Optimal majority threshold in a stochastic environment

Abstract

Within the model of social dynamics determined by collective decisions in a stochastic environment (the ViSE model), we consider the case of a homogeneous society consisting of classically rational economic agents. We obtain analytical expressions for the optimal majority threshold as a function of the parameters of the environment, assuming that the proposals are generated by means of random variables. The cases of several specific distributions of these variables are considered in more detail.

1 Introduction

Borzenko et al. (2006), the ViSE (Voting in a Stochastic Environment) model (hereinafter, the model) has been proposed. Its simplest version describes a society that consists of n classically rational economic agents who are boundedly rational egoists (hereafter, egoists). Each of them maximizes their individual utility in every act of choice, which turns out to be the most profitable noncooperative strategy. Various cooperative and egoistic strategies within the model have been studied in Borzenko et al. (2006), Chebotarev (2006), Chebotarev et al. (2009), and Malyshev and Chebotarev (2017), and altruistic strategies in Chebotarev et al. (2018b).

Each participant/agent is characterized by the current value of individual utility. A proposal of the environment is a vector of proposed utility increments of the participants. A similar model with randomly generated proposals appeared in Compte and Jehiel (2017). The society can accept or reject every proposal by means of voting, i.e., choose reform or status quo. Each agent votes for those and only those proposals that increase his/her individual utility. A proposal is accepted and implemented, i.e., the participants’ utilities are incremented in accordance with the proposal, if and only if the proportion of the society supporting this proposal is greater than a strict relative voting threshold $\alpha \in [-\frac{1}{n},1]$. Otherwise, all utilities remain unchanged. This voting procedure is called “$\alpha$-majority” [cf. Nitzan and Paroush (1982, 1984), Felsenthal and Machover (2001), Baharad et al. (2019), and O’Boyle (2009)].

The voting threshold $\alpha$ will also be called the majority threshold or, more precisely, the acceptance threshold, since $\alpha < 0.5$ is allowed.

The concept of proposal allows one to model potential changes that are beneficial for some agents and disadvantageous for others. As a result of the implementation of such a proposal, the utilities of some agents increase, while the utilities of others decrease.

The proposals are stochastically generated by the environment and put to a general vote over and over again. The subject of the study is the dynamics of the participants’ utilities as a result of this process. A similar dynamic model proposed by A. Malishevski has been presented in Mirkin (1979), Subsection 1.3 of Chapter 2. Another model whose simplest version is very close to the simplest version of the ViSE model was studied in Barberà and Jackson (2006).

Some other related voting models have been studied in the theory of legislative bargaining (see Duggan and Kalandrakis 2012), where stochastic generation of proposals has been assumed in some cases (Penn 2009; Dziuda and Loeper 2014, 2016). On other connections between the ViSE model and various comparable models, we refer to Chebotarev et al. (2018b).

In accordance with the ViSE model, the utility increments/decrements that form proposals are realizations of independent identically distributed random variables (independence is taken as a base case, models with dependent or non-identically distributed random variables can also be considered). In this paper, we present a general result applicable to any distribution that has a mathematical expectation and focus on four families of distributions: continuous uniform distributions, normal distributions (cf. Chebotarev et al. 2018a), symmetrized Pareto distributions (see Chebotarev et al. 2018b), and Laplace distributions.

Each distribution is characterized by its mathematical expectation, $\mu$ and standard deviation, $\sigma$. The ratio $\sigma /\mu$ is called the coefficient of variation of a random variable. The inverse coefficient of variation $\rho = \mu /\sigma$, which we call the adjusted (or normalized) mean of the environment, measures the relative favorability of the environment. If $\rho > 0$, then the opportunities provided by the environment are favorable on average; if $\rho < 0$, then the environment is unfavorable. We introduce the concept of optimal acceptance threshold and investigate dependence of this threshold on $\rho$ for several types of distributions.

In the present paper, we study:

• The optimal acceptance threshold for a general distribution (Sect. 2.3), i.e., the threshold that maximizes the social welfare (this generalizes Theorem 1 in Chebotarev et al. (2018a));

• Dependence of the optimal acceptance threshold on the model parameters for several specific distributions (Sects. 2.4–2.6).

• Expected utility increase for a general distribution (Sect. 3) [this generalizes Lemma 1 in Chebotarev (2006)].

2 Optimal Majority Threshold

2.1 The Model

To familiarize with the problem that the optimal majority threshold solves, let us look at the dependence of the expected utility increment of an agent on the adjusted mean of the environment $\rho$ (Chebotarev et al. 2018a).

