting Systems.” Mathematical Programming 14: 129-148. 45 Zwiebel, Jeffrey. “Block Investment and Partial Benefits of Corporate Control.” Review of Economic Studies, 62 (April): 161-185. 46 Table 1 Normalized Weights, Estimated Market Values, and Banzhaf Scores for an Illustrative Six Voter Major Rule Weighted Voting Game weights 1 2 3 4 5 6 size range of MWCs .095 normalized weights .048 .095 .143 .190 .238 .286 estimated market shares .074 .111 .111 .148 .259 .296 .037 Banzhaf scores .036 .107 .107 .179 .250 .321 47 Table 2 Values of (normalized) EU Weights, Estimated Market-Values, and Banzhaf Scores: 1958-1995 (a) 1958 1958 wts normalized wts normalized wts2* market value wts2* Banzhaf scores 1 2 2 4 4 4 .059 .118 .118 .235 .235 .235 0 .125 .125 .250 .250 .250 0 .141 .141 .240 .240 .240 0 .143 .143 .238 .238 .238 * dummy omitted (b) 1973 1973 wts normalized wts market value wts Banzhaf scores 2 3 3 5 5 10 10 10 10 .034 .052 .052 .086 .086 .172 .172 .172 .172 .049 .066 .066 .082 .082 .164 .164 .164 .164 .016 .066 .066 .091 .091 .167 .167 .167 .167 48 Table 2 (cont.) Values of (normalized) EU Weights, Estimated Market-Values, and Banzhaf Scores: 1958-1995 (c) 1981 1981 wts normalized wts market value wts Banzhaf scores 2 3 3 5 5 5 10 10 10 10 .032 .048 .048 .079 .079 .079 .159 .159 .159 .159 .04 .04 .04 .08 .08 .08 .16 .16 .16 .16 .041 .041 .041 .082 .082 .082 .158 .158 .158 .158 (d) 1986 1986 wts normalized wts market value wts Banzhaf scores 2 3 3 5 5 5 5 8 10 10 10 10 .026 .039 .039 .066 .066 .066 .066 .105 .132 .132 .132 .132 .026 .048 .048 .067 .067 .067 .067 .107 .126 .126 .126 .126 .029 .046 .046 .067 .067 .067 .067 .109 .129 .129 .129 .129 49 Table 2 (cont.) Values of (normalized) EU Weights, Estimated Market-Values, and Banzhaf Scores: 1958-1995 (e) 1995 1995 wts normalized wts market value wts Banzhaf scores 2 3 3 3 4 4 5 5 5 5 8 10 10 10 10 .023 .034 .034 .034 .046 .046 .057 .057 .057 .057 .092 .115 .115 .115 .115 .025 .036 .036 .036 .046 .046 .057 .057 .057 .057 .093 .114 .114 .114 .114 .023 .036 .036 .036 .048 .048 .059 .059 .059 .059 .092 .112 .112 .112 .112 50 Table 3 Correlations Between Actual Weights and Estimated Market-Values and Banzhaf Scores 1958 adjusted r2 between weights and best-fitting market-share values adjusted r2 between weights and Banzhaf scores adjusted r2 between bestfitting marketvalue shares and Banzhaf scores .941 1973 .995 1981 .996 1986 .997 1995 .999+ .953 .985 .996 .998 .999 .999 .980 .999+ .999 .999 51 Table 4 Differences between Maximum and Minimum MWCs and MBCs in the EU Council of Ministers, 1958-1995 year 1958 (n= 6) q=12/17 (.706) 4,4,4, 2,2, 1 1973 (n = 9) q=41/58 (.707) 10,10,10,10, 5,5, 3,3, 2 1981 (n = 10) q=45/63 (.714) 10,10,10,10, 5,5,5, 3,3, 2 1986 (n =12) q=54/76 (.711) 10,10,10,10, 8, 5,5,5,5, 3,3, 2 .71 .76 .30 .39 .05 1995 (n =15) q=62/87 (.712) 10,10,10,10, 8, 5,5,5,5, 4,4, 3,3,3 2 .71 .74 .30 .34 .03 vector of weights (normalized) minimum MWC (normalized) maximum MWC (normalized) minimum MBC (normalized) maximum MBC difference between (normalized) minimum and maximum MWC difference between (normalized) minimum and maximum MBC sum of difference between (normalized) minimum and maximum MWCs and MBCs maximum (normalized) difference between adjacent weights .75 .75 .375 .500 0 .71 .77 .31 .34 .06 .71 .73 .32 .33 .02 .125 .03 .01 .09 .04 .125 .09 .03 .14 .07 .12 .09 .08 .04 .03 52 Endnotes 1 Voting processes involving multicameral legislatures or executive vetoes on legislative decisions with the possibility of legislative override also often fall into the category of weighted voting games (Brams, 1975). 2 We would emphasize that we use the term “power” in a very limited way, only as it applies to weighted voting games. There are numerous other ways to think about power (see e.g., Champlin, 971; Bell, Edwards and Wagner, 1969). Thus, for example, we do not consider power in terms of ability to influence the decisions of others. 3 A good introduction to the topic of power scores is Felsenthal and Machover (1998); also see the various essays in Holler (1982) and Holler and Owen (2001). 4 Because power may not be proportional to weight, and depends upon the exact nature of the index used to calculate power, the implications of using some particular set of weights are difficult to understand. In the U.S., this has led to legal challenges to the use of weighted voting systems (Grofman and Scarrow, 1981). In fact, the U.S. Supreme Court has rejected use of power scores to measure representative fairness from the standpoint of one person, one vote (Grofman, 1981). 5 See below. 53 6 By a blocking coalition, we mean one large enough so that, without the defection of one or more of its members, no winning coalition is possible. 