机制设计与激励：经济学与计算机科学的信任观对比

[SQUEAKING] [RUSTLING] [CLICKING] ROBERT M. TOWNSEND: So let me just say that, shockingly, we have arrived at lecture 8, which means we're well more than halfway done with the lectures. So today is continuing the theme of trying to bring computer science and economics onto the same page. And the juxtaposition is everywhere today, so "Mechanism Design and Incentives" has to do with the way economists think about trust versus "Protocols and Notions of Trust" in computer science.

The longer title is "Private Information, Incentives to Report and Take Actions"-- as in mechanism design versus illustrative Byzantine Generals problem that utilizes a different notion of trust to emphasize the difference between economics and computer science in the way we think about algorithms, which is not to say there isn't a middle ground. I'm trying to avoid a bipolar view, but that is the way the lectures are laid out.

So the outlined is information constrained allocations. And I'll do it through an example, Although the principles generalize. And we'll think about implementing this without a planner in the jargon of economics, rather, using the tools of computer science. So that helps mitigate the bipolar view of economics versus computer science. We will utilize the tools of computer science very heavily within the mechanism design problem.

And then we'll go on to talk about algorithms for validation and so on and readdress this difference by an illustrative paper of Steve Morris and Hyun Shin on the Byzantine generals problem. So first half-- Information Constrained Allocations vis a vis this insurance example-- how to implement without a planner using the tools of computer science. So to be specific, I will talk about an agrarian economy. But that said, we can think about this notation for the same model as applying much more generally to today's financial markets.

So it's a pure exchange economy. There's only one period for now. There's two agents, labeled 1 and 2, and a K-dimensional vector of endowments. The endowment of agent 1, the villa, so to speak, is seen by that agent alone. So shocks are private to the agent. And we can parameterize this endowment, which is e1 for agent 1, and e is endowment as a function of epsilon by a simpler notation, where theta is this parameter theta, taking realizations in some larger set with probabilities p of theta.

So that's for the villa. And agent 2 is like a central monastery. Agent 2's endowment for simplicity is public. For that matter, agent 2 is going to be risk-neutral. Agent 1 is going to be risk-averse. And again, we have this K-dimensional vector, publicly observed vector of, quote, "endowments" for the monastery. So these two agents are going to agree to some rule for allocating resources-- some rule f, function f, mapping a message sent from agent one to agent 2 into positive and negative transfers of each of the items in this K-dimensional vector.

It will be in the notation as if the villa is paying a tax, as if the transfer, if positive, is going from the via to the monastery. But transfers can be negative. This is, after all, an insurance example. The agent has a random endowment, is risk-averse, and the monastery agent 2 is willing to undertake that insurance. But the problem will be how to mitigate the damage caused from the private information if you can.

So they agree to a resource allocation scheme, which works as follows. Villa 1, agent 1 waits to see the output vector parameter theta, then sends a message m to the monastery, and the allocation would be f of m. Let's focus on the decision problem of the agent about what message to send. So the agent knows theta by now prior to sending the message and chooses some message m so that the transfer from the agent to the monastery would be f of m.

So that just subtracts off of output. So what message to send? Well, let's suppose there is a unique maximizing message, and let's put a star on it. So m star of theta is the actual optimized message the agent would send given the set of messages which are possible in the allocation rule f. So this statement, 83, is a simple statement of maximization that the chosen message is at least as great in utility as choosing any other arbitrary message.

In fact, if it were unique, then this is a strict inequality for any message other than m star-- so just a statement of the maximum. And in particular, if we took m on the right-hand side to be a particular message, any message will do. 83 still holds. But let's consider m to be what the agent would have sent in the counterfactual situation where the theta were theta tilde rather other than what it really is, namely theta.

So we just substitute this chosen alternative message m equal m star of theta into equation 83. And we end up with this equality. So that inequality is a simple substitution. So now we just do some-- this is a very broad class of games. I haven't told you what the set of messages looks like. I haven't told you what f's, how they're chosen from some domain. It's a very broad mechanism design problem. But we can come up with a simpler scheme.

And now I'm about to describe this alternative scheme. And in it, the message space is now restricted to the set of possible values of theta. That doesn't mean that we're going to require the agent tell the truth. He just outputs, high. It could have been low or vice versa. Lying is perfectly possible. But he can only announce possible values of theta. And it's common knowledge what the set of realizations of theta is.

So that's enforceable. And we do something with the f. Namely, we invent a new allocation rule g. and g logically just maps these announced messages about theta into an allocation. So here, we're doing it for this maximizing message the agent would have sent if his parameter were theta tilde, which is really a composite function over two things. It's over f, the allocation rule, and over the optimized maximized strategy the agent would employ in the original game.

And we collapse that down to g. So now the agent says, I know what I would have done if I had a certain value theta tilde. So I'll just let the computer do that for me. I'll just load in my maximizing strategy and just say, here's my theta. Then the m star of theta behind the scenes would be generated, and it would be mapped under f to g. We'll just now adopt g as, in the end, the allocation rule, this alternative allocation rule in this new mechanism.

Well, again, if you start doing this substitution, this is now g of theta. This is g of theta tilde. So we have this new equation 85. And so 85 just simply says, if the parameter value were theta and you said so, that would dominate weekly in utility terms having theta and announcing something else, theta tilde. So this looks like a truth-telling constraint. But that's really very treacherous way to put it because we're not requiring truth telling.

We're inducing truth telling without loss of generality as a way to represent the outcome in a game where there's private information. And, of course, if telling the truth is what the agent does, then the allocation will be g of theta, which is exactly what would have happened before under the old allocation rule f with the optimizing behavior m star. So we're going to achieve exactly the same outcomes in this general game with f and capital M as we achieve now.

The thing is we know how to search for optimal mechanisms now because all we need to do to capture the private information is impose, without loss of generality, this constraint 85. So to find the optimized allocation rule, we do the usual thing. We're going to maximize a lambda-weighted sum of the ex ante expected utilities of the agents subject not just to the resource constraint, but to this incentive constraint. So this is lambda 1 for agent 1.

It's ex-ante, so we're taking expectations over theta with the associated probabilities. Again, by our normalization, what agent 1 gives up, agent 2

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