Direct revelation mechanism

By the revelation principle (Section 3.2), it is sufficient to consider an incentive compatible direct mechanism. In other words, regardless of the mechanism constructed by the intermediary, given the equilibrium of the mechanism, we can construct an equivalent incentive compatible direct mechanism, where the buyer and the supplier report their respective valuations to the intermediary, and the intermediary determines if the trade is to take place. If so, she determines the buyer s payment and the suppliers revenue. Otherwise, the players take their outside options in a direct matching market. Let T(/3,p, w) represents the direct revelation mechanism, where /3(s, v) is the probability that the trade will take place, p(s, v) is the expected payment to be made by the buyer to the intermediary (the asked price), and w(s, v) is the expected payment from the intermediary to the supplier (the bid price), where s and v are the valuations given by the supplier and buyer, respectively. As mentioned above, the intermediary is aware of the buyer and the supplier s outside options as random variables characterized by distributions G and F, respectively. Based on this information the intermediary establishes the buyer s virtual willingness to pay follows ... 

Let r P,p,w) represents the direct revelation mechanism, where Pi v) is the probability that buyer i will get the object, Pi v) is the expected payment from buyer i to the intermediary, and w v) is the expected payment from the intermediary to the supplier. Given mechanism F(/ ,p, w), knowledge of her own valuation Vi, and other bidders j i valuations in terms of gj vj) buyer i s expected gain from the trade is as follows ... 

McAfee (1992) proposes a double auction model that explicitly considers the role of an intermediary who intervenes in the trade and keeps track of supply and demand at asked and bid prices. Like the market specialist in NYSE, the intermediary makes a profit by regulating the trade using a certain mechanism. In the following, we describe this double auction as a direct revelation mechanism. 

Optimal Multiattribute Auctions. Che [25] has proposed optimal multiattribute auctions for the special case of seller cost functions that are defined in terms of a single unknown parameter. Che s auctions are direct-revelation mechanisms, and he considers both first-price and second-price variations. The second-price variation is exactly the one-sided VCG mechanism, and Che is able to derive the optimal scoring function (or reported cost function) that the buyer should state to maximize her payoff in equilibrium. Branco [21] extends the analysis to consider the case of correlated seller cost functions. No optimal multiattribute auction is known for a more general formulation of the problem, for example for the case of preferential-independence. 

Conitzer Sandholm [31] propose automated mechanism design, in which a computational method is used to design mechanisms with respect to highly-enumerative description of the function space and agent type space. The challenge in this automated MD approach is to develop structured representations of the problem to constrain the input size to the optimization. However, automated MD cannot solve the wider issues presented by the second two problems because it is only applicable to direct revelation mechanisms and because it continues to ignore computational considerations in the formulation of the problem. 

In a direct mechanism, the auctioneer announces how the goods will be allocated and the payments set as a function of the private information reported by the bidders. It remains then for the bidders to report (not necessarily truthfully) their private information. Once the reports are made, the auctioneer applies the announced function and the resulting allocations made and payments executed. One can think of a direct mechanism as being a sealed bid auction, but with the bids being complex enough to communicate the private information of the agents. If the direct mechanism has the property that it is in the bidders interests, suitably defined, to report their private information truthfully, call it a direct revelation mechanism. 

Given any mechanism, there is a corresponding direct revelation mechanism which will produce the same outcome as the given mechanism. Thus, for the purposes of analysis, one can restrict attention to direct revelation mechanisms. The revelation principle does pose a problem. If a direct mechanism suffices why do we observe indirect mechanisms in practice We defer a discussion of this to sections 5.3 and 6. 

By the revelation principle we can restrict attention to direct revelation mechanisms. Each agent is asked to announce her type and the auctioneer, as a function of the announcements, decides what element of F to pick and what payments each agent will make. 

In the general framework of supply chain intermediation (Section 3.2), an intermediary can be characterized in a mechanism design fi mework using the revelation principle, requiring the specification of a direct revelation mecha-... 

Revelation of the mechanisms used by rock-inhabiting microscopic life to establish itself on rock surfaces can answer a broad spectrum of fundamental questions. Although no direct analogies can be drawn between fossil organisms and their modern counterparts, understanding the rules and laws of rock-inhabiting community development on modern Earth could elucidate the principles of land colonization in the far past. 

The beauty of the revelation principle is that it allows theoretical impossibility and possibility results to be established in the space of direct mechanisms, and carried over to apply to all mechanisms. For example, an indirect mechanism can be constructed with a particular set of properties only if a direct mechanism can also be constructed with the same set of properties. 

The communication complexity of a mechanism considers the size of messages that must be sent between agents and the mechanism to implement the outcome of a mechanism. To motivate this problem, recall that mechanism design often makes an appeal to the revelation-principle and considers direct mechanisms. However, direct mechanisms require agents to report complete and exact information about their type, which is often unreasonable in problems such as combinatorial auctions. In the worst-case the VCG mechanism for a combinatorial auction requires each agent to submit 2 numbers, given M items, to report its complete valuation function. 

One can imagine a mechanism that is not direct . Consider one that involves multiple rounds where participation in one round is conditional on bids or other information revealed in previous rounds. These indirect mechanisms can be quite baroque. It is here that the revelation principle comes to our aid. 

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