Analytical Expression of the Expected Values of Capital at Voting in the Stochastic Environment

In the simplest version of the model of group decision making in the stochastic environment, the participants are segregated into egoists and a group of collectivists. A “proposal of the environment” is a stochastically generated vector of algebraic increments of participants’ capitals. The social dynamics is determined by the sequence of proposals accepted by a majority voting (with a threshold) of the participants. In this paper, we obtain analytical expressions for the expected values of capitals for all the participants, including collectivists and egoists. In addition, distinctions between some principles of group voting are discussed.

💡 Research Summary

The paper presents a stochastic voting model that captures how a society of n agents, divided into ℓ egoists and g = n − ℓ collectivists, reacts to randomly generated “environment proposals.” Each proposal is a vector (d₁,…,dₙ) of capital increments drawn independently from a normal distribution N(µ, σ²). At every discrete time step the proposal is put to a vote. A participant votes “for” if the proposal yields a positive personal increment; otherwise the vote is “against.” Egoists follow this rule individually, while the collectivist group decides according to a prescribed group‑voting principle.

Two basic group‑voting principles are examined. Principle A (majority‑within‑the‑group) declares the group’s vote “for” when the number of group members receiving a positive increment exceeds those receiving a negative increment. Principle B (sum‑positive) declares “for” when the sum of the group’s increments is positive. A third, derived principle A′ modifies the internal threshold of Principle A as a function of the overall voting threshold α and the egoist proportion β = ℓ/(2n), allowing the group to neutralize or even reverse the egoists’ advantage in certain α‑ranges.

The authors first note that each egoist’s probability of voting “for” is p = Φ(µ/σ), where Φ is the standard normal CDF. Consequently, the number of egoists voting “for” follows a binomial distribution B(ℓ, p). They introduce the notion of a “normal voting sample” and prove Lemma 1 (exact expectation of a sample element) and Lemma 2 (normal approximation of that expectation). The exact formula involves a sum over binomial coefficients (Equation 6), while the approximation replaces the binomial by a Gaussian with mean μ′ = pℓ and variance σ′² = pqℓ, yielding a compact expression (Equation 10).

Using these tools, the expected capital increment for an egoist, M(e_d_E), is derived. If the group supports the proposal, the overall acceptance condition requires more than γ n egoist “for” votes, where γ = α − (1 − 2β). If the group does not support, the condition is the usual α n. Hence
M(e_d_E) = P_G·µ⁺(γn) + (1 − P_G)·µ⁺(αn),
where µ⁺(·) denotes the expectation of a normal voting sample with the indicated threshold, and P_G is the probability that the group votes “for.” Under Principle A, P_G is the probability that more than half of the g group members receive a positive increment (a binomial tail). Under Principle B, P_G = Φ(µ√g/σ), because the sum of g independent N(µ,σ²) variables is N(µ√g, σ²). The authors also provide normal approximations (Equation 22) that replace the binomial tails by Gaussian CDFs, greatly simplifying the dependence on α, β, µ, σ, ℓ, and g.

The expected increment for a group member, M(e_d_G), is more intricate because it depends on both egoist voting outcomes and the group’s own decision rule. The authors decompose the event space into three mutually exclusive cases: (i) egoists already reach the α n threshold (proposal accepted regardless of the group), (ii) egoists reach the intermediate γ n < #for ≤ α n (the group’s vote decides), and (iii) egoists fall below γ n (proposal rejected). In case (i) the increment is simply µ. In case (ii) the increment equals µ plus an additional term that depends on the group‑voting principle: under Principle A it is µ⁺(g/2) (the expectation when the group’s internal majority is positive); under Principle B it is µ⁺(µ,σ√g,1,0), i.e., the expectation of a normal voting sample conditioned on a positive sum. The final expression (Equation 30 for Principle A, Equation 34 for Principle B) combines these contributions weighted by the probabilities P_α and P_γ (the probabilities that egoists exceed α n and γ n, respectively). Normal approximations replace these binomial probabilities by Gaussian CDFs F_α and F_γ.

The paper further discusses the impact of the derived formulas. The parameter β controls the relative size of the egoist bloc; larger β reduces the group’s ability to block proposals (P_G decreases) and raises the egoists’ expected gain, while smaller β gives the group more leverage. The voting threshold α determines how demanding the majority requirement is; higher α makes acceptance rarer, reducing expected gains for both sides but amplifying the relative advantage of the larger bloc. The normal approximations are shown to be accurate when pqℓ ≥ 9, especially near p ≈ 0.5, and remain useful for a wide range of parameter values.

Finally, Principle A′ is introduced to allow the group to adjust its internal threshold α′ as a function of α and β:
α′ = ½ − δ²/(2β) for α < β,
α′ = ½ + δ²/(2β) for α > 1 − β,
α′ = ½ otherwise,
where δ = β − α (or α − (1 − β) in the second case). By substituting α′ for the internal majority threshold in the formulas for P_G, the group can deliberately diminish the egoists’ advantage in the α‑intervals where they would otherwise dominate. This demonstrates how the design of voting rules (choice of α and group principle) can be used to steer collective outcomes.

In summary, the authors provide a complete analytical framework for the expected capital dynamics in a stochastic voting environment. They deliver exact binomial‑based expressions and tractable normal approximations for both egoist and collectivist participants under several voting principles, elucidate the roles of the key parameters (α, β, µ, σ, ℓ, g), and show how rule design can manipulate the balance between individual self‑interest and group welfare. The results have potential applications in political science, economics, and any domain where collective decisions are made under uncertainty.