# Finding conditions on unspecified CDF that permit a solution to an equation

[A duplicate thread can also be found at https://stats.stackexchange.com/questions/59450/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ]

Let $F(\alpha) := \mathbb{P}(\tilde{\alpha} \leq \alpha)$ be an arbitrary, strictly increasing and twice differentiable CDF that is defined on the interval $[0, \overline{\alpha}]$, where $\overline{\alpha}>1$ may be infinite. Moreover, let $\mathbb{E}(\tilde{\alpha}) = 1$.

Let $N \geq 2$ be a natural number, and $\delta \in (0,1)$ real.

I am looking for necessary and sufficient criteria on $F$ in order to find a solution to

$(N-1)[1-\delta+\delta F(\alpha)] - \delta \alpha F'(\alpha) =0$,

where $\alpha$ can be any real positive number not larger than $\overline{\alpha}$.

Now, it is clear that the left term, $(N-1)[1-\delta+\delta F(\alpha)]$, is strictly increasing in $\alpha$ and bounded above by $N-1$. Hence, if the probability mass is sufficiently "concentrated" in some interval (implying that $F'(.)$ is large in that interval), a solution $\alpha^*$ must exist by continuity of $F'$ and $F$.

However, it would be nice to have some sharper conditions on $F$ that are necessary or sufficient for a solution. Ideally, I'd wish to have a result that gives an upper bound on $F$'s variance, or similar.

One thing that seems problematic is that the above equation never uses the fact that $\mathbb{E}(\tilde{\alpha})=1$. I've tried to apply Markov's inequality but it didn't help me much.

Numerical simulations reveal that a solution can usually be found if the variance of the considered CDF is sufficiently low. Examples include $F$ log-normal or $F(\alpha) = (\frac{b \alpha}{b+1})^b$ for $b>1$. The uniform distribution on $[0,2]$ is a borderline case that doesn't permit a solution.

I would greatly appreciate any help or ideas to this (economics) research problem. Many thanks!

• These variables must have some meaning, or else you should simplify the expressions by using $M=N−1$ instead of $N$, for example. Why not include that meaning, so the equation you are trying to solve is intuitive? – Douglas Zare May 19 '13 at 22:32
• A nice answer has been provided here: stats.stackexchange.com/questions/59450/… Since the equation is derived from an economics paper, there is of course some interpretation to the variables. $N$ is the number of firms in a market, and $\delta$ is the rate with which firms discount future profits. Loosely speaking, finding a solution $\alpha^*$ to the above equation corresponds to the existence of a Nash-equilibrium where firms collude on a supra-competitive price level, given noisy demand following $F(.)$. – Martin May 20 '13 at 8:51
• First, it's bad to post the same question on different sites at the same time, particularly without linking in both directions. This leads to duplication of effort from the people you are asking to help you. Second, the answer on stat.stackexchange is quite incomplete. It says you can make local changes to $F′$ to get a solution without adjusting the moments by much. That's obvious. The interesting question is what conditions on the moments are sufficient. Since you accepted the trivial answer, I suppose you aren't actually interested in sufficient conditions so I won't bother to post. – Douglas Zare May 20 '13 at 21:12
• I'm not really familar with the etiquette of stackexchange or mathoverflow, so I'm sorry if I didn't adhere to the standards. I can understand the problem with duplicate threads though and will add links on both sites. I have accepted the "trivial" answer on stats.stackexchange since I didn't expect any more input to come. Obviously, this doesn't mean that I'm not interested in a different perspective, especially if you feel that the given answer is incomplete. In fact, I would already be most grateful if I was pointed to some helpful resource on the topic. Or given the sketch of a solution. – Martin May 20 '13 at 21:46