Runaway Competition: A Correction and Extension of Results for a Model of Competitive Helping

We investigate and generalize an existing model of competitive helping within a biological market, first introduced for a population of competing individuals in which one individual provides help to all others; the rest compete for the help available from this individual by providing help themselves. Our generalized model comprises two strategies in which each individual of a specific type provides the same amount of help as all other individuals of that type. Each individual's fitness function is dependent on this level of help, the cost of providing the help, and the fact that help is proportionally reciprocated by other individuals. Competitive helping occurs when individuals actively try to help more than other individuals. To assess the emergence of equilibrium help strategies as adopted by proportions of the population, we examine the competition over available help within two settings: replicator dynamics and agent-based numerical simulations. To move one step further in our generalization, we use the agent-based model to study the N-person competitive helping game, where all individuals in the population are heterogeneous with respect to help provided. Our results show that helping does not increase indefinitely with the population size, as concluded previously, and while there are some instances of an increase in help provided as a result of competition, this competition can be detrimental to all individuals and in most cases, one type simply gives up (thus evolving to a "no help" strategy). The degree to which an individual's help is reciprocated by the others in the population has strong implications in the long-term behaviour of equilibrium help levels of types of individuals (and of individuals themselves); these equilibrium help levels diverge from existing conjectures in current literature. Lastly, small amounts of passively provided (costless) help results in runaway competition among all individuals.