Decision Making using Game Theory: An introduction for managers

117The Cournot, von Stackelberg and Bertrand duopolies: an application of mixed-motive gamesIf c 1 is the marginal cost of production per tonne of linerboard for SmurWt-Stone and c 2 is the marginal cost of production per tonne for InternationalPaper, both constants for the year, how many tonnes should each Wrm producein order to maximise proWt? The New York Stock Exchange, of which both Wrmsare conscientious members, stipulates that Wrms must set their price structuresand production levels independently.SmurWt-Stone and International Paper are not strictly in competitionwith each other, but are partly competing and partly cooperating in amarket. Although neither Wrm could alone cater for the entire market,it can be assumed that each Wrm could supply any non-negative level ofoutput within reason, so the duopoly can be modelled as a mixedmotivegame. Each Wrm needs to maximise proWt subject to what themarket will take. Once SmurWt-Stone and International Paper havedecided on their respective optimal levels of production, the marketprice is made, the pay-oVs (proWts) are eVectively determined and thegame is assumed to be over. (The game is assumed to be a one-oV.)The cost to SmurWt-Stone of producing R 1 tonnes is c 1 R 1 and the costto International Paper of producing R 2 tonnes is c 2 R 2 , assuming noWxed costs. Therefore, the proWt functions are:Ψ 1 (A R)R 1 c 1 R 1Ψ 2 (A R)R 2 c 2 R 2where Ψ 1 represents SmurWt-Stone’s proWt and Ψ 2 represents that ofInternational Paper. (Notice that the proWt for each Wrm depends onthe output of the other Wrm as well as its own, since R R 1 R 2 .)Substituting for R gives:Ψ 1 (A R 1 R 2 )R 1 c 1 R 1 AR 1 R 12 R 1 R 2 c 1 R 1andΨ 2 (A R 1 R 2 )R 2 c 2 R 2 AR 2 R 22 R 1 R 2 c 2 R 2The Wrms choose their strategies independently and simultaneously, sothe concept of the Nash equilibrium oVers a solution. This involvesdrawing each Wrm’s reaction function, which is a curve that shows every