**Funded by the European Commission (PERG-GA-2008-230879) and OTKA (Hungarian Fund for Scientific Research, NF 72610**

**László Á. Kóczy**(Principal Investigator)**Amandine Ghintran**(Postdoctoral researcher, since 1-6-2010)**Fabien Lange**(Postdoctoral researcher)**Morgane Tanvé**(Postdoctoral researcher, since 22-1-2010)

- Lange F., Kóczy L. Á. (2013): Power indices expressed in terms of minimal winning coalitions, Social Choice and Welfare 41, 281-292.
- Kóczy L. Á. (2012): Beyond Lisbon: Demographic trends and voting power in the European Union Council of Ministers, Mathematical Social Sciences 63, 152-158.
- Csóka, P., Herings, P.J.J., Kóczy L. Á. (2011): Balancedness Conditions for Exact Games, Mathematical Methods of Operational Research 74, 41-52.
- Péter Csóka, P. Jean-Jacques Herings, László Á. Kóczy, Miklós Pintér:
, European Journal of Operational Research, 2010*Convex and Exact Games with Non-transferable Utility* - László Á. Kóczy:
, Group Decision and Negotiation, 19 (2010), 267-277 (see Working Paper)*Strategic Aspects of the 1995 and 2004 EU Enlargements* - Fabien Lange and Michel Grabisch:
, Discrete Mathematics, 309 (2009), 4037-4048., 2009*The interaction transform for functions on lattices* - Péter Csóka, P. Jean-Jacques Herings, László Á. Kóczy:
, Games and Economic Behavior, 67, No 1. 266-276, 2009*Stable Allocations of Risk* - László Á. Kóczy, Martin Strobel:
, Scientometrics, 2009*The Invariant Method can be Manipulated* - Fabien Lange and Michel Grabisch:
, Mathematical Social Sciences, 2009 (see Working Paper)*Values on regular games under Kirchhoff’s laws* - Kóczy Á. László:
, Közgazdasági Szemle, 56, No. 5. 422-442, 2009 (see Working Paper)*Központi felvételi rendszerek. Taktikázás és stabilitás* - Kóczy Á. László:
, Studies in Computational Intelligence 243. Towards Intelligent Engineering and Information Technology, Rudas, Imre J.; Fodor, János; Kacprzyk, Janusz (Eds.), 2009 (see Working Paper)*Measuring Voting Power: The paradox of New Members vs. The Null Player Axiom* - László Á. Kóczy:
, Games and Economic Behavior, 66, No 1. 559-565, 2009*Sequential coalition formation and the core in the presence of externalities*

- Fabien Lange, László Á. Kóczy:
, Keleti Faculty of Economics, Working Paper Series 1002, 2010*Power indices expressed in terms of minimal winning coalitions* - László Á. Kóczy:
, Keleti Faculty of Economics, Working Paper Series 1001, 2010*Voting games with endogenously infeasible coalitions* - László Á. Kóczy:
, Keleti Faculty of Economics, Working Paper Series 0905, 2009*Stationary consistent equilibrium coalition structures constitute the recursive core* - László Á. Kóczy, Alexandru Nichifor, Martin Strobel:
, Keleti Faculty of Economics, Working Paper Series 0902, 2009*Article length bias in journal rankings* - Julien Reynaud, Fabien Lange, Lukasz Gatarek:
, Keleti Faculty of Economics Working Paper Series 0808, 2008*Proximity in coalition building*

- Kóczy Á. László:
, manuscript, 2009*A magyarországi felvételi rendszerek sajátosságai* - László Á. Kóczy:
, manuscript, 2009*Stategic Power Indices: Quarrelling in Coalitions* - László Á. Kóczy, Martin Strobel:
, manuscript, 2009*The Ranking of Economics Journals by a Tournament Method* - Fabien Lange and Michel Grabisch:
, manuscript, 2009*New Axiomatizations of the Shapley interaction index for bi-capacities* - Fabien Lange and László Á. Kóczy:
, manuscript, 2009*Power indices expressed in terms of minimal winning coalitions* - Fabien Lange:
, manuscript, 2009*Random walks on set systems and games with communication graphs* - László Á. Kóczy, Martin Strobel:
, manuscript, 2008*Academic Journal Assessment by Citation Tournaments* - Péter Csóka, P. Jean-Jacques Herings, László Á. Kóczy:
, manuscript, 2008*Balancedness Conditions for Exact Games* - Fabien Lange:
, manuscript, 2009*Random walks on set systems and games with communication graphs*

