<p>The type space is the support of the belief about the types. It seems a bit weird to allow a mechanism designer to arbitrarily restrict their belief support. However, you are right: if the designer can find out more about the private types by acquiring information, this would intuitively allow them to reduce information rent, because it makes the type &quot;less private.&quot; Of course it would be best to fully unveil types such that no incentive for revelation has to be provided and full surplus extraction becomes possible. In your constructed example it may not be helpful to exclude very high types <span class="math-container">$&gt;M$</span> as more profitable deviations are often found in closer vicinity to the real type. That depends on the problem at hand of course.</p>
<p>I can't see clearly where you want to go with this. So let me suggest something else in a similar spirit: mechanism design with evidence. Here, agents can make their private information hard by providing evidence that their true type belongs to some subset of the type space. See, e.g., the work by Ben-Porath, Dekel &amp; Lipman (<a href="https://www.aeaweb.org/articles?id=10.1257/aer.104.12.3779" rel="nofollow noreferrer">AER 2014</a>, <a href="https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA14991" rel="nofollow noreferrer">Ecta 2019</a> and more), Bull and Watson (<a href="https://www.sciencedirect.com/science/article/abs/pii/S0899825606000352" rel="nofollow noreferrer">GEB 2007</a>), Hagenbach, Koessler &amp; Perez-Richet (<a href="https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA11070" rel="nofollow noreferrer">Ecta 2014</a>) and others.</p>

CC BY-SA 4.0
2020-12-16T14:40:31.233Z
These references are very informative! And I've added links to them for further reading. I just roughly read the paper by Ben-Porath, Dekel & Lipman (AER 2014) and it seems suitable to me.  I would come back to this thread after understanding these literatures :)
Mengfan Ma
CC BY-SA 4.0
2020-12-16T15:10:22.363Z
Maybe a new paper by Schweighofer-Kodritsch and Strausz is also helpful, you can see a presentation here: https://conference2.aau.at/event/4/page/28-conference-programme
Bayesian
CC BY-SA 4.0
2020-12-16T15:13:37.260Z
Another --completely different-- avenue would be information design a la Kamenica & Gentzkow: https://economics.stackexchange.com/questions/29253/why-is-it-called-bayesian-persuasion Here it is allowed to manipulate the type distribution.
Bayesian
CC BY-SA 4.0
2020-12-16T10:46:30.763Z
2020-12-16T15:04:53.167Z
2020-12-16T15:04:53.167Z
Bayesian
<p>When we are to design an IC mechanism for an auction, we usually assume that each agent <span class="math-container">$i$</span>'s type <span class="math-container">$\theta_i$</span> is draw from its type space <span class="math-container">$\Theta_i$</span>, which contains all the possible types of <span class="math-container">$i$</span>. For instance in a single item auction, we may assume that <span class="math-container">$\theta_i \in \mathcal{R}^+$</span>, i.e., <span class="math-container">$\Theta_i = \mathcal{R}^+$</span>.</p>
<p>In some scenario, prior information about the type space can be acquired by the auctioneer by some means. For example, the auctioneer may confirm that <span class="math-container">$\theta_i \le M$</span> before designing any mechanism. That is, the maximum money each agent can pay is no more than <span class="math-container">$M$</span>. The auctioneer may obtain the information from some statistic query in data market, if we have to explain how she gets it. Then we can validly assume that <span class="math-container">$\Theta_i$</span> is <span class="math-container">$[0, M]$</span> instead of <span class="math-container">$\mathcal{R}^+$</span>. Suppose some agent can only benifit himself by misreporting some <span class="math-container">$\theta_i^{\prime} &gt; M$</span>. Then we have no worry about truthfulness because now <span class="math-container">$\Theta_i = [0, M]$</span>. Note that agents would not report higher than <span class="math-container">$M$</span> because they are informed of the auctioneer's access to the statistics(privacy issue is not considered here).</p>
<p>The above example may not be well constructed, however I'm considering similar setting where  some constraints on agents' type space may make truthful mechanisms easier to get, or even make nontruthful mechanism truthful. I failed to find out any literature on this topic. I appreciate if someone can provide guidance on where to look for such research and whether this is valid as a research point.</p>

<reference-request><auctions><mechanism-design>
The influence of restricted type space on incentive compatible mechanism design?
Mengfan Ma
# The influence of restricted type space on incentive compatible mechanism design?

(Asked by: username_0 on 2020-12-16)

When we are to design an IC mechanism for an auction, we usually assume that each agent $i$'s type $\theta\_i$ is draw from its type space $\Theta\_i$, which contains all the possible types of $i$. For instance in a single item auction, we may assume that $\theta\_i \in \mathcal{R}^+$, i.e., $\Theta\_i = \mathcal{R}^+$.

In some scenario, prior information about the type space can be acquired by the auctioneer by some means. For example, the auctioneer may confirm that $\theta\_i \le M$ before designing any mechanism. That is, the maximum money each agent can pay is no more than $M$. The auctioneer may obtain the information from some statistic query in data market, if we have to explain how she gets it. Then we can validly assume that $\Theta\_i$ is $[0, M]$ instead of $\mathcal{R}^+$. Suppose some agent can only benifit himself by misreporting some $\theta\_i^{\prime} > M$. Then we have no worry about truthfulness because now $\Theta\_i = [0, M]$. Note that agents would not report higher than $M$ because they are informed of the auctioneer's access to the statistics(privacy issue is not considered here).

The above example may not be well constructed, however I'm considering similar setting where some constraints on agents' type space may make truthful mechanisms easier to get, or even make nontruthful mechanism truthful. I failed to find out any literature on this topic. I appreciate if someone can provide guidance on where to look for such research and whether this is valid as a research point.

**Tags:** reference-request, auctions, mechanism-design

---

## 1 answers

### Answer by username_1 on 2020-12-16. Score: 4 | Accepted answer

The type space is the support of the belief about the types. It seems a bit weird to allow a mechanism designer to arbitrarily restrict their belief support. However, you are right: if the designer can find out more about the private types by acquiring information, this would intuitively allow them to reduce information rent, because it makes the type "less private." Of course it would be best to fully unveil types such that no incentive for revelation has to be provided and full surplus extraction becomes possible. In your constructed example it may not be helpful to exclude very high types $>M$ as more profitable deviations are often found in closer vicinity to the real type. That depends on the problem at hand of course.

I can't see clearly where you want to go with this. So let me suggest something else in a similar spirit: mechanism design with evidence. Here, agents can make their private information hard by providing evidence that their true type belongs to some subset of the type space. See, e.g., the work by Ben-Porath, Dekel & Lipman ([AER 2014](https://www.aeaweb.org/articles?id=10.1257/aer.104.12.3779), [Ecta 2019](https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA14991) and more), Bull and Watson ([GEB 2007](https://www.sciencedirect.com/science/article/abs/pii/S0899825606000352)), Hagenbach, Koessler & Perez-Richet ([Ecta 2014](https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA11070)) and others.

> **Comments:**
> **username_0**: These references are very informative! And I've added links to them for further reading. I just roughly read the paper by Ben-Porath, Dekel & Lipman (AER 2014) and it seems suitable to me.  I would come back to this thread after understanding these literatures :)  
> **username_1**: Maybe a new paper by Schweighofer-Kodritsch and Strausz is also helpful, you can see a presentation here: https://conference2.aau.at/event/4/page/28-conference-programme  
> **username_1**: Another --completely different-- avenue would be information design a la Kamenica & Gentzkow: https://economics.stackexchange.com/questions/29253/why-is-it-called-bayesian-persuasion Here it is allowed to manipulate the type distribution.  

---

