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Coordination Failures in Immigration Policy
Working paper by Giordani , Paolo E., Ruta, Michele, 2011
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Staff Working Paper ERSD-2011-02 Date: 26 January 2011
World Trade Organization
Economic Research and Statistics Division
Coordination Failures in Immigration Policy
Paolo E. Giordani
LUISS "Guido Carli" University
World Trade Organization
Manuscript date: December 2010
Disclaimer: This is a working paper, and hence it represents research in progress. This paper
represents the opinions of the authors, and is the product of professional research. It is not meant to
represent the position or opinions of the WTO or its Members, nor the official position of any staff
members. Any errors are the fault of the author. Copies of working papers can be requested from the
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number and title.
Coordination Failures in Immigration Policy
Paolo E. Giordaniy
LUISS "Guido Carli" University
World Trade Organization
We propose a theoretical framework for analyzing the problems associated to unilat-
eral immigration policy in receiving countries and for evaluating the grounds for reform of
international institutions governing immigration. We build a model with multiple destina-
tion countries and show that immigration policy in one country is inuenced by measures
adopted abroad as migrants choose where to locate (in part) in response to di¤erences
in immigration policy. This interdependence gives rise to a leakage e¤ect of immigration
policy, an international externality well documented in the empirical literature. In this
environment, immigration policy becomes strategic and unilateral behavior may lead to
coordination failures, where receiving countries are stuck in welfare inferior equilibria.
We then study the conditions under which a coordination failure is more likely to emerge
and argue that multilateral institutions that help receiving countries make immigration
policy commitments would address this ine¢ ciency.
Keywords: Immigration policy, cross-border externalities, coordination failures, mul-
cation: F02, F22, J61
Acknowledgments: We would like to thank G. De Arcangelis, C. Devillanova, G. Facchini, F. Robert-
Nicoud, R. Staiger and seminar participants at the Geneva Trade and Development Workshop, the European
Trade Study Group Annual Meeting and the SIE Conference for comments and suggestions. Remaining errors
are our responsibility. Disclaimer: The opinions expressed in this paper should be attributed to the authors.
They are not meant to represent the positions or opinions of the WTO and its Members and are without
prejudice to Members’rights and obligations under the WTO.
yDepartment of Economics and Business, LUISS "Guido Carli" University, Viale Romania 32, 00197 Roma,
Italy (e-mail: firstname.lastname@example.org).
zEconomics Research Division, World Trade Organization, Rue de Lausanne 154, 1211 Geneva, Switzerland
What type of multilateral institutions do countries need to govern international migrations?
Several economists have recently raised this question (among others, Bhagwati, 2003, Hat-
ton, 2007, and Hanson, 2009). In particular, Hatton (2007) examines whether the basic
principles governing the World Trade Organization could improve international cooperation
on migration between sending and receiving countries.
The present work aims at contributing to this debate but takes a somewhat di¤erent ap-
proach for two reasons. In our view, a prerequisite for a precise answer to the above question
is the identi
cation of the international externalities associated with unilateral policy-making
in migration policy. In some sense, this is a key lesson that can be inferred from the economic
literature on the multilateral trading system. As Bagwell and Staiger (1999 and 2002) show,
the GATT/WTO system has e¤ectively improved international trade policy cooperation pre-
cisely because it provides a framework to neutralize a key cross-border spillover associated
with unilateral policy-making in the trade domain, the terms-of-trade externality. Second,
the scope of our analysis is di¤erent -and possibly more limited- compared to Hatton (2007).
Rather than looking at the problems of international cooperation between host and sending
economies, we focus on the interaction of immigration measures implemented by countries
that are on the receiving end of immigration. Our goal is to clearly identify the externality
associated with immigration policy in this set of countries and to investigate the welfare
implications of this economic interdependence.1
A large body of empirical literature has recently studied the long-run determinants of
immigration policy and found four key (and somehow interrelated) channels: distributional,
political economy, non-economic and international determinants. Distributional factors in-
clude the e¤ect of immigration on the labor market and on welfare systems (Borjas, 1994 and
2003, Boeri et al., 2002, Razin et al., 2002). In turn, distributional determinants are chan-
nelled into government policies through voting and/or lobbying activity by interest groups
that stand to lose or gain from immigration (Goldin, 1993, Facchini et al., 2008). Non-
economic forces, such as racism or xenophobia, may inuence voters’attitudes -and, hence,
1While we explicitly model the migration decision of foreign workers (as further discussed below, the
migratory decisions are the key transmission mechanism of policy spillovers), the welfare e¤ects of emigration
on the sending region are not analyzed in this work.
immigration policy (Dustmann and Preston, 2007, and O’Rourke and Sinnott, 2006). Finally,
and crucially for the present work, immigration policy abroad is a determinant of immigra-
tion policy at home (Timmer and Williamson, 1998, Boeri and Brücker, 2005, Hatton and
Williamson, 2005). The positive correlation between domestic and foreign measures suggests
that countries aim at anticipating an externality associated with the immigration policy of
other destination countries.
As these international determinants are a key concern of this paper, we briey review the
available evidence. In their historical account of migratory ows and immigration policy in the
NewWorld in the late 19th and early 20th century, Timmer and Williamson (1998) argue that
countries in the New World must have paid close attention to each others’policies as migrants
were pulled from and pushed toward one country in response to less or more restrictive policies
in others.2 In particular, they
nd that "Australia’s openness decreased ows to Canada,
Brazil’s pro-immigrants subsidies reduced ows to Australia, and Argentina saw an increased
share of the immigrant pie as the United States closed its doors" (Timmer and Williamson,
1998, p. 756). A second study that documents the immigration policy spillover is Boeri
and Brücker (2005) who adopt a di¤erent methodology and look at a di¤erent immigration
episode. In January 2004, the European Union enlarged to ten new member states from
Eastern and Central Europe. Transitional arrangements allowed individual EU countries to
temporarily breach the principle of free movement of people inside the Union and to impose
restrictions on immigration from the new member states. Boeri and Brücker (2005)
these arrangements a¤ected the geographical orientation of migrants from the new member
states and resulted in substantial diversion of migration ows from countries closing their
borders to countries with more open rules.
Contrasting with these developments in the empirical literature, there have been few
attempts to integrate these factors into formal models of immigration policy formation. In
particular, most existing models, such as Benhabib (1996), de Melo et al. (2001), Dolmas
and Hu¤man (2004), Ortega (2005) and Facchini and Willman (2005), incorporate some
form of domestic factors, but are silent about international determinants.3 The main reason
is that standard theory focuses on the e¤ects of immigration on a single receiving economy
2See also Hatton and Williamson (2005).
3There are some recent noticeable exceptions that are discussed at the end of the Introduction.
and considers as exogenous the migratory decision of foreign workers (see Borjas, 1995).4
These features, by construction, shut down any possible cross-border spillover created by
immigration policy. The present work contributes to
lling this gap in the formal literature
by providing a simple and tractable model of immigration policy interdependence.
In our model immigration policy and migration choices are endogenous. The set up con-
siders two regions. The receiving region is formed of a set of identical countries that choose
independently immigration policy. In order to make a convincing case of the mechanism
discussed in this paper, we rely on a general model that is broadly consistent with distrib-
utional, political economy and non-economic determinants of immigration policy (a speci
model is presented in Appendix A). Immigration is assumed to have bene
ts and costs on
host economies, so that there is an optimal number of foreign workers for each receiving
country. The sending region is populated by a set of workers who can choose whether to
migrate or not and -to a certain extent- in which country to move to. Migratory decisions
depend on the economic incentives that foreign workers face and on the policy regulating
migratory ows enacted in the receiving countries.
