Today during an informal conversation with an established business researcher, I learned such a fact:
In the classical Cournot competition model, if one player is a profit-maximizer, the other is a market-share-maximizer, then in equilibrium the market-share-maximizer actually earns a higher profit than the profit-maximizer.
I find this statement quite counterintuitive, so I attempted showing it by myself, to little avail.
The following is what I tried with the basic Cournot model:
There are two competing firms. Let price (demand) be $P(q)=a-q$ for some constant $a$, where $q$ is the total quantity equal to $q_1+q_2$, the sum of the quantities of the two firms respectively. Then profit of firm $i$ is $\pi_i=q_i(P(q_1+q_2)-c)$ where $c$ is the constant unit cost.
Assume firm $1$ maximizes profit: $$ \frac{\partial \pi_1}{\partial q_1}=P(q_1+q_2)-c-q_1=a-2q_1-q_2-c. \tag{1}$$
Firm $2$ maximizes market share $\frac{q_2}{q_1+q_2}$: $$\frac{\partial \left(\frac{q_2}{q_1+q_2}\right)}{\partial q_2}=\frac{q_1+q_2-q_2}{(q_1+q_2)^2}=\frac{q_1}{(q_1+q_2)^2}>0 $$ as long as $q_1>0$. So firm $2$'s best response is its whole capacity whenever firm $1$ produces a positive amount. It is reasonable to assume that firm $2$ wouldn't produce so much to drive the equilibrium price to below its cost $c$ (when it would make a loss); in this case, $q_2^*=a-q_1-c$.
Setting $(1)=0$ we get $a-2q_1^*-(a-q_1^*-c)-c=0$, or $q_1^*=0$. Hence both firms make zero profits. Indeed, the market-share-maximizer earns no less profit than the profit maximizer, but this result is rather trivial and I don't think it is what this researcher meant.
Did I miss something here? It is completely possible that I misinterpreted what this researcher meant during the conversation. So my question is, if everything I've derived here is correct, is there some other interpretation or variant of the "fact" I quoted that is actually true and interesting? Or is there a related similar result that is true and interesting?
I think that your colleague meant that the market share maximizer also take into account marginal cost, which you had omitted in the above model. In my opinion, if one substitutes market shares instead quantities in the model, result do not change. The reason is that market shares are measures that are based in the quantities traded so, in fact, these are only a linear reparametrization.