Existence of market sharing equilibrium in hotelling model I'm reading a paper on Competition in Two-Sided Markets. The model is a Hotelling-type model, with consumers on some interval, choosing their preferred firm based on price and 'distance' to the different firms (see below). In it, there's an assumption made that guarantees that there's no single monopoly that serves all consumers, which amounts to the condition:

$$4t_1t_2&gt;(\alpha_1\alpha_2)^2$$ 

But I'm not sure how this is derived. Annoyingly, the paper says: "it turns out that the necessary and sufficient condition for a market sharing equilibrium is $4t_1t_2&gt;(\alpha_1\alpha_2)^2$", without providing the derivation. I'd appreciate a hint on how to derive this, or an actual derivation of this condition.





These screenshots are from page 8 from Competition in Two-Sided Markets.
Market sharing simply means that both platforms have some market share. So the condition can probably be obtained by finding the closed-form solution for market shares (not only implicitly as in (7)) and imposing the condition that all market shares are positive. This is akin to each platform having an interior solution.

I am not sure which Hessian you are referring to, but my guess is that you mean the hessian of a firm's optimization problem (choosing both of their prices) taking the prices of the other platform as given. In that case, if the Hessian is negative definite, regardless of the other firm's prices, an interior solution exists.