I'm currently working with the Bertrand Network mathematical model. The model basically allows multiple sellers to compete in different market shares at the same time.
The utility function for each seller is:
$$u_i(x)=x(\alpha_i + \sum_{j \in N(i)}\beta_{i,j} \overline{F}_j(x))$$
Where $N(i)$ is the set of neighbors of $i$ (his competitors), $\beta_{i,j}$ is the size of the market share of the seller $i$ and $j$ and $\overline{F}_j(x)$ is the inverse $C.D.F$ function that says the chance of the firm $j$ to choose the price of $x$ or higher. Last but not least, $\alpha_i$ is the size of the captive market of $i$.
The article states the following:
A sketch solution (of an equilibrium) specifies a sketch and, additionally, it defines values $F_r(t)$ for every seller $r$ and point $t$ in the set $T$ of boundary points of the sketch. These values must satisfy the following linear program (LP1) in the variables $\{u_i\}_{i\in[n]}$, $\{F_r(t)\}_{r \in [n],t \in T}$ (observe that the values of $t \in T$ are not variables).
The model is long to be explained and I'll try to provide an example. Consider the following competition among three firms:
Firm $f_2$ compete against the firm $f_1$ and $f_3$ at the same time for the market share $\beta_{1,2}$ and $\beta_{2,3}$ respectively. Firm $f_1$ and $f_3$ are not competitors.
From the utility formula, we have:
$$u_1(x)=x(\alpha_1 + \beta_{1,2} \overline{F}_2(x))$$ $$u_2(x)=x(\alpha_2 + \beta_{1,2} \overline{F}_1(x) + \beta_{2,3} \overline{F}_3(x))$$ $$u_3(x)=x(\alpha_3 + \beta_{2,3} \overline{F}_2(x))$$
It is known (from the article) that there is an equilibrium for this competition, with $\alpha_1 > \alpha_2 > \alpha_3 > 0$ and $1 = t_1 > t_2 > t_3 > 0$ and $t_k \in [0,1]$:
$$u_1(t_1)=u_1(x) \forall x \in [t_2,t_1]$$ $$u_2(t_1)=u_2(x) \forall x \in [t_3,t_1]$$ $$u_3(t_1)=u_3(x) \forall x \in [t_3,t_2]$$
Now, the linear program that must be satisfied in this equilibrium:
$$u_1=t_1(\alpha_1 + \beta_{1,2} \overline{F}_2(t_1))$$ $$u_1=t_2(\alpha_1 + \beta_{1,2} \overline{F}_2(t_2))$$ $$u_2=t_1(\alpha_2 + \beta_{1,2} \overline{F}_1(t_1) + \beta_{2,3} \overline{F}_3(t_1))$$ $$u_2=t_2(\alpha_2 + \beta_{1,2} \overline{F}_1(t_2) + \beta_{2,3} \overline{F}_3(t_2))$$ $$u_2=t_3(\alpha_2 + \beta_{1,2} \overline{F}_1(t_3) + \beta_{2,3} \overline{F}_3(t_3))$$ $$u_3=t_2(\alpha_3 + \beta_{2,3} \overline{F}_2(t_2))$$ $$u_3=t_3(\alpha_3 + \beta_{2,3} \overline{F}_2(t_3))$$
$$u_1 \geqslant t_3(\alpha_1 + \beta_{1,2} \overline{F}_2(t_3))$$ $$u_3 \geqslant t_1(\alpha_3 + \beta_{2,3} \overline{F}_2(t_1))$$
$$\overline{F}_1(t_3) = 1$$ $$\overline{F}_2(t_3) = 1$$ $$\overline{F}_3(t_3) = 1$$ $$\overline{F}_2(t_1) = 0$$ $$\overline{F}_3(t_1) = 0$$ $$\overline{F}_1(t_1) > 0$$
$$\overline{F}_3(t_1) < \overline{F}_3(t_2)$$ $$\overline{F}_1(t_2) < \overline{F}_1(t_3)$$ $$\overline{F}_2(t_1) < \overline{F}_2(t_2)$$ $$\overline{F}_2(t_2) < \overline{F}_2(t_3)$$ $$\overline{F}_3(t_2) < \overline{F}_3(t_3)$$
What I'm looking for are values for $t_k$ and $\overline{F}_i(t_k) \in [0,1]$ that will satisfy the constraints above but I'm not sure what is the objective function in this case. I haven't worked with Linear Programs before.
My question is: since I must find the values for $t_k$ and $\overline{F}_i(t_k)$ that combined will give the best utility for every seller, is it correct to consider the objective function as follows?
$$maximize\ u = \frac{u_1(t_1) + u_1(t_2)}{2} + \frac{u_2(t_1) + u_2(t_2) + u_2(t_3)}{3} + \frac{u_3(t_2) + u_3(t_3)}{2}$$
I'm considering that, if competitors are trying to optimize their utilities, the sum of their best utilities would lead to a global utility that I have to maximize, but I'm not sure if I can do it, I believe I'm losing some basic concept of Linear Programming. Remember that $u_1(t_1) = u_1(t_2)$ and so on.
I'm a bit lost, I'd appreciate any hints, names or articles that I should read to see if I'm indeed following the right direction.
Thanks in advance.