Storable votes with a “pay as you win” mechanism

Abstract

This paper introduces a new storable vote mechanism (Storable Votes-Pay as you win, SV-PAYW) where a fixed number of votes is cast among different alternatives, and the votes spent (and redistributed) on each election depend only on the number cast for the winning alternative. The mechanism is expected to deliver more enfranchisement, efficiency and a reduction of uncertainty and strategic behavior with respect to previously known voting systems. To compare the pure storable votes with the SV-PAYW implementations, two key characteristics are monitored: the “enfranchisement gap”, which measures the proportionality between political influence and electoral victories, and the “efficiency ratio”, which assesses the utility derived from the allocation of electoral victories on a scale from random allocation (zero) to the social optimum (one). SV-PAYW consistently outperforms pure storable votes in terms of enfranchisement in all cases. Additionally, as a general rule (there are some exceptions), the “efficiency ratio” tends to be higher for SV-PAYW, hovering around 0.7.

Social optimum equation transformation

$$\begin{aligned} w_{A}\sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}\gamma _{ij}p_{ij}u_{i}^{A}+(1-w_{A})\sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}(1-\gamma _{ij})p_{ij}u_{j}^{B} \end{aligned}$$

(A1)

Some terms re-arrangement leaves the following expression:

$$\begin{aligned} \sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}\left[ w_{A}\gamma _{ij}p_{ij}u_{i}^{A}+(1-w_{A})(1-\gamma _{ij})p_{ij}u_{j}^{B}\right] \end{aligned}$$

(A2)

Then:

$$\begin{aligned} \sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}\left[ w_{A}\gamma _{ij}p_{ij}u_{i}^{A}+p_{ij}u_{j}^{B}-w_{A}p_{ij}u_{j}^{B}-\gamma _{ij}p_{ij}u_{j}^{B}+w_{A}\gamma _{ij}p_{ij}u_{j}^{B}\right] \end{aligned}$$

(A3)

The terms without controls are grouped on one side and the terms that include controls are grouped on the other:

$$\begin{aligned} \sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}\left[ p_{ij}u_{j}^{B}-w_{A}p_{ij}u_{j}^{B}\right] +\sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}\left[ w_{A}\gamma _{ij}p_{ij}u_{i}^{A}-\gamma _{ij}p_{ij}u_{j}^{B}+w_{A}\gamma _{ij}p_{ij}u_{j}^{B}\right] \end{aligned}$$

(A4)

And simplifying the last term:

$$\begin{aligned} \sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}\left[ p_{ij}u_{j}^{B}-w_{A}p_{ij}u_{j}^{B}\right] + \sum _{i=1}^{I^{A}}\sum _{j=1}^{I^{B}}\gamma _{ij}p_{ij}(w_{A}u_{i}^{A}-(1-w_{A})u_{j}^{B}) \end{aligned}$$

(A5)