Let $\varvec{\zeta }=(\zeta _1,\ldots ,\zeta _n)$ denote a random proposal on some step. Its component $\zeta _i$ is the proposed utility increment of agent i. The components $\zeta _1,\ldots ,\zeta _n$ are independent identically distributed random variables. $\zeta$ will denote a similar scalar variable without reference to a specific agent. Similarly, let $\varvec{\eta }=(\eta _1,\ldots ,\eta _n)$ be the random vector of actual increments of the agents on the same step. If $\varvec{\zeta }$ is adopted, then $\varvec{\eta }=\varvec{\zeta }$; otherwise $\varvec{\eta }=(0,\ldots ,0)$. Consequently,

$$\begin{aligned} \varvec{\eta }=\varvec{\zeta }I(\varvec{\zeta }, \alpha n), \end{aligned}$$

(1)

where¹

$$\begin{aligned} I(\varvec{\zeta }, \alpha n) = {\left\{ \begin{array}{ll} 1, &{}\#\{k:\zeta _k> 0, k = 1,...,n\} > \alpha n \\ 0, &{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

(2)

and $[\alpha n] \le n$, $[\alpha n] = n-1$ corresponds to unanimity, $[\alpha n] = -1$ and $[\alpha n] = n$ to accept and reject of proposal without voting, respectively.²

Equation (1) follows from the assumption that each agent votes for those and only those proposals that increase his/her individual utility.

Let $\eta$ be a random variable similar to every $\eta _i,$ but having no reference to a specific agent. We are interested in the expected utility increment of an agent, i.e. $\mathrm{E}(\eta ),$ where $\mathrm{E}(\cdot )$ is the mathematical expectation.

Consider an example. For 21 participants and $\alpha =0.5$, the dependence of $\mathrm{E}(\eta )$ on $\rho =\mu /\sigma$ is presented in Fig. 1, where proposals are generated by the normal distribution.

Fig. 1

Expected utility increment of an agent: 21 agents; $\alpha = 0.5$; normal distribution

Figure 1 shows that for $\rho \in (-0.85,-0.266),$ the expected utility increment is an appreciable negative value, i.e., proposals approved by the majority are, on average, unprofitable and impoverishing for the society. This part of the curve is called a “pit of losses.” For $\rho < -0.85,$ the negative mean increment is very close to zero, since the proposals are extremely rarely accepted.

2.2 A Voting Sample

By a “voting sample” of size n with absolute voting threshold $n_0$ we mean the vector of random variables $(\zeta _1 I(\varvec{\zeta }, n_0),\ldots ,\zeta _n I(\varvec{\zeta }, n_0))$, where $\varvec{\zeta }=(\zeta _1,\ldots ,\zeta _n)$ is a sample from some distribution and $I(\varvec{\zeta }, n_0)$ is defined by (2). According to this definition, a voting sample vanishes whenever the number of positive elements of sample $\varvec{\zeta }$ does not exceed the threshold $n_0$.

The lemma on “normal voting samples” obtained in Chebotarev (2006) can be generalized as follows.

Theorem 1

Let $\varvec{\eta }=(\eta _1,\ldots ,\eta _n)$ be a voting sample from some distribution with an absolute voting threshold $n_0 \in \{-1,0,\ldots ,n\}$. Then, for any $k=1,\ldots ,n,$

$$\begin{aligned} \mathrm{E}(\eta _k) = \sum \limits _{x=n_0+1}^{n} \left( (E^+ + E^-)\frac{x}{n} - E^-\right) \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x}, \end{aligned}$$

(3)

where $E^- = \big |\mathrm{E}(\zeta \; | \; \zeta \le 0)\big |, E^+ = \mathrm{E}(\zeta \; | \; \zeta > 0),$ $p = P\{\zeta > 0\} = 1 - F(0), q = P\{\zeta \le 0\} = F(0),$ $\zeta$ is the random variable that determines the utility increment of any agent in a random proposal, and $F(\cdot )$ is the cumulative distribution function of $\zeta .$

Proof

According to the formula of total probability for mathematical expectations, we have

$$\begin{aligned} \mathrm{E}(\eta _k) = 0 \cdot P\{n^+ \le n_0\} \; + \sum \limits _{x=n_0+1}^{n} \mathrm{E}(\eta _k \; | \; n^+ = x) P\{n^+ = x\} \end{aligned}$$

(4)

for any $k =1,\ldots ,n$ (hereafter we use $\eta$ instead of $\eta _k$). Furthermore,

$$\begin{aligned} \mathrm{E}(\eta \; | \; n^+ = x)= & {} \mathrm{E}(\eta \; | \; n^+ = x, \; \eta> 0) \, P\{\eta > 0 \; | \; n^+ = x \} \nonumber \\&+\, \mathrm{E}(\eta \; | \; n^+ = x, \; \eta \le 0) \, P\{\eta \le 0 \; | \; n^+ = x \}, \end{aligned}$$

(5)

$$\begin{aligned} P\{\eta > 0 \; | \; n^+ = x \} = \frac{x}{n}, \; \end{aligned}$$

(6)

and

$$\begin{aligned} P\{\eta \le 0 \; | \; n^+ = x \} = 1 - \frac{x}{n}. \end{aligned}$$

(7)

