Uncertainty, instrument choice, and the uniqueness of Nash equilibrium: microeconomic and macroeconomic examples
This paper contains two examples of static, symmetric, positive-sum games with two strategic players and a play by nature: (1) a microeconomic game between duopolists with joint costs facing uncertain demands for differentiated goods and (2) a macroeconomic game between two countries' with inflation-bias preferences confronting uncertain demands for moneys. In both examples, each player can choose either of two variables as an instrument, and reaction functions are linear in the chosen instruments. With no uncertainty, there are four (Nash) equilibria, one for each possible instrument pair, because each player is indifferent between instruments given the instrument choice and instrument value of the other player. With uncertainty in the form of an additive disturbance, there are fewer equilibria because each player is not indifferent between instruments. These results are in accordance with the logic of Poole (1970) and Weitzman (1974) as explained by Klemperer and Meyer (1986) using examples of differentiated duopoly games with independent costs. In their main example with linear reaction functions, there is always a unique equilibrium. In contrast, in each of our examples with uncertainty, there is a unique equilibrium for some parameter values, but there are two equilibria for others. It is somewhat surprising that in both the Klmperer and Meyer example and our examples with unique equilibria, for some parameter values with the smallest amount of uncertainty the symmetric instrument pair chosen in the unique equilibrium is the one that yields the lower payoff with no uncertainty.
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