<p>The simplest example I could cook up. I might have messed up somewhere.</p>

<p>Consider the provision of a public good with two agents, <span class="math-container">$n=2$</span>. Let the type space be such that <span class="math-container">$\theta \in \{0,1\}$</span>, and both types are equally likely. Either the good is provided, <span class="math-container">$k=1$</span>, and total cost <span class="math-container">$c=2/3$</span> occurs, or the good is not provided, <span class="math-container">$k=0$</span>, which is costless.
Let 
<span class="math-container">$$v_i (k; \theta_i) = k  (\theta_i - \frac{c}{2}).$$</span>
That is, either the good is not provided and nobody pays or it is provided and both share the cost. Of course transfers can be used to compensate.
It is efficient to provide if at least one agent values the good, i.e., has a type of 1. AGV leads to this efficient rule <span class="math-container">$k^* (1,1)=k^* (0,1)=k^* (1,0)=1$</span> and <span class="math-container">$k^* (0,0)=0$</span>.</p>

<p>Then,
<span class="math-container">$$\xi_i (1) = \frac{1}{2} \left( v_j(k^*(1,0);0) + v_j(k^*(1,1);1) \right)
= \frac{1}{2} (0+1 - c) = \frac{1}{6},$$</span>
<span class="math-container">$$\xi_i (0) = \frac{1}{2} \left( v_j(k^*(0,0);0) + v_j(k^*(0,1);1) \right)
= \frac{1}{2} (0+1- c/2) = \frac{1}{3},$$</span></p>

<p>Now, suppose <span class="math-container">$(\theta_i,\theta_j)=(0,1)$</span>. Then,
<span class="math-container">$$v_i(1;0)+\xi_i(0) = 0 - \frac{1}{3}  + \frac{1}{3}= 0 &lt; \xi_j (1) = \frac{1}{6}.$$</span>
So, we don't have ex-post IC in that case.</p>

CC BY-SA 4.0
2019-06-23T13:21:53.043Z
Actually I have one more question. Does this property with violating ex post IR holds when $\forall k$ we have $v_i(k;\theta_i) \ge0 $? And what might be the function that shows that AGV satisfies ex post IR under aforementioned assumption?
Daniel Alexsandrovich
CC BY-SA 4.0
2019-06-23T17:11:02.563Z
See Theorem 24 here https://www.isid.ac.in/~dmishra/gmdoc/lectmd14.pdf
There exists an efficient, Bayesian incentive compatible, and interim individ-ually rational mechanism which balances budget if and only if the modified pivotal mechanismruns an expected surplus.
Bayesian
CC BY-SA 4.0
2019-06-22T19:55:02.720Z
2019-06-22T20:53:44.630Z
2019-06-22T20:53:44.630Z
Bayesian
<p>I have the following question. I consider AGV mechanism which is as follows:
<span class="math-container">$u_i(x,\theta_i) = v_i(k,\theta_i) + t_i $</span> is an utility function where <span class="math-container">$x = (k,t_1,...,t_n)$</span> vector of alternatives. There are N agents. We choose <span class="math-container">$k^*$</span> such that <span class="math-container">$\sum v_i(k^*(\theta) ,\theta_i) \ge \sum v_i(k^*,\theta_i) $</span> and take 
<span class="math-container">$$t_i(\theta) = E_{\tilde{\theta}_{-i}}\Big[ \sum_{i \neq j}v_j(k^*(\theta_i,\tilde{\theta}_{-i}),\tilde{\theta}_j)\Big] + h_i(\theta_{-i}) = \xi_i(\theta_i) + h_i(\theta_{-i}). $$</span>
Also we define
<span class="math-container">$$h_i(\theta_{-i}) = - \frac{1}{N-1} \sum_{i\neq j}\xi_j(\theta_j)$$</span></p>

<p>Now I want to check whether is it true or not that this mechanism satisfies ex post individual rationality. There are some sources claiming that AGV mechanism violates ex post IR but satisfies ex ante IR.</p>

<p>Ex post IR is the following property:
Mechanism <span class="math-container">$\phi$</span> satisfies ex post IR iff <span class="math-container">$\forall \ \theta, \forall \ i $</span> we have: <span class="math-container">$v_i(k;\theta_i)+t_i\ge 0$</span>.</p>

<p>So here we have:
<span class="math-container">$$v_i(k;\theta_i) + \xi_i(\theta_i) \ge \frac{1}{N-1}\sum_{j\neq i} \xi_j(\theta_j)$$</span> 
In general there are now reasons for this inequality to hold. But what might be an example that this inequality is violated?</p>