The recent extension of the European Union and in particular the introduction of new voting weights in the European Union Council of Ministers has prompted a renewed discussion of power in the European Union and voting power in general. Countries wanted to be sure that their power does not decrease as a result of the new entrants more than necessary. New entrants, like Hungary were especially eager to see the new allotment of votes. It is well known, however, that voting power does not equal to voting shares (for an example consider a two-party parliament with 75%-25% division of the votes: clearly the first party will have all the power) and what is the power of voters is then another question that the scientific literature offers a number of answers to. The large number of answers already hints that the issue is not simple and the discussion is far from over. The unprecedented number of publications around the issue (An incomplete list: Hosli, 1993; Johnston, 1995; Garrett and Tsebelis, 1999; Holler and Widgrén, 1999; Felsenthal and Machover, 2001; Laruelle and Widgrén, 2004; Nurmi and Meskanen, 2004; Berg, 2004) shows both the theoretical and the political interest in the topic. Yet, not all are in favour of the indices used or in fact at all using power indices. In this project we provide solutions to some of the criticisms, best summarised by Albert (2003).

**Voting power and game theory. **

Since its break-in into economics, game theory occupies a more and more prominent role. Economists were happy to embrace a mathematical model that helped them to turn the field into a formal science. In this process the story often gets blurred and it is questionable how much is left of the original ideas in the application's application. Game theory deals with rational players who act strategically to maximise their utility. While power indices originate from an application of the Shapley-value (one of the best known concepts in cooperative game theory) to simple games (Shapley and Shubik, 1954), the indices are no more than statistical measures of random voting by the players. In the first part of the project we expand these voting situations in a way that gives voters some control over their power. We study properties of strategic power indices and apply them to the aforementioned application of the Council of Ministers. The generalisation of the idea to values, such as the Shapley value is straightforward, how the usual tale about voters arriving randomly goes through is yet to be seen.

**Voting with preferences. **

Unfortunately the applicability of the model is limited by the fact that power indices are a priori measures of voting power (Felsenthal and Machover, 2004). They measure voting power before the issues to be voted upon or the preferences of players are known. These conditions are not satisfied in many of the more common voting situations, for instance in national parliaments, where the ideology or political agenda of the parties is well-known, making certain coalitions more, others less likely. This criticism is not new, but spatial voting provides an answer (Hinich, 1976; Holler and Widgrén, 1999; Napel and Widgrén, 2002). In this model players are represented by vectors expressing their positions on a number of issues represented by the dimensions of the vectorspace. We also assume that players can form coalitions, but only convex coalitions are possible. For instance, if the space is simply a line, with the only dimension expressing the position in terms of left or right, only connected coalitions are permitted: we assume that players are more willing to form coalitions with players that are closer to them. The given geometry of the voters puts a restriction on the possible sets of coalitions and therefore also on winning coalitions, but it is perfectly possible to define power indices on this system, too.

Bilbao et al. (1998) have calculated power measures for games on convex geometries. We intend to generalise our strategic model to such games, too and it will be particularly interesting if we also allow players to choose or change their location in space. We expect results supporting the observation that parties crowd in the middle: the difference between programs of central-left and central-right parties is marginal.

Communication networks is a further generalisation of this model. Two players who do not communicate can only form a coalition if there are other players via whom they can.

**The power of the minority **

Another problem with power indices is their black-or-white treatment of coalitions: a coalition is winning or loosing. According to this a coalition controlling 51 seats out of 100 is just as winning as the one controlling 99. If these numbers are voting weights and it is the prime minister and the leader of the opposition who vote, the statement is probably correct. In many of the applications of this theory, the voters are not machines, but 51 actual people sitting there and pressing buttons and then it is clear why would the leader of the first coalition feel its position weaker. Some of the voters supporting its coalition might simply be ill or, worse, follow their personal views rather than party directives and occasionally vote against their own party. Probabilistic models of voting are known (Palfrey, 2002), but they deal with the weighted voters (that is, the parties) rather than the members of the parliament. We believe our approach is original. It would result voting games that are not simple, that is, where winning coalitions can have different values. Moreover, the difference between a winning and losing coalition is blurred and the transition is continuous (a coalition with 50 seats having half of the power).

