A DYNAMIC VARIANT OF THE BATTLE OF THE SEXES

This paper proposes a differential game to introduce dynamic interacti ons into the well known battle of the sexes: the husband prefers one a ctivity ('boxing') and the wife another ('ballet'). Although the game is played presumably non-cooperatively, both are interested and willin g to invest into the development and maintenance of their personal rel ationship which is only possible if they spend some time together. We show for the symmetric version of the game that non-cooperative commit ment strategies (i.e. open-loop strategies) lead to more 'harmony' tha n linear-Markov-perfect strategies where egoistic behaviour is much mo re pronounced. These linear-Markov-perfect Strategies constrain the se t of smooth, stable but nonlinear-Markov strategies in such a manner t hat all these nonlinear-Markov-perfect strategies lead to less egoism and more harmony. Furthermore, particular nonlinear strategies may ind uce two steady states (depending on history) where one of the two stea dy states may result in a level of harmony that exceeds the cooperativ e outcome: under these circumstances, non-cooperation docs not harm th e public good harmony, in stark contrast to a play in linear-Markov-pe rfect strategies. Finally, numerical simulations of asymmetric situati ons confirm the results that were analytically obtained for the symmet ric game.