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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!

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  • $\begingroup$ Arrow and Debreu have no uniqueness proof and their assumptions do not guarantee uniqueness. $\endgroup$ Apr 6, 2022 at 17:15
  • $\begingroup$ @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? $\endgroup$
    – dodo
    Apr 6, 2022 at 17:17
  • $\begingroup$ Under some additional smoothness assumptions one gets local uniqueness "generically". But local uniqueness is very different from uniqueness. $\endgroup$ Apr 6, 2022 at 20:05

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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.

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.

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  • $\begingroup$ Ok so mathematically $\oplus$ is also vector addition, BUT the economic interpretation of $\oplus$ and $+$ are very different? $\endgroup$
    – dodo
    Apr 7, 2022 at 15:00
  • $\begingroup$ I don't think there is a concensus on the meaning of $\oplus$ in a formula. $\endgroup$
    – tdm
    Apr 8, 2022 at 8:18

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