The Dynamics of Bertrand Price Competition With Cost‐reducing Investments

Abstract

Abstract:We extend the classic Bertrand duopoly model of price competition to a dynamic setting where competing duopolists invest in a stochastically improving production technology to "leapfrog" their rival and attain temporary low cost leadership. We find a huge multiplicity of Markov perfect equilibria (MPE) and show that when firms move simultaneously the set of all MPE payoffs is a triangle that includes monopoly payoffs and a symmetric zero mixed strategy payoff. When firms move asynchronously, the set of MPE payoffs is strictly within this triangle, but there still is a vast multiplicity of MPE, most of which involve leapfrogging.

Has Dynamic Programming Improved Decision Making?

“…It is not clear that, despite the superhuman performance of AlphaZero, the training converges to Nash equilibrium strategies, because it could cycle as in the well-known fictitious play algorithm for finding Nash equilibria (Brown 1951). Iskhakov et al (2018), using the recursive lexicographical search (RLS) algorithm of Iskhakov et al (2015), characterize all equilibria to a class of dynamic directional games with Bertrand price competition and leapfrogging investments that have many fewer states than chess, yet have billions of equilibria. It is not clear that RL algorithms trained on copies of themselves will also result in strategies that are approximate best responses to other strategies, and it may be possible to train new generations of algorithms to exploit weaknesses in AlphaZero, similar to what was done in the evolutionary tournament of competing trading strategies in the double auction market, as reported by Miller et al (1993).…”