Ron Holzman

Aggregation of binary evaluations

We study a general aggregation problem in which a society has to determine its position (yes/no) on each of several issues, based on the positions of the members of the society on those issues. There is a prescribed set of feasible evaluations, i.e., permissible combinations of positions on the issues. This framework for the theory of aggregation was introduced by Wilson and further developed by Rubinstein and Fishburn. Among other things, it admits the modeling of preference aggregation (where the issues are pairwise comparisons and feasibility reflects rationality), and of judgment aggregation (where the issues are propositions and feasibility reflects logical consistency). We characterize those sets of feasible evaluations for which the natural analogue of Arrows impossibility theorem holds true in this framework.

Bundling equilibrium in combinatorial auctions

This paper analyzes ex post equilibria in the VCG combinatorial auctions. If Σ is a family of bundles of goods, the organizer may restrict the bundles on which the participants submit bids, and the bundles allocated to them, to be in Σ .T heΣ -VCG combinatorial auctions obtained in this way are known to be truth-telling mechanisms. In contrast, this paper deals with non-restricted VCG auctions, in which the buyers choose strategies that involve bidding only on bundles in Σ , and these strategies form an equilibrium. We fully characterize those Σ that induce an equilibrium in every VCG auction, and we refer to the associated equilibrium as a bundling equilibrium. The main motivation for studying all these equilibria, and not just the domination equilibrium, is that they afford a reduction of the communication complexity. We analyze the tradeoff between communication complexity and economic efficiency of bundling equilibrium.

Approximations of pseudo-Boolean functions ; applications to game theory

This paper studies the approximation of pseudo-Boolean functions by linear functions and more generally by functions of (at most) a specified degree. Here a pseudo-Boolean function means a real valued function defined on {0,1}n, and its degree is that of the unique multilinear polynomial that expresses it; linear functions are those of degree at most one. The approximation consists in choosing among all linear functions the one which is closest to a given function, where distance is measured by the Euclidean metric onR2n. A characterization of the best linear approximation is obtained in terms of the average value of the function and its first derivatives. This leads to an explicit formula for computing the approximation from the polynomial expression of the given function. These results are later generalized to handle approximations of higher degrees, and further results are obtained regarding the interaction of approximations of different degrees. For the linear case, a certain constrained version of the approximation problem is also studied. Special attention is given to some important properties of pseudo-Boolean functions and the extent to which they are preserved in the approximation. A separate section points out the relevance of linear approximations to game theory and shows that the well known Banzhaf power index and Shapley value are obtained as best linear approximations of the game (each in a suitably defined sense).ZusammenfassungIn dieser Arbeit wird die Approximation Pseudo-Boolescher Funktionen lineare Funktionen bzw. allgemeiner durch Funktion kleiner/gleich eines festen Grades studiert. Dabei ist eine Pseudo-Boolesche Funktion eine reellwertige Funktion definiert auf {0,1}n und ihr Grad ist jener des eindeutig bestimmten multilinearen Polynoms, durch die sie dargestellt werden kann. Lineare Funktionen sind jene mit einem Grad kleiner oder gleich Eins. Die Approximation besteht darin, daß unter allen linearen Funktionen jene mit dem kleinsten Euklidischen Abstand inR2n gewählt wird. Es wird eine Charakterisierung der besten linearen Approximation durch den Mittelwert der Funktion und ihrer ersten Ableitungen angegeben. Sie führt auf eine explizite Formel, um die Approximation aus der Polynomform gegebenen Funktion zu berechnen. Diese Ergebnisse werden später verallgemeinert, um Approximationen höheren Grades zu behandeln. Ferner werden Ergebnisse bezüglich des Zusammenhanges von Approximationen verschiedenen Grades gewonnen. Im linearen Fall wird auch eine im gewissen Sinne eingeschränkte Approximationsaufgabe behandelt. Spezielle Betrachtung wird jenen wichtigen Eigenschaften Pseudo-Boolescher Funktionen gezollt, die bei der Approximation erhalten bleiben. Ein weiterer Abschnitt zeigt die Relevanz linearer Approximationen in der Spieltheorie auf und zeigt Verbindungen zwischen den hier erzielten Ergebnissen und dem wohlbekannten Banzhaf Index auf.

Aggregation of binary evaluations with abstentions

A general model of aggregation of binary evaluations over interrelated issues, introduced by Wilson and further studied by Rubinstein and Fishburn and by the authors, is extended here to allow for abstentions on some of the issues. It is shown that the same structural conditions on the set of feasible evaluations that lead to dictatorship in the model without abstentions, lead to oligarchy in the presence of abstentions. Arrows impossibility theorem for social welfare functions, Gibbards oligarchy theorem for quasi-transitive social decision functions, as well as some apparently new theorems on preference aggregation, are obtained as corollaries.

Network structure and strong equilibrium in route selection games

Abstract In a route selection game on a network, every player chooses a route from the origin to the destination, which are common to all players. Costs are assigned to road segments in the form of monotone nondecreasing functions of the number of players who use them. Each player incurs a total cost equal to the sum of the costs of the road segments in his route. It is known that such a game always has a Nash equilibrium in pure strategies. Here we obtain a structural characterization of those networks for which a strong equilibrium is guaranteed to exist regardless of the cost assignment. The route selection games based on networks in this class enjoy more stability as well as other desirable properties of equilibrium regarding uniqueness and efficiency.

Load Balancing in Quorum Systems

This paper introduces and studies the question of balancing the load on processors participating in a given quorum system. Our proposed measure for the degree of balancing is the ratio between the load on the least frequently referenced element and on the most frequently used one. We give some simple sufficient and necessary conditions for perfect balancing. We then look at the balancing properties of the common class of voting systems and prove that every voting system with odd total weight is perfectly balanced. (This holds, in fact, for the more general class of ordered systems.) We also give some characterizations for the balancing ratio in the worst case. It is shown that for any quorum system with a universe of size

Characterization of ex post equilibrium in the VCG combinatorial auctions

n

Impartial Nominations for a Prize

, the balancing ratio is no smaller than

Fractional Kernels in Digraphs

1/(n-1)

Aggregation of binary evaluations for truth-functional agendas

, and this bound is the best possible. When restricting attention to nondominated coteries (NDCs), the bound becomes