Resumen de Three essays on collusion-proof mechanism design
Collusion is a serious concern in real-life allocation problems. Auctions, for example, often create incentives for the bidders to compete less fiercely so as to increase their profits. This thesis contributes to the theoretical literature on how to design collusion-proof mechanisms. Each of the three chapters addresses this question in a specific economic environment, namely multidimensional bargaining (Chapter 1), multi-unit procurement (Chapter 2) and single-good allocation (Chapter 3). A common feature, apart from collusion, is the presence of asymmetric information. In other words, the key problem is that the mechanism designer (e.g. the auctioneer) does not know the agents' preferences (e.g. the bidders' values) and is concerned that they might coordinate their strategies.
As is conventional in the mechanism design literature, we assume the agents' utility functions to be quasilinear in money. The implementation concept is pure dominant-strategy equilibrium. Accordingly, we focus on deterministic mechanisms that are strategy-proof in the sense that each agent's optimal decision is independent of the other agents' strategies. A mechanism is considered to be collusion-proof if no group of agents find it profitable to jointly deviate from their dominant strategies. Exactly what constitutes a joint deviation hinges on how sophisticated a collusive arrangement can be. The central notion studied in this thesis assumes that the colluding agents cannot reallocate among each other after the mechanism has been executed. In particular, they are not able to exchange money. This limited form of collusion is characteristic of a weak cartel. The corresponding concept of collusion-proofness is usually referred to as group strategy-proofness in the literature. Chapter 2 additionally considers strong cartels, whose members can reallocate among each other. This capability increases the scope for collusive behavior and, therefore, restricts the set of collusion-proof mechanisms. In what follows, we explain the specific contribution of each paper in more detail.
Chapter 1: Multidimensional Bargaining and Posted Prices Coauthored with Angel Hernando-Veciana, Chapter 1 studies one of the canonical models of the mechanism design literature: bilateral trade between a seller and a buyer. While the standard setup features a single indivisible good, we allow the two agents to exchange several divisible goods. Owning a bundle of goods is valuable to each agent. The mechanism designer knows that these values are increasing and concave functions but is ignorant of their exact shape. To elicit this information, the designer resorts to a revelation mechanism.
We characterize the set of deterministic mechanisms that provide each agent with a dominant strategy, ensure voluntary participation and deter collusion. There turns out to be a unique type of mechanism that satisfies these three properties (plus an innocuous technical condition). The designer exogenously specifies a set of tradable bundles and a pair of prices for each bundle. The tradable bundles must be totally ordered. The shape of each agent's price function must be such that it induces single-plateaued preferences over bundles. Both agents then announce their optimal bundles and, generally, trade the smaller of the two.
A characteristic feature of these mechanisms is that the prices are exogenous, or "posted". This is entirely due to collusion-proofness, which restricts the impact an agent can have on the other agent's allocation. The same restriction arises if the mechanism is required to be budget balanced, that is, if the seller's and the buyer's payments must coincide. Exploiting this connection, we derive a similar posted-price characterization when collusion-proofness is replaced by budget balance. The resulting set of mechanisms is smaller than before because budget balance forces the mechanism's range to be linear. In other words, the designer merely specifies a baseline bundle and its price. The agents can only trade certain multiples of the baseline bundle. This second result is important because budget balance is a natural restriction in many settings where the designer is unable to subsidize trade or absorb a budget surplus. We provide a unified proof of both results via the property on non-bossiness, which is implied by both collusion-proofness and budget balance.
Chapter 2: Collusion-proof Mechanisms for Multi-Unit Procurement Chapter 2 considers a multi-unit procurement model. A principal wants to buy a discrete number of homogeneous units from a finite set of agents. Providing units is costly for each agent. The cost increases convexly in the number of units provided. The exact shape of these cost functions is unknown to the principal. Real-life examples of this model include electricity markets and business-to-business supply chains.
Our first result characterizes the set of mechanisms robust to the formation of strong cartels. That is, no group of agents should benefit from coordinating their strategies and then reallocating among each other. This requirement is met only by the small class of "fixed-price mechanisms": the principal sets an exogenous price per unit, and then each agent provides his optimal quantity at that price. This finding illustrates that only rigid mechanisms fully prevent collusion.
In a fixed-price mechanism, the quantity procured is endogenous: the larger the agents' costs, the less units they provide. This endogeneity is undesirable if---as is often the case in practice---the principal needs a predetermined quantity. Unfortunately, this shortcoming persists if the principal is concerned with weak rather than strong cartels. As our second result shows, every group-strategy-proof mechanism that induces voluntary participation will sometimes procure no units at all. In other words, the principal cannot set a quantity target.
If the agents cannot reallocate, collusion-proof mechanisms can be more flexible than fixed-price mechanisms. Our third result characterizes the shape of group-strategy-proof mechanisms under an appealing fairness condition: the price per unit should be uniform across agents. We show that, instead of a fixed price, these mechanisms can involve as many prices as there are agents in the economy. The principal first offers the largest price. This price is selected if each agent is willing to provide a positive number of units. On the other hand, if x agents decline to supply any quantity, the principal reduces the price to the (1+x)-th largest. This price is selected if all but x agents wish to provide a positive number of units. If y additional agents now refuse to chip in, the principal reduces the price to the (1+x+y)-th largest, and so on. Finally, each agent supplies his optimal quantity at the selected price. In the special case that the principal offers the same price in each round, this type of mechanism boils down to a fixed-price mechanism. Offering different prices allows the principal to respond to the expected composition of agents present in each round. We show by example that these price adjustments can in fact be optimal for the principal.
Chapter 3: Collusion-proof and Fair Auctions Chapter 3 analyzes the standard auction setting, in which a finite set of agents compete for a single indivisible good. We study whether it is possible to design mechanisms that are both collusion-proof and fair. Collusion-proofness refers to weak cartels. Fairness means that any two agents with the same value for the good should end up with the same utility. Trivially, a mechanism is collusion-proof and fair if it never assigns the good and charges the same lump-sum payment to each agent. Since such mechanisms are obviously unappealing, we focus on non-trivial mechanisms. That is, the agents' utilities should be sensitive to the reports made.
There do exist non-trivial mechanisms that satisfy either collusion-proofness or fairness. The second-price auction, for example, is fair but not collusion-proof: if the second-highest bidder underreports his value, his utility will be unchanged, but the winner will be better off. On the other hand, suppose the auctioneer offers the good at a fixed price to one agent after another until one of them accepts. Although collusion-proof, this mechanism is not fair because the agents ordered first are treated favorably.
We show that the existence of collusion-proof and fair mechanisms depends on the structure of the value domain. Collusion-proofness and fairness are compatible if and only if the maximum of the value domain exists and is bounded away from the next largest value. In particular, this condition is satisfied if there are finitely many values, but it is violated under the standard assumption that the value domain is an interval. The intuition is that collusion-proofness and fairness are opposed to each other. Collusion-proofness requires the mechanism to be rather rigid, whereas fairness calls for flexibility. The two properties clash if and only if the value domain is sufficiently rich.
