Decision Making using Game Theory: An introduction for managers

142Repeated gamesBUPALargesubsidyModeratesubsidySmallSubsidyLargesubsidy20, 20 40, 10 0, 0GHGModeratesubsidy10, 40 30, 30 0, 0Smallsubsidy0, 0 0, 0 25, 25Figure 7.2The NHS subsidy game with multiple Nash equilibria.equilibrium, BUPA in the above scenario, is given by the expression:40 20(t R 1)So breaking the cooperative equilibrium yields a beneWt to the deviantof £30m, or 30/t R per iteration. According to the deWnition of boundedrationality then, the cooperative outcome is an equilibrium as long as:ε30/t RObviously, if t R is very large, this condition is always satisWed, since(30/t R ) 0, and the cooperative outcome (30, 30) becomes prominentat least in the early stages of the game.There are inherent diYculties, however, contained within the notionof bounded rationality. Friedman (1986), for example, argues thatbounded rationality implies that players only calculate optimal strategiesfor a limited number of iterations and that therefore the gamebecomes shorter and the result of backward induction more likely.Avoiding the paradox of backward induction: multiple Nash equilibriaWith multiple Nash equilibria, there is no unique prediction concerningthe last iteration, so players have a credible threat with which toinduce other players to play the cooperative solution. Consider thefollowing game, a variation of the one in Example 7.1, but one in which