Using the fact that by the independence of the components of the proposal, for all $x>n_0$ we have $\mathrm{E}(\eta \; | \; n^+ = x, \; \eta> 0) = \mathrm{E}(\eta \; | \; \eta> 0) = \mathrm{E}(\zeta \; | \; \zeta > 0) = E^+$ and similarly $\big |\mathrm{E}(\eta \; | \; n^+ = x, \; \eta \le 0)\big | = \big |\mathrm{E}(\eta \; | \; n^+>n_0, \; \eta \le 0)\big | = \big |\mathrm{E}(\zeta \; | \; \zeta \le 0)\big | = E^-$ and substituting (5)–(7) and $P\{n^+=x\} = \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x}$ (where p is the probability that a proposal component is positive and $q = 1- p$) into (4) we get (3). $\square$

Corollary 1

Let $\varvec{\eta }=(\eta _1,\ldots ,\eta _n)$ be a voting sample from some distribution with an absolute voting threshold $n_0 = 0$. Then, for any $k=1,\ldots ,n,$

$$\begin{aligned} \mathrm{E}(\eta _k) = \mu + E^-(1-p)^n, \end{aligned}$$

(8)

where $\mu = \mathrm{E}(\zeta )$ and the other notations are defined in Theorem 1.

Proof

Using the properties of the binomial distribution we get

$$\begin{aligned} \mathrm{E}(\eta _k)&= \sum \limits _{x=1}^{n} \left( (E^+ + E^-)\frac{x}{n} - E^-\right) \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x} \\&= (E^+ + E^-) \sum \limits _{x=0}^{n} \frac{x}{n} \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x} - E^- \sum \limits _{x=1}^{n} \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x} \\&= (E^+ + E^-) p - E^- \left( 1 - \begin{pmatrix} n \\ 0 \end{pmatrix} p^0 q^{n}\right) \\&= p E^+ - (1-p) E^- + E^-(1-p)^n. \end{aligned}$$

Since $p E^+ - (1-p) E^- = \mu ,$ we have (8). $\square$

2.3 A General Expression for the Optimal Voting Threshold

For each specific environment, there is an optimal acceptance threshold³$\alpha _0$ that provides the highest possible expected utility increment $\mathrm{E}(\eta )$ of an agent.

The optimal acceptance threshold for the normal distribution as a function of the environment parameters has been studied in Chebotarev et al. (2018a). This threshold turns out to be independent of the size of the society n.

Voting with the optimal acceptance thresholds always yields positive expected utility increments and so it is devoid of “pits of losses.”

The following theorem provides a general expression for the optimal voting threshold, which holds for any distribution that has a mathematical expectation.

Theorem 2

In a society consisting of egoists, the optimal voting threshold is

$$\begin{aligned} \alpha _0 = \left( 1+\frac{E^+}{E^-}\right) ^{-1}, \end{aligned}$$

(9)

where $E^- = \big |\mathrm{E}(\zeta \; | \; \zeta \le 0)\big |, E^+ = \mathrm{E}(\zeta \; | \; \zeta > 0),$ and $\zeta$ is the random variable that determines the utility increment of any agent in a random proposal.

In terms of the value $R = \frac{E^+}{E^-},$ which we call the win/loss magnitude ratio,  Eq. (9) takes the form

$$\begin{aligned} \alpha _0 = \left( 1+R\right) ^{-1}. \end{aligned}$$

Proof

Consider the expected social welfare increase when some proposal is adopted:

$$\begin{aligned} n^+ E^+ - (n - n^+) E^-, \end{aligned}$$

where $n^+$ is the number of positive components in a proposal.

This expression is positive if and only if $\frac{n^+}{n} > \frac{E^-}{E^+ + E^-}.$

We obtained an analytical expression for the expected utility increment as the sum (3) in Theorem 1. Let us consider the sign of the sum terms. They are positive if and only if $\frac{x}{n} > \frac{E^-}{E^+ + E^-}.$

Therefore, the voting threshold $\alpha _0 = \frac{E^-}{E^+ + E^-} = \left( 1+\frac{E^+}{E^-}\right) ^{-1}$ allows us to take exactly all positive terms getting the maximum sum. Consequently, this threshold is optimal for society and for each agent due to uniformity.⁴$\square$

Note that if a threshold $\alpha$ is optimal and $[\alpha _1 n]=[\alpha n],$ then $\alpha _1$ is also an optimal threshold.

Let $\bar{\alpha }_0$ be the center of the half-interval of optimal acceptance thresholds for fixed $n,\sigma$, and $\mu$. Then this half-interval is $[\bar{\alpha }_0-\tfrac{1}{2n}, \bar{\alpha }_0+\tfrac{1}{2n}[$. Figures 2, 3, 4, 5 and 6 show the dependence of $\bar{\alpha }_0$ on $\rho =\mu /\sigma$ for several distributions used for the generation of proposals.

As one can observe for various distributions, outside the segment $\rho \in [-0.7,\, 0.7]$, if an acceptance threshold is close to the optimal one and the number of participants is appreciable, then the proposals are almost always accepted (to the right of the segment) or almost always rejected (to the left of this segment). Therefore, in these cases, the issue of determining the exact optimal threshold loses its practical value.

2.4 Proposals Generated by Continuous Uniform Distributions

Let $-a < 0$ and $b > 0$ be the minimum and maximum values of a continuous uniformly distributed random variable, respectively.