This model can also be studied empirically. Prof. Aleskerov of Moscow has a detailed dataset of voting in the Duma that he already offered to make available in case I can work on this topic. Similar data are likely to be available for other national parliaments, too.

My main goal is to assess the robustness of key concepts in the measurement of voting power and to apply this to actual problems, such as the qualified majority voting in the European Union Council of Ministers. Game theory, as a means to solve economic problems has been widely recognised (among others by the 1994 and 2005 Nobel Memorial Prizes in Economics) and many of the highly regarded economics journals welcome publications from this field even if the contents are mostly theoretical. As a consequence I expect to be able to publish the findings in journals en par with my existing publications.

In the following I elaborate on the three subprojects and outline the timing of the proposed work.

**Introduction**

The first power measure that we know of is due to Penrose (1946), but the idea did not really catch on and very much the same concept was reinvented twice by Banzhaf (1965) and Coleman (1971). Measuring voting power is often seen as a subfield in cooperative game theory because the first index that got critical acclaim is the application of the Shapley value (Shapley, 1953) to simple (TU-) games: the Shapley-Shubik index (Shapley and Shubik, 1954). While these original models have since been enriched in many ways, classical game theory deals with rational agents who maximise their payoff choosing certain strategies. In cooperative games, however these strategies are implicit, and the payoff is also given as for instance coalitional payoffs that the coalition must share then. The Shapley value calculates the expected marginal contribution of players. Assuming superadditivity, in particular that the addition of a player will increase a coalition's worth by more than what the player could get on its own, the story behind this value is that players arrive randomly and receive their contribution to the coalition's payoff. The generalisation to voting games is straightforward: Here losing coalitions are worth nothing, while winning coalitions get a payoff of 1. The player who turns the losing coalition into a winning one, gets the credit of winning. The more often a player is pivotal, the higher its power. Contrary to this the Banzhaf index assumes that coalitions form with equal probability and there are several variants of it: absolute and normalized measures, the Banzhaf index and several power indices that introduce modifications on how coalitions and pivotal players are taken into account (Johnston, 1978; Holler and Packel, 1983; Deegan and Packel, 1978).

**Quarrelling**

Now, once again, we must emphasise that game theory deals with rational, payoff maximising players. While with the Shapley value the idea of "fairness" is still acceptable; we may say that a player should get its marginal contribution, if a power index shows a player's power it is only a correct and sensible measure of power if the player cannot manipulate it. Unfortunately power indices do not pass this test. Since Kilgour (1974) we know that the Paradox of Quarrelling Members might occur (Brams, 2003): Two players may benefit from quarrelling, that is, from refusing to form coalitions where the other is also present. As we have said the strategies in cooperative games are implicit and are represented by the coalitions formed. Then of course a player cannot be forced to participate in a coalition, even if that is a winning one. By refusing to participate in coalitions a player can increase its power, it is in its “power" to have more power.

Observe that the Shapley value is also subject to such manipulations, but on the other hand for instance the core is not: by refusing certain coalitions, a player can only be worse off in this case.

Elaborating on this idea we introduce strategic power indices by extending the game with strategies for the players by which they have some influence on the coalitions they are willing to join. There are several possibilities to implement this idea and in the project we will investigate these possibilities, discuss the different model's features, pros and cons. In general we consider a two-stage game with a noncooperative choice of acceptable coalitions and a second, implicit cooperative stage where the powers are determined. As the noncooperative game might have multiple (Nash) equilibria even in pure strategies we must look at possible selections of such equilibria. Our aim is to select an equilibrium that is unique and therefore allows us to have a well-defined strategic power index and on the other hand coincides with standard power indices in games where strategic considerations do not play a role - and we believe there will still be many such games, perhaps even the majority of voting games will belong to this group, at least for some power indices.

