In this section we establish university attendance which depends on the admission standard/threshold set by each university. Using [1] we rewrite the payoff functions of the two universities as
${\mathrm{\pi}}_{i}({x}_{Aj},{x}_{Bj})=W\left({n}_{i}\left({x}_{Aj},{x}_{Bj}\right),f{n}_{i}\left({x}_{Aj},{x}_{Bj}\right)-c\left({n}_{i}\left({x}_{Aj},{x}_{Bj}\right)\right),{\mathrm{\Theta}}_{i}\left({x}_{Aj},{x}_{Bj}\right)\right)-{\mathrm{\Phi}}_{i},$[2]where ${n}_{i}\left({x}_{Aj},{x}_{Bj}\right)$ and ${\mathrm{\Theta}}_{i}\left({x}_{Aj},{x}_{Bj}\right)$ are the number of students admitted and the average quality of students at university $i,$ $i=A,B$, respectively, given the admission standards $\left({x}_{Aj},{x}_{Bj}\right)$ for locations $j=1,2$. Notice that we have replaced research expenditure ${R}_{i}$ with ${R}_{i}=f{n}_{i}\left({x}_{Aj},{x}_{Bj}\right)-c\left({n}_{i}\left({x}_{Aj},{x}_{Bj}\right)\right)$, i. e. the difference between total revenue from tuition fees and teaching costs.

It is clear that $\mathrm{\partial}{\mathrm{\Theta}}_{i}/\mathrm{\partial}{x}_{ij}>0:$ an increase in a university’s admission standard increases the average ability of that university’s students. According to the timing of the game, universities set their admission standards following their decision on the opening (or not) of a branch campus. Therefore there are four distinct possibilities, depending on the first-stage subgames:

1.

Neither university opens a branch campus, hence the relevant standards to set are ${x}_{A1}$ and ${x}_{B1}$.

2.

University $A$ opens a branch campus; the standards to be set are ${x}_{A1}$, ${x}_{A2}$ and ${x}_{B1}$.

3.

University $B$ opens a branch campus; the standards to be set are ${x}_{A1}$, ${x}_{B1}$ and ${x}_{B2}$.

4.

Both universities open a branch campus; the relevant standards are ${x}_{A1}$, ${x}_{A2}$, ${x}_{B1}$ and ${x}_{B2}$.

A natural question arising here is whether the standards chosen will be symmetric, meaning that the universities set the same admission standards, or asymmetric, where universities choose different admission standards. A preliminary result is the following.

**Lemma 1::** *Symmetric standards cannot be part of an equilibrium with positive student demand*.

**Proof:** *See Appendix A.1. ■*

Lemma 1 establishes that an equilibrium requires setting different admission standards.

In relation to subgame 4 (see above) we clarify that one can distinguish the following possibilities: (a) University $A$ sets a higher standard in both countries, (b) University $B$ sets a higher standard in both countries, (c) University $A$ sets a higher standard in country 1, (d) University $A$ sets a higher standard in Country 2. Cases (a) and (b) and cases (c) and (d) are clearly specular. In what follows, we focus on equilibria where university $A$ sets the higher standard in every country in which it operates, and we call university $A$ “elitist”.
We therefore state,

**Assumption 1::** *${x}_{Aj}>{x}_{Bj}$.*

The following propositions describe university attendance of students from Country 1 and 2, respectively.

**Proposition 1::** *Consider students living in Country 1. Let university* $i\in \left\{A,B\right\}$ *set standard* ${x}_{i1}$*, and Assumption 1 hold. A student attends university* $A$ *if* $\mathrm{\theta}\in \left[{x}_{A1},1\right]$ *and university* $B$ *if* $\mathrm{\theta}\in \left[{x}_{B1},{x}_{A1}\right).$

**Proof:** *See the Appendix A.2. ■*

The next proposition establishes university attendance of students from Country 2. For these students, university attendance depends on (i) whether or not one or two branch campuses are opened, and (ii), in the case where only university $B$ opens a branch campus, whether $t$ is greater or not than ${t}^{\ast}$ (see Definition 1). Indeed, a student with high ability may be admitted to university $A,$ but the mobility costs are high so that the student may prefer to attend the branch campus of university $B.$ The equivalent of Proposition 1 for students of Country 2 follows.

**Proposition 2::** *Consider students living in Country 2. Let university* $i\in \left\{A,B\right\}$ *set standard* ${x}_{ij}$ *in their site in Country* $j$ *with* ${x}_{Aj}>{x}_{Bj}$ *(assumption 1)*.

*(i)*

**No BC**: a student attends university if she is privileged: in particular, university $A$ if $\mathrm{\theta}\in \left[{x}_{A1},1\right]$ and university $B$ if $\mathrm{\theta}\in \left[{x}_{B1},{x}_{A1}\right).$

*(ii)*

**University** $A$ **operates BC**: a student attends university $A$ if $\mathrm{\theta}\in \left[{x}_{A2},1\right]$ and university $B$ if privileged and $\mathrm{\theta}\in \left[{x}_{B1},{x}_{A1}\right).$

*(iii)*

**Universities** $A$ **and** $B$ **operate BC**: a student attends university $A$ if $\mathrm{\theta}\in \left[{x}_{A2},1\right]$ and university $B$ if $\mathrm{\theta}\in \left[{x}_{B2},{x}_{A2}\right).$

*(iv)*

**University** $B$ **operates BC, and** $t\le {t}^{\ast}$: a student attends university $A$ if privileged and $\mathrm{\theta}\in \left[{x}_{A1},1\right]$ and university $B$ either if non-privileged and $\mathrm{\theta}\in \left[{x}_{A1},1\right]$ or, if not privileged and $\mathrm{\theta}\in \left[{x}_{B2},{x}_{A1}\right).$

*(v)*

**University** $B$ **operates BC, and** $t>{t}^{\ast}$: a student never attends university $A,$ and attends university $B$ if $\mathrm{\theta}\in \left[{x}_{B2},1\right].$

Propositions 1 and 2 allow us to simplify the interaction between universities and students so that, in the first stage, the strategy space is binary, and consists of the decision of whether to open (or not) a branch campus in country 2 and in the second stage, the strategy space is given by the admission standards, ${x}_{Aj},{x}_{Bj}\in [0,1]$. The equilibrium concept is subgame perfect equilibrium by backward induction. Figure 1 depicts the timing of the game.

Figure 1: The simplified game.

To make further progress and obtain explicit closed-form solutions for the remainder of the paper we specify the functional relationships as follows:
$U({x}_{ij},\mathrm{\theta})=w{x}_{ij}+\mathrm{\theta},$[3]
$W\left({n}_{i},{R}_{i},{\mathrm{\Theta}}_{i}\right)=\sum _{j}{n}_{ij}\left(w{x}_{ij}+f-c{n}_{ij}\right),$[4]
$c\left({n}_{ij}\right)=c\sum _{j}{n}_{ij}^{2},$[5]
$G(\mathrm{\theta})=\mathrm{\theta},$[6]where $S\left({x}_{ij}\right)=w{x}_{ij},$ $w>0$ is a graduate’s wage. The specifications chosen (eqs [3], [4], [5] and [6]) respect the properties of the associated general functions. In particular, note that the average ability of students increases with ${x}_{ij}$. Finally, assuming a uniform distribution for ability is standard in the literature of university competition (Del Rey, 2001; De Fraja and Iossa, 2002; Cesi and Paolini, 2013 and Carroni, Cesi and Paolini 2015).

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