If the world had only a single receiving country, a host government could easily select a
policy that supports the e¢ cient level of immigration -that is, the level that optimally trades
o¤ the costs and bene
ts of immigration. Governments, however, do not act in a vacuum:
immigration policy in one country alters the migratory choices of foreign workers and, hence,
the ows of migrants into other destinations (the immigration policy spillover). Note
that this externality is created by the international mobility of prospective migrants. When
foreign workers choose, not only whether to migrate or not, but also where to migrate (i.e. the
destination country), policy restrictions (liberalizations) in one country increase (decrease)
migratory ows in other receiving economies, as a larger number of migrants will target the
country with lower restrictions. In other words, the costs and bene
ts of immigration in any
host economy are, in part, determined by the policy stance of other receiving countries. This
international externality lowers the ability of national governments to optimally manage their
In this interdependent environment, coordination failures can materialize that lead to
4Few theoretical contributions have considered the interdependence between immigration policy in the host
economy and immigration decisions. See, in particular, Bellettini and Berti Ceroni (2007), Bianchi (2007) and
Giordani and Ruta (2010).
ine¢ cient equilibria. The choice of immigration policy is strategic and de
nes a symmetric,
simultaneous game among all destination countries from which multiple symmetric policy
equilibria emerge that can be Pareto-ranked. The "cooperative solution", that is, the immi-
gration policy associated with the optimal number of migrants for each country, is only one
in the continuum of Nash equilibria of this policy game. Coordination failures in immigra-
tion policy may arise because, for each policy maker, expectations on the behavior of the
governments of other destination economies are critical in the determination of the policy
outcome of the receiving region. For instance, if any one government expects that others will
strengthen immigration barriers, then it will
nd it convenient to restrict its policy stance
to neutralize the negative externality of an excessive inux of migrants, thus triggering a
series of restrictive measures. Too little immigration will result relative to the e¢ cient level
for the overall destination region. Similarly, beliefs of immigration liberalizations by other
receiving economies will trigger a reduction in restrictions that will result in a Pareto-inferior
equilibrium characterized by too much immigration.
Once we identify the problem that characterizes immigration policy in this framework,
we discuss two further issues. First, we analyze the problem of equilibrium selection and
show that coordination failures in immigration policy are not only possible, but they are
also likely to emerge in presence of uncertainty on the policy strategy of other receiving
governments. The game-theoretic literature has proposed alternative equilibrium re
for coordination games admitting a multiplicity of equilibria. These re
nements stress the
fact that players may coordinate on a strategy which is less risky, even if Pareto-dominated.5
In particular, we characterize the immigration policy equilibrium that is robust to strategic
uncertainty (Andersson et al., 2010) and show that the Pareto-e¢ cient equilibrium is not
robust -i.e. that the unilateral policy outcome may well support ine¢ ciently low or high
The second issue that we investigate is how an increase in the international mobility of mi-
grants (for instance due to technological innovations, such as improvements in transportation
and communication means) a¤ects the "likelihood" of a coordination failure. We
an increase in migrants’mobility does not change the e¢ cient policy for the receiving region,
5The classic work is Harsanyi and Selten (1988) on the risk-dominant equilibrium. Cooper et al. (1990 and
1992) and van Huyck et al. (1990) nd that coordination failures are likely to arise in experimental settings.
For a survey of the empirical literature see Cooper (1999), chapter 1.
but it expands the set of equilibria (a measure of the indeterminacy of equilibria) and alters
the robust equilibrium, as it increases each policy maker’s uncertainty about other govern-
ments’strategies. Intuitively, both
ndings can be rationalized as an increased international
mobility of migrants magni
es the cross-border externality associated with immigration pol-
icy. This suggests that the "globalization" may be amplifying the chances of coordination
failures across destination countries, thus augmenting the need for policy coordination in the
While we leave a further discussion of the policy implications of our model to the conclu-
sion, some preliminary considerations can be put forward. First, while both trade policy and
immigration policy are characterized by a cross-border externality, the immigration policy
game has radically di¤erent features. Trade policy interactions determine a (terms-of-trade
driven) prisoner’s dilemma situation, while interactions in the domain of immigration pol-
icy lead to a trust dilemma, a coordination problem where governments achieve e¢ cient
policies only if they make mutually consistent decisions. Second, while the trade policy game
leads to too little trade, coordination failures in immigration policy may determine either too
little or too much immigration from the perspective of the receiving world. Third, multilat-
eral institutions should help countries escape ine¢ cient equilibria. This theory suggests that
immigration policy commitments (that can be credibly enforced) can provide a coordination
device to receiving countries.
Our work is related to several recent studies. The literature on asylum seeking has mod-
elled the spillover e¤ect in national refugee laws and emphasized that coordination problems
may emerge in this context (Hatton, 2004, Facchini, Lorz and Willmann, 2006). The paper
by Bubb, Kremer and Levine (2007), in particular, is closely related to ours. They show
that restricting refugee law in some host countries (i.e. increasing the standard of proof to
distinguish between refugees and migrants) may induce other host economies to do the same
and that this may lead to a multiplicity of equilibria in refugee law. Recent formal work
on immigration policy emphasizes di¤erent channels of international interdependence in this
domain. Brücker and Schröder (2010) build a model where a destination country’s e¤ort
to improve the skill composition of its immigration pool induces other host economies to
adopt similar immigration reforms. De la Croix and Docquier (2010)
nd positive external-
ities in immigration policy in a model where host economies have an aversion to the global
inequality created by barriers to international labor movements. Relative to these papers
there are two main innovations in our work. First, the immigration policy spillover is the
result of an endogenous response of potential migrants to di¤erences in immigration policy
in host economies. Second, we formally analyze the determinants of multiple equilibria in
immigration policy and study the issue of equilibrium selection.
The paper is organized as follows. Section 2 introduces the multiple-country framework.
In this setting, we formalize the immigration decision of foreign workers and the immigration
policy spillover. In Section 3 we prove the existence of a multiplicity of Nash equilibria and
carry out the comparative statics analysis. Section 4 studies the issue of equilibrium selection
under strategic uncertainty. A concluding section discusses the implications of this model for
the design of international institutions governing immigration.
2 A Multiple-Country Model of Immigration Policy
In this section we introduce a model of immigration policy with a sending region, populated
by F workers, and a receiving region composed of m countries, indexed by h = 1; :::;m. Each
host economy has identical fundamentals, but decides immigration policy independently of
the other destination countries. Foreign workers can choose whether to migrate or not and
where to locate in the receiving region. This setting is su¢ cient to determine the interna-
tional spillover characterizing immigration policy and, hence, the type of strategic problem
associated with unilateral immigration policy in the receiving world.
Immigration has bene
ts and costs for the host economies. De
ne the welfare of the
generic economy h in the receiving region as a continuous function in the number of immi-
grants in the country, Wh (Ih), and suppose that this function admits one and only one
maximum at Ih = Î. This value of immigration is the one which optimally trades o¤ costs
ts of migrants for the host economy. While we are agnostic about the source of
these costs and bene
ts, a standard model of immigration policy that supports this structure
is presented in Appendix A.
