Our model has to depict the basic issues related to the European immigration policy we have discussed in the introduction. First of all, external border enforcement is a public good, and there exists a conflict over its funding. No supranational authority can enforce a scheme of taxes and subsidies, thus countries interact strategically and nobody is forced to contribute. As a consequence, a contribution game fits very well this situation.

Our players are a central country (*C*) and a border country (*B*). *B* shares its border with an emigration country and provides immigration control for *C* as well. Both countries can contribute to curb immigration, but they have different immigration targets and face possibly different costs of contributing.

Before introducing our contribution game we have to clarify how immigration control is obtained.

### 2.1 Immigration control

Each country would like to halt immigration once its target is achieved. However, restricting immigration is expensive: it requires resources to enforce the border, screen immigrants, contrast illegal inflows and so on. A convenient way to summarize these actions is describing immigration restriction as an output produced through the resources *C* and *B* are willing to spend in order to achieve their targets.

We define with *g*
_{
B
} and *g*
_{
C
} the contributions by *B* and *C* respectively. Let *M* be the inflow of immigrants into the federation, which is given by

M=\stackrel{\u0304}{M}-d({g}_{B}+{g}_{C})\phantom{\rule{2em}{0ex}}0<d<1,

(1)

Where \stackrel{\u0304}{M}>0 depicts the inflow into the federation in case of no restriction (i. e. *g*
_{
B
}=*g*
_{
C
}=0). This kind of “production function” fits the idea that the amount of restriction is proportional to the resources used.^{9}

### 2.2 Payoffs

As we have pointed out in the introduction, the peculiarity of immigration control is that the country’s desired quota acts as a bliss point. Thus, payoffs include a quadratic loss function with respect to the national target, given by 0\le {M}_{C}^{\ast}<\stackrel{\u0304}{M} and 0\le {M}_{B}^{\ast}<\stackrel{\u0304}{M} respectively.

Since we are interested in studying how equilibrium contributions are determined, we consider the targets as exogenously given. In our view, this fits well a situation in which member countries try to defend their national interests within a federal assembly like the European Parliament.

We also assume perfect information on the destination chosen by immigrants: both countries know how many immigrants are willing to settle in *C* and how many immigrants are willing to settle in *B*. This assumption is by no means restrictive: since mobility within the federation is unrestrained, it only indicates that C and B know how the population inflow is going to be shared. Countries are usually well informed about their relative attractiveness for the immigrants.

Finally, *C* and *B* bear a quadratic cost to collect the resources needed to enforce the border.^{10} As a consequence, we write the utilities as follows:

\begin{array}{ll}{U}_{C}& =-\frac{1}{2}{(M-{M}_{C}^{\ast})}^{2}-\frac{1}{2}{g}_{C}^{2}\phantom{\rule{2em}{0ex}}\end{array}

(2)

\begin{array}{ll}{U}_{B}& =-\frac{1}{2}{(M-{M}_{B}^{\ast})}^{2}-\frac{\pi}{2}{g}_{B}^{2}.\phantom{\rule{2em}{0ex}}\end{array}

(3)

Note that since *M* is the inflow into the federation, {M}_{C}^{\ast} and {M}_{B}^{\ast} denote the federal target preferred by C and B respectively.^{11}

The parameter *π*≥1 allows for the possibility that the disutility of gathering the resources needed to curb immigration is relatively higher for B.

*π* is introduced for sake of generality, because it gives us the opportunity to analyze a cost asymmetry: in principle, it is quite possible that C and B bear different costs to gather the same contribution. This might happen, for instance, when B faces a fiscal crisis, as it is currently the case of Italy and Spain.

By substituting (1) into (2) and (3) we can rewrite the payoffs:

\begin{array}{ll}{U}_{C}& =-\frac{1}{2}{(\stackrel{\u0304}{M}-d({g}_{B}+{g}_{C})-{M}_{C}^{\ast})}^{2}-\frac{1}{2}{g}_{C}^{2}\phantom{\rule{2em}{0ex}}\end{array}

(4)

\begin{array}{ll}{U}_{B}& =-\frac{1}{2}{(\stackrel{\u0304}{M}-d({g}_{B}+{g}_{C})-{M}_{B}^{\ast})}^{2}-\frac{\pi}{2}{g}_{B}^{2}\phantom{\rule{2em}{0ex}}\end{array}

(5)

We are now going to solve the model under sequential and simultaneous decisions. In order to avoid redundancies the main properties of the results are discussed at the end of this section.

