To save or not to save: The Fisher game

We examine the Fisher market model when buyers, as well as sellers, have an intrinsic value for money. We show that when the buyers have oligopsonistic power they are highly incentivized to act strategically with their monetary reports, as their potential gains are unbounded. This is in contrast to the bounded gains that have been shown when agents strategically report utilities [5]. Our main focus is upon the consequences for social welfare when the buyers act strategically. To this end, we define the Price of Imperfect Competition (PoIC) as the worst case ratio of the welfare at a Nash equilibrium in the induced game compared to the welfare at a Walrasian equilibrium. We prove that the PoIC is at least 1/2 in markets with CES utilities with parameter 0 ≤ ρ ≤ 1 – this includes the classes of Cobb-Douglas and linear utility functions. Furthermore, for linear utility functions, we prove that the PoIC increases as the level of competition in the market increases. Additionally, we prove that a Nash equilibrium exists in the case of Cobb-Douglas utilities. In contrast, we show that Nash equilibria need not exist for linear utilities. However, in that case, good welfare guarantees are still obtained for the Nash dynamics of the game.