2.1 Migratory Choices and the Policy Spillover
We now introduce the migratory choice of foreign workers. Immigration is a non-reversible
decision. Each migrant faces a psychological cost to leave her own country, i, which is
uniformly distributed in [0; ], where is normalized to 1. The government in h can set
up an immigration policy which is parametrized by a cost borne by immigrants once in the
new country, h 2 R+. This parameter can be interpreted in several ways, from the cost of
bureaucratic procedures that each immigrant faces in the host economy to laws that a¤ect
the life of immigrants in the host country, such as the number of years to obtain voting rights
Migrants are internationally mobile, in the sense that in a world formed of several po-
tential host economies they have some freedom in choosing their destination. Clearly, the
international mobility of migrants is limited by a series of factors in addition to immigration
restrictions in the receiving world, including primarily geographical distance, but possibly
other factors such as technology (e.g. communication technologies) or cultural diversity (e.g.
adaptability to di¤erent cultures).6
We capture the limited international mobility of migrants by assuming that foreign work-
ers are of two kinds. A fraction F , with 2 (0; 1), can decide freely which receiving country
to move to in the set m ("free foreign workers"). The remaining fraction (1)F are instead
constrained in their choice ("constrained foreign workers"). For reasons of symmetry, we fur-
ther suppose that each receiving country can attract at most (1)F=m constrained foreign
workers; that is, potential migrants of "constrained type" are distributed uniformly across
the receiving region. The parameter captures the international mobility of migrants. A
higher value of , that is, an increase in the set of "free foreign workers", can be motivated
by several factors that reduce the (non-policy) constraints to the migrants’mobility, such as
an improvement in transportation or telecommunication technologies.
A constrained foreign worker in the pool (1 )F=m, indexed by i, will migrate to h if
and only if
bh (h) i 0; (1)
6See, among others, Belot and Hatton (2008) and Grogger and Hanson (2008).
where bh is the endogenous net bene
t that the foreign worker receives if she migrates to
country h, assumed to be a twice continuously di¤erentiable and decreasing function of h.
It is immediate to
nd the threshold value of the psychological cost (such that all those below
that value are willing to migrate) as
h = bh (h) :
Given that i is distributed uniformly in [0; 1], the number of constrained migrants to h will
be bh (h) (1)F=m.
The number of free foreign workers potentially entering each country h is instead given
by the whole pool of free foreign workers, F . In addition to satisfying condition (1), free
foreign workers will also compare the payo¤ obtained by migrating to country h to the one
obtained by migrating to any other receiving country (denoted by h).8 Free foreign workers
will target country h if
bh (h) > bh
() h < h
Therefore, policy di¤erences in the destination world a¤ect migration choices. Speci cally,
the number of free foreign workers actually migrating to h is 0 if h > h (crowding out),
and bh (h)F if h < h (crowding in). Finally, if h = h, free migrants are indi¤erent
and distribute symmetrically across the receiving region, that is, bh (h)F=m for any h.
As a result, immigration ows to country h are a function of h’s immigration policy as
well as of the measures imposed in the rest of the destination countries. The total number
of (constrained plus free) migrants to country h can then be described as Ih
bh (h)Fh, where
F [(1) =m+] F if h < h
(1)F=m F if h > h
F=m ~F if h = h:
This e¤ect of immigration policy abroad on the ow of migrants into the host economy
is the key cross-border externality in this model and the mechanism of economic interdepen-
7The fact that a tightenening of immigration policy reduces the foreign workers’bene
ts from immigration
is a desirable feature for any reasonable model of immigration policy. The model developed in appendix has
8The absence of asymmetric equilibria (which will be proven in Appendix C) allows us to simplify the
notation: bh and h denote the (identical) bene t and policy set in all m 1 countries other than h.
dence that we highlight. Importantly, the theory closely captures the essential international
policy spillover emphasized in the empirical literature discussed in the Introduction.
Two related considerations seem relevant. The
rst is on the interpretation of parameter
in the model. If is equal to zero (i.e. no international mobility of foreign workers),
then ~F = F = F and there is no policy spillover. As increases, di¤erences in immigration
policies among destination countries have a larger e¤ect on the ow of migrants. In other
words, can be interpreted as an elasticity -i.e. the responsiveness of migrants to policy
di¤erences. Factors such as improvements in transportation and communication technologies
or proximity are likely to increase this elasticity and hence magnify the size of the immigration
The second consideration relates to the size of this international externality. In their
study, Timmer and Williamson (1998)
nd that the e¤ect of the immigration policy spillover
is statistically signi
cant but small, while Boeri and Brücker (2005) show that policy dif-
ferences increased by up to
ve times immigration to more open EU members compared to
the counterfactual of free mobility in the EU. The two studies need not be in contradiction
as they are consistent with di¤erent values of in the model. The international mobility
of migrants from the Old to the New World in the 19th century was limited by distance
and technological factors compared to modern immigration from Eastern to Western Europe.
This is consistent with a higher value of in the latter immigration episode and, hence, with
a stronger policy externality.
3 Multiple Policy Equilibria and Coordination Failures
Given the international externality created by the migratory behavior of foreign workers, we
now characterize the equilibrium immigration policies by studying the strategic interaction
among receiving countries. Formally, this interaction can be represented as a symmetric
coordination game among the governments of the m destination countries, each deciding
its own immigration policy in a non-cooperative fashion. As we will see, this game admits a
continuum of symmetric, Pareto-rankable, Nash equilibria. In particular, we prove that there
exists an interval of immigration policies
such that, if all countries but h select any
policy in that interval, country h will
nd it best to do the same. We also show that there
exists a pay-o¤ dominant equilibrium belonging to that interval, and that such equilibrium
is associated with policy ̂ which, if implemented by all host countries, is able to "attract"
the optimal number of migrants, Î, for all of them. This policy is the one solving equation
Î = bh (̂) ~F . All other equilibria around this optimal policy equilibrium are instead sub-
optimal and represent a coordination failure among the receiving countries driven by the
immigration policy spillover.
A coordination failure arises in this game because immigration policies across receiving
countries are strategic complements. To give an intuition, start from the optimal policy
equilibrium, ̂. If all other countries but h restrict their policy above ̂, country h is better
o¤ following this restriction rather than su¤ering the “crowding in”of migrants that would
result from sticking to ̂. This incentive continues up to policy . Symmetrically, if all
other countries but h loosen up their policy below ̂, country h is better o¤ by implementing
this softer policy stance rather than su¤ering a “crowding out”of migrants. This incentive
continues up to policy . A strategic complementarity across host countries is thus responsible
for the positive co-movement of immigration policies documented in the data.
