Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics

Arnold, Barry C.; Sarabia, José Marı́a

doi:10.1007/978-3-319-93773-1

Cited by 45 publications

(58 citation statements)

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“…Precisely, the Lorenz curve

L_{X}

associated with the nonnegative random variable

X

is defined at probability level

p \in (0, 1)

L_{X} (p) = \frac{\int_{0}^{p} F_{X}^{- 1} (ξ) normald ξ}{\int_{0}^{1} F_{X}^{- 1} (ξ) normald ξ} = \frac{normalE (][, X normalI (][, X \leq F_{X}^{- 1} (p)))}{normalE [X]} .

where

normalI [\cdot]

is the indicator function (equal to 1 if the condition appearing within the brackets is fulfilled and to 0 otherwise). We refer the interested reader to Arnold and Sarabia (2018) for an extensive presentation of Lorenz curves and to Denuit and Vermandele (1999) for applications to insurance.…”

Section: Self‐governing or No‐insurance‐company Modelmentioning

confidence: 99%

Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models

Denuit

Robert

2021

Risk Manage Insurance Review

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This paper purposes to formalize the three business models dominating peer‐to‐peer (P2P) property and casualty insurance: the self‐governing model, the broker model, and the carrier model. The former one develops outside the insurance market whereas the latter ones may originate from the insurance industry, by partnering with an existing company or by issuing a new generation of participating insurance policies where part of the risk is shared within a community and higher losses, exceeding the community's risk‐bearing capacity are covered by an insurance or reinsurance company. The present paper proposes an actuarial modeling based on conditional mean risk sharing, to support the development of this new P2P insurance offer under each of the three business models. In addition, several specific questions are also addressed in the self‐governing model. Considering an economic agent who has to select the optimal pool for a risk to be shared with other participants, it is shown that uniform comparison of the Lorenz or concentration curves associated to the respective total losses of the pools under consideration allows the agent to decide which pool is preferable.

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“…Precisely, the Lorenz curve

L_{X}

associated with the nonnegative random variable

X

is defined at probability level

p \in (0, 1)

L_{X} (p) = \frac{\int_{0}^{p} F_{X}^{- 1} (ξ) normald ξ}{\int_{0}^{1} F_{X}^{- 1} (ξ) normald ξ} = \frac{normalE (][, X normalI (][, X \leq F_{X}^{- 1} (p)))}{normalE [X]} .

where

normalI [\cdot]

Section: Self‐governing or No‐insurance‐company Modelmentioning

confidence: 99%

Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models

Denuit

Robert

2021

Risk Manage Insurance Review

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“…Proposition 3.1. (e.g., Arnold [4], Arnold-Sarabia [5], and Tamir [46]) Let Q be an arbitrary subset of R N and assume that Q admits a least majorized element. Then, an element of Q is least majorized in Q if and only if it is decreasingly minimal in Q.…”

Section: Definition and Notationmentioning

confidence: 99%

Egalitarian Solution for Games With Discrete Side Payment

Otsuka

2021

JORSJ

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In this paper, we study the egalitarian solution for games with discrete side payment, where the characteristic function is integer-valued and payoffs of players are integral vectors. The egalitarian solution, introduced by Dutta and Ray in 1989, is a solution concept for transferable utility cooperative games in characteristic form, which combines commitment for egalitarianism and promotion of indivisual interests in a consistent manner. We first point out that the nice properties of the egalitarian solution (in the continuous case) do not extend to games with discrete side payment. Then we show that the Lorenz stable set, which may be regarded as a variant of the egalitarian solution, has nice properties such as the Davis-Maschler reduced game property and the converse reduced game property. For the proofs we utilize recent results in discrete convex analysis on decreasing minimization on an M-convex set investigated by Frank and Murota.

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“…This means that uncertainty is increased by moving some probability weight from a more likely option to a less likely option. It turns out that this simple idea leads to a concept widely known as majorization [ 27 , 29 , 30 , 31 , 32 , 33 ], which has roots in the economic literature of the early 19th century [ 26 , 34 , 35 ], where it was introduced to describe income inequality, later known as the Pigou–Dalton Principle of Transfers . Here, the operation of moving weight from a more likely to a less likely option corresponds to the transfer of income from one individual of a population to a relatively poorer individual (also known as a Robin Hood operation [ 30 ]).…”

Section: Decision-making With Limited Resourcesmentioning

confidence: 99%

Bounded Rational Decision-Making from Elementary Computations That Reduce Uncertainty

Gottwald

Braun

2019

Entropy

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In its most basic form, decision-making can be viewed as a computational process that progressively eliminates alternatives, thereby reducing uncertainty. Such processes are generally costly, meaning that the amount of uncertainty that can be reduced is limited by the amount of available computational resources. Here, we introduce the notion of elementary computation based on a fundamental principle for probability transfers that reduce uncertainty. Elementary computations can be considered as the inverse of Pigou-Dalton transfers applied to probability distributions, closely related to the concepts of majorization, T-transforms, and generalized entropies that induce a preorder on the space of probability distributions. As a consequence we can define resource cost functions that are order-preserving and therefore monotonic with respect to the uncertainty reduction. This leads to a comprehensive notion of decision-making processes with limited resources. Along the way, we prove several new results on majorization theory, as well as on entropy and divergence measures.

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Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics

Cited by 45 publications

References 0 publications

Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models

Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models

Egalitarian Solution for Games With Discrete Side Payment

Bounded Rational Decision-Making from Elementary Computations That Reduce Uncertainty

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