Limitations of Incentive Compatibility on Discrete Type Spaces

In the design of incentive compatible mechanisms, a common approach is to enforce incentive compatibility as constraints in programs that optimize over feasible mechanisms. Such constraints are often imposed on sparsified representations of the type spaces, such as their discretizations or samples, in order for the program to be manageable. In this work, we explore limitations of this approach, by studying whether all dominant strategy incentive compatible mechanisms on a set T of discrete types can be extended to the convex hull of T.Dobzinski, Fu and Kleinberg (2015) answered the question affirmatively for all settings where types are single dimensional. It is not difficult to show that the same holds when the set of feasible outcomes is downward closed. In this work we show that the question has a negative answer for certain non-downward-closed settings with multi-dimensional types. This result should call for caution in the use of the said approach to enforcing incentive compatibility beyond single-dimensional preferences and downward closed feasible outcomes.

Mechanism design without quasilinearity

This paper studies a model of mechanism design with transfers where agents' preferences need not be quasilinear. In such a model, (i) we characterize dominant strategy incentive compatible mechanisms using a monotonicity property, (ii) we establish a revenue uniqueness result (for every dominant strategy implementable allocation rule, there is a unique payment rule that can implement it), and (iii) we show that every dominant strategy incentive compatible, individually rational, and revenue‐maximizing mechanism must charge zero payment for the worst alternative (outside option). These results are applicable in a wide variety of problems (single object auction, multiple object auction, public good provision, etc.) under suitable richness of type space. In particular, our results are applicable to two important type spaces: (a) type space containing an arbitrarily small perturbation of quasilinear type space and (b) type space containing all positive income effect preferences.

Approximately Maximizing the Broker's Profit in a Two-sided Market

We study how to maximize the broker's (expected) profit in a two-sided market, where she buys items from a set of sellers and resells them to a set of buyers. Each seller has a single item to sell and holds a private value on her item, and each buyer has a valuation function over the bundles of the sellers' items. We consider the Bayesian setting where the agents' values/valuations are independently drawn from prior distributions, and aim at designing dominant-strategy incentive-compatible (DSIC) mechanisms that are approximately optimal. Production-cost markets, where each item has a publicly-known cost to be produced, provide a platform for us to study two-sided markets. Briefly, we show how to covert a mechanism for production-cost markets into a mechanism for the broker, whenever the former satisfies cost-monotonicity. This reduction holds even when buyers have general combinatorial valuation functions. When the buyers' valuations are additive, we generalize an existing mechanism to production-cost markets in an approximation-preserving way. We then show that the resulting mechanism is cost-monotone and thus can be converted into an 8-approximation mechanism for two-sided markets.

Multi-Unit Bilateral Trade

We characterise the set of dominant strategy incentive compatible (DSIC), strongly budget balanced (SBB), and ex-post individually rational (IR) mechanisms for the multi-unit bilateral trade setting. In such a setting there is a single buyer and a single seller who holds a finite number k of identical items. The mechanism has to decide how many units of the item are transferred from the seller to the buyer and how much money is transferred from the buyer to the seller. We consider two classes of valuation functions for the buyer and seller: Valuations that are increasing in the number of units in possession, and the more specific class of valuations that are increasing and submodular.Furthermore, we present some approximation results about the performance of certain such mechanisms, in terms of social welfare: For increasing submodular valuation functions, we show the existence of a deterministic 2-approximation mechanism and a randomised e/(1 − e) approximation mechanism, matching the best known bounds for the single-item setting.

Autocratic Mechanisms: A Form of Dictatorship in Constrained Combinatorial Auctions

We characterize the space of deterministic, dominant-strategy incentive compatible, individually rational, and Pareto-optimal combinatorial auctions where efficiency is not required. We examine a model with two players and k nonidentical items (2k outcomes), multidimensional types, private values, non-negative prices, and quasilinear preferences for the players with one relaxation — the players are subject to publicly-known budget constraints. We show that if it is publicly known that the players value the bundles more than the smaller of their budgets then the studied space includes one type of mechanism: autocratic mechanisms (a form of dictatorship). Furthermore, we prove that there are families of autocratic mechanisms that uniquely fulfill the basic properties of deterministic, dominant-strategy incentive compatible, individually rational, and Pareto-optimal. Interestingly the above basic properties are a weaker requirement than it may initially appear, as the property of Pareto optimality in our model of budget-constrained players and non-negative prices do not coincide with welfare maximization, i.e., efficiency as such is a much weaker requirement.

Dictatorial Mechanisms in Constrained Combinatorial Auctions

We study the possibility space of deterministic, dominant-strategy incentive compatible, individually rational, and Pareto efficient combinatorial auctions in a model with two players and two nonidentical items (four outcomes). Our model has multidimensional types, private values, nonnegative prices, and quasilinear preferences for the players with one relaxation – the players are subject to publicly known budget constraints. We show that the space we study essentially includes one type of mechanisms: autocratic mechanisms (a form of dictatorship). Furthermore, we prove that there are families of autocratic mechanisms that uniquely fulfill the basic properties of deterministic, dominant-strategy incentive compatible, individually rational, and Pareto efficient. The mechanisms in the autocratic families are identical except for two to three price parameters that differentiate them.