<p>In a <a href="http://www.sciencedirect.com/science/article/pii/0022053174900751" rel="nofollow">famous paper from 1974</a>, Hal Varian proved (in Theorem 2.3) that:</p>

<blockquote>
  <p>In an economy with homogeneous divisible googs, if all agents have monotonic and <a href="https://en.wikipedia.org/wiki/Convex_preferences" rel="nofollow">convex preferences</a>, then there exists an allocation which is both Pareto-efficient and <a href="https://en.wikipedia.org/wiki/Envy-freeness" rel="nofollow">envy-free</a>.</p>
</blockquote>

<p>He said (after Theorem 2.8) that, if preferences are not convex, then Pareto-efficient envy-free allocations might not exist. As an example, he brings the following economy: there are two agents with the same preferences:</p>

<p>$$u_1(x,y) = u_2(x,y) = \min(x,y)$$</p>

<p>and the total bundle is $(2,1)$.</p>

<p>I do not see how this is a negative example. If we give each agent a bundle of $(1,0.5)$, then the division is envy-free, since the value of every agent is exactly 0.5. It is also Pareto-efficient, since to increase the utility of one agent, it is necessary to give him some y from the other agent, but this hurts the utility of the other agent.</p>

<p>Am I missing something?</p>

CC BY-SA 3.0
2015-11-29T21:12:53.730Z
The preferences $$u_1(x,y) = u_2(x,y) = \min(x,y)$$ are convex. They are not strictly convex. I also don't see how a counterexample could be constructed using them.
Giskard
CC BY-SA 3.0
2015-11-29T21:15:50.273Z
Also, given these utility functions not any bundle is prefered to the $0$ bundle, so this seems quite strange.
Giskard
CC BY-SA 3.0
2015-11-30T02:26:51.963Z
Maybe there is a typo in the paper, and the function should be $\max$ instead of $\min$. Does it make sense?
Erel Segal-Halevi
CC BY-SA 3.0
2015-11-30T07:45:27.067Z
That makes sense. Equitable allocations for that function provide 1 utility for each player, but that cannot be efficient as $(2,0)$, $(0,1)$ is feasible. And the utility function is not convex. I think you are right.
Giskard
CC BY-SA 3.0
2015-11-30T07:48:16.210Z
@denesp OK, good to know!
Erel Segal-Halevi
<pareto-efficiency><fairness>
An economy with no Pareto-efficient envy-free allocations
Erel Segal-Halevi
# An economy with no Pareto-efficient envy-free allocations

(Asked by: username_0 on 2015-11-29)

In a [famous paper from 1974](http://www.sciencedirect.com/science/article/pii/0022053174900751), Hal Varian proved (in Theorem 2.3) that:

> In an economy with homogeneous divisible googs, if all agents have monotonic and [convex preferences](https://en.wikipedia.org/wiki/Convex_preferences), then there exists an allocation which is both Pareto-efficient and [envy-free](https://en.wikipedia.org/wiki/Envy-freeness).

He said (after Theorem 2.8) that, if preferences are not convex, then Pareto-efficient envy-free allocations might not exist. As an example, he brings the following economy: there are two agents with the same preferences:

$$u\_1(x,y) = u\_2(x,y) = \min(x,y)$$

and the total bundle is $(2,1)$.

I do not see how this is a negative example. If we give each agent a bundle of $(1,0.5)$, then the division is envy-free, since the value of every agent is exactly 0.5. It is also Pareto-efficient, since to increase the utility of one agent, it is necessary to give him some y from the other agent, but this hurts the utility of the other agent.

Am I missing something?

**Tags:** pareto-efficiency, fairness
> **Comments:**
> **username_1**: The preferences $$u_1(x,y) = u_2(x,y) = \min(x,y)$$ are convex. They are not strictly convex. I also don't see how a counterexample could be constructed using them.  
> **username_1**: Also, given these utility functions not any bundle is prefered to the $0$ bundle, so this seems quite strange.  
> **username_0**: Maybe there is a typo in the paper, and the function should be $\max$ instead of $\min$. Does it make sense?  
> **username_1**: That makes sense. Equitable allocations for that function provide 1 utility for each player, but that cannot be efficient as $(2,0)$, $(0,1)$ is feasible. And the utility function is not convex. I think you are right.  
> **username_0**: @denesp OK, good to know!  

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## 0 answers

