Reducing courtship time promotes marital bliss: The Battle of the Sexes game revisited with costs measured as time lost

Abstract

Classic bimatrix games, that are based on pair-wise interactions between two opponents belonging to different populations, do not consider the cost of time. In this article, we build on an old idea that lost opportunity costs affect individual fitness. We calculate fitnesses of each strategy for a two-strategy bimatrix game at the equilibrium distribution of the pair formation process that includes activity times. This general approach is then applied to the Battle of the Sexes game where we analyze the evolutionary outcome by finding the Nash equilibria (NE) of this time-constrained game when courtship and child rearing costs are measured by time lost. While the classic Battle of the Sexes game has either a unique strict NE (specifically, all males exhibit Philanderer behavior and either all females are Coy or all are Fast depending on model parameters), or a unique interior NE where both sexes exhibit mixed behavior, including time costs for courtship and child rearing changes this prediction. First, (Philanderer, Coy) is never a NE. Second, if the benefit of having offspring is independent of parental strategies, (Philanderer, Fast) is the unique strict NE but a second stable interior NE emerges when courtship time is sufficiently short. In fact, as courtship time becomes shorter, this mixed NE (where most males are Faithful and the Coy female population is increasing) attracts almost all initial population configurations. Third, this latter promotion of marital bliss also occurs when parents who share in child rearing receive a higher benefit from their offspring than those that don't. Finally, for courtship time of moderate duration, the same phenomenon occurs when the population size increases.

Using chemical reaction network theory to show stability of distributional dynamics in game theory

“…Stability of group distribution dynamics for two strategy games. Now suppose the population has 2 phenotypes, Cooperate (C) and Defect (D), and that individuals are either single or in groups of size m. 7 The prototypical example of interactions within a group is the m−player public goods game (PGG) where each group member contributes a portion of their identical endowment to the public good and receives an equal share of the total contribution multiplied by an enhancement factor r. In the classic set-up for the one-shot PGG, individuals are always in groups that play the game once. Under the usual assumption that 1 < r < m, it is best from an individual perspective to contribute nothing given the choices of the other members of his/her group (i.e., mutual Defection is the only Nash equilibrium (NE)) [e.g., 8,23,18].…”

“…The first example (Section 3.1) is the Battle of the Sexes (BoS) game [9] where pair interaction times and times to care for offspring or for courtship are strategy dependent. Cressman and Křivan [7] assumed that the resulting distributional equilibrium is unique and stable and they analysed the Nash equilibria of the game as a function of interaction times. Again, using chemical reaction theory (CRNT) we prove that the distributional equilibrium is stable.…”

“…The BoS game is somewhat particular in that it assumes that some interaction times are equal to zero (e.g., Philanderer males do not interact with Coy females or Philanderer males do not spend any time caring for offspring) which simplifies the distributional dynamics. As a second example, we consider a generalised BoS game of [7] where we assume all pair interaction times as well as times to recover for disbanded individuals before pairing again can be positive. Then we show that this additional "mixing" prevents us to apply the Deficiency Zero Theorem for the resulting distributional dynamics.…”

“…The reaction network (7) Reaction network (8) defines a kinetics matrix K which has initial complexes in columns and final complexes in rows and the entries of the matrix are the corresponding reaction rates. With elements of C and S listed in the same order as above, the kinetics matrix is…”

“…When the kinetics matrix K and stoichiometric matrix Y as well as the vector ψ are extended to games with a general number m of strategies, f in (9) becomes the (m + m 2 )−dimensional vector field for the distributional dynamics (2). To apply CRNT to this MAK system, we must consider the set of reactions in (7) given by…”