# The Sound of Silence: Minorities, Abstention and Democracy

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## Abstract

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## 1. Introduction

## 2. The Model

- a
- Initial Conditions ($t=0$). At the beginning of each simulation run the N voters are distributed in the preference space, adopting a procedure inspired by the preferential attachment algorithm of Barabasi and Albert [31], that we call “preferential displacement”. Specifically, we insert each new voter in a random position of the entire space with a probability of $30\%$, while the same voter is randomly displaced within the $\delta $-neighborhood of an existing voter i (i.e., within a circular domain centered in $({x}_{i,1},{x}_{i,2})$ with radius equal to $\delta $) with a probability of $70\%$. The latter voter is selected with a probability ${\pi}_{i}\left(\delta \right)$ proportional to the fraction of other voters already included in the same neighbourhood. In other words, ${\pi}_{i}\left(\delta \right)={n}_{i}/{\sum}_{j}{n}_{j}$, being ${n}_{i}$ the number of voters present in the $\delta $-neighborhood of i and ${\sum}_{j}{n}_{j}$ the total number of voters already displaced in the preference space. At variance with the standard uniform initial conditions, typical of the $HK$ model, this process will produce an asymmetric distribution of voters, which will tend to visibly aggregate inside a certain (randomly selected, different for each simulation run) region in the preference space. The preferential displacement, on one hand, makes the initial conditions intuitively more realistic, since it is likely that voters’ opinions about the policy under discussion could concentrate from the beginning around a specific combination of target and tools; on the other hand, it will also result to be essential for the Majority formation, as explained later (see step c).Another important feature of the model is that not all the agents are necessarily involved in the voting process: as a matter of fact, a certain percentage ${p}_{A}$ of them can decide to abstain from the beginning. These ${N}_{A}$ abstained voters neither move nor interact with the other active voters, thus remaining in their initial positions in the preference space for the whole simulation. Since, as discussed before, we would like to model two types of voter abstention, the strategic one (swing voter’s curse) and the non-strategic one (protest abstention), two different ways of selecting the abstained voters in the preference space will be implemented: strategic abstainers will be randomly selected among the N voters, while non-strategic ones will be chosen at margins of the preference space, surrounding active voters. As we will show, though non interacting with the active voters, abstainers will still influence the dynamics of the debate phase just thanks to their presence and to their different placement in the preference space.In panels 1a and 2a of Figure 1 we show an example of initial conditions for two different simulation runs, with $N=500$ voters each and with a percentage ${p}_{A}=20\%$ of abstention. Voters are arranged according with the preferential displacement procedure: together with the 400 active voters (red points), one can find the ${N}_{A}=100$ abstainers (gray points) who either are randomly distributed in the preference space, if their abstention is strategic (1a), or surround the active voters, if their abstention is non-strategic (2a).
- b
- Debate Process ($t\in (0,{t}_{D}]$). In the debate phase, starting at $t>0$, we implement a standard (discrete) $HK$ dynamics in two-dimensions involving only the $N-{N}_{A}$ active voters. Each active voter i-th is endowed with a compatibility domain, $B({\mathbf{x}}_{i},\epsilon )$, defined as a circle centered on her preference profile ${\mathbf{x}}_{i}$, with radius equal to the so-called confidence bound, $\epsilon \in [0,1]$, which in this phase is the same for all the voters. The political debate is modeled as the parallel update, at each time step $t+1$, of all the voters’ profiles, so that each of them becomes equal to the average of the preference profiles of all the active voters included within her compatibility domain at time t. In other words:$${\mathbf{x}}_{i}(t+1)=\frac{{\sum}_{j:\parallel {\mathbf{x}}_{i}\left(t\right)-{\mathbf{x}}_{j}\left(t\right)\parallel <\epsilon}{a}_{ij}{\mathbf{x}}_{j}\left(t\right)}{{\sum}_{j:\parallel {\mathbf{x}}_{i}\left(t\right)-{\mathbf{x}}_{j}\left(t\right)\parallel <\epsilon}{a}_{ij}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(i=1,\cdots ,N)$$In order to calibrate the confidence bound, we introduce a parameter $f\in (0,1)$, called “political fragmentation”. By defining $\epsilon \left(f\right)=\frac{{\epsilon}_{c}}{2}(2-f)$, the confidence bound can vary below its critical threshold in the interval $[\frac{{\epsilon}_{c}}{2},{\epsilon}_{c}]$, thus producing few clusters when $f\to 0$ and many clusters when $f\to 1$. In presence of non-strategic abstention, the confidence bound is further reduced according with the following expression: $\epsilon (f,{p}_{A})=\epsilon \left(f\right)\frac{100-{p}_{A}}{100}$, in order to take into account the reduction of the preference space occupied by active voters. The size and the symmetry of the clusters of voters obtained at the end of the debate phase depend on both their initial distribution over the preference space and the degree of abstention. In panels 1b and 2b of Figure 1 we show the aspect of the preference space at time ${t}_{D}$ for the two considered simulation runs, where we set $f=0.5$: the system has reached its steady state with all the active voters collapsed in several clusters with different sizes for both types of abstention.
- c
- Majority Formation ($t\in ({t}_{D},{t}_{M}]$). After the debate phase, we need some of the newly formed clusters to merge in order to realize some majority. Such a goal can be realized by rescaling, at $t={t}_{D}$, the confidence bound of voters in each cluster as ${\epsilon}_{i}=\epsilon (f,{p}_{A})+[0.5-\epsilon (f,{p}_{A})]\frac{{N}_{C}}{N}$, being ${N}_{C}$ the size of the cluster to which voter i belongs. In other words, voters belonging to larger clusters will interact with a greater number of other voters, and viceversa. The process goes on until a new steady state is reached at $t={t}_{M}$, where some clusters have collapsed to form the relative majority M, whose position ${\mathbf{x}}_{M}$ is represented by blue points in panels 1c and 2c of Figure 1; the remaining clusters (orange points) are considered together to form the relative minority m (green point), whose position ${\mathbf{x}}_{m}$ in the preference space coincides with the weighted average of the positions of all the clusters contributing to it. Of course we must require that the size ${N}_{M}$ of majority is always greater than the size ${N}_{m}$ of minority, i.e., ${N}_{M}>{N}_{m}$: as anticipated, we verified that this can be realized only with “preferential displacement” initial conditions, which moves the center of gravity of preferences away from the geometric center of the space, thus helping the majority formation (anyway, in case this condition should be still not fulfilled, we stop the simulation and repeat the run starting from new initial conditions).
- d
- Final Decision ($t\in ({t}_{M},{t}_{F}]$). At the beginning of this last phase the confidence bound of all the voters is set at ${\epsilon}_{i}=0.5$, well above the critical threshold. Therefore, the system quickly moves towards a steady state where all the active voters have collapsed into a single cluster, which represents the final decision emerging (through the voting process) from the compromise between their respective preferences within the two coalitions. The position ${\mathbf{x}}_{F}$ of such a cluster, reached at time $t={t}_{F}$, typically lies somewhere between the positions of the majority and the minority (red point in panels 1d and 2d of Figure 1), depending on both the relative size of the two coalitions and their relative positions at $t={t}_{M}$ within the preference space.

