A two-stage Bertrand–Edgeworth game

Abstract

In our investigation we are expanding a Bertrand-Edgeworth duopoly into a two-stage game in which during the first stage the firms can select their rationing rule. We will show that under certain conditions the efficient rationing rule is an equilibrium action of the first stage.

Asymmetric price adjustments: A supply side approach

“…As long as the demand is inelastic, this formulation captures any combined rationing rule, including the proportional rationing rule (e.g., Tasnádi 1999). The second line of equation (1) identifies the tie-breaking rule that is used, among others, in Davidson and Deneckere (1986) and Kreps and Scheinkman (1983).…”

Bertrand–Edgeworth equilibrium with a large number of firms

“…Papers in the literature that adopt a strongly manipulable TBR include, for example, Hellwig (1986, 1993), Osborne and Pitchik (1986), Maskin (1986) (the first example provided by him) and Tasnádi (1999b). Appropriately extending the TBRs in these papers to the present context, one can write that…”

“…3 We then claim that the restrictions on r i (p i , p, n) are satisfied by a parametric class of rationing rules (though not the proportional one). Using the combined rationing rule introduced by Tasnádi (1999b), suppose…”

“…For intermediate values of λ, other rationing rules emerge (see Tasnádi (1999b) for an interpretation). Clearly, if d(p i ) is concave then r i (p i , p, n) is decreasing and concave in p i .…”

Bertrand–Edgeworth duopoly with linear costs: A tale of two paradoxes

“…the rationing rule) draws heavily on the combined rationing rule introduced by Tasnádi (1999b). Clearly, for λ = 1 we have the efficient rationing rule, whereas for λ = 0 we have the proportional rationing rule (see Tirole (1988) and Vives (1999) for a discussion of these two rationing rules).…”