  We examine the tuning of cooperative behavior in repeated multi-agent games
using an analytically tractable, continuous-time, nonlinear model of opinion
dynamics. Each modeled agent updates its real-valued opinion about each
available strategy in response to payoffs and other agent opinions, as observed
over a network. We show how the model provides a principled and systematic
means to investigate behavior of agents that select strategies using
rationality and reciprocity, key features of human decision-making in social
dilemmas. For two-strategy games, we use bifurcation analysis to prove
conditions for the bistability of two equilibria and conditions for the first
(second) equilibrium to reflect all agents favoring the first (second)
strategy. We prove how model parameters, e.g., level of attention to opinions
of others (reciprocity), network structure, and payoffs, influence dynamics
and, notably, the size of the region of attraction to each stable equilibrium.
We provide insights by examining the tuning of the bistability of mutual
cooperation and mutual defection and their regions of attraction for the
repeated prisoner's dilemma and the repeated multi-agent public goods game. Our
results generalize to games with more strategies, heterogeneity, and additional
feedback dynamics, such as those designed to elicit cooperation.