CC BY-SA 4.0
2019-06-22T17:41:50.357Z
You are right. AGV is ex-ante IR (agents would agree to play before knowing $\theta_i$), but it is neither interim (once they know their own type) nor ex-post IR (after the outome is realized).
Bayesian
CC BY-SA 4.0
2019-06-22T18:07:00.413Z
but how can I show it accurately?
Daniel Alexsandrovich
CC BY-SA 4.0
2023-03-01T09:40:06.170Z
**What does it mean that AGV is ex-ante individually rational?** Where is this claim coming from?
Does not individual rationality require a definition of what happens if an agent does not participate?
Endre Csóka
<mechanism-design>
AGV mechanism and Individual rationality
Daniel Alexsandrovich
# AGV mechanism and Individual rationality

(Asked by: username_0 on 2019-06-22)

I have the following question. I consider AGV mechanism which is as follows:
$u\_i(x,\theta\_i) = v\_i(k,\theta\_i) + t\_i $ is an utility function where $x = (k,t\_1,...,t\_n)$ vector of alternatives. There are N agents. We choose $k^\*$ such that $\sum v\_i(k^\*(\theta) ,\theta\_i) \ge \sum v\_i(k^\*,\theta\_i) $ and take
$$t\_i(\theta) = E\_{\tilde{\theta}\_{-i}}\Big[ \sum\_{i \neq j}v\_j(k^\*(\theta\_i,\tilde{\theta}\_{-i}),\tilde{\theta}\_j)\Big] + h\_i(\theta\_{-i}) = \xi\_i(\theta\_i) + h\_i(\theta\_{-i}). $$
Also we define
$$h\_i(\theta\_{-i}) = - \frac{1}{N-1} \sum\_{i\neq j}\xi\_j(\theta\_j)$$

Now I want to check whether is it true or not that this mechanism satisfies ex post individual rationality. There are some sources claiming that AGV mechanism violates ex post IR but satisfies ex ante IR.

Ex post IR is the following property:
Mechanism $\phi$ satisfies ex post IR iff $\forall \ \theta, \forall \ i $ we have: $v\_i(k;\theta\_i)+t\_i\ge 0$.

So here we have:
$$v\_i(k;\theta\_i) + \xi\_i(\theta\_i) \ge \frac{1}{N-1}\sum\_{j\neq i} \xi\_j(\theta\_j)$$
In general there are now reasons for this inequality to hold. But what might be an example that this inequality is violated?

**Tags:** mechanism-design

> **Comments:**
> **username_1**: You are right. AGV is ex-ante IR (agents would agree to play before knowing $\theta_i$), but it is neither interim (once they know their own type) nor ex-post IR (after the outome is realized).  
> **username_0**: but how can I show it accurately?  
> **username_2**: **What does it mean that AGV is ex-ante individually rational?** Where is this claim coming from? Does not individual rationality require a definition of what happens if an agent does not participate?  

---

## 1 answers

### Answer by username_1 on 2019-06-22. Score: 3 | Accepted answer

The simplest example I could cook up. I might have messed up somewhere.

Consider the provision of a public good with two agents, $n=2$. Let the type space be such that $\theta \in \{0,1\}$, and both types are equally likely. Either the good is provided, $k=1$, and total cost $c=2/3$ occurs, or the good is not provided, $k=0$, which is costless.
Let
$$v\_i (k; \theta\_i) = k (\theta\_i - \frac{c}{2}).$$
That is, either the good is not provided and nobody pays or it is provided and both share the cost. Of course transfers can be used to compensate.
It is efficient to provide if at least one agent values the good, i.e., has a type of 1. AGV leads to this efficient rule $k^\* (1,1)=k^\* (0,1)=k^\* (1,0)=1$ and $k^\* (0,0)=0$.

Then,
$$\xi\_i (1) = \frac{1}{2} \left( v\_j(k^\*(1,0);0) + v\_j(k^\*(1,1);1) \right)
= \frac{1}{2} (0+1 - c) = \frac{1}{6},$$
$$\xi\_i (0) = \frac{1}{2} \left( v\_j(k^\*(0,0);0) + v\_j(k^\*(0,1);1) \right)
= \frac{1}{2} (0+1- c/2) = \frac{1}{3},$$

Now, suppose $(\theta\_i,\theta\_j)=(0,1)$. Then,
$$v\_i(1;0)+\xi\_i(0) = 0 - \frac{1}{3} + \frac{1}{3}= 0 < \xi\_j (1) = \frac{1}{6}.$$
So, we don't have ex-post IC in that case.

> **Comments:**
> **username_0**: Actually I have one more question. Does this property with violating ex post IR holds when $\forall k$ we have $v_i(k;\theta_i) \ge0 $? And what might be the function that shows that AGV satisfies ex post IR under aforementioned assumption?  
> **username_1**: See Theorem 24 here https://www.isid.ac.in/~dmishra/gmdoc/lectmd14.pdf There exists an efficient, username_1 incentive compatible, and interim individ-ually rational mechanism which balances budget if and only if the modified pivotal mechanismruns an expected surplus.  

---