- For the majority component (M) it is the metric distance between the final position (i.e., the representative preference profile) of the majority cluster and that of the final decision: ${\mathrm{GPS}}_{M}=\parallel {\mathbf{x}}_{M}-{\mathbf{x}}_{F}\parallel $;
- For the minority component (m) it is the metric distance between the final position (i.e., the representative preference profile) of the minority cluster and that of the final decision: ${\mathrm{GPS}}_{m}=\parallel {\mathbf{x}}_{m}-{\mathbf{x}}_{F}\parallel $;

## 3. Simulation Results

## 4. Conclusive Remarks

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Subsequent phases of a single simulation run ($N=500$, ${p}_{A}=20\%$, $f=0.5$). Column 1: strategic abstention; Column 2: non-strategic abstention. Panels (

**a**): random initial conditions with preferential displacement (abstained voters in gray, active voters in red). Panels (

**b**): in red are visible the stationary clusters formed by active voters at the end of the debate phase. Panels (

**c**): some clusters have collapsed in the majority coalit ion (blue point) while the others (orange points), considered all together, represent the minority coalition (green point). Labels indicate the size of both clusters and coalitions. Panels (

**d**): all the existing clusters have collapsed in only one (red point), representing the position of the final decision. See text for more details.

**Figure 2.**Strategic abstention—In correspondence of each fragmentation level and each degree of abstention, the following four panels are reported: (

**a**) majority gps distribution; (

**b**) minority gps distribution; (

**c**) majority size distribution; (

**d**) minority size distribution. Output data are collected over 1000 simulation runs. Scale of y-axis goes from 0 to 50 for GPS distributions and from 0 to 30 for size ones. The absolute majority threshold at $N/2=250$ voters is reported in panels (

**c**,

**d**) as a vertical dashed line. See text for more details.

**Figure 3.**Non-strategic abstention—In correspondence of each fragmentation level and each degree of abstention, the following four panels are reported: (

**a**) majority gps distribution; (

**b**) minority gps distribution; (

**c**) majority size distribution; (

**d**) minority size distribution. Output data are collected over 1000 simulation runs. Scale of y-axis goes from 0 to 50 for GPS distributions and from 0 to 30 for Size ones. The absolute majority threshold at $N/2=250$ voters is reported in panels (

**c**,

**d**) as a vertical dashed line. See text for more details.

**Figure 4.**Strategic abstention—Average gps as a function of political fragmentation and degree of abstention, for majority (

**top left**panel) and minority (

**top right**panel). The same quantities disaggregated for cases when absolute or simple majorities are reached, are also reported in the middle and bottom panels, respectively. With a few exceptions, the effects of abstention on minority seems always positive in terms of preference representation (average gps reduction), for any level of fragmentation. Disaggregated data show an apparent countertrend (an increasing minority gps) which can be explained through statistical considerations about the relative proportion of simulation runs giving rise to absolute and simple majorities. See text for more details.

