<p>Consider a mechanism design problem in which the agents have type spaces that are uncountably infinite. I have a couple of questions which are numbered throughout this post.</p>
<p>The revelation principle states that any outcome that can be achieved by an indirect mechanism (where agents report something other than their types) can be achieved by a direct mechanism where agents report their type truthfully. This seems to introduce some practical issues when the type spaces are infinite.</p>
<p><strong>(Q1)</strong> Does the revelation principle still apply when agents are unable to actually report their types to the mechanism?</p>
<p>Regardless, I understand that the revelation principle still has value from an analysis perspective. Presumably, one can <em>envision</em> a situation in which the mechanism has access to the true infinite types of each agent, and evaluate outcomes in this situation. Applying the contrapositive of the revelation principle: if we can't achieve a desired outcome in this situation, then there is no indirect mechanism in which the desired outcome can be achieved.</p>
<p>Back to the practical issue at hand. When it comes to implementation, we cannot ask agents to report their infinite types. Agents in any real implementation would need to report something smaller than their type space (that is, in an indirect mechanism).</p>
<p><strong>(Q2)</strong> If we show that a desirable outcome is possible by a truthful direct mechanism, then there is <em>some</em> indirect mechanism that can also reach this outcome. How do we know <em>which</em> indirect mechanism can achieve this outcome?</p>

CC BY-SA 4.0
2021-05-03T19:20:55.047Z
I do not understand Q1. How are they unable to report their type? I understand that there are direct mechanisms where the action space is smaller than the type space, but this is irrelevant to the "ideal" indirect mechanism of the revelation principle.
Giskard
CC BY-SA 4.0
2021-05-03T19:22:30.710Z
I also do not understand Q2 :) What do you mean by "*If we show that a desirable outcome is possible by a truthful direct mechanism, then there is some indirect mechanism that can also reach this outcome.*"? Would you be satisfied by relabeling the action space, so that instead of types $t_1,t_2,\dots$ we have actions $a_1,a_2,\dots$? This seems fairly straightforward.
Giskard
CC BY-SA 4.0
2021-05-03T19:29:17.420Z
@Giskard Say if each agent's type was an infinitely long sequence of real numbers (just for illustration purposes). There's no way the agent can tractably report such a sequence.
Erik M
CC BY-SA 4.0
2021-05-03T19:51:43.417Z
So? The revelation princible is not about tractability? I am sorry, I do not understand your underlying problem.
Giskard
CC BY-SA 4.0
2021-05-04T09:58:22.297Z
@Giskard I think the idea is that, if my type is $0.18472948294...$ (infinite sequence of numbers with no 'clever' abbreviation), then I cannot report my type since that would take infinite time.
afreelunch
CC BY-SA 4.0
2021-05-04T10:19:01.650Z
1. How did you learn your type in finite time? 2. The revelation principle is not about tractability, it does not matter if reporting is practically possible.
Giskard
CC BY-SA 4.0
2021-06-15T14:20:26.560Z
@afreelunch If this was the problem, then even indirect mechanisms face it. For instance, in a first-price auction with 2 bidders and iid uniformly distributed values you would bid half of your value. It would also take infinite time to bid 0.18472948294.../2.
Bayesian
<game-theory><mechanism-design>
Revelation principle when the type space is infinite
Erik M
# Revelation principle when the type space is infinite

(Asked by: username_0 on 2021-05-03)

Consider a mechanism design problem in which the agents have type spaces that are uncountably infinite. I have a couple of questions which are numbered throughout this post.

The revelation principle states that any outcome that can be achieved by an indirect mechanism (where agents report something other than their types) can be achieved by a direct mechanism where agents report their type truthfully. This seems to introduce some practical issues when the type spaces are infinite.

**(Q1)** Does the revelation principle still apply when agents are unable to actually report their types to the mechanism?

Regardless, I understand that the revelation principle still has value from an analysis perspective. Presumably, one can *envision* a situation in which the mechanism has access to the true infinite types of each agent, and evaluate outcomes in this situation. Applying the contrapositive of the revelation principle: if we can't achieve a desired outcome in this situation, then there is no indirect mechanism in which the desired outcome can be achieved.

Back to the practical issue at hand. When it comes to implementation, we cannot ask agents to report their infinite types. Agents in any real implementation would need to report something smaller than their type space (that is, in an indirect mechanism).

**(Q2)** If we show that a desirable outcome is possible by a truthful direct mechanism, then there is *some* indirect mechanism that can also reach this outcome. How do we know *which* indirect mechanism can achieve this outcome?

**Tags:** game-theory, mechanism-design

> **Comments:**
> **username_1**: I do not understand Q1. How are they unable to report their type? I understand that there are direct mechanisms where the action space is smaller than the type space, but this is irrelevant to the "ideal" indirect mechanism of the revelation principle.  
> **username_1**: I also do not understand Q2 :) What do you mean by "*If we show that a desirable outcome is possible by a truthful direct mechanism, then there is some indirect mechanism that can also reach this outcome.*"? Would you be satisfied by relabeling the action space, so that instead of types $t_1,t_2,\dots$ we have actions $a_1,a_2,\dots$? This seems fairly straightforward.  
> **username_0**: @username_1 Say if each agent's type was an infinitely long sequence of real numbers (just for illustration purposes). There's no way the agent can tractably report such a sequence.  
> **username_1**: So? The revelation princible is not about tractability? I am sorry, I do not understand your underlying problem.  
> **username_2**: @username_1 I think the idea is that, if my type is $0.18472948294...$ (infinite sequence of numbers with no 'clever' abbreviation), then I cannot report my type since that would take infinite time.  
> **username_1**: 1. How did you learn your type in finite time? 2. The revelation principle is not about tractability, it does not matter if reporting is practically possible.  
> **username_3**: @username_2 If this was the problem, then even indirect mechanisms face it. For instance, in a first-price auction with 2 bidders and iid uniformly distributed values you would bid half of your value. It would also take infinite time to bid 0.18472948294.../2.  

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## 0 answers

