On the costly voting model: the mean rule (Q2241250)

The following model of policy choosing by a group of people is discussed in the article. There are \(n\) players; each player \(i=1,...,n\) has the preferred policy \(x_i\in [0,1]\) and values the policy \(x\in [0,1]\) as \(v(x-x_i)\), where \(v:[-1,1]\to\mathbb {R}\) is a symmetric (\(v(x)=v(-x)\)) and concave function, e.g., \(v(x)=-|x|\), \(v(x)=-x^2\). (So \(x_i\) is most valued by \(i\) among all \(x\).) Every player can either attend or not attend a committee that chooses a compromise policy in the policy space \([0,1]\). The cost of participation in a committee \(X\subset \{1,...,n\}\) is \(c(|X|)\), \(c:\{1,...,n\}\to\mathbb {R}^{+}\), where \(|X|\) denotes the number of members of \(X\). (So the cost of participation in a committee depends only on its size.) The compromise policy established by the committee \(X\) is given by the mean value \(m(X) = \sum_{i\in X} x_i/|X|\), \(m: 2^{\{1,...,n\}}\setminus\{\emptyset\}\to[0,1]\). The payoff of the player \(i\) under the formation of the committee \(X\) is \(v(x_i-m(X))-c(|X|)\cdot 1\{i\in X\}\) where \(1\{i\in X\} =1\) when \(i\in X\) and \(0\) otherwise. (\(X\) can be considered nonempty as there is no incentive for all players to abstain from the participation in any committee under the assumption that \(v(0)>c(1)\).) A committee is in equilibrium if no one profits by joining or leaving it. (So we deal with a pure Nash equilibrium.) Assume that the players are ordered according to their growing policies \(0=x_1<x_2<...<x_n=1\). A universal collection of committees is an ascending sequence \(\{1\}=X_1\subset X_2\subset ... \subset X_n=\{1,...,n\}\) with \(X_{k+1}=X_k\cup\{i\}\), where \(i\) is the player outside \(X_k\) with the policy most distanced to the policy of the committee \(X_k\), i.e., \(|x_i-m(X_k)| = \max_{j\not\in X_k} |x_j-m(X_k)|\). The main theorem of the article states that given a universal collection of committees \(X_k\), \(k=1,...,n\) (which always exists and generically is unique) for any cost function \(c\) one can always find \(k\) such that \(X_k\) is in equilibrium. The size of the committee in equilibrium is discussed for the concrete valuation function \(v(x)=-|x|\) and either a constant cost \(c(|X|)=C\) or a shared cost \(c(|X|)=C/|X|\). Also, general criteria are provided for whether a one-person committee can be in equilibrium; they involve the relation between the valuation \(v\) and cost \(c\). These criteria support a common wisdom that costly participation discourages participating in committees.