**Figure 5.**Non-strategic abstention—Average gps as a function of political fragmentation and degree of abstention, for majority (

**top left**panel) and minority (

**top right**panel). The same quantities disaggregated for cases when absolute or simple majorities are reached, are also reported in the middle and bottom panels, respectively. The effects of abstention on minority seems always positive in terms of preference representation (gps reduction), for any level of fragmentation. No countertrends in disaggregated data is observed for this type of abstention. See text for more details.

**Table 1.**The four main columns, further classified according to values of fragmentation, report the effects of both types of abstention (for increasing values of ${p}_{A}$) on, respectively, the size ${N}_{M}$ of reached majority, the ratio between the sizes of majority and minority (${N}_{M}/{N}_{m}$), the size ${N}_{m}$ of the minority, and the number of clusters forming the minority. Notice that, regardless of fragmentation, ${N}_{M}/{N}_{m}$ decreases for strategic abstention, but increases for the non-strategic one (except for $50\%$). Values are always averaged over 1000 simulation runs. See text for more details.

av size maj | maj/min ratio | av size min | av min clusters | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

strategicabstention | $\mathit{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | $\mathit{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | $\mathit{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | $\mathit{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | ||||||||

0.1 | 0.5 | 0.9 | 0.1 | 0.5 | 0.9 | 0.1 | 0.5 | 0.9 | 0.1 | 0.5 | 0.9 | |

$0\%$ | 393 | 375 | 340 | 3.67 | 3.00 | 2.13 | 107 | 125 | 160 | 2 | 4 | 11 |

10% | 342 | 331 | 301 | 3.17 | 2.78 | 2.02 | 108 | 119 | 149 | 2 | 5 | 12 |

20% | 300 | 285 | 262 | 3.00 | 2.48 | 1.90 | 100 | 115 | 138 | 2 | 5 | 12 |

30% | 259 | 245 | 224 | 2.85 | 2.33 | 1.78 | 91 | 105 | 126 | 2 | 5 | 13 |

40% | 215 | 204 | 187 | 2.53 | 2.13 | 1.65 | 85 | 96 | 113 | 3 | 6 | 14 |

50% | 177 | 166 | 153 | 2.42 | 1.98 | 1.58 | 73 | 84 | 97 | 3 | 6 | 14 |

av size maj | maj/min ratio | av size min | av min clusters | |||||||||

nonstrategicabstention | $\mathbf{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | $\mathbf{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | $\mathbf{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | $\mathbf{f}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$value | ||||||||

0.1 | 0.5 | 0.9 | 0.1 | 0.5 | 0.9 | 0.1 | 0.5 | 0.9 | 0.1 | 0.5 | 0.9 | |

0% | 393 | 375 | 340 | 3.67 | 3.00 | 2.13 | 107 | 125 | 160 | 2 | 4 | 11 |

10% | 362 | 357 | 315 | 4.11 | 3.84 | 2.33 | 88 | 93 | 135 | 2 | 4 | 11 |

20% | 329 | 333 | 287 | 4.63 | 4.97 | 2.54 | 71 | 67 | 113 | 2 | 3 | 9 |

30% | 294 | 293 | 247 | 5.25 | 5.14 | 2.40 | 56 | 57 | 103 | 2 | 3 | 8 |

40% | 292 | 292 | 248 | 5.03 | 5.03 | 2.43 | 58 | 58 | 102 | 2 | 3 | 8 |

50% | 208 | 187 | 157 | 4.95 | 2.97 | 1.69 | 42 | 63 | 93 | 3 | 4 | 10 |

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**MDPI and ACS Style**

Biondo, A.E.; Pluchino, A.; Zanola, R. The Sound of Silence: Minorities, Abstention and Democracy. *Entropy* **2022**, *24*, 56.
https://doi.org/10.3390/e24010056

**AMA Style**

Biondo AE, Pluchino A, Zanola R. The Sound of Silence: Minorities, Abstention and Democracy. *Entropy*. 2022; 24(1):56.
https://doi.org/10.3390/e24010056

**Chicago/Turabian Style**

Biondo, Alessio Emanuele, Alessandro Pluchino, and Roberto Zanola. 2022. "The Sound of Silence: Minorities, Abstention and Democracy" *Entropy* 24, no. 1: 56.
https://doi.org/10.3390/e24010056