**A model**

In the following we briefly outline one of the possible models: quarrelling in coalitions. The game consists of two stages: The first stage is a noncooperative one, where the set of feasible winning coalitions is determined. Based on this, there is a second, implicit, cooperative stage, where a power index is used to determine power. Power is payoff, therefore players will reject coalitions in stage 1 to maximize their power in the second stage, thus we look for subgame perfect equilibria.

Kilgour (1974) discussed the possibility of two players quarrelling. In this version of the model we generalize this idea to multiplayer coalitions and allow any member of the coalition to start the quarrelling. In other words a player can start quarrelling in any coalition C he finds unattractive. If C is blocked, any coalition containing C is also blocked: the expansion of the coalition, the arrival of new members does not reconcile the quarrelling players. It is important to emphasize that quarrelling destroys the possibility of forming a particular coalition, it is an irreversible step. Therefore, when studying equilibria, we need only to consider deviations in the direction of removing further winning coalitions. As a consequence the situation where no winning coalitions survive is a Nash equilibrium, although probably not the one one would be interested in.

We therefore select a particular Nash equilibrium: starting from the set of all winning coalitions, winning coalitions are blocked (by profitable blocks) one-by-one. When no further blocks are possible, we arrive to a Nash equilibrium. We suspect that the intersection of such Nash equilibria is either empty or one of these Nash equilibria. The latter is of course special in the sense that it contains the coalitions that are surely not blocked (when starting from all coalitions being feasible). If the intersection is an empty set, on the other hand, this is a warning sign: if players base their actions on the power index used in the game, they may end up in an impasse, where no decisions are possible any more. It is slightly worrying that using the most popular power indices: the Shapley-Shubik or the Banzhaf power indices this is exactly the situation we find for the EU Council of Ministers.

**Renegotiation**

Rejecting winning coalitions seems like a waste, almost paradoxical: While such a block might increase the power of one of the players, it will certainly lead to an overall loss of power. It will be interesting to consider the possibility to renegotiate after this stage. While in the basic model it is crucial that a block is irreversible, this modification makes sense. It is only in the standard literature for power indices that the same credit must be given for a large and a tiny pivotal player. Diermeier and Merlo (2004), Gelman et al. (2004) Fréchette et al. (2005) (and references therein), find that proportional distribution of power behaves surprisingly well both in empirical tests and coalition formation models, suggesting that assigning power proportionally is a common practice if not the standard. While this is just one of the alternatives the question is there, what is the power index that is immune against strategic blocks?

**Farsight**

A farsighted, dynamic cooperative game might also answer these questions and many would find such a model more appropriate: Without farsight it is possible that a player rejects a coalition and thereby prompts a sequence of further blocks that make him end up with very little power. This player could “foresee" these reactions. Note that the usual criticisms of farsightedness do not apply here: as power is only calculated in the second stage there is no payoff to collect along the way. Adding preferences to the model enriches it in another dimension. While the techniques are the same, results will most likely be much more difficult to obtain. In such models already the calculation of the indices is a difficult task. While for standard power indices analytical results are available here these are not really expected. Our aims are accordingly more modest: we develop a model that, when used for a particular example, provides basis for numerical calculations and in addition to this hope to obtain qualitative conclusions, at least for small games. Our last subproject tries to do justice to the minorities. In this part of the project we want to attach graded values to coalitions: the value of a coalition being the probability that it is winning.

**The power of the minority**

This approach is rather different from power indices in general where a coalition is either winning or losing. It is also different from fuzzy coalitional games (Butnariu and Klement, 1993) where it is the membership of a player in a coalition that is graded. We believe it is an original approach and we have found no literature on this or similar ideas. The advantage is of course that once the values are available, values, such as the Shapley value (and perhaps generalisations of power indices) can be used to calculate power in this non-simple game.

The first task is therefore to develop some formal model to capture this idea. The next task is then to refine or modify the model until a form is available that is compact enough to be attractive. Then we hope to obtain some characterisation results and hopefully some interesting results from empirical investigations.

Our basic observation is that a priori measures of voting power are no more than statistical measures of random voting and therefore power indices lack the game theoretical features attributed to them. Similar criticisms have been voiced by Albert (2003), but while that article was written in a discussion for or against the (existing) indices, we do not only criticise them, but also provide modifications that are immune to the aforementioned weaknesses. Shapley (1962) has already pointed out that in voting games “the acquisition of power is the payoff," in the 40 years that passed since then, it was not recognised that if voting games are indeed games, and players are rational, they will want to actively increase their payoff, that is: their power. We plan to study different models, which are suited for different voting situations.

