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# Recent working papers

[1.] Claim games for estate division problems (version 2017)

Hans Peters, Marc Schröder, Dries Vermeulen

**Abstract** The estate division problem considers the issue of dividing an estate when the sum of entitlements is larger than the estate. This paper studies the estate division problem from a non-cooperative perspective. The integer claim game introduced by O’Neill (1982) and extended by Atlamaz et al (2011) is generalized by specifying a sharing rule to divide every interval among the claimants. We show that for all problems for which the sum of entitlements is at most twice the estate the existence of a Nash equilibrium is guaranteed for a general class of sharing rules. Moreover, the corresponding set of equilibrium payoffs is independent of which sharing rule in the class is used. Well-known division rules that always assign a payoff vector in this set of equilibrium payoffs are the adjusted proportional rule, the random arrival rule and the Talmud rule. For all other estate division problems there is a natural sharing rule for which no Nash equilibrium exists. We provide a class of sharing rules for which an equilibrium always exists.

[2.] Condorcet Consistency and the strong no show paradoxes (2018)

Laura Kasper, Hans Peters, Dries Vermeulen

**Abstract **We consider voting correspondences that are, besides Condorcet Consistent, immune against the two strong no show paradoxes. That is, it cannot happen that if an additional voter ranks a winning alternative on top then that alternative becomes losing, and that if an additional voter ranks a losing alternative at bottom then that alternative becomes winning. This immunity is called the Top Property in the first case and the Bottom Property in the second case. We identify the maximal voting correspondence

satisfying Condorcet Consistency and the Top Property. This voting correspondence also satisfies the Bottom property. It contains the Minimax Rule but it is strictly larger. In particular, voting functions (single-valued voting correspondences) that are Condorcet Consistent and immune against the two paradoxes must select from this maximal correspondence, and we demonstrate several ways in which this can or cannot be done. We also consider a weaker version of Condorcet Consistency.

[3.] Choice on the simplex domain (2017)

Walter Bossert, Hans Peters

**Abstract **One unit of a good has to be divided among a group $N$ of individuals 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. The resulting rules are what we refer to as $N$-path choice functions. If there are only three individuals, the monotonicity property can be weakened. We also consider the issue of rationalizability and show that, for the three-agent case, excluding cycles of length three in the revealed preference relation implies the strong axiom of revealed preference, that is, the exclusion of cycles of any length.

[4.] Information aggregation with continuum of types (2017)

Irem Bozbay, Hans Peters

**Abstract** We consider an information aggregation problem where a group of voters wants to make a `yes’ or `no’ decision over a single issue. Voters have state-dependent common preferences, but hold possibly conflicting private information about the state in the form of types (signals). We assume that types are distributed from a state-dependent continuous distribution. In this model, Bayesian equilibrium voting and efficient voting coincide, and informative voting means that a voter votes in favor of the issue if and only if the signal exceeds a cut-point level. Our main result is an answer, in the form of a condition on the parameters of the model, to the question when informative voting is efficient.

[5.] Random social choice functions for single-peaked domains on trees (2018)

Hans Peters, Souvik Roy, Soumyarup Sadhukan

**Abstract **Finitely many agents have single-peaked preferences on a finite set of alternatives structured as a tree. Under a richness condition on the domain we characterize all unanimous and strategy-proof random social choice functions. These functions are uniquely determined by the values they assign to preference profiles where all peaks are on leafs of the tree.

[6.] Self-implementation of social choice correspondences in strong equilibrium (2018)

Bezalel Peleg, Hans Peters

**Abstract** A social choice correspondence is self-implementable in strong equilibrium if it is implementable in strong equilibrium by a social choice function selecting from the correspondence itself as a game form. We characterize all social choice correspondences implementable this way by an anonymous social choice function satisfying no veto power, given that the number of agents is large relative to the number of alternatives. It turns out that these are exactly the social choice correspondences resulting from feasible elimination procedures as introduced in Peleg (1978).

[7.] 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.

[8.] Set and revealed preference axioms for multi-valued choice (2018)

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 a weaker version of it, as well as on the weak axiom of revealed preference for sets. We establish connections between these conditions and their relations with collections of choice sets, called weak and strong sets, that partially or completely determine the choice correspondences satisfying the above independence properties.

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

Hans Peters

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