Reciprocity in the shadow of threat

Abstract

This paper considers a partial equilibrium model of conflict where two agents differently evaluate a contested stake. Differently from common contest models, agents have the option of choosing a second instrument to affect the outcome of the conflict. The second instrument is assumed to capture positive investments in ‘conflict management’—labeled as ‘talks’. The focus is on the asymmetry in the evaluation of the stake: whenever the asymmetry in the evaluation of the stake is large there is no room for cooperation and a conflict trap emerges; whenever the degree of asymmetry falls within a critical interval, cooperation seems to emerge only in the presence of a unilateral concession; as the evaluations of the stake converge, only reciprocal concessions can sustain cooperation. Finally the concept of entropy is applied to measure conflict and conflict management.

Appendix

To check whether the critical points presented in (8) constitute a Nash Equilibrium I have to compute the Hessian matrices for both agents.

I start considering the payoff function of agent 1 evaluated at the critical points for agent 2, namely π ^rc₁ (g ₁, g ^*₂ , h ₁, h ^*₂ ). It becomes:

$$ \begin{aligned}& \pi^{{\rm rc}}_{1} {\left({g_{1}, g^{*}_{2}, h_{1}, h^{*}_{2}} \right)} = \frac{g_{1} x{\left({\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}}{\delta^{6} x^{2} + g_{1} {\left({\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}} + \frac{\delta^{3} s_{2} x - {\left({\delta^{2} + 1} \right)}^{2} {\left({h_{1} + s_{2} + g_{1}} \right)}}{{\left({\delta^{2} + 1} \right)}^{2}}. \\& \\\end{aligned} $$

and the Hessian matrix is given by:

$$ \begin{aligned} H_{1} &= {\left({\begin{array}{*{20}c}\frac{\partial \pi^{{\rm rc}}_{1}}{\partial g_{1} g_{1}}& \frac{\partial \pi ^{{\rm rc}}_{1}}{\partial h_{1} g_{1}} \\\frac{\partial \pi ^{{\rm rc}}_{1}}{\partial g_{1} h_{1}}& \frac{\partial \pi ^{{\rm rc}}_{1}}{\partial h_{1} h_{1}} \\\end{array}} \right)}\\ &= {\left({\begin{array}{*{20}c}- \frac{2\delta^{6} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{4} {\left({h_{1} + 1} \right)}^{2}}{{\left[{\delta^{6} x^{2} + g_{1} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}} \right]}^{3}}& \frac{\delta^{6} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left[{\delta^{6} x^{2} - g_{1} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}} \right]}}{{\left[{\delta^{6} x^{2} + g_{1} {\left({\delta ^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}} \right]}^{3}} \\\frac{\delta^{6} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left[{\delta^{6} x^{2} - g_{1} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}} \right]}}{{\left[{\delta^{6} x^{2} + g_{1} {\left({\delta ^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}} \right]}^{3}}& - \frac{2\delta^{6} g^{2}_{1} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{4}}{{\left[{\delta ^{6} x^{2} + g_{1} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{1} + 1} \right)}} \right]}^{3}} \\ \end{array}} \right)} \\\end{aligned} $$

Then, substitute the critical values h ^*₁ , g ^*₁ . The matrix becomes:

$$ H_{1} = {\left({\begin{array}{*{20}c} - \frac{2{\left({\delta^{2} + 1} \right)}}{\delta^{2} x}& \frac{{\left({\delta^{2} + 1} \right)}{\left({\delta^{2} - 1} \right)}}{\delta^{2} x} \\ \frac{{\left({\delta^{2} + 1} \right)}{\left({\delta^{2} - 1} \right)}}{\delta^{2} x}& - \frac{2{\left({\delta^{2} + 1} \right)}}{\delta^{2} x} \\\end{array}} \right)} $$

Let H _1k denote the kth order leading principal submatrix of H ₁ for k = 1,2. The determinant of the kth order leading principal minor of H _1k is denoted by | H _2k |. The leading principal minors alternate signs as follows:

$$ {\left| {H_{{11}}} \right|} < 0 $$

$$ {\left| {H_{{12}}} \right|} > 0 $$

Then I compute the payoff function for agent 2 π ^rc₂ (g ^*₁ , g ₂, h ^*₁ , h ₂),

$$ \begin{aligned}& \pi^{{\rm rc}}_{2} {\left({g^{*}_{1}, g_{2}, h^{*}_{1}, h_{2}} \right)} = \frac{g_{2} \delta x{\left({\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}}{\delta^{4} x^{2} + g_{2} {\left({\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}} + \frac{\delta^{2} s_{1} x - {\left({\delta^{2} + 1} \right)}^{2} {\left({h_{2} + s_{1} + g_{2}} \right)}}{{\left({\delta^{2} + 1} \right)}^{2}}. \\& \\\end{aligned} $$

and the Hessian matrix is given by:

$$ \begin{aligned} H_{2} &= {\left({\begin{array}{*{20}c}\frac{\partial \pi^{{\rm rc}}_{2}}{\partial g_{2} g_{2}}& \frac{\partial \pi ^{{\rm rc}}_{2}}{\partial h_{2} g_{2}} \\\frac{\partial \pi ^{{\rm rc}}_{2}}{\partial g_{2} h_{2}}& \frac{\partial \pi ^{{\rm rc}}_{2}}{\partial h_{2} h_{2}} \\\end{array}} \right)} \\ & = {\left({\begin{array}{*{20}c}- \frac{2\delta^{5} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{4} {\left({h_{2} + 1} \right)}^{2}}{{\left[{\delta^{4} x^{2} + g_{2} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}} \right]}^{3}}& \frac{\delta^{5} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left[{\delta^{4} x^{2} - g_{2} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}} \right]}}{{\left[{\delta^{4} x^{2} + g_{2} {\left({\delta ^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}} \right]}^{3}} \\ \frac{\delta^{5} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left[{\delta^{4} x^{2} - g_{2} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}} \right]}}{{\left[{\delta^{4} x^{2} + g_{2} {\left({\delta ^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}} \right]}^{3}}& - \frac{2\delta^{5} g^{2}_{2} x^{3} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{4}}{{\left[{\delta ^{6} x^{2} + g_{2} {\left({\delta^{4} + 2\delta^{2} + 1} \right)}^{2} {\left({h_{2} + 1} \right)}} \right]}^{3}} \\\end{array}} \right)} \\\end{aligned} $$

Substitute g ^*₂ and h ^*₂ the matrix becomes:

$$ H_{2} = {\left({\begin{array}{*{20}c}- \frac{2{\left({\delta^{2} + 1} \right)}}{\delta x}& \frac{{\left({\delta^{2} + 1} \right)}{\left({1 - \delta^{2}} \right)}}{\delta^{3} x} \\ \frac{{\left({\delta^{2} + 1} \right)}{\left({1 - \delta^{2}} \right)}}{\delta^{3} x}& - \frac{2{\left({\delta^{2} + 1} \right)}}{\delta x} \\\end{array}} \right)} $$

Also in this case, let H _2k denote the kth order leading principal submatrix of H ₂ for k = 1,2. The determinant of the kth order leading principal minor of H ₂ is denoted by | H _2k |. The leading principal minors alternate in sign as follows:

$$ {\left| {H_{{21}}} \right|} < 0 $$

$$ {\left| {H_{{22}}} \right|} > 0 \Leftrightarrow \delta > {\sqrt 3}/3 $$

Since the Hessian matrix for agent 2 is not negative semidefinite it is clear that to have an interior solution for an equilibrium the value of δ must be larger than a critical value. That is, the asymmetry in the evaluation of the stake must not be too large.