7 Alternatively (a la Zwicker and Taylor, 1997) we might specify a range of values for each actor that provide highly constrained bounds on the amount of the bribe each is likely to receive for her vote relative to the amount received by other actors, but we will not pursue that approach here. (See brief further discussion below.) 8 In further work not reported here we apply our market share model to the weighted voting game that will be in place in 2005, after the next EU recent enlargement is implemented. 9 Moreover, our market value approach may avoid some of the many paradoxes associated with power scores, e.g., that in which giving a member more weight actually reduces her power (Saari and Sieberg, 2001). 10 This index exists in both a normalized and a non-normalized form. The non-normalized form simply counts how many times a given player is decisive. The normalization forces the sum of voter scores to add to one. 11 See also Luce and Raiffa, 1957; Straffin, 1977; Owen, 1995a. 54 12 By the quota of a weighted voting game we mean the number of votes necessary for a coalition to be winning. 13 For certain games with very large numbers of players it can be shown that the differences between Shapley-Shubik and Banzhaf weights can be quite considerable (Owen, 1995b). 14 A minimal winning coalition is one in which the removal of any single actor will change the coalition from winning to losing. 15 They have also been defended on exactly the same grounds, by asserting that they are based on fully general assumptions and can thus provide a plausible "baseline. " While the concept of a priori power is appealing to the extent that one could use a priori power scores as building blocks in a larger model that then incorporates key aspects of the social context, we are highly skeptical of this argument in favor of an a prioristic approach. When we do have contextual information, it replaces rather than builds upon the calculations based on a prioristic assumptions. Consider contextual information on the likelihood of agreement between actors and/or their likelihoods of coalescing. If one knew that actor A and actor B agreed 75% of the time, instead of the 50% assumed in the Banzhaf calculations, then one would need to recalculate power scores from scratch based upon the new assumptions. There is no way to take the original power scores, combine them with the contextual information, and output a posteriori power 55 scores. Similarly for the Shapley-Shubik measures. Indeed, rather than regarding a prioristic measures as acontextual and general, it is at least as sensible to regard them as simply being based upon inherently implausible assumptions about the political context. 16 For reasons we do not understand, despite the recent interest in viewing political coalitions within an ideological and issue framework to estimate voting power, as far as we are aware, the Shapley-Owen index, directly intended for this purpose, has not been made use of by authors other than Owen and his collaborators. 17 By making other assumptions about the nature of the location of the bills the committee can be expected to vote upon relative to the status quo for each, alternative measures of decisiveness can be generated. 18 Connected coalitions are ones where, if actors can be arrayed along a single policy or issue or ideological or other dimension in terms of their “ideal points” for most preferred outcomes on the continuum, then, when any two actors are found in a coalition, any actors intermediate in ideal points on the line segment joining the two actors must also be in the coalition. Grofman (1982) has generalized the concept of connectedness for situations where actors may be thought of as embedded in a multidimensional policy/issue space, what he refers to as k-connectedness. 19 We believe that, in this context, the application of the idea of clone independence is original 56 with the present authors, but we may be in error about this claim. 20 See also Maschler, Peleg and Shapley (1979). 21 Our approach is thus analogous to identifying a market clearing price structure. 22 This assertion parallels a point made by Bossert, Brams and Kilgour (2002: 185) that it is quite reasonable for cooperative and non-cooperative game theoretic approaches to give different answers in many situations since the assumptions of each approach can be quite different. Thus, the search for a single model (or a single approach) can be misguided. A related point is made by Wuffle (1999) who asserts that, for any given substantive topic, T, there is no such thing as “the rational model” of T, only “a rational choice model” of T. The devil is in the details. 