Abstract
The objective of this paper is to derive some integer-majorization results for variable-sum comparisons. We use an axiomatic framework to establish equivalence between several intuitively reasonable conditions. © 2011 Elsevier Inc. All rights reserved.
Elsevier has been kind enough to allow reprinting of this article published in Journal of Economic Theory, Volume, 147, 2012. Elsevier’s permission in this context is acknowledged with sincere gratitude.
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Notes
- 1.
- 2.
For simplicity, we restrict attention to a fixed population set up. But our results can be extended easily to the variable population case under the assumption of replication invariance of the evaluation function in the spirit of Dasgupta et al. (1973).
- 3.
This implication can be obtained as follows: note that the NIME axiom posits that
$$ E^{n} (x_{1} , \ldots ,x_{j} + 1, \ldots ,x_{i} , \ldots ,x_{n} ) - E^{n} (x_{1} , \ldots ,x_{j} , \ldots ,x_{i} , \ldots ,x_{n} ) $$$$ \ge E^{n} (x_{1} , \ldots ,x_{j} , \ldots ,x_{i} + 1, \ldots ,x_{n} ) - E^{n} (x_{1} , \ldots ,x_{j} , \ldots ,x_{i} , \ldots x_{n} ) $$while if \( x = T_{\text{MIME}} (y) \), then the conditions \( E^{n} (x) \ge E^{n} (y) \) is
$$ E^{n} (y_{1} , \ldots ,y_{j} + 1, \ldots ,y_{i-1} , \ldots ,y_{n} ) \ge E^{n} (y_{1} , \ldots ,y_{j} , \ldots ,y_{i} , \ldots, y_{n} ) $$Subtracting \( E^{n} (y_{1} , \ldots ,y_{j} , \ldots ,y_{i} - 1 \ldots ,y_{n} ) \) from both sides of the second inequality, we obtain the NIME axiom requirement if \( y_{i} - 1 = x_{i} \) and \( y_{k} = x_{k} \) for all \( k \ne i \).
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Acknowledgements
For comments and suggestions, we are grateful to two referees, an associate editor of this journal, Vincenzo Denicolò and participants of the JET Symposium on “Inequality and Risk”, Paris, June 25–26, 2010. Chakravarty thanks the Bocconi University, Milan, Italy, for support. Financial support from the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (Prin 2007) is gratefully acknowledged by Claudio Zoli.
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Chakravarty, S.R., Zoli, C. (2019). Stochastic Dominance Relations for Integer Variables. In: Chakravarty, S. (eds) Poverty, Social Exclusion and Stochastic Dominance. Themes in Economics. Springer, Singapore. https://doi.org/10.1007/978-981-13-3432-0_13
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