# Taking a mechanism as given, what goal functions can we implement?

Let $\theta_i$ be $i$'s private information about the state of the world, $\theta \in \Theta$, where $\theta = \times_i \theta_i$. Each agent reports $\hat{\theta_i}$ to a social planner, so that $\hat{\theta}$ is the reported state.

Typically, in mechanism design, we take a goal function, $f(\theta): \theta \rightarrow X$, where $X$ is the set of all possible outcomes, as given and look for mechanisms that implement it. By the revelation principle, we know to look for mechanisms under which incentives are such that, $\forall i$, $\hat{\theta_i}=\theta_i$. This usually manifests itself in incentive compatibility constraints.

However, there's a different (but related) question that could be asked. Suppose the social planner will directly make the choices, a vector $\mathbf{c}$, to maximize the social planner's utility function, $U(\theta,\mathbf{c})$, which is common knowledge. Individual agents have utility functions $u_i(\theta,\mathbf{c})$. Suppose also the social planner assumes the agents are truthful, so that $\theta = \hat{\theta}$. Then the social planner will choose $\mathbf{c}$ to solve: $$\max_{\mathbf{c}}\:\: U(\hat{\theta},\mathbf{c})$$ Under what conditions on $U$ and $u_i$ ($\forall i$) will the social planner's assumption that $\theta = \hat{\theta}$ be valid?

Is anybody aware of papers that consider mechanism design from this angle?

A spiel on why this question is worth asking: In some environments, seemingly reasonable goal functions (or correspondences) yield impossibility results (i.e. no mechanism can implement it) or otherwise deeply unsatisfying results (e.g. the only mechanisms that can implement it are dictatorial). The classical example is, of course, Arrow's Impossibility Theorem. But if a client tells us her goal function, it's not all that helpful to simply respond to her: "Sorry, it can't be implemented." Rather, you might want to suggest what goal functions could be implemented. In the particular question above, I'm essentially taking a simple mechanism as given (the mechanism is simply that agents make their reports and the client/planner maximizes her objective taking those reports as true), and then asking for what objective functions, $U(\theta,\mathbf{c})$, that truthful reporting assumption can be valid. Note also that the $U(\theta,\mathbf{c})$ implicitly defines a goal function: $$x \in f(\theta) \iff x \in \arg \max_\mathbf{c} \:U(\theta,\mathbf{c}).$$

• Do you know the implementability literature as summarised, for example, here: scholar.harvard.edu/maskin/files/… ? – Ubiquitous Oct 22 '15 at 17:39
• @Ubiquitous To some degree, but not as well as I would like. Thank you for that helpful reference. In that literature, they tend to ask the question: For what social choice rules (or, equivalently, goal functions) does there exist a mechanism that implements them? The one I'm asking is related but distinct: What social choice rules can a given mechanism implement? There may well be some answers to my question in that literature, so I'll certainly invest some time into it. – Shane Oct 22 '15 at 17:56
• Actually the implementation theory literature is very useful and pertinent in the following sense: If I know that from that literature that a given SCR is not implementable (using whatever standard I might choose), then I also know (trivially) that it won't be implementable by the mechanism I describe. In other words, that literature provides a (too loose) bound on the SCRs I can implement with the proposed simple mechanism. – Shane Oct 22 '15 at 18:23