Let us now de
ne the payo¤ function of the government of generic country h as a func-
tion of its immigration policy, h, and of the policy strategy followed by all other receiv-
ing countries, h, h
. This payo¤ function can be found by substituting for
= bh (h)Fh (where Fh is given in (2)) into the welfare function, Wh (Ih). Intu-
itively, this payo¤ function is a step function which depends on whether h is higher, lower
or equal to h. If it is lower, country h will experience a crowding in (Fh = F ), and the
payo¤ function is obtained by substituting for Ih = bh (h)F into Wh (Ih). If it is higher,
country h will experience a crowding out (Fh = F ), and expression Ih = bh (h)F is instead
substituted into Wh (Ih). Finally, if it is equal, migrants distribute equally across the host
region (Fh = ~F ), and the payo¤ function is obtained by substituting for Ih = bh (h) ~F into
Wh (Ih). We can then write9
Wh (h; F ) if h > h
if h = h
if h < h:
This payo¤ function is drawn in Figure 1. The solid curve in Figure 1 represents country
h’s welfare when its immigration policy is equal to the one implemented in the rest of the
receiving region (Wh
). The dashed curve captures h’s welfare when its policy stance
is more restrictive than abroad (Wh (h; F )), while the dotted curve represents the opposite
). Each of them is assumed to be twice continuously di¤erentiable and
strictly concave in h (in the standard model of immigration policy developed in appendix
we study the conditions for which this is the case -see Appendix B). Note that the optimal
number of migrants for country h is unambiguously given by Î and, hence, the three functions
have the same maximum. However, the policy delivering this level of immigration depends
on whether this policy is higher, lower or equal to the one implemented abroad. In Figure 1,
we have called these policy values respectively ̂,̂, ̂.10
INSERT FIGURE 1 HERE
Before proving the existence of the continuum of policy equilibria, we provide a simple
intuition of this result. Along the interval
it is Wh
Wh (h; F ) ;Wh
Assume that all other countries set up a policy h 2
. Then, it is easy to show
that, for country h, any payo¤ associated with h 6= h is lower than the one associated
with h = h. In fact, suppose country h sets up a policy h lower than h. Then, for
9The pay-o¤ function is not continuously di¤erentiable, which prevents us from using the standard tools of
di¤erential calculus to nd the best-response functions and the Nash equilibria of the game. Note also that,
albeit more complicated, this function resembles the pay-o¤ function of a Bertrand competition game with
homogeneous goods, in which each rm’s pro t depends on whether its price is higher, lower or equal to the
one set up by its rivals (see for instance Tirole, 1988, pp. 209-211). In particular, the policy game described
in this paper shares many features with price competition games where rms’costs are assumed to be convex
(Dastidar, 1995, Weibull, 2006).
10 It is easy to prove that ̂ < ̂ < ̂. Moreover, de
ning "open door" policy (od) and "closed door" policy
(cd) as the policies which induce, respectively, all foreign workers and no foreign worker to emigrate to h,
it is possible to prove that Wh
, while Wh
(as depicted in Figure 1). These proofs are available upon request from the authors.
any h < h, function Wh
lies uniformly below Wh
, that is to say, any
h < h is associated with lower welfare than h = h. On the other hand, suppose
country h chooses a policy h higher than h. Then, for any h > h, function Wh (h; F )
lies uniformly below Wh
. As a result, whatever h 2
, country h’s best
response is h = h.11
Policy ̂ belongs to the interval
and is a Nash equilibrium. Indeed, it is the pay-
o¤ dominant Nash equilibrium in that, if all other countries set up ̂, country h is able to
attract the optimal number of migrants Î by adopting the same policy, h = ̂. Equilibria
surrounding the optimal policy equilibrium are Pareto-inferior outcomes which result from
a coordination failure driven by the international policy spillover associated with migrants’
mobility across the receiving region. A graphical representation of the set of equilibria is
provided in Figure 2. We can enunciate the following
INSERT FIGURE 2 HERE
Proposition 1 There exist a lower and an upper threshold, and , such that any symmetric
guration of immigration policies, (1; :::; m) = (
; :::; ), for which 2 ; , is a
Nash equilibrium of the game. The optimal policy equilibrium h = ̂ 8h belongs to the set
of symmetric Nash equlibria. All other equilibria are sub-optimal and are Pareto-ranked by
the distance from ̂.
Proof. In Appendix C
The logic of the proof is simple and consists of exploiting some of the properties of the three
welfare functions Wh
,Wh (h; F ) ;Wh
(such as continuity and strict concav-
ity) to prove that, for any h 2
and for any h, it isWh
Wh (h; F ) ;Wh
11This reasoning only applies along the interval
. Suppose for instance h > . In this case, country
h’s best response would be to slightly undercut policy h. This softer policy (implying a crowding in country
h), would be associated with a higher welfare, that is, Wh
. An analogous reasoning
applies to policy values below .
As a result, along that interval enacting the same policy as the rest of the region will be better
than enacting any other policy above or below that policy.12
Another way to look at this set of Nash equilibria is by drawing the reaction curves of
the host countries in the immigration policy game. The reaction function of generic country
h is drawn as the black line in space h; h in Figure 3. For any h 2
best response is h = h. Hence, along that interval, the reaction curve is a 45 degree line
(as in Bryant’s (1983) game). For any h < ̂ (h > ̂), country h’s best response is to
set up ̂ (̂) -as that policy allows country h to attract the optimal number of migrants, Î.
Hence, the reaction curve is a horizontal line along that policy value. When h is any value
inside the interval [̂; ), country h’s best response is to set up a slightly (by a however small
") tougher immigration policy, and the best response is drawn as the solid black line slightly
above the 45 degree line. Finally, when h 2 [; ̂), country h’s best response is to set up a
slightly (by a however small ") softer immigration policy, and the best response is drawn as
the solid black line slightly below the 45 degree line.13 The best-response function of country
h can then be written as
̂ if h < ̂
h + " if h 2 [̂; )
h if h 2
h " if h 2 (; ̂]
̂ if h > ̂
The reaction curve of country h is the mirror image of the one of country h and is depicted as
the light grey line in Figure 3. They overlap along the interval
, which then constitutes
the measure of equilibria, while no intersection occurs when h is lower than or higher than
12Note that removing the assumption of symmetric fundamentals would not alter the logic of this result.
Speci cally, if countries had asymmetric fundamentals, the policy attracting the optimal number of migrants
would di¤er across host economies. However, starting from this optimal policy con guration, strategic com-
plementarities still characterize immigration policy.
13 In rigourous mathematical terms the best response function is not de
ned when h belongs to [̂; ) or
to [; ̂), the reason being that we have de
ned the policy variable as a continuous variable. With an abuse
of notation we write h " instead of ? in the expression for the best-response function (4), as if variable
were de ned as a discrete variable which could only take multiple values of an indivisible ". This is because
we here privilege intuition to rigour. Of course, nothing substantial changes.
INSERT FIGURE 3 HERE
The above discussion illustrates the key problem associated with immigration policy when
the receiving economy is formed by multiple countries: coordination failures can arise in this
environment. The economy can be stuck in an inferior Nash equilibrium where restrictions
to immigration are either ine¢ ciently high (h 2 (̂; ] 8h) or ine¢ ciently low (h 2 [; ̂)
8h), and hence destination countries fail to attract the "right" number of foreign workers.
The reason for this ine¢ ciency is the international spillover created by immigration policy,
which in turn results from the international mobility of migrants (i.e. their ability to choose
their destination in addition to whether they want to migrate or not).
Starting at an ine¢ cient equilibrium, no country can improve its welfare with unilateral
immigration policy initiatives, but all receiving economies could be made better o¤ under an
agreement that called for mutual policy adjustments. In this respect, immigration policy has
much to learn from trade policy, even if the structure of the immigration and the trade policy
game is quite di¤erent. In particular, most authors consider current immigration policy in
advanced economies as too restrictive.14 As this model shows, excessive restrictions can be
the result of a coordination failure among receiving countries. No country would unilaterally
choose to loosen up its policy stance as this would result in an inux of migrants beyond its
e¢ cient point. While each government, acting independently, is powerless to coordinate the
policy choices of the others, an international agreement could provide governments with an
avenue to commit to a reduction of immigration restrictions and escape from a coordination
However, the model also makes quite clear the di¤erences in the economic problem facing
trade and immigration policy makers. Di¤erently from the trade context, in the immigration
policy game among receiving countries the key element is con
dence rather than conict. As
it is well known from the trade literature (Bagwell and Staiger, 1999 and 2002), excessive
trade restrictions can be the result of a terms-of-trade driven prisoner’s dilemma situation.
