Proportional rules for state contingent claims


We consider rationing problems where the claims are state contingent. Before the state is realized individuals submit claims for every possible state of the world. A rule distributes resources before the realization of the state of the world. We introduce two natural extensions of the proportional rule in this framework, namely, the ex-ante proportional rule and the ex-post proportional rule, and then we characterize them using standard axioms from the literature.

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  1. 1.

    They used “Weak Sequential Core” as the stability criterion which was defined in Habis and Herings (2011).

  2. 2.

    By lottery we mean probability distribution over states of the world to be realized in the stage two.

  3. 3.

    The standard rationing problem is defined as (Nxt) where N is a finite set of agents, x is a claim vector \(x=(x_{i})_{i\in N}\ge 0\) such that \( \sum \nolimits _{i\in N}x_{i}{\ge t}\) and \(t\ge 0\) is the resource to be shared among the agents. A rationing rule \(\varphi \) assigns a vector of shares \(\varphi (N,x,t)\in {\mathbb {R}} _{+}^{N}\) to every rationing problem such that \(\sum \nolimits _{i\in N}\varphi _{i}(N,x,t)=t\).

  4. 4.

    \( \Delta ^{|S|-1}\) denotes a \(|S|-1\) dimensional simplex.

  5. 5.

    More precisely this is a restricted domain of problems where N and S are fixed so a better notation would be \({\mathcal {D}}(N,S).\) However, for notational simplicity we use \({\mathcal {D}}\) since it does not raise any confusion.

  6. 6.

    We use the notation \(x_{Ts}{:}{=}\sum _{i\in T}(x_{is})\), where \(T\subseteq N.\)

  7. 7.

    Note that the standard non-bossy axioms protect the other individuals from any unilateral change of report by an individual whereas NARAS axiom protects those individuals only against a specific change of the report, i.e., the reallocated report with the same expected value.


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Corresponding author

Correspondence to Sinan Ertemel.

Additional information

We would like to thank our advisor, Hervé Moulin, for several helpful comments and remarks. Discussions with Anna Bogomolnaia, Youngsub Chun, Ruben Juarez, Juan Moreno-Ternero, Arunava Sen, and William Thomson have been of great help. The valuable comments of the Associate Editor and two anonymous referees have greatly improved our paper. We thank Graham Brownlow and David Seymour for their help with proof-reading. Sinan Ertemel gratefully acknowledges support from “TÜBİTAK 2232 Grant 115C030”. Rajnish Kumar acknowledges the British Council grant “UGC-UKIERI 2016-17-059”.

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Ertemel, S., Kumar, R. Proportional rules for state contingent claims. Int J Game Theory 47, 229–246 (2018).

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  • Rationing
  • Proportional rule
  • State contingent claims
  • No advantageous reallocation

JEL Classification

  • C71
  • D63
  • D81