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Let $F(p)$ denote the distribution of prices in a market, $\pi(p, F)$ are profits choosing $p$ given distribution $F$. $E\pi(p,F)$ is defined to be

$$ E \pi(p, F) = R(p) \psi(p, F)$$

where $R(p) = p D(p)$ is the revenue per customer, and $\psi(p, F)$ denotes the expected share of customers that a seller with price $p$, given distribution $F$ will face.

Definition (2.1) in Stahl (1996) (just below eq. 2.3) defines the symmetric Nash equilibrium as:

$$ E \pi(p,F) \leq \max_p p E \pi(p, F) \, \forall p$$

My question is why $(2.1)$ is defined as such? I would have thought that a Nash equilibrium requires choice being optimal, hence no incentive to change profits:

$$ E \pi(p,F) \leq \max_p E \pi(p,F) \, \forall p$$

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  • $\begingroup$ To me it looks like a typo. as R(p) = p D(p) we already have the p in expectation. Have you tried checking the proofs? How it is used in there? $\endgroup$
    – The Almighty Bob
    Commented Mar 31, 2015 at 8:33
  • $\begingroup$ @TheAlmightyBob On page 350 he says "[...] These are necessary conditions that stem from the requirement that $E\pi(\bar p, F) \geq E\pi(p, F)$". This almost makes me think it is a typo, but (i) there's no maximization operator on the RHS here, so I may just overlook something (ii) it would be a typo in the fundamental definition of the paper, and that's somewhat hard to believe. $\endgroup$
    – FooBar
    Commented Mar 31, 2015 at 15:03
  • $\begingroup$ I have to say that I thought this was a typo too. The inequality sign is a bit confusing, because the condition holds with equality almost everywhere. The condition $E\pi(p,F)=\max_p E\pi(p,F)\forall p$ looks much more like a standard mixed strategy equilibrium condition. The $E\pi(p,F)= E\pi(\overline{p},F)\forall p$ condition makes sense because the way one often solves for these distributions is to set profits from an arbitrary $p$ to those from the maximum of the support. Doing this ensures that $F(\overline{p})=1$ on the RHS so there is only one $F$ term (on the LHS) to solve for. $\endgroup$
    – Ubiquitous
    Commented Apr 1, 2015 at 7:12

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