In contrast, the analysis of this section shows that receiving countries face a trust dilemma,
a coordination problem which leads to multiple equilibria. While in both situations govern-
14 In particular, several authors
nd that there would be global gains from lowering immigration restrictions
that limit the movement of workers from low-income to high-income countries. See Clemens, Montenegro and
Pritchett (2008), Hanson (2008) and Rosenzweig (2007).
ments can be stuck at an ine¢ cient equilibrium, a key issue of the immigration policy game is
equilibrium selection (an issue that does not emerge in a prisoner’s dilemma situation, where
there is only one equilibrium). Governments may coordinate on the ine¢ cient equilibrium as
this is the one that is associated with policy choices that are less "risky", an issue that will
be addressed in Section 4.
3.1 Migrants’International Mobility and Coordination Failures
An important question is how the set of equilibria is a¤ected by the underlying parameters
of the model. In particular, in this subsection we study the e¤ect on the receiving countries
of a change in the international mobility of foreign workers (). As discussed above, this
parameter captures the responsiveness of migrants to di¤erences in the policy stance and is
determined by factors, such as technology, that are likely to change over time.
We begin by stating the following
Proposition 2 An increase in international mobility of foreign workers () expands the set
of symmetric Nash equilibria, while it does not a¤ect the Pareto dominant equilibrium. That
> 0 and
= 0 :
Proof. In Appendix D.
To grasp the intuition, recall that function Wh
is not a¤ected by changes in
as the adoption of the same policy across the receiving region neutralizes the spillover e¤ect.
Therefore, the solid curve in Figure 2 does not move as varies. On the other hand, following
an increase in , functionWh (h; F ) shifts leftward and functionWh
Intuitively, workers’ international mobility is responsible for the cross-border externality,
which is the source of the equilibrium multiplicity. An increase in international mobility
implies a more powerful externality and an ever expanding measure of policy equilibria.
Taken together, Propositions 1 and 2 have two implications. First, higher realizations
of parameter , by expanding the set of equilibria, worsen the problem of coordination
failure and indeterminacy. Second, as the "new" equilibria are more distant from the optimal
policy, they are associated with lower welfare for the receiving region. To put it di¤erently,
this result suggests that the new wave of globalization, driven by a fall in transportation and
communication costs, may be exacerbating coordination failures and increasing the gains
from immigration policy coordination for all receiving countries.
4 Selection of Equilibria under Strategic Uncertainty
The previous section illustrates the possibility of coordination failures in immigration policy
due to the presence of multiple Nash equilibria in the immigration policy game. Whether
coordination failures actually occur depends on which equilibrium policy makers coordinate.
As shown above, the "payo¤-dominant" equilibrium, the one associated with policy h = ̂
8h, is in the set of equilibria. This, however, is not necessarily the equilibrium that players
Experimental evidence on coordination games quite convincingly rejects the view that
coordination problems will not occur in simple strategic interactions (Cooper et al., 1992, Van
Huyck et al., 1990). One possible rationalization of this evidence is that payo¤-dominance is
not the only basis for coordination, and that players may converge towards other equilibria
which present alternative salient features. An alternative proposed in this literature is that
players coordinate towards the "risk-dominant" equilibrium (Harsanyi and Selten, 1988), the
key insight being that a strategy may be preferred over the other if it is less risky in the face
of strategic uncertainty.15
An evolution of this equilibrium selection criterion, which applies to games characterized
by a continuous space of strategies (as the immigration policy game introduced in Section 3),
is the robustness to strategic uncertainty (Andersson et al., 2010). Whenever a game admits
a continuum of equilibria, even the slightest uncertainty about the opponents’ strategies
might lead each player to deviate from any given policy equilibrium. It is then "arguably
reasonable to require equilibria to be robust to small amounts of uncertainty about other
players’strategies" (Andersson et al., 2010, p.2).16
15Whether players are more likely to coordinate towards payo¤-dominant or risk-dominant equilibria is the
focus of empirical literature (for a survey see Cooper, 1999). The experimental evidence in Cooper et al.
(1992) shows that risk-dominance can provide a better guide to equilibrium selection than payo¤-dominance.
cally, Andersson et al. (2010) show that there is only one equilibrium surviving the robustness
test in a price competition game with a continuous strategy space and admitting a continuum of equilibria
(Dastidar, 1995). Abbink and Brandts (2008) and Argenton and Muller (2009) provide experimental evidence
in favor of this "robust equilibrium".
In this subsection we prove that there is a unique equilibrium which is robust to strate-
gic uncertainty and show that the robust equilibrium is di¤erent from the payo¤-dominant
equilibrium. This result reveals that coordination failures in immigration policy are not only
possible but also likely to emerge.
rst formally characterize strategic uncertainty in the immigration policy game. Fol-
lowing Andersson et al. (2010), we model strategic uncertainty by assuming that the proba-
bilistic belief of policy maker h about the action of any other government j in the receiving
economy is given by:
~hj = j + t"hj ;
where t 2 R+ and "hj hj are statistically independent noise terms. The distribution hj
belongs to an arbitrary family of probability distributions with non decreasing hazard rate
The introduction of this noise de
nes a new, "perturbed", game. Intuitively, the robust
equilibrium is an equilibrium of this perturbed game when the noise tends to zero. More
formally, if an equilibrium strategy pro
le (r; :::; r) is the unique limit to any sequence of
equilibria indexed by t as t ! 0, that strategy pro
le is robust to strategic uncertainty. In
the next proposition we prove that such limit exists and is unique.
Proposition 3 There exists a unique equilibrium which is robust to strategic uncertainty.
This equilibrium, (r; :::; r), is de
ned by Wh (r; F ) =Wh
8h and is Pareto-inferior
to the payo¤ dominant equilibrium (̂; :::; ̂).
Proof. In Appendix E.
Policy r is the one for which the incentives to restrict or loosen the immigration policy
stance for each strategically uncertain government in the receiving region exactly o¤set each
other. As shown in Figure 4, the robust equilibrium of the immigration policy game corre-
sponds to the point where the functions Wh (h; F ) (the dashed curve) and Wh
dotted curve) intersect. In this point, denoted by A, the expected welfare loss associated
with a policy higher or lower than the rest of the host region tends to zero.
INSERT FIGURE 4 HERE
As there is a continuous strategy space, for any policy h, government h’s subjective
probability that any other government will choose exactly the same policy is zero. Hence,
with probability one, policy h will either be the lowest or not. In the
rst case, country h
will experience a crowding in, in the second it will experience a crowding out. In Figure 4, for
any policy h 2 (r; ], a government facing uncertainty on the strategies of other receiving
governments has an incentive to lower its immigration restrictions. The reason being that
the welfare if other countries’policies are less stringent (Wh (h; F )) is lower than the welfare
if other receiving countries’immigration policies are more restrictive than the one set up in
). Conversely, for h 2
, every government has an incentive to raise
restrictions as the expected welfare under a crowding in (Wh
) is lower than under a
crowding out of migrants (Wh (h; F )). Only for policy
r welfare under crowding in and
crowding out are equal and a government has no incentive to alter its policy stance.
A comparison of the robust and Pareto-dominant equilibria sheds light on two issues.
