Relative Entropy in Determining Degressive Proportional Allocations

The principle of degressively proportional apportionment of goods, being a compromise between equality and proportionality, facilitates the application of many different allocation rules. Agents with smaller entitlements are more interested in an allocation that is as close to equality as possible, while those with greater entitlements prefer an allocation as close to proportionality as possible. Using relative entropy to quantify the inequity of allocation, this paper indicates an allocation that neutralizes these two contradictory approaches by symmetrizing the inequities perceived by the smallest and largest agents participating in the apportionment. First, based on some selected properties, the set of potential allocation rules was reduced to those generated by power functions. Then, the existence of the power function whose exponent is determined so as to generate the allocation that symmetrizes the relative entropy with respect to equal and proportional allocations was shown. As a result, all agents of the apportionment are more inclined to accept the proposed allocation regardless of the size of their entitlements. The exponent found in this way shows the significant relationship between the problem under study and the well-known Theil indices of inequality. The problem may also be seen from this viewpoint.

[1]  D. Felsenthal,et al.  The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes , 1998 .

[2]  The Desired Political Entropy , 1969 .

[3]  W. R. Thompson.  On the Theory of Apportionment , 1935 .

[4]  Frank Lad,et al.  Extropy: a complementary dual of entropy , 2011, ArXiv.

[5]  William Thomson,et al.  Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey , 2003, Math. Soc. Sci..

[6]  L. Penrose.  The Elementary Statistics of Majority Voting , 1946 .

[7]  Robert A. Agnew,et al.  Optimal Congressional Apportionment , 2008, Am. Math. Mon..

[8]  Janusz Lyko,et al.  Remarks on Unrounded Degressively Proportional Allocation , 2018, Trans. Comput. Collect. Intell..

[9]  Nicola Maaser,et al.  A note on the direct democracy deficit in two-tier voting , 2012, Math. Soc. Sci..

[10]  Linus Schrage,et al.  The apportionment problem and the European Parliament , 1977 .

[11]  Sergio Hernández,et al.  A Brief Review of Generalized Entropies , 2018, Entropy.

[12]  W. Thomson.  Consistent Allocation Rules , 1996 .

[13]  A. Charman.  The Census and the Second Law: An Entropic Approach to Optimal Apportionment for the U.S. House of Representatives , 2017, 1712.09440.

[14]  José Martínez-Aroza,et al.  Spline methods for degressive proportionality in the composition of the European Parliament , 2012, Math. Soc. Sci..

[15]  Karol Życzkowski,et al.  Jagiellonian Compromise an alternative voting system for the Council of the European Union , 2007 .

[16]  Application of Hamilton's and Divisor Methods to Degressively Proportional Allocation Functions☆ , 2014 .

[17]  Martin Zachariasen,et al.  The allocation between the EU member states of the seats in the European Parliament Cambridge Compromise , 2011 .

[18]  Jean-François Laslier,et al.  Optimal Apportionment , 2013, Journal of Political Economy.

[19]  Luc Lauwers,et al.  Minimally Disproportional Representation: Generalized Entropy and Stolarsky Mean-Divisor Methods of Apportionment , 2008 .

[20]  Henri Theil,et al.  Technical Note - The Allocation of Power that Minimizes Tension , 1971, Oper. Res..

[21]  H. Young,et al.  Distributive justice in taxation , 1988 .

[22]  José Martínez-Aroza,et al.  Several methods for degressively proportional allotments. A case study , 2008, Math. Comput. Model..