Cycles of maximin and utilitarian policies under the veil of ignorance

Abstract

A conceptual and mathematical model of a social community behavior in a choice situation under a veil of ignorance, where two alternative policies—Rawlsian maximin and Harsanyian utilitarianism—can be implemented through the aggregation of individual preferences over these two policies, is constructed and investigated. We first incorporate in our conceptual model psychological features such as risk-aversion and prosocial preferences that likely underlie choices of welfare policies. We secondly develop and select the mathematical model presented it by means of an autonomous system of ordinary differential equations. A qualitative analysis of this system global phase-plane behavior shows possible tendencies of community development under social choices over Rawlsian or utilitarian societies depending on psychological parameters such as risk aversion and prosocial preferences.

Appendices

Appendix 1: The SIRS model

When $\gamma_{2} = 0$, the equations of the system (2) turn into:

$$ \left\{ \begin{array}{lll} \frac{{dX_{t} }}{dt} = - \left( {\beta_{2} - \beta_{1} } \right)X_{t} Y_{t} + \gamma_{1} Z_{t} ,\,\,X\left( {t_{0} } \right) = X_{0} , \hfill \\ \frac{{dY_{t} }}{dt} = \left( {\beta_{2} - \beta_{1} } \right)X_{t} Y_{t} - \left( {\nu_{2} - \nu_{1} } \right)Y_{t} ,\,\,Y\left( {t_{0} } \right) = Y_{0} , \hfill \\ \frac{{dZ_{t} }}{dt} = \left( {\nu_{2} - \nu_{1} } \right)Y_{t} - \gamma_{1} Z_{t} ,\,\,Z\left( {t_{0} } \right) = Z_{0} . \hfill \\ \end{array} \right. $$

(3)

Since the total population does not change, it has the place:

$$ \frac{{dX_{t} }}{dt} + \frac{{dY_{t} }}{dt} + \frac{{dZ_{t} }}{dt} = 0. $$

To determining the global phase-plane behavior of the model (3) we use qualitative methods. The model (3) has two equilibrium points:

$$ \begin{aligned} \bar{X}_{1} & = N,\,\bar{Y}_{1} = 0,\bar{Z}_{1} = 0; \\ \bar{X}_{2} & = \frac{{\nu_{2} - \nu_{1} }}{{\beta_{2} - \beta_{1} }},\quad \bar{Y}_{2} = \gamma_{1} \frac{{N - \bar{X}_{2} }}{{\nu_{2} - \nu_{1} + \gamma_{1} }},\quad \bar{Z}_{2} = \frac{{\left( {\nu_{2} - \nu_{1} } \right)\bar{Y}_{2} }}{{\gamma_{1} }}. \\ \end{aligned} $$

Appendix 2: The SIRS-type model

Recall that $\gamma_{2} \ne 0.$ Taking into account that $Z\left( t \right) = N - X\left( t \right) - Y\left( t \right),X\left( {t_{0} } \right) = X_{0} ,Y\left( {t_{0} } \right) = Y_{0} ,$ and denoting $\beta = \beta_{2} - \beta_{1} ,\,\nu = \nu_{2} - \nu_{1} ,$ we rewrite the initial system (2) as:

$$ \left\{\begin{array}{ll} \frac{{dX_{t}}}{dt} = - \beta X_{t} Y_{t} + \gamma_{1} \left( {N - X_{t} - Y_{t} } \right) - \gamma_{2} X_{t}, \\ \frac{{dY_{t}}}{dt} = \beta X_{t} Y_{t} - \nu Y_{t} \left( {N - X_{t} - Y_{t} } \right).\end{array} \right. $$

(4)

This system has the following equilibrium points, denoted as above E1, E2, and E3:

$$ \bar{X}_{E1} = 0,\,\bar{Y}_{E1} = N, $$

(5)

$$ \bar{X}_{E2} = \frac{1}{2}N + \frac{a - \sqrt b }{2\beta \nu },\,\bar{Y}_{E2} = \frac{\nu }{2(\beta + \nu )}N - \frac{a - \sqrt b }{2\beta (\beta + \nu )}, $$

(6)

$$ \bar{X}_{E3} = \frac{1}{2}N + \frac{a + \sqrt b }{2\beta \nu },\,\overline{Y}_{E3} = \frac{\nu }{2(\beta + \nu )}N - \frac{a + \sqrt b }{2\beta (\beta + \nu )}, $$

(7)

where $a = \beta \gamma_{1} + \beta \gamma_{2} + \gamma_{2} \nu$ and $b = \left( {a + N\beta \nu } \right)\,{\kern 1pt}^{2} - 4N\beta^{2} \nu \gamma_{1} ,$ such that $\beta \ne 0,\,\nu \ne 0,\,\beta + \nu \ne 0$ and $b \ge 0.$

To classify stability characteristics of the equilibrium points (5)–(7) of the system (4), and thus the phase-plane behavior, we have to find out the nature of the eigenvalues

$$ \lambda_{1,2} = \frac{1}{2}\left( {{\text{Tr}}\,J\left( {\bar{X},\bar{Y}} \right) \pm \sqrt {{\text{disc}}\,J\left( {\bar{X},\bar{Y}} \right)}}\right) $$

of the Jacobian of the system:

$$ J\left( {\bar{X},\bar{Y}} \right) = \left( {\begin{array}{*{20}l} { - \beta \bar{Y} - \gamma _{1} - \gamma _{2} } \hfill & { - \beta \bar{X} - \gamma _{1} } \hfill \\ {\left( {\beta + \nu } \right)\bar{Y}} \hfill & {\beta \bar{X} + 2\nu \bar{Y}} \hfill \\ \end{array}}\right), $$

where:

$$ {\text{Tr}}\,J\left( {\bar{X},\bar{Y}} \right) = \left( {2\nu - \beta } \right)\bar{Y} + \beta \bar{X} - \gamma_{1} - \gamma_{2} $$

and:

$$ \begin{aligned} {\text{disc}}\,J\left( {\bar{X},\bar{Y}} \right) & = \left( {\left( {2\nu - \beta } \right)\bar{Y} + \beta \bar{X} - \gamma_{1} - \gamma_{2} } \right)^{2} \\ & \quad - 4\left( {\left( {\left( {\beta + \nu } \right)\bar{Y}} \right)\left( {\beta \bar{X} + \gamma_{1} } \right) - \left( {\beta \bar{Y} + \gamma_{1} + \gamma_{2} } \right)\left( {\beta \bar{X} + 2\nu \bar{Y}} \right)} \right). \\ \end{aligned} $$

To conclude, due to the parameters values the eigenvalues $\lambda_{1,2}$ can be real or complex with positive or negative real part. This can cause the stable or unstable behavior of the system (4).