First, under strategic uncertainty, the immigration policy equilibrium is distinct from the
one that maximizes welfare for the entire host region. The policy strategy robust to strategic
uncertainty can be more or less stringent than the optimal policy depending on the funda-
mentals of the economy (which determine the shapes of the two curves in Figure 4). In this
model, where immigration has both bene
ts and costs for the host economy, the presence of
an immigration policy spillover may, therefore, induce countries to select an excessively re-
strictive or loose policy. In other words, this model suggests that coordination failures driven
by the immigration policy spillover can give rise to both a "race to the top" and a "race to
the bottom" in immigration restrictions in receiving countries (see, for instance, Boeri and
Second, an increase in the international mobility of migrants () does not a¤ect the
optimal immigration policy but alters the robust equilibrium. Intuitively, the expected welfare
loss associated with both a crowding in and a crowding out increases with the size of the
immigration policy spillover (the dotted and the dashed curves in Figure 4 move further
apart, while the position of the solid curve is not a¤ected by changes in , see Proposition
2). Therefore, globalization, in the sense of an increase in the international mobility of
migrants, exacerbates the strategic uncertainty by increasing the set of Nash equilibria, and
has an ambiguous e¤ect on the robust equilibrium on which governments coordinate.
5 Conclusions and Policy Implications
This paper has examined receiving countries’motives in setting immigration policy and how
the institutional framework, particularly the absence of e¤ective coordination mechanisms,
translates these motives into policy outcomes. The analysis shows that policy at home is
inuenced by measures adopted abroad. The reason is that migrants choose where to locate,
in part in response to immigration policies in host economies. In the model, the international
mobility of migrants gives rise to a policy spillover e¤ect which rationalizes the evidence in
recent empirical studies on immigration. In this interdependent environment, immigration
policy becomes strategic and unilateral behavior may well lead to coordination failures, where
receiving countries are stuck in a welfare inferior equilibrium. The theory also shows that
ine¢ cient policy equilibria are more likely to emerge when governments are uncertain about
the immigration policy of other receiving countries and when the international mobility of
migrants is stronger.
In the rest of this section, we discuss some implications of this model and come back to the
initial question of this paper on the economic rationale for international institutions governing
rst implication of the model is that the type of coordination problem facing
immigration policy makers is di¤erent from the one facing trade policy makers. While both
unilateral trade and immigration policies may lead to an ine¢ cient equilibrium, the nature
of this equilibrium in the two cases is not the same. In the trade policy game, the
policy outcome (i.e. trade openness) is not an equilibrium of the game as each government
has an incentive to impose restrictions when the others choose free trade. Instead, the key
element of coordination failures in the immigration policy game is the lack of con
the policy choice of other governments, not an inherent policy conict as in trade policy. In
other words, it is the inability of policy makers to commit to the e¢ cient immigration policy
vis-à-vis other countries that constrains e¢ cient outcomes in this domain.
A second implication of the model is that, di¤erently from other contributions in the
literature (Hatton, 2007, Hanson, 2009), it provides a clear economic rationale for interna-
tional institutions dealing with immigration. Starting from an ine¢ cient policy equilibrium,
no receiving country could improve its welfare through unilateral policy actions. Speci
if restrictions in advanced economies are excessively high (as most experts believe), a looser
policy in any single country would result in a large inux of migrants that would lower its
welfare. An international agreement that called for mutual policy adjustments in receiving
economies (i.e. a joint reduction of restrictions) could improve upon the initial ine¢ ciency
by neutralizing the immigration policy spillover. While not the only possible rationale, this
provides a strong case for the need of multilateral institutions in the immigration domain.
In facts, several international agreements and organizations aim at coordinating immigra-
tion policy. For instance, the stated objective of the International Organization for Migration
is "to promote international cooperation on migration issues". Moreover, a forum for dialogue
on migration and immigration policy is also provided by other international institutions, such
as the OECD. While these arrangements help coordination through dialogue and the dissem-
ination of information among receiving countries, they do generally not envisage an e¤ective
enforcement mechanism. This implies that the uncertainty on other governments’strategies
still characterizes policy makers’decisions, which can lead to coordination failures.
An exception is Mode 4 of the General Agreement on Trade in Services (GATS), which
provides an opportunity to WTO Members to take on commitments regarding the tempo-
rary presence of "natural persons" from a di¤erent Member who supply a service.17 While
GATS Mode 4 has a limited scope, the binding nature of commitments within the WTO,
backed up by the enforcement mechanism provided by its dispute settlement system, is an
appealing feature of this system.18 In this sense, expanding the scope of Mode 4 may be in
the interest of receiving countries. However, one should be aware of the di¢ culties of this
process. Immigration policy is not limited to border measures, but includes a large number
17Natural persons falling within the scope of Mode 4 include independent contractual service suppliers and
natural persons employed by service suppliers (WTO, 2004). Speci cally, Mode 4 concerns a narrow (and not
clearly de ned) subset of temporary migration, as it excludes coverage of access to labour market, citizenship
and employment on a permanent basis (see WTO Annex on Movement of Natural Persons).
18The size and scope of Mode 4 movements are an issue of current debate and negotiation. While a number
of WTO Members have undertaken Mode 4 commitments that cover short-term employees (the US binding
of 65.000 H-1B visas is a noteworthy example), the overall degree of Mode 4 commitments are low. WTO
Members have generally granted access to selected categories of highly skilled persons linked to a commercial
presence, such as managers, executives and specialists. The Hong Kong Ministerial declaration in December
2005 called for a new impetus on Mode 4 commitments (e.g. an extension of the categories of natural persons
included in the commitments and of the permitted duration of stay), but improvements in the ongoing Doha
negotiations have been so far slow to materialize (see Carzaniga, 2009).
of behind-the-border measures that a¤ect the welfare of foreign workers in the host economy.
As the trade experience shows, regulating this policy can be extremely challenging.
Another implication of this analysis is that the extent of the coordination problem depends
on the magnitude of the policy spillover e¤ect. In the model this is captured by the size of the
parameter -i.e. the international mobility of migrants. While in the paper we emphasized
technology as a determinant of this parameter, other factors can inuence the responsiveness
of foreign workers to immigration policy di¤erences. In particular, receiving countries that are
more strongly interconnected, because of geographic proximity, common cultural background,
or because they have formed an economic union, will experience stronger immigration policy
spillovers and are, therefore, more likely victims of coordination failures. Institutions that
allow for e¤ective coordination (or, the creation of a single immigration policy) are more
valuable in these circumstances. This provides formal support to the frequent calls in the
policy debate for a single immigration policy in an integrated area such as the European
Union (Boeri and Brücker, 2005). Similarly, a greater involvement of the States of the US
(and, hence, a more limited role of the federal government) in immigration policy -implicit
in the law passed in the State of Arizona in 2010- may lead to welfare reducing coordination
failures within the US, as the choice of one State will inevitably a¤ect others through location
decisions of foreign workers within the US and trigger a series of policy responses in other
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A A Standard Model of Immigration Policy
This section introduces a speci
c model of immigration policy with standard features which
may help give more structure to the reasoning developed in the main text. In particular,
Subsection A.1 opens the "black box" of the welfare function we de
ned at the beginning of
Section 2, Wh (Ih). Subsection A.2 does the same for the migratory choice, by specifying the
t from immigration.