Corollary 2

The optimal majority/acceptance threshold in the case of proposals generated by the continuous uniform distribution on the segment $[-a, b]$ with $-a<0$ and $b>0$ is

$$\begin{aligned} \alpha _0 = \left( 1+\frac{b}{a}\right) ^{-1}. \end{aligned}$$

(10)

Indeed, in this case, $E^- = \frac{a}{2}, E^+ = \frac{b}{2},$ and $R = \frac{b}{a},$ hence, (9) provides (10).

Fig. 2

The center $\bar{a}_0$ of the half-interval of optimal majority/acceptance thresholds (a “ladder”) for $n=5$ and the optimal threshold (10) as functions of $\rho$ for continuous uniform distributions

If b approaches 0 from above, then $\alpha _0$ approaches 1 from below, and the optimal voting procedure is unanimity. Indeed, positive proposed utility increments become much smaller in absolute value than negative ones, therefore, each participant should be able to reject a proposal.

As $-a$ approaches 0 from below, negative proposed utility increments become much smaller in absolute value than positive ones. Therefore, a “coalition” consisting of any single voter should be able to accept a proposal. In accordance with this, the optimal relative threshold $\alpha _0$ decreases to 0.

Corollary 3

In terms of the adjusted mean of the environment $\rho =\mu /\sigma ,$ it holds that for the continuous uniform distribution,

$$\begin{aligned} \alpha _0 = {\left\{ \begin{array}{ll} 1, &{}\rho \le -\sqrt{3},\\ \frac{1}{2} \left( 1 - \frac{\rho }{\sqrt{3}} \right) , &{}-\sqrt{3}< \rho < \sqrt{3},\\ 0, &{}\rho \ge \sqrt{3}. \end{array}\right. } \end{aligned}$$

(11)

This follows from (10) and the expressions $\mu = \frac{-a+b}{2}$ and $\sigma = \frac{b+a}{2\sqrt{3}}.$ It is worth mentioning that the dependence of $\alpha _0$ on $\rho$ is linear, as distinct from (10).

Figure 2 illustrates the dependence of the center of the half-interval of optimal majority/acceptance thresholds versus $\rho =\mu /\sigma$ for continuous uniform distributions in the segment $\rho \in [-2,\, 2]$.

Fig. 3

The center $\bar{a}_0$ of the half-interval of optimal majority/acceptance thresholds (a “ladder”) for $n=21$ and the optimal threshold (12) as functions of $\rho$ for normal distributions

2.5 Proposals Generated by Normal Distributions

For normal distributions, the following corollary holds.

Corollary 4

The optimal majority/acceptance threshold in the case of proposals generated by the normal distribution with parameters $\mu$ and $\sigma$ is

$$\begin{aligned} \alpha _0 = F(\rho ) \left( 1 - \frac{\rho F(-\rho )}{f(\rho )} \right) , \end{aligned}$$

(12)

where $\,\rho = \mu /\sigma ,$ while $\,F(\cdot )$ and $f(\cdot )\,$ are the standard normal cumulative distribution function and density, respectively.

Corollary 4 follows from Theorem 2 and the facts that $E^- = -\sigma \left( \rho - \frac{f(\rho )}{F(-\rho )}\right)$ and $E^+ = \sigma \left( \rho + \frac{f(\rho )}{F(\rho )}\right) ,$ which can be easily found by integration. Note that Corollary 4 strengthens the first statement of Theorem 1 in Chebotarev et al. (2018a).

Figure 3 illustrates the dependence of the center of the half-interval of optimal majority/acceptance thresholds versus $\rho =\mu /\sigma$ for normal distributions in the segment $\rho \in [-2.5,\, 2.5]$.

We refer to Chebotarev et al. (2018a) for some additional properties (e.g., the rate of change of the optimal voting threshold as a function of $\rho$).

Fig. 4

The center $\bar{a}_0$ of the half-interval of optimal majority/acceptance thresholds (a “ladder”) for $n=131$ (odd) and the optimal threshold (13) as functions of $\rho$ for symmetrized Pareto distributions with $k = 8$

Fig. 5

The center $\bar{a}_0$ of the half-interval of optimal majority/acceptance thresholds (a “ladder”) for $n=130$ (even) and the optimal threshold (13) as functions of $\rho$ for symmetrized Pareto distributions with $k = 8$

2.6 Proposals Generated by Symmetrized Pareto Distributions

Pareto distributions are widely used for modeling social, linguistic, geophysical, financial, and some other types of data. The Pareto distribution with positive parameters k and a can be defined by means of the function $P\{\xi > x\} = \left( \frac{a}{x}\right) ^{k},$ where $\xi \in [a,\infty )$ is a random variable.

The ViSE model normally involves distributions that allow both positive and negative values. Consider the symmetrized Pareto distributions [see Chebotarev et al. (2018b) for more details]. For its construction, the density function $f(x) = \frac{k}{x} \left( \frac{a}{x}\right) ^{k}$ of the Pareto distribution is divided by 2 and combined with its reflection w.r.t. the line $x = a$.