23 For example, if we assigned weights of one 9 and four 4s, or weights of one15 and four 4s, we get the same MWCs as previously. These weighted voting games all allow the “big guy” and one “little guy: or all four little guys to make a decision. Yet, these relative weights are very different from one and different from our original representation. But more importantly for present purposes, these two later representations do not have the property that all minimal winning coalitions have the same total value. With one 9 and four 4s, the mixed coalition adds up to 13, while the little guy coalition adds to 16. With one 15 and four 4s, the mixed coalition adds to 19, and the little guy coalition adds to 16. So, these weights do not meet our condition. 57 24 Moreover, for a game with homogeneous weights, if we either assume no dummies or that all dummies have a weight of zero, then a game with weights that are exact multiples of those weights also gives rise to the same set of MWCs, i.e., the weighted voted game with A given a weight of 3k and with four actors with weights of 1k, with a quota of 4k, will generate the same set of MWCs as the homogeneous game with weights of (3, 1, 1, 1, 1) with a quota of 4. 25 A game is decisive if the complement of any losing coalition is winning. 26 If we focus on minimal winning coalitions we are implicitly excluding dummies, who would be assigned a value of zero. 27 This result follows from results in Gurk and Isbell (1959). The solution to this set of equations gives us, i.a., the nucleolus of the game. We are indebted to Guillermo Owen (personal communication, 2004) for calling the Gurk and Isbell reference to our attention. 28 See discussion of the complications caused by taking transaction costs into account later in the text. 29 Elsewhere we have generated some theorems about conditions sufficient to guarantee the existence of a set of homogeneous weights . These results are available upon request from the authors. . 58 30 This result follows from results in Gurk and Isbell (1959). 31 Even if we have homogeneous weights, this does not guarantee that the solution to the relevant system of equations will be unique when the game is not decisive. Consider the homogeneous four voter game with weights 1, 2, 2 and 3 and a quota of 5 votes. To assign consistent weights so that each minimum winning coalition gets the same payoff we must have B = C and D = A + B. One possibility is (1/8, ¼, ¼, 3/8), but another is (0, 1/3, 1/3. 1/3). And, there are an infinite number of other feasible weight assignments. We are indebted to Guillermo Owen (personal communication, 2004) for calling this example to our attention. In this example, the second solution given is the nucleolus (see Maschler, 1992). 32 The computer program we use to calculate optimal weights automatically excludes dummies. While our estimates of optimal weights are not the same with and without dummies, given our ideas about the bribing process as resulting in zero values for dummies, we believe the calculations with the dummies excluded are the correct ones. 33 Alternatively, we could ask, for each actor, what the maximal and minimal values that might be expected from the coalition structure of the weighted voting game. This notion of a range of market shares is found in Taylor and Zwicker (1997), under the name market interval, an approach whose inspiration they attribute to Young (1978) Usually (but not always) market intervals will give us reasonable (and sometimes even reasonably tight) bounds on the range of bribes that an actor might be 59 expected to demand and have some chance of getting what s/he asked for. The intuition Taylor and Zwicker (1997: 26-27) give for their market interval calculations (is a formal representation in terms of an n-stage reasoning process of an argument quite close to that given in the text below in which actors calculate reasonable values for themselves in the context of their expectations about what other actors will regard as reasonable payoffs for themselves. Indeed, the mathematical connection between the two approaches is so close that, for homogeneous games, the player values that form what Taylor and Zwicker (1997: 50-52) call a bribery equilibrium appear such that each will be within kε of our (approximate) market share values, where k is some (small) integer and ε is near zero; while for nonhomogeneous games the market weight intervals they calculate will, we conjecture, include the estimated values we give for approximate market share values based on our optimization algorithm (see Taylor and Zwicker, 1997: 52). 