Mechanisms

Note: A lot of what follows here is a simple presentation of material detailed in other sources, of which I will, naturally, mention here my own book, co-written with four graduate students of mine (well, they were all graduate students during the writing): A Toolbox for Economic Design, by Dimitrios Diamantaras, Karen A. Campbell, Emina I. Cardamone, Scott Deacle, and Lisa A. Delgado, Palgrave Macmillan, 2009. I am also developing presentation slides on this material for use in my graduate microeconomics class, and I will be posting the slides here as they become available.

Economists have a technical meaning for the word mechanism. It refers to a set of rules that prescribe how agents can interact in some economic context. The context is called a domain.

Start with a description of an economic context. Perhaps you are interested in an auction? Then your domain is a description of the objects to be auctioned and the agents who will bid. Or maybe you care to study a pure example of economic exchange. Then your context is a description of the trading agents and the goods that are available.

For the theory to have grist for its mill, at this stage your economic context/domain should be described in detail, but I skip this in this introduction.

OK, so now you have a domain and would like to get the agents in that context to behave in such a way as to achieve some social goal for them. You do not impose the goal, you take it as given and just ask how to achieve it. However, you are smart enough to realize that the agents will act according to their incentives, and not according to your desire to achieve the goal.

This is where game theory comes in. You want to predict how the agents will play (call them players now), and you’re going to set things up so that they will have certain actions at their disposal and will know how the actions they each take combine to form consequences for each of them, and the consequences come from the domain that’s relevant for the problem. Once their preferences regarding the possible consequences are present (and they are the moment the agents engage in the mechanism), the mechanism becomes a properly specified game. “All” you need now is a theory of how agents play games like this one.

The quotes are there for a reason though—and if you know game theory, you share the irony they denote. There are many theories of how players in a game play the game. This immediately creates many flavors of the theory of mechanism design.

The basic distinction between these theories revolves around what players know. Ideally, we would like a theory that gives a definite prediction of what the players will do in any game, and one such theory kind of exists (if you allow it to not always produce a prediction, that is). This theory says that each player looks for a strategy that’s the best in terms of consequences for her, independently of what the other players choose; call such a strategy a dominant strategy for this player. So this theory tells us that players play their dominant strategies.

If you start with any game chosen at random, this theory has a big problem: chances are the game has at least one player who has no dominant strategy. This is because what’s the best consequence for this player changes depending on what strategies the other players choose.

But wait! When we design a mechanism for our players to participate in, can’t we make it so that it always gives every player a dominant strategy? After all, our designed mechanism does not result in some random game. We can do better. Can we?

The answer is unfortunately that we can’t in general. If our domain is wide enough to encompass all kinds of preferences that the mechanism participants may have, the famous Gibbard-Satterthwaite theorem shows that, no matter what mechanism we design, there will be at least one economy that belongs in our domain such that, if that economy is the one in which the mechanism actually operates, then at least one of the players will find that playing as we, the designers, want, is not his or her dominant strategy. So there will be circumstances in which at least one player will have an incentive to not play as desired by the designer.

Oh, I almost forgot: there is an escape from this conclusion. If what we are trying to achieve by the mechanism we design is to make one person a dictator, so that his or her choice always prevail, then there is no problem in designing a mechanism in which every agent has the incentive to play as we want. Naturally! In such a mechanism, the dictator is the only one with moves powerful enough to affect the outcome, and why would he not play in accordance to the designer’s desire, which is to always serve the dictator’s interest? But we can hardly accept dictatorship as an acceptable resolution of the problem that the Gibbard-Satterthwaite theorem presents.

<More soon, including lecture notes. The first set of lecture notes, on social choice theory, is now available via this post, and the second set, on dominant strategy implementation, is available via this post.>

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