The density of the resulting distribution with mode (and median) $\mu$ is

$$\begin{aligned} f(x) = \frac{k}{2a} \left( \frac{|x-\mu |}{a} + 1\right) ^{-(k+1)}. \end{aligned}$$

For symmetrized Pareto distributions with $k>2,$ the following result holds true.

Corollary 5

The optimal majority/acceptance threshold in the case of proposals generated by the symmetrized Pareto distribution with parameters $\mu ,$ $\sigma ,$ and $k>2$ is

$$\begin{aligned} \alpha _0 = \frac{1}{2}\left( 1+\mathrm{sign}(\rho )\frac{1-(k-2)\hat{\rho }-(1+\hat{\rho })^{-k+1}}{1+k\hat{\rho }}\right) \end{aligned}$$

(13)

where $\,\rho = \frac{\mu }{\sigma }$, $C = \sqrt{\frac{(k-1)(k-2)}{2}} = \frac{a}{\sigma }$, and $\hat{\rho }=|\rho /C|=|\mu /a|$.

Corollary 5 follows from Theorem 2 and the facts (their proof is given below) that:

$E^- = \sigma \left( \frac{C + \rho }{k-1}\right) ,$ $E^+ = \frac{\sigma }{1 - \frac{1}{2}\left( \frac{C}{C + \rho } \right) ^k} \left( \rho + \left( \frac{C}{C + \rho } \right) ^k \frac{C + \rho }{2(k-1)}\right)$ whenever $\mu > 0;$

$E^- = -\frac{\sigma }{1 - \frac{1}{2}\left( \frac{C}{C - \rho } \right) ^k} \left( \rho - \left( \frac{C}{C - \rho } \right) ^k \frac{C - \rho }{2(k-1)}\right) ,$ $E^+ \; = \; \sigma \left( \frac{C - \rho }{k-1}\right)$ whenever $\mu \le 0.$

The “ladder” and the optimal acceptance threshold curve for symmetrized Pareto distributions are fundamentally different from the corresponding graphs for the normal and continuous uniform distributions. Namely, $\alpha _0(\rho )$ increases in some neighborhood of $\rho = 0$.

As a result, $\alpha _0(\rho )$ has two extremes. This is caused by the following peculiarities of the symmetrized Pareto distribution: an increase of $\rho$ from negative to positive values decreases $E^+$ and increases $E^-$. By virtue of (9), this causes an increase of $\alpha _0.$

This means that the plausible hypothesis about the profitability of the voting threshold raising when the environment becomes less favorable (while the type of distribution and $\sigma$ are preserved) is not generally true. In contrast, for symmetrized Pareto distributions, it is advantageous to lower the threshold whenever a decreasing $\rho$ remains close to zero (an abnormal part of the graph).

Figures 4 and 5 illustrate the dependence of the center of the half-interval of optimal voting thresholds versus $\rho =\mu /\sigma$ for symmetrized Pareto distributions with $k=8.$

Proof of Corollary 5

Let $F(\cdot )$ and $f(\cdot )\,$ be the cumulative Pareto distribution function and the Pareto density, respectively; $\,\rho =\mu /\sigma ,$ and $C \sigma = \sigma \sqrt{\frac{(k-1)(k-2)}{2}} = a.$

Let $\mu > 0$. Then

$$\begin{aligned} E^-&= -\frac{1}{F(0)} \int _{-\infty }^{0} x \frac{k}{2 C \sigma } \left( \frac{-x + C \sigma + \rho \sigma }{C \sigma } \right) ^{-(k+1)} dx\\&= -\frac{1}{\frac{1}{2} \left( \frac{C}{C + \rho }\right) ^k} \frac{k}{2 C \sigma } \left( \frac{(C \sigma )^{k+1} (-x + C \sigma + \rho \sigma )^{-k} (kx - C \sigma - \rho \sigma ) }{(k-1)k} \right) \bigg |_{-\infty }^{0} \\&= \sigma \left( \frac{C + \rho }{k-1}\right) ;\\ E^+&= \frac{1}{1 - F(0)} \int _{0}^{\mu } x \frac{k}{2 C \sigma } \left( \frac{-x + C \sigma + \rho \sigma }{C \sigma } \right) ^{-(k+1)} dx \\&\quad + \; \; \frac{1}{1 - F(0)} \int _{\mu }^{\infty } x \frac{k}{2 C \sigma } \left( \frac{x + C \sigma - \rho \sigma }{C \sigma } \right) ^{-(k+1)} dx \\&= \frac{1}{1 - \frac{1}{2} \left( \frac{C}{C + \rho }\right) ^k} \frac{k}{2 C \sigma } \left( \frac{(C \sigma )^{k+1} (-x + C \sigma + \rho \sigma )^{-k} (kx - C \sigma - \rho \sigma ) }{(k-1)k} \right) \bigg |_{0}^{\rho \sigma } \\&\quad - \,\frac{1}{1 - \frac{1}{2} \left( \frac{C}{C + \rho }\right) ^k} \frac{k}{2 C \sigma } \left( \frac{(C \sigma )^{k+1} (x + C \sigma - \rho \sigma )^{-k} (kx + C \sigma - \rho \sigma ) }{(k-1)k} \right) \bigg |_{\rho \sigma }^{\infty } \\&= \frac{\sigma }{1 - \frac{1}{2}\left( \frac{C}{C + \rho } \right) ^k} \left( \rho + \left( \frac{C}{C + \rho } \right) ^k \frac{C + \rho }{2(k-1)}\right) . \end{aligned}$$

