Decision Making using Game Theory: An introduction for managers

119The Cournot, von Stackelberg and Bertrand duopolies: an application of mixed-motive gamesR N2 A c 1 2c 23The total output at the Nash equilibrium is therefore:R N R N2 R N2 2A c 1 c 23and the maximum price that can be aVorded by the market is:P(R) A R N A 2A c 1 c 23 A c 1 c 23This solution, being a Nash equilibrium, means that both Wrms arecontent operating on these parameters, since neither feels a tendency todeviate.Figure 6.11 represents the Cournot duopoly diagrammatically – thescales assume that c 1 and c 2 are of the same order of magnitude, thoughnot necessarily equal – and further analysis is interesting, if a littlecomplicated.The two reaction function lines, having negative gradients, demonstratethe fact that the optimal level of production (and hence proWt)for one Wrm is negatively related to the levels of production of theother. The semicircles on Figure 6.11 represent a pair of the Wrms’iso-proWt curves, which plot the diVerent combinations of productionlevels that yield the same proWt for each Wrm in turn. Not surprisingly,proWts are greatest when each Wrm is the sole supplier, in which case itsupplies (A c 1 )/2or(A c 2 )/2 tonnes of linerboard to the market –the intercept value of the respective reaction functions.The sets of iso-proWt curves for SmurWt-Stone and InternationalPaper are centred, respectively, at the Nash equilibrium coordinates(A 2c 1 c 2 )/3 and (A c 1 2c 2 )/3 and these production levelsrepresent the solution. The further away a semicircle is from its axis –from its monopolistic situation – the lower the Wrm’s proWt. At theNash equilibrium, marked N on Figure 6.11, both Wrms are maximisingproWts simultaneously, so the equilibrium is unique and the tangents to