<p>Convexity (quasi-concavity) of preferences is important for both Nash Eq and exitence of general equilibrium. Without this assumption, the best response correspondences are not necessarily convex valued and this is needed in order to apply Kakutani's fixed point theorem <a href="https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem" rel="nofollow noreferrer">wiki</a>.</p>
<p>For general equilibrium, preferences are over bundles <span class="math-container">$q \in \mathbb{R}^n_+$</span>. So convexity of preferences requires that for all <span class="math-container">$\alpha \in [0,1]$</span>:
<span class="math-container">$$
q_1 \succeq q_2 \to \alpha q_1 + (1-\alpha) q_2 \succeq q_2 
$$</span>
For Nash equilibrium (in mixed strategies) preferences are over lotteries <span class="math-container">$\ell$</span> (non-negative vectors whose components add up to one). Here convexity requires that for all <span class="math-container">$\alpha \in [0,1]$</span>:
<span class="math-container">$$
\ell_1 \succeq \ell_2 \to \alpha \ell_1 + (1-\alpha) \ell_2 \succeq \ell_2.
$$</span></p>
<blockquote>
<p>Nash convexity: <span class="math-container">$p\succ q$</span> implies <span class="math-container">$ap \oplus (1−a)q \succ q$</span>. Here the <span class="math-container">$\oplus$</span> is the probability mixture operation, not a typical real-vector addition as probability is involved! For example, <span class="math-container">$0.5 \times$</span> Four Apples <span class="math-container">$+ 0.5 \times$</span> Two Apples does not equal three apples.</p>
</blockquote>
<p>Here <span class="math-container">$p$</span> and <span class="math-container">$q$</span> should be lotteries, not bundles. So <span class="math-container">$\oplus$</span> is indeed the usual vector addition.</p>

CC BY-SA 4.0
2022-04-07T15:00:34.950Z
Ok  so mathematically $\oplus$ is also vector addition, BUT the economic interpretation of $\oplus$ and $+$ are very different?
dodo
CC BY-SA 4.0
2022-04-08T08:18:24.613Z
I don't think there is a concensus on the meaning of $\oplus$ in a formula.
tdm
CC BY-SA 4.0
2022-04-07T09:00:40.057Z
2022-04-07T09:00:40.057Z
tdm
<p>Arrow Debreu (AD) uses the convex preference (A4 among their four assumptions, also see the assumption IIIc in AD 1954 ECTA) to make general equilibrium (GE) exist, unique, and well-behave.</p>
<p>What remains unclear (for me) is the <strong>role of convex preference in Nash Equilibrium</strong>: how is the role of convex preference in NE similar or different from the role in GE?</p>
<p>I thought convexity is also important in Nash Equilibrium because a non-concave utility function might give two local optima in the best response function.</p>
<p><strong>However,</strong> I think the Nash convexity is very different from the AD convexity:</p>
<p>AD convexity: <span class="math-container">$x\succ y$</span> implies <span class="math-container">$ax+(1-a)y\succ y$</span>. The addition &quot;+&quot; here can be normal addition between real numbers (or vectors). For example two dollars plus one dollar equals three dollars; three apples plus one apple equals four apples.</p>
<p>Nash convexity: <span class="math-container">$p\succ q$</span> implies  <span class="math-container">$ap\oplus(1-a)q\succ q$</span>. Here the <span class="math-container">$\oplus$</span> is the probability mixture operation, not a typical real-vector addition as probability is involved! For example, <span class="math-container">$0.5\times\text{Four Apples}+0.5\times \text{Two Apples}$</span> does not equal three apples.</p>
<p>So, is convex preference important in Nash Equilibrium, just like the importance in general equilibrium, and is my understanding of Nash Equilibrium correct?</p>
<p>Any reference/explanations mentioning the important role of convex preference in NE will help!</p>