Similarly, $E^- = -\frac{\sigma }{1 - \frac{1}{2}\left( \frac{C}{C - \rho } \right) ^k} \left( \rho - \left( \frac{C}{C - \rho } \right) ^k \frac{C - \rho }{2(k-1)}\right)$ and $E^+ \; = \; \sigma \left( \frac{C - \rho }{k-1}\right)$ whenever $\mu \le 0$. $\square$

Fig. 6

The center $\bar{a}_0$ of the half-interval of optimal majority/acceptance thresholds (a “ladder”) for $n=11$ and the optimal threshold (14) as functions of $\rho$ for Laplace distributions

2.7 Proposals Generated by Laplace Distributions

The density of the Laplace distribution with parameters $\mu$ (location parameter) and $\lambda > 0$ (rate parameter) is

$$\begin{aligned} f(x) = \frac{\lambda }{2} \exp {\left( -\lambda |x-\mu |\right) }. \end{aligned}$$

For Laplace distributions, the following corollary holds.

Corollary 6

The optimal majority/acceptance threshold in the case of proposals generated by the Laplace distribution with parameters $\mu$ and $\lambda$ is

$$\begin{aligned} \alpha _0=\frac{1}{2}\left( 1+\mathrm{sign}(\rho )\frac{1-\sqrt{2}|\rho |-\exp {\left( -\sqrt{2}|\rho |\right) }}{1+\sqrt{2}|\rho |}\right) . \end{aligned}$$

(14)

Corollary 6 follows from Theorem 2 and the facts (their proof is similar to the proof of Corollary 5) that:

$E^- = \frac{1}{\lambda },$ $E^+ = \frac{2\mu + \frac{e^{-\lambda \mu }}{\lambda }}{2-e^{-\lambda \mu }}$ whenever $\mu > 0;$

$E^- = -\frac{2\mu - \frac{e^{\lambda \mu }}{\lambda }}{2-e^{\lambda \mu }},$ $E^+ = \frac{1}{\lambda }$ whenever $\mu \le 0.$

In Lemma 3 of Chebotarev et al. (2018b), it was proved that the symmetrized Pareto distribution with parameters k, $\mu$, and $\sigma$ tends, as $k \rightarrow \infty$, to the Laplace distribution with the same mean and standard deviation.

Notice that the abnormal part of the curve $\alpha _0(\rho )$ for symmetrized Pareto distributions becomes smaller with the growth of k. Figure 6 shows that it vanishes in the case of Laplace distribution. Now we verify this using the first derivative of $\alpha _0$ with respect to $\rho$. It is

$$\begin{aligned} \frac{d\alpha _0}{d\rho } = \frac{ e^{-\sqrt{2}|\rho |} (\sqrt{2} + |\rho |) - \sqrt{2}}{(1+\sqrt{2}|\rho |)^2}. \end{aligned}$$

(15)

Note that this derivative is non-positive and is equal to zero only when $\rho = 0.$ The increasing part of the curve $\alpha _0(\rho )$ for the symmetrized Pareto distribution degenerates to a single point as this distribution converges to the Laplace distribution. It is the point, where the first derivative (15) of $\alpha _0$ is equal to zero.

Fig. 7

The first derivative of $\alpha _0$ with respect to $\rho$ (15) as a function of $\rho$ for Laplace distributions

Figures 6 and 7 show the dependence of the center of the half-interval of optimal voting thresholds versus $\rho =\mu /\sigma$ and the dependence of the first derivative of $\alpha _0$ with respect to $\rho$ versus $\rho$ for Laplace distributions, respectively.

We summarize the results of the above corollaries in Tables 1 and 2.

Table 1 Probabilities of positive and negative proposals for several distributions

Table 2 Expected win and loss for several distributions

3 Expected Utility Increment

Let $Y_n \sim$ Bin(n, p), where Bin(n, p) is the binomial distribution with parameters n and p. Let $F_n(k) = \sum \limits _{x=0}^{k} \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x}$ be the cumulative distribution function of $Y_n$. Let $G_n(k) = 1 - F_n(k) = \sum \limits _{x=k+1}^{n} \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x}$. According to a relationship between the binomial distribution and the Beta distribution we have

$$\begin{aligned} G_n(k) = B(p \; | \; k+1, n - k), \end{aligned}$$

(16)

where $B( \cdot \; | \; k + 1, n - k)$ is the cumulative distribution function of Beta distribution with $k+1$ and $n-k$ degrees of freedom. We also put $G_n(m) = 1$, when $m < 0$.

