Social leverage, a core mechanism of cooperation. Locality, assortment, and network evolution

Abstract

Spontaneous emergence of institutionalized cooperation in ubiquitous social dilemmas still is a field of highest relevance in behavioral and organizational economic research. In contrast to the theoretical prediction of defective behavior, manifold forms and degrees of cooperation exist in reality. We explain the emergence of general cooperation, even in one-shot interactions with strangers, from local interactions in a network through what we call a social-leverage mechanism. By this, more agents than just the two interaction partners get involved in an interaction, particularly common acquaintances. We analyze the social-leverage mechanism and conditions of cooperation under locality, common acquaintanceship, related assortativity, as well as “weak” and “strong ties” in the social network. We trace the co-evolution of the network, its structural dynamics, and stable cooperative equilibria. Our model relates to the tradition of emergent coordination in decentralized, non-Walrasian search, coordination and exchange systems (e.g., Diamond 1984; Axtell 2005) and in purely local interactions on, e.g., ring networks (e.g., Albin and Foley J Econ Behav Organ 18(1), 27–51, 1992). We conclude that, under social leverage, just local interaction may generate and stabilize general cooperation. We consider this the emergence of an exchange and trade system (a market) and relate this to empirical cases of the emergence of exchange cultures in the early Silk Road (seventh–ninth century) and in contemporary African countries, when formal enforcement through states and courts are largely lacking. We conclude applications and policy implications for cutting-edge techno-organizational areas of AI and platform economies.

Appendix Proof of Lemma 4.1

If player i chooses defection, his expected payoffs are

$$ {\displaystyle \begin{array}{c}R+\delta \left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)+{\theta}_{ik}\right)R+\dots +{\delta}^{t-1}\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R+{\delta}^tT+{\delta}^{t+1}0\\ {}+\dots \\ {}=\frac{\Big(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)R\left(\ 1-{\delta}^t\ \right)}{1-\delta }+R+{\delta}^tT.\end{array}} $$

If player i chooses cooperation, his expected payoffs are

$$ {\displaystyle \begin{array}{c}R+\delta \left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R+\dots +{\delta}^{t-1}\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R+{\delta}^{t+1}\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R+\dots \\ {}=\frac{\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R}{\ 1-\delta }+R.\end{array}} $$

Thus, cooperation is sustained, if

$$ {\displaystyle \begin{array}{c}\frac{\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R}{\ 1-\delta }+R>\frac{\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R\left(\ 1-{\delta}^t\ \right)}{1-\delta }+R+{\delta}^tT\\ {}\to \delta >{\delta}^1\equiv \frac{T-\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right)R}{T}\in \left(0,1\right).\end{array}} $$

Similarly for agents j, k:

$$ {\displaystyle \begin{array}{l}\delta >{\delta}^2\equiv \frac{T-\left(\mathrm{\mathbb{P}}\left(k,j\right)+\mathrm{\mathbb{P}}\left(k,i\right)\right)R}{T}\in \left(0,1\right)\\ {}\to \delta >{\delta}^3\equiv \frac{T-\left(\mathrm{\mathbb{P}}\left(j,i\right)+\mathrm{\mathbb{P}}\left(j,k\right)\right)R}{T}\in \left(0,1\right).\end{array}} $$

Cooperation is an equilibrium in the group if

$$ {\displaystyle \begin{array}{l}\delta >\max\ \left\{{\delta}^1,{\delta}^2,{\delta}^3\right\},\\ {}\delta >\frac{T-\min \Big[\left(\mathrm{\mathbb{P}}\left(i,j\right)+\mathrm{\mathbb{P}}\left(i,k\right)\right),\left(\mathrm{\mathbb{P}}\left(i,\kern0.5em j\right)+\mathrm{\mathbb{P}}\left(j,k\right)\right),\left(\mathrm{\mathbb{P}}\left(i,k\right)+\mathrm{\mathbb{P}}\left(j,k\right)\right]R}{T}.\end{array}} $$