34 If the Banzhaf scores shown in Table 1 were themselves to used as weights, they would describe a slightly different game from the original game; i.e. they do not yield the same set of MWCs. In the original game, {5,3,2,1} is a MWC, but the Banzhaf scores for these actors add to exactly .50, not enough for the quota. A slight modification of these scores (increasing the value for the 1 and decreasing it for the 4) reproduces the original game. When that modification is made, the range in values of the MWCs is more than .10 for these values, which is greater than for the original normalized weights. 35 We know that the set of scores with minimal discrepancies must form a compact connected convex 60 region in the space. It will be useful for future work to set limits on how small this convex region will be. If it is very small, the weights may practically be treated as if it they fall at a unique point. Our investigation of the degree of deviations from homogeneity for non-homogeneous weights suggests that the bribe-worthiness scores we calculate using our optimization program are probably not unique, but that the different sets of scores with identical minimal ranges are likely to be very close together. Analytical work remains to be done to make more precise this Anear uniqueness@ result Taylor and Zwicker (1997: 47-50) look at what appears to us to be a related problem, and show that there exists a limit equilibrium of their market value intervals to a set of point values. 36 We are indebted to Guillermo Owen (personal communication, 2004) for calling this work to our attention. 37 However, we will also get very high correlations when the set of weights can be partitioned into two subsets with high variance between the subsets and low variance within each subset. This condition, or something close to it, obtains for the EU data we have been looking at in this paper. 38 The question of sensitivity to alternative weight assignments is a potentially important issue. Elsewhere we have developed some theorematic results about the maximum adjustments to the elements of a given set of weights that are possible without changing the nature of the MWCs and MBCs in the new game. We show that, for certain not unreasonable assumptions about the smoothness of the weights, we can expect that, as the number of actors increases, the calculations we get about the differences between minimum and maximum minimal winning and minimal blocking coalitions will be 61 insensitive to which of the possible weight assignments we use, since the maximum feasible differences should shrink toward zero. 39 Closely related empirical findings are given in Pajala (2004). 40 If bribery is to be of a given coalition, we might think of this as analogous to purchasing an entire meal from a prix fixe menu with an already specified set of items that can be purchased as a package, and with some items found in more than one package. 41 We may think of this piscatorial perspective as treating bribery more like choice from an a la carte menu than from a pre-specified package of entries. 42 We can thus neglect “fixed votes” since they do not affect the fundamental structure of our results. 43 We might think, though, for transaction costs reduction reasons, if no other, that potential bribers would begin by trying to bribe the more highly weighted actors. See later discussion. 44 An observation along these lines about the limitations of existing bargaining models is made by Maschler (1992: 616) who observes that ”‘if time is money,’ communication barriers should perhaps enter the construction of the characteristic function.” 45 In one dimension, as noted earlier, a connected coalition is one where any coalition containing 62 any two actors must also contain all actors located “between” them on the continuum (Axelrod, 1970). A k-connected coalition is one where any coalition defining a k-dimensional simplex must contain all actors within that simplex (Grofman, 1982). 46 For modeling along those lines see Snyder (1981); cf. Dal Bo (2002). 47 Felsenthal and Machover (2001) distinguish between p-power and i-power, with the former measured by the traditional indices or variants thereof, but the latter requiring new indices intended to measure how influential an actor potentially could be in affecting/changing the policy choices of a coalition in which s/he may find herself. We can model the differential costs born by actors in terms of a simple unidimensional model in which the further the outcome is from the voter's own ideal point the more s/he must be bribed to vote opposite to her own preferences. This allows us to modify r our bribe-oriented approach in a way that is similar in spirit to the recent work on power indices that explicitly take into account (ideological) proximities among the actors. We hope to pursue an extension of our model to the case of ipower in future work 48 See e.g., Snyder, 1991; Groseclose, 1996; Groseclose and Snyder, 1996, 2000; Banks, 2000. 63