A generic receiving country, denoted by h ("home"), is populated by Nh ("native") work-
ers, each of whom supplies one unit of labor inelastically, and by Kh capitalists, each of
whom is endowed with one unit of capital. A
nal good is produced competitively via a
constant-return-to-scale technology in labor and capital:
Yh = K
Lh is the sum of natives and immigrants working in country h, that is, Lh = Nh+Ih, where Ih
denotes the endogenous number of migrants. The
nal good is the numeraire in the receiving
economy, and its price is normalized to one. As the product market is competitive, input
factors are paid their marginal productivities:
wh = (1 )
and rh =
Country h has a welfare system that, de facto, redistributes income from capitalists to
cally, the policy consists of a
xed lump-sum transfer
h to (native and
foreign) workers which is
nanced through a proportional tax h 2 [0; 1] on the capital rent.
This simple formulation captures the idea that welfare spending in h depends on the number
of migrants.19 A balanced government budget implies
h (Nh + Ih) ;
and hence the tax rate on capital income, as a function of the number of immigrants is
19The welfare system is assumed to be pre-existent to immigration. This is reasonable when the size of the
migrant labor force is low relative to the size of the native population. When this is not the case, one can
think that welfare and immigration policy are jointly determined (see for instance Casella, 2005 and Armenter
and Ortega 2010).
h (Ih) =
h (Nh + Ih)
A.1 The Optimal Number of Immigrants
We introduce a general representation of the Home government preferences over immigrants,
which includes both the case where policy makers maximize the host economy’s national
welfare as well as the general possibility that governments are also motivated by the distrib-
utional e¤ects of immigration among natives. We assume that agents use their (disposable)
income to purchase the
nal good and have a linear utility function in consumption. Let us
ne the objective function of the government as a function of the number of immigrants as
Wh (Ih) [rh (Ih) Kh
h (Nh + Ih)] + (1 ) [wh (Ih) +
where we used the above balanced budget condition (5) to substitute for h and where
2 [0; 1] is the political bias (i.e. the weight on the utility of capitalists). This formulation
includes as a special case national income maximization for = 1=2.
The optimal number of migrants in country h, denoted by Î, is the one which maximizes
condition (6).20 The FOC of this problem is
= (1 )
Nh + Ih
(1 ) Nh
Nh + Ih
h = 0:
A number Î solving the FOC above is a maximum if the second derivative, evaluated in Î, is
strictly negative, that is, if
= (1 )
Nh + Î
Nh + Î
(1 ) Nh
Nh + Î
(1 ) Nh
Nh + Î
In Appendix B we provide a su¢ cient condition for Î to be the global maximum, that is, the
only politically optimal number of migrants for the host country.
20We refer to Î as the "optimal" number of migrants. Needless to say, this is the "politically-optimal" level of
immigration, as it maximizes the government’s objective function, and it corresponds to the "socially-optimal"
number of migrants only in the special case in which = 1=2.
A.2 The Migratory Choice
We can now be more precise about the net bene
ts from migration for foreign workers (bh)
and thus about their migratory choice. Bene
ts from migration are given by wh +
is, the salary plus the welfare transfer. Costs are given by h + i, that is, the policy plus
the psychological cost. If we normalize the wage rate in the sending region (w) to zero, a
constrained foreign worker i will migrate to h if and only if
h h i 0; (7)
from which we can determine the threshold value of the psychological cost as
h = wh +
Thus the number of constrained migrants (as function of h) will be h(1)F=m.
On the other hand, a free foreign worker migrates to h if condition (7) holds, and if the
payo¤ in h is higher than any other payo¤ in the rest of the host region, that is, if21
h h i > wh +
h h i () h < h:
Immigration ows to country h are function of h’s immigration policy as well as of the
measures imposed in the rest of the destination countries. In particular, the number of
free foreign workers actually migrating to h is 0 if h > h (crowding out), and hF if
h < h (crowding in). Finally, if h = h, free migrants distribute symmetrically across
the receiving region, that is, hF=m for any h.
The total number of migrants to country h can then be described as Ih = hFh, where
h = wh +
h h and Fh is given in (2).
21Since countries are symmetric by assumption, it is h = h. Moreover, it is easy to show that in this
model dwh=dh 2 (0; 1). Thus the following condition holds true.
B Characterization of the Pay-O¤ Function
The payo¤ function, which is de
ned only implicitly in the main text (expression (3)), can
here be characterized explicitly as
Nh + hFh
h [Nh + hFh]
Nh + hFh
where h and Fh are functions of both h and h, and Fh is de
ned in (2). Thus, we can
draw three distinct welfare functions depending on whether h is higher, lower or equal to
h. If h = h, the welfare function, Wh
, is given by (8) but where Fh = ~F . If
h > h (crowding out), the welfare function Wh (h; F ) is given by (8) but where Fh = F .
Finally, if h < h (crowding in), then the welfare function, Wh
, is still given by
(8) but where Fh = F . The next subsection invetigates the conditions under which the three
welfare functions are strictly concave.
B.1 Strict Concavity of the Welfare Functions
Consider welfare function
Wh (h; F ) =
Nh + hF
h (Nh + hF )
Nh + hF
rst derivative can be written as
@Wh (h; F )
(1 ) Nh
where Lh = Nh + hF . It is now possible to calculate the second derivative as
@2Wh (h; F )
(1 ) Nh
(1 ) Nh
(1 ) Nh
1 2KhLh FL2h
where again Lh = Nh + hF . Simple algebra shows that the following condition on the
parameters of the model ensures that @2Wh (h; F ) =@
h is strictly lower than zero for any
value of h:
> H(F ) 1 +
(1 + )
1 + h
We assume that this condition is satis
ed. Given that H(F ) is a decreasing function in F
(and given that F < ~F < F ), condition (9) also ensures the strict concavity of the other two
welfare functions, Wh
C Proof of Proposition 1
Welfare function Wh (h; F ) admits a global maximum in ̂, and it is continuous, strictly
decreasing and strictly concave in the interval
. Welfare function Wh
a global maximum in ̂, and it is continuous, strictly increasing and strictly concave in the
. Since Wh
, and since ̂ < ̂, then the two curves
must cross once and only once in the interval
. Denote this intersection point by .
Moreover, for exactly the same reasons, it must be that Wh
Wh (h; F ) for any
. An entirely analogous reasoning holds for the other interval,
exists a unique value 2
, such that Wh
, and it must be that
for any h 2
As a result, for any h 2
it is Wh
;Wh (h; F ), and hence
whatever h 2
, country h’s best response is h = h. Thus, any policy h 2
8h is a symmetric Nash equilibrium of the game.
Since < ̂ < , policy ̂ is a Nash equilibrium of the game. Moreover, it is immediate
to show that it is the optimal policy equilibrium: when all other countries h set up ̂, then
h = ̂ is the policy which maximizes welfare function Wh
, as it allows country h to
attract the optimal number of migrants Î. Moreover, since Wh
is strictly increasing
in h in the interval
and strictly decreasing in the interval [̂; ], it is also immediate
to verify that the e¢ ciency loss is greater, the higher the distance from the optimal policy.
This proves that the equilibria are Pareto-ordered by the distance of from ̂.
Notice that any h < as well as any h > 8h are not Nash equilibria. If all other
countries set up policy h in the interval
, then Wh (h; F ) Wh
country h’s best response is to set up a slightly tighter policy. As a result, there are no
equilibria below . If instead h < ̂, country h’s best response is simply h = ̂. The
same reasoning applies for h > .
