Multilateral limit pricing in price-setting games

In this paper, we characterize the set of pure strategy undominated equilibria in differentiated Bertrand oligopolies with linear demand and constant unit costs when firms may prefer not to produce. When all firms are active, there is a unique equilibrium. However, there is a continuum of non-equivalent Bertrand equilibria on a wide range of parameter values when the number of firms (n) is more than two and n⁎∈[2,n−1] firms are active. In each such equilibrium, the firms that are relatively more cost or quality efficient limit their prices to induce the exit of their rival(s). When n≥3, this game does not need to satisfy supermodularity, the single-crossing property, or log-supermodularity. Moreover, the best responses might have negative slopes. Our main results extend to a Stackelberg entry game where some established incumbents first set their prices, and then a potential entrant sets its price.