Adib Bagh

Variational convergence: Approximation and existence of equilibria in discontinuous games

We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.

How to Price Discriminate When Tariff Size Matters

Firms that serve a large market with many diverse consumer types use discriminatory or nonlinear pricing to extract higher revenue, inducing consumers to separate by self-selecting from a large number of tariff options. But the extent of price discrimination must often be tempered by the high costs of devising and managing discriminatory tariffs, including costs of supporting consumers in understanding and making selection from a complex menu of choices. These tariff design trade-offs occur in many industries where firms face many consumer types and each consumer picks the number of units to consume over time. Examples include wireless communication services, other telecom and information technology products, legal plans, fitness clubs, automobile clubs, parking, healthcare plans, and many services and utilities. This paper evaluates alternative ways to price discriminate while accounting for both revenues and tariff management costs. The revenue-maximizing menu of quantity-price bundles can be very or infinitely large and hence not practical. Instead, two-part tariffs 2PTs, which charge a fixed entry fee and a per-unit fee, can extract a large fraction of the optimal revenue with a small menu of choices, and they become more attractive once the costs of tariff management are factored in. We show that three-part tariffs 3PTs, which use an additional instrument, the “free allowance,” are an even more efficient way to price discriminate. A relatively small menu of 3PTs can be more profitable than a menu of 2PTs of any size. This 3PT menu can be designed with less information about consumer preferences relative to the menu of two-part tariffs, which, in order to segment customers optimally, needs fine-grained information about preferences. Our analysis reveals a counterintuitive insight that more-complex tariffs need not always be more profitable; it matters whether the complexity is from many choices or more pricing instruments.

Convergence of set-valued mappings: Equi-outer semicontinuity

The concept of equi-outer semicontinuity allows us to relate the pointwise and the graphical convergence of set-valued-mappings. One of the main results is a compactness criterion that extends the classical Arzelà-Ascolì theorem for continuous functions to this new setting; it also leads to the exploration of the notion of continuous convergence. Equi-lower semicontinuity of functions is related to the outer semicontinuity of epigraphical mappings. Finally, some examples involving set-valued mappings are re-examined in terms of the concepts introduced here.

Doing More with Less: Price Discrimination with Three-Part vs. Two-Part Tariffs

Multi-part tariffs, such as menus of two-part tariffs and three-part tariffs, are widely used in industry, especially for pricing of information goods, online services, telecommunications products, etc. This paper examines the effectiveness of these price structures for price discriminating between heterogeneous customers. We show that a relatively small menu of three-part tariffs (3PTs) can be more profitable and, sometimes, socially more desirable than a larger menu (more items) of two-part tariffs (2PTs). Often, a single three-part tariff can beat a menu of multiple two-part tariffs. Moreover, this 3PT menu has lower hidden costs - it can be designed with less information about consumer preferences, relative to the menu of two-part tariffs. The 3PT structure not only produces higher profit, but has lower managerial and decision complexity.

Existence of equilibria in constrained discontinuous games

We establish the existence of pure strategy equilibria in games with discontinuous payoffs where the set of feasible actions of each player varies, also in a discontinuous fashion, as a function of the actions of the other players. Such games are used in modeling abstract economies and other games where players share common constraints. Our approach circumvents the difficulties that arise from the presence of discontinuities by modifying the original problem and allowing the players to use strategies that possibly lie outside their feasible sets. We then show that each modified game has

An Ergodic Theorem for Random Lagrangians with an Application to Stochastic Programming

\varepsilon

Pure strategy equilibria in Bertrand games with discontinuous demand and asymmetric tie-breaking rules

ε-equilibria points. Under certain conditions, and as the extent of modification becomes smaller and smaller and