Incentive ratio and market equilibrium

Abstract

The thesis considers Fisher market with Constant Elasticity of Substitution (CES) utilities, including Linear utility, Leontief utility and Cobb-Douglas utility, and studies strategic behaviors of individual buyers in market equilibria. The notion of incentive ratio is introduced to capture the extent to which utilities can be increased by strategic behaviors of individuals. Formally, in a given market, the incentive ratio of any given buyer is defined to be his maximum possible utility obtainable by employing strategic behaviors divided by his utility when bidding truthfully, given any fixed bids of all other buyers. The incentive ratio of the whole market is defined to be the maximum incentive ratio of all buyers. Note that the ratio is at least one which happens for an incentive compatible mechanism. While buyers may get larger utilities when they behave strategically, it is interesting to know the range of its improvement. This thesis shows that the benefits of strategic behaviors may be quite limited. For Leontief market, the incentive ratio is at most 2, and we give an example to show that this upper bound is tight. Moreover, the incentive ratio of Leontief market is insensitive to market size. For Linear market, the thesis presents a constructive proof for the incentive ratio of 2-buyer market. For Cobb-Douglas market, the thesis derive the best response strategy of the buyers and the su±cient and necessary condition for bidding truthful to be a dominant strategy for all buyers. We also prove the incentive ratio is independent of the number of buyers. Meanwhile, simulation results show that the incentive ratio of Cobb-Douglas market is indeed small. In addition, the thesis obtains a matching exponential algorithmic bound for TUCKER in the oracle model, in all dimensions. The lower bound is based on a reduction from the Direction Preserving Zero Point (DPZP) to TUCKER. DPZP is a new mathematical structure introduced recently as an appropriate technical tool to formulate the more traditional fixed point computational problems. The upper bound requires a combinatorial parity lemma that relates the existence of the complementary edges on the boundary with certain parity properties of the fully-colored simplices on the boundary of the hypergrid. The connection also allows us to derive an immediate proof that TUCKER is PPAD-complete for all constant dimensions.