By-product mutualism and the ambiguous effects of harsher environments – A game-theoretic model

• We provide a game-theoretic model of by-product mutualism.
• Using this model, we study the effect of harsher environments on cooperation.
• In variants of the model, harsh environments are interpreted in different ways.
• Harsher environments may either encourage or discourage cooperation.

We construct two-player two-strategy game-theoretic models of by-product mutualism, where our focus lies on the way in which the probability of cooperation among players is affected by the degree of adversity facing the players. In our first model, cooperation consists of the production of a public good, and adversity is linked to the degree of complementarity of the players׳ efforts in producing the public good. In our second model, cooperation consists of the defense of a public, and/or a private good with by-product benefits, and adversity is measured by the number of random attacks (e.g., by a predator) facing the players. In both of these models, our analysis confirms the existence of the so-called boomerang effect, which states that in a harsh environment, the individual player has few incentives to unilaterally defect in a situation of joint cooperation. Focusing on such an effect in isolation leads to the “common-enemy” hypothesis that a larger degree of adversity increases the probability of cooperation. Yet, we also find that a sucker effect may simultaneously exist, which says that in a harsh environment, the individual player has few incentives to unilaterally cooperate in a situation of joint defection. Looked at in isolation, the sucker effect leads to the competing hypothesis that a larger degree of adversity decreases the probability of cooperation. Our analysis predicts circumstances in which the “common enemy” hypothesis prevails, and circumstances in which the competing hypothesis prevails.