Finally, a simple contradiction argument (drawn from Amir et al., 1996) proves that
asymmetric equilibria do not exist in this policy game. Let (1; 2; 3:::; h; :::; m) be an
asymmetric equilibrium (thus with at least two ’s being distinct). Assume then, w.l.o.g.,
that 1 = maxh fhg and 2 = minh fhg so that 1 > 2. Since the game is symmetric,
every permutation of (1; 2; 3:::; h; :::; m) is also an equilibrium. Consider for instance
(1; 2; 3:::; h; :::; m) and (2; 1; 3:::; h; :::; m). The fact that both of them are equilib-
ria implies that country 1 strictly weakens its immigration policy from 1 to 2 as the other
countries restrict theirs from (2; 3:::; h; :::; m) to (1; 3:::; h; :::; m), which contradicts
the fact that country 1’s best-response is nondecreasing (see expression (4)).
D Proof of Proposition 2
The lower threshold is by de
nition the immigration policy such thatWh
where, remind, ~F F=m and F (1)F=m. Let us de
Wh ; F Wh ; ~F = 0
as the implicit function of with respect to . It holds that
We show that d=d < 0.
The numerator writes as
The second term is null as Wh
does not depend on . The rst term can be shown
to be negative. In fact
First note that @Wh
=@I is positive since in point welfare increases with the number
of immigrants. Second, since I = b
F , it is @I=@F > 0. Finally, it holds @F=@ =
F=m < 0. Then it is dWh
=d < 0 and hence dG=d < 0.
As regards the denominator in (10), it is
as in point it is dWh
=dh < 0 and dWh
=dh > 0. This proves that d=d <
The proof that d=d > 0 is entirely analogous and is then omitted. Finally, it is trivial
to show that the optimal policy is not a¤ected by an increase in international mobility.
Notice that functionWh
does not depend on , and thus ̂, as a solution to equation
=dh = 0, will not depend on either.
E Proof of Proposition 3
The proof will follow closely the argument developed in Andersson et al. (2010) for the price
competition game. Let t 2 R+ and suppose that the government of each country h holds a
probabilistic belief about any other government j’s policy of the following form:
~hj = j + t"hj ; (11)
for some statistically independent noise terms "hj hj . Distribution hj belongs to an
arbitrary family of cumulative distribution functions, D : R ! [0; 1], characterized by
non-decreasing hazard rate function.22
Given that the support of random variable ~hj is
, this variable distributes
according to the following cumulative distribution function (c.d.f.):
Dthj (x) =
The introduction of uncertainty de nes a new, "perturbed", game. For any t 2 R+,
a strategy pro
le (1; 2; 3:::; h; :::; m) is a t-equilibrium if, for each player h, the
strategy h maximizes h’s expected payo¤ under the probabilistic belief of the form given in
(11). Notice that a t equilibrium is simply a Nash equilibrium of this perturbed game.
A strategy pro le r is (strictly) robust to strategic uncertainty if, for any collection of
c.d.f’s hj 2 , there exists a sequence of t equilibria,
with tk ! 0, such that
tk ! r as k !1. We now apply these de nitions to our policy game.
For any policy h that the policy maker of country h decides to implement, her subjective
probability that any other policy maker will choose exactly the same policy is zero. Hence,
with probability one, her policy will either be the lowest or not. In the
rst case, country
h will experience a "crowding in", in the second it will experience a "crowding out". Under
strategic uncertainty, country h will then select that policy which maximizes the following
expected payo¤ function:
th () =
241 hjhjt hj
35 Wh h; F + (12)8<:1Y
241 hjhjt hj
359=; Wh (h; F ) :
22Many common distributions, such as the normal or the exponential distribution, present this feature. This
is the only assumption we impose on this arbitrary family of distribution functions, and it is useful to provide
easily a su¢ cient condition for the uniqueness of the robust equilibrium.
The expression above represents the expected payo¤ of country h when setting up policy
h. In particular, the
rst term is equal to the probability that h is lower than any other
policy j 8j times the payo¤ associated with the resulting "crowding in". The second term
is instead given by the probability that h is higher than at least one j times the payo¤
associated with the resulting "crowding out". The FOC for the maximization problem writes
24 hjcdjt hjhjt
where hj () = 0hj (). It can be proven that the objective function (12) is strictly concave
along the interval
(to make the argument developed here less burdensome, this proof
is provided separately in Subsection E1). Hence, every solution to the FOC above in that
interval is a t-equilibrium.
Consider any sequence htki1k=1 ! 0 and de
ne limk!1 kh h (for the Bolzano-Weierstrass
theorem this limit exists and belongs to the interval
). We now investigate what
happens to the solution to the FOC when t tends to zero. In particular, we prove that
htki1k=1 ! 0 implies Wh
h; F ), whose solution is
The FOC can be rearranged as follows:
375+ @Wh(kh;F)@kh @Wh(kh;F)@kh
Suppose by contradiction that, when t tends to zero, it isWh
6=Wh (h; F ). Then,
in order for the FOC to be satis
ed when htki1k=1 ! 0, it must necessarily be that
The expression above says that, in the limit, the sum of the instantaneous probabilities that
any j is equal to h must tend to zero. This is true if
j 6= h 8j. We now show that this
The fact that Wh
6= Wh (h; F ) implies that, for the generic government h,
is either higher or lower than Wh (h; F ). Suppose for instance it is higher (the
reasoning under the opposite case in which Wh
< Wh (
h; F ) is entirely analogous
and is omitted). If that is the case then, in order for h to be a best response, it must
necessarily be that r < h 6 j for any j. But since j > r, then also for country j it
must be that Wj
j ; F
j ; F
, and thus j 6 h for any h. The two implications
are true only when h =
j , which contradicts the above statement. As a result, in order for
the FOC to be true, it must necessarily be that Wh
= Wh (
h; F ) whose solution is
r for any h. This completes the proof.
E.1 Concavity of the Objective Function
We here prove that a su¢ cient condition for function (12) to be strictly concave along the
is that distribution hj has non-decreasing hazard rate. The FOC can equiv-
alently be written as
= @Wh(h;F )@h
24 hjcdjt hjhjt
24 hjcdjt hjhjt
35 = 0:
It is now easy to show that function @th () =@h is strictly decreasing in h. Starting from
rst addend, its derivative has the following expression and is negative:
24 hjcdjt hjhjt
24 hjcdjt hjhjt
where D () stands for "derivative with respect to h". For practical reasons, the signs of the
four terms are denoted under each of them. In particular, while the signs of the last three
terms are self-apparent, the
rst term is negative whenever h > ̂.
Turning to the second addend, the expression under the product operator is decreasing
(as the cumulative distribution hj () is an increasing function of h). The derivative of
=@h with respect to h is also strictly decreasing by assumption. The di¤erence
Wh (h; F ) is instead increasing in h, at least for any h < ̂. Finally, notice
that the function under the sum operator is non-decreasing to the extent that the hazard
is assumed to be non-decreasing. Given that
a decreasing hazard rate implies that the function under the sum operator is decreasing as
( )FW hh ~,m ( )FW hh ,m( )FW hh ,m
Figure 1: The payo¤ function of receiving country h depending on whether h is higher, lower
or equal to h.
The span of equilibria
Figure 2: The policy equilibria of the game.
h-mm m m̂m̂
Figure 3: The best-response functions of country h (in black) and of country h (in grey).
( )FW hh ~,m ( )FW hh ,m( )FW hh ,m
Figure 4: The equilibrium robust to strategic uncertainty in point A.