We also introduce the idea of a gradual membership of a coalition in the set of winning coalitions, that is, a non-binary payoff in voting games. As a particularly interesting and perhaps unusual result this model enables us to calculate the power of an opposition, of a losing coalition as of course all, but the 0 coalition are also winning with some –possibly infinitesimally small- probability.

Our contributions are theoretical and elementary. We address some fundamental issues concerning power indices. Considering the general commotion over power indices we hope not only theorists will be interested in our findings, but they can also be used in applications. Journals that act as fora for discussions on power indices and in particular: power in the European Union are en par with top economics journals in terms of impact factors. We hope our papers will only enhance these.

...

As we have already explained in Section 1, the latest years saw an explosion of research on power indices. Most of the published papers are however of different nature from ours.

Three branches must be mentioned: Firstly, there is widespread interest in evaluating power in the European Union Council of Ministers. Many of the papers study different possible allocations of voting weights, and discuss the implications as given by the Shapley or the Banzhaf indices. These papers are applied papers, that take the indices as given and their focus is on the applications. There is a literature originating from social choice theory and although it is often very theoretical, the issues are normative, discussing topics as equity and fairness. Finally, prompted by the first group of papers there is an extensive discussion whether it is appropriate to use power indices for this application or at all.

The present proposal belongs to the third group, although of course we expect results for the popular applications. However, unlike most of the literature that focuses on axioms and paradoxes to evaluate and (re)define power indices using the classical setting (mostly following Felsenthal and Machover, 1995 and 1998) we look at the way power indices are defined in general and criticise the fundamentals. As such, our views are similar to those of Albert (2003), but we also provide remedies by introducing the strategic model.

Measuring power is an important and interesting topic attracting many researchers with strikingly different backgrounds. We believe this “hype” will not be over as long as a further extension of the EU is likely at all. With Turkey interested in membership, in particular due to its size, we predict this scientific discussion not to be over any time soon.

When we mean discussion, we indeed mean an extensive conversation involving dozens of authors, who know each other’s work well and comment them in the form of scientific papers. I was very much pleased to see that presenting my work at some conferences already attracted some positive remarks from well-known figures of the field. Based on that I am convinced my results will be publishable in some of the leading journals of the field and will attract numerous citations.

Formal science has the disadvantage that its arguments are somewhat more difficult to convey to the general public or even the decision makers. The strong media presence of qualified majority voting (in connection to the EU Council of Ministers, of course) indicates both the profound importance of the issues at hand together with the acceptance of the power index approach as the solution to it and the media’s increasing efforts to present scientific ideas to the public.

The impact of original ideas to a field with such media attention can vary. However, as our model does not deal with mere obscure mathematical details, but discusses fundamental issues about the (potential) strategic behaviour in coalition formation, something that is in the interest of voters in weighted voting situations to consider it is reasonable to hope that the field in the broad sense will be receptive to these ideas. The consequences of our criticisms are far-reaching affecting some theories from economics to political science raising numerous new questions about voting, democracy and public choice.

...

The proposed team consists of the principal investigator, and two postdocs. The tasks of the researchers are well defined:

- The principal investigator coordinates the work of the research team and is responsible for the general execution of the project with particular attention to the main, theoretical results.
- Postdoc 1 obtained his (or her) PhD in political science or political economy, is familiar with the political science literature. At the same time he is prepared to learn the power index approach and to work using mathematical models. His task is to enhance the communication with politics: both identify potential applications, interpret our conclusions for these applications and generally popularize (present at scientific meetings and publish) our results in the political science community using the appropriate language.
- Postdoc 2 has a strong mathematical background, primarily in discrete mathematics, combinatorics and statistics, having obtained his (or her) PhD in one of these fields. Familiarity with the problem itself is preferred, but not essential. Postdoc 2 will supply many of the proofs, which will likely use advanced mathematics beyond the level usually used in economic theory. Postdoc 2 has a crucial share in the part using gradual coalitions producing the backbone of the mathematical model and developing many of the properties.

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