### 2.3 Results: sequential decisions

In the case of sequential decisions, both *C* and *B* could have the right to move first. We are now going to explore both cases.

#### 2.3.1 *C* moves first

Assume for the moment that *C* is the leader and *B* is the follower. We solve the game by backwards induction. The best response of *B* to *C* is

{\stackrel{\u0304}{g}}_{B}=\frac{d(\stackrel{\u0304}{M}-{M}_{B}^{\ast})-{d}^{2}{g}_{C}}{\pi +{d}^{2}}.

(6)

By substituting (6) into (4) we can write the leader’s problem:

\underset{{g}_{C}}{\text{max}}{U}_{C}=-\frac{1}{2}{\left[\stackrel{\u0304}{M}-d\left({g}_{C}+\frac{d(\stackrel{\u0304}{M}-{M}_{B}^{\ast})-{d}^{2}{g}_{C}}{\pi +{d}^{2}}\right)-{M}_{C}^{\ast}\right]}^{2}-\frac{1}{2}{g}_{C}^{2}

which yields

{g}_{C}^{\ast}=\frac{{\Delta}_{C}(\pi +{d}^{2})\mathrm{\pi d}-\pi {d}^{3}{\Delta}_{B}}{{\pi}^{2}{d}^{2}+{(\pi +{d}^{2})}^{2}}

(7)

where {\Delta}_{C}\equiv (\stackrel{\u0304}{M}-{M}_{C}^{\ast}), and {\Delta}_{B}\equiv (\stackrel{\u0304}{M}-{M}_{B}^{\ast}) measure the desired entry restriction.

By substituting (7) into (6) we get

{g}_{B}^{\ast}=\frac{{\Delta}_{B}(\pi +{d}^{2}+\pi {d}^{2})d-\pi {d}^{3}{\Delta}_{C}}{{\pi}^{2}{d}^{2}+{(\pi +{d}^{2})}^{2}}

(8)

we therefore have obtained the equilibrium contributions of both players when *C* moves first.

These contributions are positive under the following conditions:

\begin{array}{ll}{g}_{C}^{\ast}& >0\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{1em}{0ex}}\frac{{\Delta}_{C}}{{\Delta}_{B}}>\frac{{d}^{2}}{\pi +{d}^{2}}\phantom{\rule{2em}{0ex}}\end{array}

(9)

\begin{array}{ll}{g}_{B}^{\ast}& >0\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{1em}{0ex}}\frac{{\Delta}_{C}}{{\Delta}_{B}}<\frac{\pi +{d}^{2}+\pi {d}^{2}}{\pi {d}^{2}}\phantom{\rule{2em}{0ex}}\end{array}

(10)

Since *Δ*
_{
C
} and *Δ*
_{
B
} measure the restriction desired by *C* and *B* respectively, we define the ratio \frac{{\Delta}_{C}}{{\Delta}_{B}} as the “relative restriction” desired by *C*.

\frac{{\Delta}_{C}}{{\Delta}_{B}}>1

means that *C* likes more restriction relative to *B*. The opposite occurs when \frac{{\Delta}_{C}}{{\Delta}_{B}}<1. Conditions (9) and (10) indicate that for a player to contribute positively his desired relative restriction must be sufficiently high. This will be crucial in the rest of the paper.

Now we are going to present the results when *B* is the leader.

#### 2.3.2 *C* moves second

When *B* moves first, the best response function of *C* is

{\stackrel{\u0304}{g}}_{C}=\frac{d(\stackrel{\u0304}{M}-{M}_{C}^{\ast})-{d}^{2}{g}_{B}}{1+{d}^{2}}.

