On the Construction of High Dimensional Simple Games

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function $\chi\colon\{0,1\}^n\rightarrow \{0,1\}$. However, its naive encoding needs $2^n$ bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using $n$ weights and one threshold. For heterogeneous agents, one can represent $\chi$ as an intersection of $k$ threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring $k\ge 2^{\frac{n}{2}-1}$ and provided a construction guaranteeing $k\le {n\choose {\lfloor n/2\rfloor}}\in 2^{n-o(n)}$. The magnitude of the worst-case situation was thought to be determined by Elkind et al.~in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number $k$ for a subclass of voting systems. As an application, we give a construction for $k\ge 2^{n-o(n)}$, i.e., there is no gain from a representation complexity point of view.Comment: 13 pages, 1 tabl

The institutional determinants of property regime change in new democracies: the Russian Federation, Hungary, and Czechoslovakia

Is there a relationship between the design of democratic institutions and optimal collective decisions? Optimum decisions are defined as achieving goals important to the transition such as deep and equitable property reforms. Democratic institutions refer to first-order institutions of governance and the electoral rules for choosing leaders. Overseeing both are the written or parchment constitutions. Constitutions are designed to distribute power among actors, generate efficiency, and govern the interactions among actors. My findings showed that constitutional designs intentionally and sometimes with unanticipated consequences can result in highly cooperative, competitive, or conflictual struggles by political actors over high-stakes distributive issues such as privatization. I argued that chief executives as the institutional setting varied were more or less likely to successfully conduct radical and equitable property reforms. Using distributive theories of institutions, testing of the simpler bivariate model of executive power proved to be indeterminate, except in isolated instances of decision making. Various configurations of executive power assumed importance in a more complex model that incorporates additional variables -- the power distribution governing executive/legislative relations and multicameral institutions. In the Russian case, a powerful executive,in the context of the asymmetric distribution of legislative and non-legislative powers between the branches, promoted property reforms but also conflict between the branches. In Hungary, a powerful executive by virtue of a monopoly over the constitutional powers could act freely and decisively, but only insofar privatization policies were deemed effective. Further, executive power was limited by the powerful Hungarian Constitutional Court. In the Czechoslovak case, the even distribution of powers appeared to most optimize collective decision making. The power and efficiency properties of its moderately consensual institutions intertwined to create institutional incentives that were more likely to produce low-cost and sustainable privatization policies. Rather than serving as a rival account,the unity of purpose argument complimented and extended the power distribution argument. Efficiency effects can arise from the design of electoral rules and subsequent party system formation. The findings supported this interpretation and the role of partisan parliamentary conditions in the formation of winning legislative coalitions in parliament

Extensions of the shapley value in weighted voting systems

The present work reviews the concept of values in the theory of games with particular reference to political games. A model based on the Shapley value concept is developed and applied to simulated and practical voting situations. In particular it is shown how numerical expressions can be obtained for the values of each group or party given their sizes and with knowledge of their previous voting patterns. Data based on the Nigerian political set up as well as other political systems, including the U.N., E.E.C. etc. was used for calculating the values of the different participants

The Spatial Analysis of Elections and Committees: Four Decades of Research

It has been more than thirty five years since the publication of Downs's (1957) seminal volume on elections and spatial theory and more than forty since Black and Newing (1951) offered their analysis of majority rule and committees. Thus, in response to the question "What have we accomplished since then?" it is not unreasonable to suppose that the appropriate answer would be "a great deal." Unfortunately, reality admits of only a more ambiguous response

The Constitutional Choice of Bicameralism

We take a garden-variety instance of distributive politics-- a divide-the-cake stage game -- and explore dynamic extensions in different institutional settings: (i) repeated play of the stage game in a simultaneous-term unicameral legislature; (ii) repeated play in a staggered-term unicameral legislature; and (iii)repeated play in a bicameral setting of one staggered-term and one simultaneous-term legislative chamber. We are then able to entertain decisions taken at "the constitutional moment" regarding which institutional forms to employ.institutions; distributive politics; repeated divide-the-cake games; bicameralism

A MATHEMATICAL ANALYSIS OF CONFLICTS IN VOTING SYSTEMS

Game Theory applied to voting gives rise to the study of many different topics. In this thesis we present some results in order to evaluate the power share inside a Parliament, to analyze the criteria for the assessment of the voters\u2019 preferences and to provide more efficient ways to compute the existing instruments. The first part of the thesis deals with the issue of power in a voting system. In order to better analyze it, we investigate the way of combining a communication structure with the already existing indices of power. Moreover, we present the possibility of assuming some coalitions as less probable, but not infeasible, as the political scenario is complex enough to admit the possibility of very unlikely coalitions to form. Then, we add another important aspect to the problem; the alliances inside a decisional situation are not stable, but may evolve with time, mainly due to the fact that each agent aims at getting a higher power. We refer to an already existing instrument which, theoretically, may perfectly describe this situation. Unfortunately, the current computational complexity does not permit to adopt it in many real cases. Another important topic is the evaluation of the power to block instead of the power to win. We provide an index which evaluates the veto power and which can be extended in order to catch other characteristics of the game; for example, the probability that a party takes a particular decision, or that not every member of a party is present at the vote. The second part of this thesis considers a previous step of the formation of a democracy: the evaluation of how much the resulting Parliament reflects the preferences expressed by the electors. We think that the goodness of a Parliament mainly depends on the power share between the parties and not only on the number of seats. Finally, the last part shows a new exact method to improve the efficiency of the evaluation of one of the existing indices of power, the Public Good index

The Spatial Analysis of Elections and Committees: Four Decades of Research

It has been more than thirty five years since the publication of Downs's (1957) seminal volume on elections and spatial theory and more than forty since Black and Newing (1951) offered their analysis of majority rule and committees. Thus, in response to the question "What have we accomplished since then?" it is not unreasonable to suppose that the appropriate answer would be "a great deal." Unfortunately, reality admits of only a more ambiguous response

Contributions to Game Theory and Management. Vol. III. Collected papers presented on the Third International Conference Game Theory and Management.

The collection contains papers accepted for the Third International Conference Game Theory and Management (June 24-26, 2009, St. Petersburg University, St. Petersburg, Russia). The presented papers belong to the field of game theory and its applications to management. The volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments.