The Design

Game theory is usually read forward: given the rules, what will the players do? Mechanism design turns the telescope around. Given the outcome we want, what rules would lead players pursuing their own interests to produce it? The difficulty is already inside the question. The players know things the designer cannot simply inspect: what an object is worth to them, what a task would cost, what they would accept instead. A procedure that asks for these facts changes the advantage of reporting them. If a buyer’s reported enthusiasm raises the price he must pay, discretion begins to look profitable. The designer must therefore build a reason for honesty into the rules, making the information needed for the desired outcome information that its holder is willing to supply.

The second-price auction shows how a small alteration of the rule can supply that reason. One indivisible object goes to the highest bidder, who pays the second-highest bid. Assume private values known to their holders, money entering payoffs one for one, no collusion or outside consequences, and an auctioneer who follows the announced rule. In an invented example, the object is worth ten to you and the highest other bid is seven. Reporting ten wins a surplus of three; shading your bid to six loses it. Raising the report cannot improve the price while you still win. If the highest other bid were eleven, reporting ten would sensibly lose, while reporting twelve would buy a loss of one. More generally, a lower report can forgo a profitable purchase and a higher one can create an unprofitable purchase. Truthful bidding is weakly dominant: for any fixed other bids, a false report offers no higher payoff, though it can sometimes yield the same one.¹

The revelation principle extends the search for truthful rules without promising that every desired outcome is attainable. Begin with a mechanism and a particular equilibrium in which each type chooses some strategy. Construct a direct procedure that asks each player for his type, plays the old strategy assigned to that report, and carries out the old outcome. Truthful reports reproduce the selected equilibrium. If pretending to be another type now brought a profitable deviation, choosing that other type’s old strategy would already have been a profitable deviation before; the original behaviour would not have been an equilibrium. In the Bayesian version, the comparison uses expected payoffs given the player’s information and the equilibrium conduct of others. This is weaker than the auction’s guarantee against every possible opposing bid. The principle permits a search among truthful direct mechanisms while preserving the relevant incentive requirement; it does not silently replace one requirement with the other.²

Making honesty an equilibrium still leaves the designer with a demanding set of conditions to satisfy. The allocation must be physically possible, participants must find the offered terms at least as good as their available alternatives, and each relevant misreport must fail to improve the payoff under the chosen incentive standard. The procedure must also be one that can actually be carried out. Myerson’s account gives the mediator confidential reports and a committed rule for using them; changing who hears a report, or allowing the mediator to revise the promised terms after learning it, can change the game. The revelation principle is powerful because it brings these requirements into a common description. It lets us ask which allocations and transfers can satisfy them together. It supplies no general assurance that players will select the truthful equilibrium whenever one exists, or that an institution can safely dispense with the commitments on which the construction rests.³

A small invented trade problem shows how sensible demands can collide. A buyer values one object at either two or four; its owner values keeping it at either one or three. Each knows his own value. Require trade whenever the buyer’s value exceeds the owner’s, truthfulness to be a dominant strategy, and voluntary participation at every realised pair of values. Require also that the buyer’s payment go exactly to the seller, with no outside subsidy or deduction. When values are two and one, the price cannot exceed two. A buyer worth four, facing that same seller, could report two and still receive the object, so his truthful price cannot be higher. But when values are four and three, the seller must receive at least three. A seller worth one, facing that same buyer, could report three and still sell, so his truthful receipt cannot be lower. At the pair four and one, the same transfer must therefore be no more than two and no less than three. These requirements cannot all be met.⁴

The contradiction identifies a choice the design must make, rather than an instruction to abandon exchange. A fixed price can make acceptance straightforward for buyer and seller while missing some trades that would have benefited both. Outside funds can loosen a payment constraint, while creating another question about who supplies them. A weaker incentive requirement changes what must be checked and which beliefs the result depends upon. Even the judgment that a design is efficient needs its point of view stated. Before anyone learns his value, a procedure can look attractive in expectation; after learning it, some types may prefer another arrangement. Myerson’s trade example distinguishes these evaluations and finds that rules can rank differently under them. An institution’s merits therefore include the opportunities it makes possible, the information and commitments it requires, and the people whose alternatives it treats as decisive.⁵

There is a quiet politics in this engineering, because the objective has to be chosen before the mechanism can be called good. A sale may be designed to place an object with whoever values it most, to obtain revenue for its owner, or to open opportunities for participants who would otherwise be excluded. Those purposes can demand different rules and distribute the gains differently. The mathematics can test whether a rule serves a specified purpose under specified conditions; it cannot confer a right to choose that purpose on whoever happens to possess the mathematics. The earlier plate’s reversal consequently reaches beyond the auction room. Rules are made by someone, and the capacity to make them is itself a form of power. A serious account of design must ask who sets the end, whose interests enter its measure, and who can require the designer to answer for the game the others must play.