Now we can prove the following theorem.

Theorem 3

Let $\varvec{\eta }=(\eta _1,\ldots ,\eta _n)$ be a voting sample from some distribution with an absolute voting threshold  $n_0 \in \{1,\ldots ,n-1\}$. Then for any $k=1,\ldots ,n$ it holds that

$$\begin{aligned} \mathrm{E}(\eta _k) = p (E^+ + E^-) B(p \; | \; n_0, n - n_0) - E^- B(p \; | \; n_0 + 1, n - n_0), \end{aligned}$$

(17)

where $E^- = \big |\mathrm{E}(\zeta \; | \; \zeta \le 0)\big |, E^+ = \mathrm{E}(\zeta \; | \; \zeta > 0),$ $B(\cdot \; | \; m, l)$ is the cumulative distribution function of Beta distribution with m and l degrees of freedom, $p = P\{\zeta > 0\} = 1 - F(0),$ $\zeta$ is the random variable that determines the utility increment of any agent in a random proposal, and $F(\cdot )$ is the cumulative distribution function of $\zeta .$

Corollary 1 in Sect. 2.2 extends the result of Theorem 3 to $n_0=0$. It is easy to prove that $\mathrm{E}(\eta _k) = \mu$ for $n_0=-1$ and $\mathrm{E}(\eta _k) = 0$ for $n_0=n.$

Proof

Let us prove the following equation:

$$\begin{aligned} \sum \limits _{x=k}^{n} x \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x} = np \cdot G_{n-1}(k-2). \end{aligned}$$

(18)

Denoting $y=x-1$ we have

$$\begin{aligned} \sum \limits _{x=k}^{n} x \begin{pmatrix} n \\ x \end{pmatrix} p^x q^{n-x}&= np \sum \limits _{x=k}^{n} x \frac{(n-1)!}{(n-x)!x!} p^{x-1} q^{n-x} \\&= np \sum \limits _{x=k}^{n} \frac{(n-1)!}{((n-1)-(x-1))!(x-1)!} p^{x-1} q^{(n-1)-(x-1)} \\&= np \sum \limits _{x=k}^{n} \begin{pmatrix} n-1 \\ x-1 \end{pmatrix} p^{x-1} q^{(n-1)-(x-1)} \\&= np \sum \limits _{y=k-1}^{n-1} \begin{pmatrix} n-1 \\ y \end{pmatrix} p^{y} q^{(n-1)-y} \\&= np \cdot G_{n-1}(k-2). \end{aligned}$$

Now we get (17) applying the definition of $G_n(k)$, (16), and (18) to (3). $\square$

The formula (17) can be rewritten in terms of the regularized incomplete beta function.⁵

Corollary 7

Let $\varvec{\eta }=(\eta _1,\ldots ,\eta _n)$ be a voting sample from some distribution with an absolute voting threshold  $n_0 \in \{1,\ldots ,n-1\}$. Then for any $k=1,\ldots ,n$ it holds that

$$\begin{aligned} \mathrm{E}(\eta _k) = \mu I_p(n_0, n - n_0) + \frac{E^- p^{n_0} (1-p)^{n-n_0}}{n_0 B(n_0,n-n_0)}, \end{aligned}$$

(19)

where $I_p(n_0, n - n_0)$ and $B(n_0,n-n_0)$ are the regularized incomplete beta function and beta function, respectively, $\mu = \mathrm{E}(\zeta ),$ and the other notations are defined in Theorem 3.

Proof

Using the properties of the Beta distribution and incomplete beta function we get

$$\begin{aligned} \mathrm{E}(\eta _k)&= p (E^+ + E^-) B(p \; | \; n_0, n - n_0) - E^- B(p \; | \; n_0 + 1, n - n_0) \\&= p (E^+ + E^-) I_p(n_0, n - n_0) - E^- \left( I_p(n_0, n - n_0) - \frac{p^{n_0} (1-p)^{n-n_0}}{n_0 B(n_0, n - n_0)} \right) \\&= \left( p E^+ - (1-p) E^- \right) I_p(n_0, n - n_0) + \frac{E^- p^{n_0} (1-p)^{n-n_0}}{n_0 B(n_0,n-n_0)} \\&= \mu I_p(n_0, n - n_0) + \frac{E^- p^{n_0} (1-p)^{n-n_0}}{n_0 B(n_0,n-n_0)}. \end{aligned}$$

$\square$

Theorem 3 allows one to obtain a specific expression for $\mathrm{E}(\eta _k$) for each distribution generating proposals by applying specific forms of $E^+$, $E^-$, p, and q (Tables 1, 2). For example, for continuous uniform distributions, the following corollary holds.

