Balázs Sziklai

Characterization sets for the nucleolus in balanced games

We provide a new modus operandi for the computation of the nucleolus in cooperative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually essential and dually saturated coalitions determine both the core and the nucleolus in monotonic games whenever the core is non-empty. We show how these two sets are related with the existing characterization sets. In particular we prove that if the grand coalition is vital then the intersection of essential and dually essential coalitions forms a characterization set itself. We conclude with a sample computation of the nucleolus of bankruptcy games - the shortest of its kind.

Fair Apportionment in the View of the Venice Commission's Recommendation

In this paper we analyze the consequences of the fairness recommendation of the Venice Commission in allocating voting districts among larger administrative regions. This recommendation requires the size of any constituency not to differ from the average constituency size by more than a fixed limit. We show that this minimum difference constraint, while attractive per definition, is not compatible with monotonicity and Hare-quota properties, two standard requirements of apportionment rules. We present an algorithm that efficiently finds an allotment such that the differences from the average district size are lexicographically minimized. This apportionment rule is a well-defined allocation mechanism compatible with and derived from the recommendation of the Venice Commission. Finally, we compare this apportionment rule with mainstream mechanisms using real data from Hungary and the United States.

Monotonicity and Competitive Equilibrium in Cake-cutting

We study the monotonicity properties of solutions in the classic problem of fair cake-cutting – dividing a single heterogeneous resource among agents with subjective utilities. Resource- and population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all of them are better-off (if there is more to share) or all are worse-off (if there is less to share). We formally introduce these concepts to the cake-cutting setting and present a meticulous axiomatic analysis. We show that classical cake-cutting protocols, like the Cut and Choose, Banach-Knaster, Dubins–Spanier and many other fail to be monotonic. We also show that, when the allotted pieces must be contiguous, proportionality and Pareto-optimality are incompatible with each of the monotonicity axioms. We provide a resource-monotonic protocol for two players and show the existence of rules that satisfy various combinations of contiguousness, proportionality, Pareto-optimality and the two monotonicity axioms.

Traffic routing oligopoly

The purpose of this paper is to introduce a novel family of transferable utility games related to congested networks. We assume that players are traffic coordinators, who explicitly route their deliveries in the network. The costs of the players are determined by the total latency of the deliveries, which in turn can be calculated by the edge latency functions. Since the edge latency functions assign a latency value to the total flow on the corresponding edge, as cooperating players redesign their routing in order to minimize their overall cost, outsiders will be affected as well. This gives rise to externalities therefore the resulting game is described in partition function form. We show that cooperation may imply both negative and positive externalities in the defined game. We assume that coalitions may determine their routing according to different predictive strategies. We show that the increasing order of predictive strategies may converge to a Nash equilibrium (NE), although convergence is not guaranteed, even if a unique NE exists. Furthermore we analyze the superadditivity and stability properties of the game, and show that subadditivity may arise and the recursive core may be empty if the latency functions are not monotone or not continuous.

On the core and nucleolus of directed acyclic graph games

We introduce directed acyclic graph (DAG) games, a generalization of standard tree games, to study cost sharing on networks. This structure has not been previously analyzed from a cooperative game theoretic perspective. Every monotonic and subadditive cost game—including monotonic minimum cost spanning tree games—can be modeled as a DAG-game. We provide an efficiently verifiable condition satisfied by a large class of directed acyclic graphs that is sufficient for the balancedness of the associated DAG-game. We introduce a network canonization process and prove various structural results for the core of canonized DAG-games. In particular, we characterize classes of coalitions that have a constant payoff in the core. In addition, we identify a subset of the coalitions that is sufficient to determine the core. This result also guarantees that the nucleolus can be found in polynomial time for a large class of DAG-games.

THE NUCLEOLUS OF THE BANKRUPTCY PROBLEM BY HYDRAULIC RATIONING

In this note, we give a straightforward and elementary proof of a theorem by Aumann and Maschler stating that in the well-known bankruptcy problem, the so-called CG-consistent solution described by the Talmud represents the nucleolus of the corresponding coalitional game. The proof nicely fits into the hydraulic rationing framework proposed by Kaminski. We point out further interesting properties in connection with this framework.