(11)

In order to solve the leader’s problem, we now substitute the best response function of *C* (11) into (5) and we find the equilibrium contribution of *B* ({g}_{B}^{\ast \ast}). Then, we plug {g}_{B}^{\ast \ast} into (11) and we solve for the follower’s contribution ({g}_{C}^{\ast \ast}).

The equilibrium contributions are

\begin{array}{ll}{g}_{C}^{\ast \ast}& =\frac{{\Delta}_{C}({d}^{2}+\pi +\pi {d}^{2})d-{d}^{3}{\Delta}_{B}}{{d}^{2}+\pi {(1+{d}^{2})}^{2}}\phantom{\rule{2em}{0ex}}\end{array}

(12)

\begin{array}{ll}{g}_{B}^{\ast \ast}& =\frac{{\Delta}_{B}(1+{d}^{2})d-{d}^{3}{\Delta}_{C}}{{d}^{2}+\pi {(1+{d}^{2})}^{2}}.\phantom{\rule{2em}{0ex}}\end{array}

(13)

The conditions for having positive contributions are summarized below:

\begin{array}{ll}{g}_{C}^{\ast \ast}& >0\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{1em}{0ex}}\frac{{\Delta}_{C}}{{\Delta}_{B}}>\frac{{d}^{2}}{{d}^{2}+\pi +\pi {d}^{2}}\phantom{\rule{2em}{0ex}}\end{array}

(14)

\begin{array}{ll}{g}_{B}^{\ast \ast}& >0\mathit{\text{for}}\frac{{\Delta}_{C}}{{\Delta}_{B}}<\frac{1+{d}^{2}}{{d}^{2}}.\phantom{\rule{2em}{0ex}}\end{array}

(15)

Finally, we are going to solve the simultaneous game.

### 2.4 Results: simultaneous decisions

In a simultaneous game, the best response functions for *C* and *B* are, respectively, (11) and (6), and the solutions are

\begin{array}{ll}{\stackrel{~}{g}}_{C}& =\frac{{\Delta}_{C}(\pi +{d}^{2})d-{d}^{3}{\Delta}_{B}}{{d}^{2}+\pi +\pi {d}^{2}}\phantom{\rule{2em}{0ex}}\end{array}

(16)

\begin{array}{ll}{\stackrel{~}{g}}_{B}& =\frac{{\Delta}_{B}(1+{d}^{2})d-{d}^{3}{\Delta}_{C}}{{d}^{2}+\pi +\pi {d}^{2}}.\phantom{\rule{2em}{0ex}}\end{array}

(17)

These contributions are positive under the following conditions:

\begin{array}{ll}{\stackrel{~}{g}}_{C}& >0\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{1em}{0ex}}\frac{{\Delta}_{C}}{{\Delta}_{B}}>\frac{{d}^{2}}{{d}^{2}+\pi}\phantom{\rule{2em}{0ex}}\end{array}

(18)

\begin{array}{ll}{\stackrel{~}{g}}_{B}& >0\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{1em}{0ex}}\frac{{\Delta}_{C}}{{\Delta}_{B}}<\frac{1+{d}^{2}}{{d}^{2}}\phantom{\rule{2em}{0ex}}\end{array}

(19)

By observing (7), (8), (12), (13), (16) and (17) it is evident that the equilibrium contribution of each player is decreasing with respect to the desired immigration restriction of the other player. In other words, in all cases the contribution of *C* is decreasing with *Δ*
_{
B
}, and the contribution of *B* is decreasing with *Δ*
_{
C
}.

To understand intuitively this result, suppose then that *B* prefers strict border enforcement and *C* is relatively open. As a consequence the ratio \frac{{\Delta}_{C}}{{\Delta}_{B}} is low, and *C* has an incentive to free ride, because *B* will provide enough immigration control for both countries. This conveys the essential insight that, in order for both countries to contribute, the national targets {M}_{C}^{\ast} and {M}_{B}^{\ast} must not be too different. This result extends earlier findings by Varian (1994)^{12}, and it has crucial consequences that we are going to discuss in the rest of the paper.

Before proceeding to compare the outcomes under the sequential and the simultaneous regimes, it is indispensable to understand when contributions are positive and when there exists joint contribution.