Essays in collective decision making

Abstract

Four different problems in collective decision making are studied, all of which are either formulated directly in a game-theoretical context or are concerned with neighboring research areas. The rst two problems fall into the realm of cooperative game theory. In the first one, a decomposition of transferable utility games is introduced. Based on that decomposition, the structure of the set of all transferable utility games is analyzed. Using the decomposition and the notion of minimal balanced collections, a set of necessary and sufficient conditions for a transferable utility game to have a singleton core is given. Then, core selective allocation rules that, when confronted with a change in total cost, not only distribute the initial cost in the same manner as before, but also treat the remainder in a consistent way are studied. Core selective rules which own a particular kind of additivity that turns out to be relevant in this context are also characterized. In the second problem, different notions of merge proofness for allocation rules pertaining to transferable utility games are introduced. Relations between these merge proofness notions are studied, and some impossibility as well as possibility results for allocation rules are established, which are also extended to allocation correspondences. The third problem deals with networks. A characterization of the Myerson value with two axioms is provided. The first axiom considers a situation where there is a change in the value function at a network g along with all networks containing g. At such a situation, the axiom requires that this change is to be divided equally between all the players in g who are not isolated. The second axiom requires that if the value function assigns zero to each network, then each player gets zero payo at each network. Modifying the rst axiom, along a characterization of the Myerson value, a characterization of the position value is also provided. Finally, the fourth problem is concerned with social choice theory which deals with collective decision making in a society. A characterization of the Borda rule for a given set of alternatives with a variable number of voters is studied on the domain of weak preferences, where indi erences between alternatives are allowed at agents' preferences. A new property, which we refer to as degree equality, is introduced. A social choice rule satis es degree equality if and only if, for any two pro les of two nite sets of voters, equality between the sums of the degrees of every alternative under these two pro les implies that the same alternatives get chosen at both of them. The Borda rule is characterized by the conjunction of faithfulness, reinforcement, and degree equality on the domain of weak preferences.