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The Strong, the Weak and the Cunning: Power and Strategy in Voting Games

Final Report Summary - SWEAC (The Strong, the Weak and the Cunning: Power and Strategy in Voting Games)


The aim of this project has been to develop and refine methods to measure the allocation of power in voting situations. In our research plan we have outlined three particular areas. While many of the results are yet to be published, we are proud to say that the project has been successful both in producing interesting results in all these areas and to produce high quality international publications. In the following we outline the results per area. In the text we omit citations to the general literature - for that we refer the reader to our papers.

Strategic voting games

Despite the sound mathematical foundations of power indices, all the widely used power indices exhibit properties considered strange or counter-intuitive. The strange features are known as the paradoxes of voting power in the literature. This line of research has focussed on the paradox of quarrelling members: the observation that two voters may mutually benefit from refusing to cooperate. We have introduced strategic voting games, where the individual players unilaterally initiate a quarrel to increase their power: At first it may sound paradoxical that someone's power can increase by refusing to participate in majority coalitions. While this general observation is not so surprising, the exact specification of the model including the existence of the corresponding noncooperative equilibrium is less straightforward. We have introduced two alternative models, where, despite relatively minor differences the conclusions are surprisingly different. In either case we present sufficient conditions for the existence of a unique strategic modification of a power index. We find that only minimal winning coalitions might sustain, surplus coalitions are subject to quarrelling (while some minimal winning coalitions might be, too). We were also interested in other paradoxes and in the possibility to modify power indices in such a way that the mathematical foundations are still there, but their predictions are less surprising. Looking into this problem we have found that the paradox of new members contradicts the null player property. This paradox states that a voter's power may increase when new voters are admitted into the voting body and so each member's voting share decreases. The null player property states that a voter must have zero power if its addition to a subset of the voters never turns the coalition into a winning coalition and therefore never contributes to the decision making. Interestingly the ``power index'' where power equals to the share of voting weights does not exhibit any of the paradoxical properties. This index gives the natural, instinctive allocation of power. When we talk about paradoxes, then perhaps we only criticise that the predictions differ from this allocation. The effect of quarrelling works via the externalities and so the game can be seen as a coalition formation (or rather: a coalition destruction) game with externalities. For these games we had already defined the core and now we have been working on its implementation. We provide results for totally balanced games, while in another paper we promise a completely general characterisation on the set of games in partition function form. A better understanding of coalitional games with externalities will allow us to define the equilibrium of the quarrelling game in a more direct and more general way. This, however remains an open question. We have also looked at applying the model to particular voting situations where the quarrelling could make sense. It turned out, however, that, apart from fairly small games, the problem is very complex. We have introduced a way to compute power indices directly from the minimal winning coalitions, skipping the step of generating the whole set of winning coalitions thereby conceptually and usually also computationally simplifying the calculations.


Tournaments are directed graphs, where the the direction of an arc tells us the winner of a particular pairwise comparison. Here the objective is not to determine an allocation, but a winner, or more generally the ranking of the nodes. We have introduced a method to rank academic journals using a tournament approach and provide a ranking of the main academic journals in economics. In another paper we show that the acclaimed invariant method can be manipulated: a journal can improve its position in the ranking by making a citation to another journal. Finally we investigate the property of article splitting -- the length of a typical paper should not influence the quality ranking -- and introduce a modified version of the invariant method that is invariant to article splitting too.

Gradual coalitions

In the third and last subproject we are interested in the same type of results, but in a more general setting where the membership of players in the coalitions is not certain and/or the value of coalitions is a random variable. We have introduced generalised weighted voting games. These games are motivated by weighted voting situations via representatives, where, due to illness or other engagement a voting block, such as a party cannot always rely on the full number of representatives and hence the full weight in the voting game. Such a model fits very well voting in a national parliament described in another paper. Applications We have applied the theory of power indices to demographic data from the European Statistical Office to find that Lisbon Treaty does not only increase the decision probability in the EU Council of Ministers, but the benefits are not shared equally. The biggest losses are suffered by medium-sized countries with shrinking populations, with Hungary in the bottom 3 when ranked by the change in power. Another set of results calculates power distributions in international organisations, focussing on the Bretton Woods institutions and finds that if switching between country groups were permitted, Hungary would be better off joining the Scandinavian group. Still on the topic of international organisations, we introduced a foresighted model to explain strategic differences between the EFTAn and Eastern extensions of the European Union