Corollary 8

Let $\varvec{\eta }=(\eta _1,\ldots ,\eta _n)$ be a voting sample from the continuous uniform distribution on the segment $[-a, b]$ with $-a<0$ and $b>0$ with an absolute voting threshold $n_0 = \in \{1,\ldots ,n-1\}$. Then for any $k=1,\ldots ,n$ it holds that

$$\begin{aligned} \mathrm{E}(\eta _k) = \frac{b}{2} B\left( \frac{b}{a+b} \; \big \vert \; n_0, n - n_0\right) - \frac{a}{2} B\left( \frac{b}{a+b} \; \big \vert \; n_0 + 1, n - n_0\right) , \end{aligned}$$

(20)

where $B(\cdot \; | \; m, l)$ is the cumulative distribution function of Beta distribution with m and l degrees of freedom.

4 Comparison of the Expected Utility Increments

Chebotarev et al. (2018b), the issue of correct location-and-scale standardization of distributions for the analysis of the ViSE model has been discussed. An alternative (compared to using the same mean and variance) approach to standardizing continuous symmetric distributions was proposed. Namely, distributions similar in position and scale must have the same $\mu$ and the same interval (centered at $\mu$) containing a certain essential proportion of probability. Such a standardization provides more similarity in the central region and the same weight of tails outside this region.

In what follows, we apply this approach for the comparison of the expected utility for several distributions. Namely, for each distribution, we find the variance such that the first quartiles (and thus, all quartiles because the distributions are symmetric) coincide for zero mean distributions, where the first quartile, $Q_1$, splits off the “left” 25% of probability from the “right” 75%.

For the normal distribution, $Q_1 \approx -0.6745 \sigma _N$, where $\sigma _N$ is the standard deviation.

For the continuous uniform distribution, $Q_1 = -\frac{\sqrt{3}}{2} \sigma _U$, where $\sigma _U$ is its standard deviation.

For the symmetrized Pareto distribution, $Q_1 = C(1-2^\frac{1}{k})\sigma _P$, where $\sigma _P$ is the standard deviation and $C = \sqrt{\frac{(k-1)(k-2)}{2}}$. This follows from the equation

$$\begin{aligned} F_P(Q_1) = \frac{1}{2}\left( \frac{C}{C-\frac{Q_1}{\sigma _P}}\right) ^{k} = \frac{1}{4}, \end{aligned}$$

where $F_P(\cdot )$ is the corresponding cumulative distribution function.

For the Laplace distribution, $Q_1 = \frac{-\ln {2}}{\lambda } = -\sigma _L\frac{\ln {2}}{\sqrt{2}}$, where $\sigma _L$ is the standard deviation.

Consequently, $\sigma _U \approx 0.7788\sigma _N$, $\sigma _P \approx 1.6262\sigma _N$ for $k = 8$, and $\sigma _L \approx 1.3762\sigma _N$.

Fig. 8

Expected utility increment of an agent as a function of $\mu$ with a majority/acceptance threshold of $\alpha = \frac{11}{21}$ for several distributions: black line denotes the symmetrized Pareto distribution, black dotted line the normal distribution (with $\sigma _N=1$), gray line the continuous uniform distribution, and gray dotted line the Laplace distribution

Fig. 9

Expected utility increment of an agent as a function of $\mu$ when the optimal majority/acceptance thresholds are used for several distributions: black line denotes the symmetrized Pareto distribution, black dotted line the normal distribution (with $\sigma _N=1$), gray line the continuous uniform distribution, and gray dotted line the Laplace distribution

Figures 8 and 9 show the dependence of the expected utility increment of an agent on the mean $\mu$ of the proposal distribution for several distributions (normal, continuous uniform, symmetrized Pareto, and Laplace) for the majority threshold $\alpha = \frac{11}{21}$ and the optimal acceptance threshold, respectively. They are obtained by substituting the parameters of the environments into (17), (11)–(14). Obviously, the optimal acceptance threshold excludes “pits of losses” because the society has the option to take insuperable threshold of 1 and reject all proposals.

Fig. 10

The optimal majority/acceptance threshold as function of $\mu$ for several distributions: black line denotes the symmetrized Pareto distribution, black dotted line the normal distribution, gray line the continuous uniform distribution, and gray dotted line the Laplace distribution

Figure 10 illustrates the dependence of the optimal majority threshold on $\mu$ for the same list of distributions. It helps to explain why for $\alpha = \frac{11}{21},$ the continuous uniform distribution has the deepest pit of losses (because of the biggest difference between the actual and optimal thresholds), and why the symmetrized Pareto and Laplace distributions have no discernible pit of losses (because those differences are the smallest).

5 Conclusion

In this paper, we obtained general expressions for the expected utility increase and the optimal voting threshold (i.e., the threshold that maximizes social/individual welfare) as functions of the parameters of the stochastic proposal generator in the assumptions of the ViSE model. These expressions were given more specific forms for several types of distributions.

Estimation of the optimal majority/acceptance threshold seems to be a solvable problem in real situations. If the model is at least approximately adequate and one can estimate the type of distribution and $\rho =\mu /\sigma$ by means of experiments, then it is possible to obtain an estimate for the optimal acceptance threshold using the formulas provided in this paper.

We found that for some distributions of proposals, the plausible hypothesis that it is beneficial to increase the voting threshold when the environment becomes less favorable is not generally true. A deeper study of this issue should be the subject of future research.