# AGV mechanism and Individual rationality

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?

• 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). Commented Jun 22, 2019 at 17:41
• but how can I show it accurately? Commented Jun 22, 2019 at 18:07
• 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? Commented Mar 1, 2023 at 9:40

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.
• 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? Commented Jun 23, 2019 at 13:21