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# Recent working papers

[1.] Choice on the simplex domain (2019)

Walter Bossert, Hans Peters

**Abstract **One unit of a good has to be divided among a group of agents who each are entitled to a minimal share, and these shares sum up to less than one. The

associated set of choice problems consists of the unit simplex and all its full-dimensional subsimplices with the same orientation. We characterize all choice rules that are independent of irrelevant alternatives, continuous, and monotonic. We also consider the issue of rationalizability and show that in general, excluding cycles of any fixed length does not imply the strong axiom of revealed preference, that is, the exclusion of cycles of any length. For continuous three-agent choice rules, however, excluding cycles of length three implies the strong axiom of revealed preference.

[2.] Unanimous and strategy-proof probabilistic rules for single-peaked preference profiles on graphs (2019)

Hans Peters, Souvik Roy, Soumyarup Sadhukan

**Abstract **Finitely many agents have single-peaked preferences on a finite set of alternatives structured by a connected graph. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles where all peaks are on leafs of the tree, and thus extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leafs.

Finally, the two results are combined to obtain a general characterization for every connected graph.

[3.] Characterizing NTU-bankruptcy rules using bargaining axioms (2018)

Bas Dietzenbacher, Hans Peters

**Abstract** This paper takes an axiomatic bargaining approach to bankruptcy problems with nontransferable utility by characterizing bankruptcy rules in terms of properties from

bargaining theory. In particular, we derive new axiomatic characterizations of the proportional rule, the truncated proportional rule, and the constrained relative equal awards rule using properties which concern changes in the estate or the claims.

[4.] Set and revealed preference axioms for multi-valued choice (2019)

Hans Peters, Panos Protopapas

**Abstract** We consider choice correspondences for arbitrary sets of alternatives, and focus on the condition of independence of irrelevant alternatives and on two weaker versions of it, as well as on the weak axiom of revealed preference for sets, in contrast to revealed preference for singletons. We establish the connection between the condition of independence of irrelevant alternatives and so-called strong sets.

[5.] Power and the Shapley value (2018)

Hans Peters

**Abstract **This chapter reviews some results from the literature in which power indices related to the Shapley value are developed. First, a class of power indices for so-called effectivity functions is axiomatically characterized, based on [1]. As a special case, the Owen-Shapley spatial power index as proposed in [2] is obtained. Second, following [3], a class of power indices for situations in which subsets of players control other players is described. Examples of such situations include financial structures in which firms and other shareholders exercise control through shares in each other. Third, following [4], a class of power indices is considered where relations between the players are determined via a directed graph.

[1] Karos D, Peters H (2018) Effectivity and power. Games and Economic Behavior 108:363-378

[2] Owen G, Shapley LS (1989) Optimal location of candidates in ideological space. International Journal of Game Theory 18:339–356

[3] Karos D, Peters H (2015) Indirect control and power in mutual control structures. Games and Economic Behavior 92:150–165

[4] Peters H, Timmer J, van den Brink, R (2016) Power on digraphs. Operations Research and Decisions 26:107–125

[6.] Self-implementation of social choice correspondences in Nash equilibrium (2019)

Saptarshi Mukherjee, Hans Peters

** Abstract **A social choice correspondence is Nash self-implementable if it can be implemented in Nash equilibrium by a social choice function that selects from it as the game form. We provide a fairly complete characterization of all unanimous and anonymous social choice correspondences that can be Nash implemented by an anonymous selection. In particular, such social choice correspondences include the top correspondence and are included in the Pareto correspondence.