Bertrand Equilibria and Sharing Rules

We analyze how sharing rules affect Nash equilibria in Bertrand games, where the sharing of profits at ties is a decisive assumption. Necessary conditions for either positive or zero equilibrium profits are derived. Zero profit equilibria are shown to exist under weak conditions if the sharing rule is “sign-preserving”. For Bertrand markets we define the class of “expectation sharing rules”, where profits at ties are derived from some distribution of quantities. In this class the winner-take-all sharing rule is the only one that is always sign-preserving, while for each pair of demand and cost functions there may be many others. JEL: C72, D43, L13