Equal, proportional, and mixed sharing of cooperative production under the threat of sabotage

Abstract

This paper analyzes the consequences of a parametrized class of sharing rules on the propensity of individuals to sabotage each other in a cooperative production framework. The considered sharing rules include equal and proportional sharing as special cases and are parametrized with respect to their sensitivity to relative input contributions. This parameter affects the equilibrium provision of productive individual labor (that increases the respective individual input contribution) but also the propensity to sabotage others (which decreases the input contributions of sabotaged individuals). The theoretical analysis shows that sharing rules in which more weight is put on equal sharing induce zero sabotage in equilibrium; however, they might also lead to inefficient underproduction. In contrast, sharing rules that are highly sensitive with respect to relative input contributions lead to destructive sabotage activities and moreover to inefficient overproduction.

Appendix

Proof of Lemma 1

Claim 1: $\mathbf s^e =\mathbf 0$.

The proof of the claim follows by contradiction. Suppose that an allocation is efficient where at least one individual $j\in N$ exerts positive amounts of sabotage activity. Denote this allocation as $(\mathbf x',l',s' )$ where for at least one $k\ne j$ $s'_{jk}>0$. This implies that $l'_k>0$ because otherwise sabotaging agent $k$ would not have any affect (because $\frac{\partial r^k}{\partial s_{jk}}=0$ for $l_k=0$ by assumption). However, this allocation is strictly dominated by an allocation where individual $j$ refrains from sabotage and the additional produced output is, for instance, equally distributed to all individuals. Formally such an allocation can be expressed as $(\hat{\mathbf{x }},\mathbf{l }',\hat{\mathbf{s }})$ where ${\hat{\mathbf{s }}}=({\hat{\mathbf{s }}_\mathbf{j }}=\mathbf{0 },\mathbf{s'_{-j} })$ and $\hat{x}_i=x'_i+[f(\hat{R})-f(R')]/n>x'_i$. The last inequality follows from the fact that $\hat{R}=\sum _{i\in N}r(l'_i,\hat{\mathbf{s }}^\mathbf{i })>\sum _{i\in N}r(l'_i,\mathbf s'^i )=R'$ because $r(l'_k,\hat{\mathbf{s }}^\mathbf k )>r(l'_k,\mathbf s'^k )$ and $r(l'_i,\hat{\mathbf{s }}^\mathbf i )=r(l'_i,\mathbf s'^i )$ for all $i\ne k$. Note that feasibility still holds because $\sum _{i\in N}\hat{x}_i=f(\hat{R})$.

This implies that $u(\hat{\mathbf{x }}_\mathbf{i },\mathbf{l }'_\mathbf{i },\hat{\mathbf{s }}_\mathbf{i })>u(\mathbf {x'_i,l',s'_i} )$ for all $i\ne j$ because $\hat{x}_i>x'_i$ and $\hat{\mathbf{s }}_i=s'_i$ but also that $u(\hat{\mathbf{x }}_\mathbf{j },\mathbf{l }'_\mathbf{j },\hat{\mathbf{s }}_\mathbf{j })>u(\mathbf x'_j,l'_j,s'_j )$ because $\hat{x}_j>x'_j$ and $\hat{s}_{jk}=0<s'_{jk}$. Hence, an allocation $(\mathbf x',l',s' )$ where some individual exerts positive amounts of sabotage activity cannot be pareto-efficient because $(\hat{\mathbf{x }},\mathbf{l }',\hat{\mathbf{s }})$ is preferred by all individuals.

Claim 2: $(\mathbf x^e,l^e )$ satisfies Eqs. (2) and (3).

A pareto-efficient allocation in this set-up can be expressed as the set of feasible allocations that maximizes the following weighted linear social welfare function $\sum _{i\in N}\lambda _i u(x_i,l_i)$ where $\lambda _i>0$ for all $i\in N$ [by Negishi’s theorem, see Proposition 16.E.1 and 16.E.2 in Mas-Colell et al. (1995)]. The first order conditions of an (interior) solution to this maximization problem can be formulated as in Eqs. (2) and (3). As the maximization problem is well behaved (positive amounts of sabotage activities are ruled out by Claim 1), first order conditions are also sufficient. $\square$