Stackelberg leader in a collective action model
Within a mathematical modelling framework, we analyze the conditions allowing a self-governedcollective to achieve Stackelberg equilibrium. It is assumed that members of the collective generate common income through individual effort, which income is then distributed among all members of the collective according to their predetermined share. Effort invested by each agent wields (imposes) a positive influence on the marginal income resulting from the effort invested by any other agent. Each member of the collective aims to maximize their individual gain. Within a model built on the most general principles, it is shown that a Stackelberg equilibrium outcome is preferable over Nash equilibrium. The model, utilizing the special case of income function and private costs functions; helps identify the correlation between the agents’ efforts and their individual characteristics such as the agent’s share in the income, income elasticity by effort, and subjective valuation of private costs. It is shown that additional income generatedby the move (transition) from Nash to Stackelberg equilibrium depends only on the elasticity of income vs. the leader’s effort, and the sum of elasticity indexes for all members of the collective. We introduce the definition and the conditions for the existence of a distinctive agent, who acts as a Stackelberg leader and ensures maximum individual gain for each member of the collective (including their own). The absence of a distinctive agent in a collective gives rise to the Stackelberg leadership problem, as each member of the collective is only able to obtain maximum gains when acting as a follower.