CC BY-SA 4.0
2022-04-06T17:15:52.307Z
Arrow and Debreu have no uniqueness proof and their assumptions do not guarantee uniqueness.
Michael Greinecker
CC BY-SA 4.0
2022-04-06T17:17:23.430Z
@MichaelGreinecker Ok so I am not very clear about the role of convex preference in both GE and NE. I need to update the question. I thought (strictly) concave smooth utility gives unique smooth demand function, and a smooth demand function gives locally unique GE? Or shall I say that the GE is well-behaved because of the assumptions made by AD?
dodo
CC BY-SA 4.0
2022-04-06T20:05:42.907Z
Under some additional smoothness assumptions one gets local uniqueness "generically". But local uniqueness is very different from uniqueness.
Michael Greinecker
<microeconomics><game-theory><reference-request><utility><nash-equilibrium>
Convex Preference in Nash Equilibrium
dodo
# Convex Preference in Nash Equilibrium

(Asked by: username_0 on 2022-04-06)

Arrow Debreu (AD) uses the convex preference (A4 among their four assumptions, also see the assumption IIIc in AD 1954 ECTA) to make general equilibrium (GE) exist, unique, and well-behave.

What remains unclear (for me) is the **role of convex preference in Nash Equilibrium**: how is the role of convex preference in NE similar or different from the role in GE?

I thought convexity is also important in Nash Equilibrium because a non-concave utility function might give two local optima in the best response function.

**However,** I think the Nash convexity is very different from the AD convexity:

AD convexity: $x\succ y$ implies $ax+(1-a)y\succ y$. The addition "+" here can be normal addition between real numbers (or vectors). For example two dollars plus one dollar equals three dollars; three apples plus one apple equals four apples.

Nash convexity: $p\succ q$ implies $ap\oplus(1-a)q\succ q$. Here the $\oplus$ is the probability mixture operation, not a typical real-vector addition as probability is involved! For example, $0.5\times\text{Four Apples}+0.5\times \text{Two Apples}$ does not equal three apples.

So, is convex preference important in Nash Equilibrium, just like the importance in general equilibrium, and is my understanding of Nash Equilibrium correct?

Any reference/explanations mentioning the important role of convex preference in NE will help!

**Tags:** microeconomics, game-theory, reference-request, utility, nash-equilibrium

> **Comments:**
> **username_1**: Arrow and Debreu have no uniqueness proof and their assumptions do not guarantee uniqueness.  
> **username_0**: @username_1 Ok so I am not very clear about the role of convex preference in both GE and NE. I need to update the question. I thought (strictly) concave smooth utility gives unique smooth demand function, and a smooth demand function gives locally unique GE? Or shall I say that the GE is well-behaved because of the assumptions made by AD?  
> **username_1**: Under some additional smoothness assumptions one gets local uniqueness "generically". But local uniqueness is very different from uniqueness.  

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## 1 answers

### Answer by username_2 on 2022-04-07. Score: 1 | Accepted answer

Convexity (quasi-concavity) of preferences is important for both Nash Eq and exitence of general equilibrium. Without this assumption, the best response correspondences are not necessarily convex valued and this is needed in order to apply Kakutani's fixed point theorem [wiki](https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem).

For general equilibrium, preferences are over bundles $q \in \mathbb{R}^n\_+$. So convexity of preferences requires that for all $\alpha \in [0,1]$:
$$
q\_1 \succeq q\_2 \to \alpha q\_1 + (1-\alpha) q\_2 \succeq q\_2
$$
For Nash equilibrium (in mixed strategies) preferences are over lotteries $\ell$ (non-negative vectors whose components add up to one). Here convexity requires that for all $\alpha \in [0,1]$:
$$
\ell\_1 \succeq \ell\_2 \to \alpha \ell\_1 + (1-\alpha) \ell\_2 \succeq \ell\_2.
$$

> Nash convexity: $p\succ q$ implies $ap \oplus (1−a)q \succ q$. Here the $\oplus$ is the probability mixture operation, not a typical real-vector addition as probability is involved! For example, $0.5 \times$ Four Apples $+ 0.5 \times$ Two Apples does not equal three apples.

Here $p$ and $q$ should be lotteries, not bundles. So $\oplus$ is indeed the usual vector addition.

> **Comments:**
> **username_0**: Ok  so mathematically $\oplus$ is also vector addition, BUT the economic interpretation of $\oplus$ and $+$ are very different?  
> **username_2**: I don't think there is a concensus on the meaning of $\oplus$ in a formula.  

---

