Laboratory Federalism: The Open Method of Coordination (OMC) as an Evolutionary Learning Process

LABORATORY FEDERALISM: THE OPEN METHOD OF

COORDINATION (OMC) AS AN EVOLUTIONARY LEARNING

PROCESS

ANA B. ANIAUniversity of Vienna

ANDREAS WAGENERUniversity of Hannover

AbstractIn view of the concept of laboratory federalism, the OpenMethod of Coordination (OMC), adopted by the EU asa mode of governance, can be interpreted as an imitativelearning dynamics of the type considered in evolutionarygame theory. Its iterative design and focus on good prac-tice are captured by the behavioral rule “imitate the best.”In a redistribution game with utilitarian governments andmobile welfare recipients, we compare the outcomes of im-itative behavior (long-run evolutionary equilibria) and de-centralized best-response behavior (Nash equilibria). Thelearning dynamics leads to coordination on a strict subsetof Nash equilibria, favoring policy choices that can be sus-tained by a simple majority of Member States.

1. Introduction

A major argument in the debate over centralization versus decentralizationin federal systems is that decentralization is more conducive to good policiesas it allows for local experimentation, mutual learning, and the diffusion of

Ana B. Ania, Department of Economics, University of Vienna, Oskar-Morgenstern-Platz1, 1090 Vienna, Austria ([email protected]). Andreas Wagener, In-stitute of Social Policy, University of Hannover, Koenigsworther Platz 1, 30167 Hannover,Germany ([email protected]).

We thank Carlos Als-Ferrer, Simon Weidenholzer, Guttorm Schjelderup, Peyton Young,Myrna Wooders, an associate editor, two anonymous referees, and seminar participants inBayreuth, Hannover, Munich, Innsbruck, Trento, Zuerich, Maastricht, and Budapest forhelpful comments and suggestions.

Received October 1, 2010; Accepted October 5, 2011.

C© 2013 Wiley Periodicals, Inc.Journal of Public Economic Theory, 16 (5), 2014, pp. 767–795.

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768 Journal of Public Economic Theory

Figure 1: The iterative OMC process.

best practice. As stated in the concept of laboratory federalism (Oates 1999),reasonable policies are likely to be adopted in decentralized systems evenwhen knowledge about the economic and social environments and the poli-cies best suited to them is only limited; jurisdictions introduce policy inno-vations that, once experienced to be successful, are copied by others. Theseideas are also at the basis of the Open Method of Coordination (OMC) adoptedby the European Union (EU) as a mode of governance based on voluntarycooperation of Member States.

The OMC aspires to be a “means of spreading best practice and achiev-ing greater convergence towards the main EU goals” (European Council2000). To this aim, policies are examined and jointly evaluated so as “to de-velop a mutual learning process” (EU Commission 2011). Figure 1 illustratesthe process of benchmarking and of knowledge dissemination. After agree-ing on a set of EU-wide common objectives and indicators, Member States in-dividually design and implement national policies. These policies are jointlyevaluated and compared. Best-performing policies are identified and theiradoption is encouraged (but not forced). The process is then iterated.

Kerber and Eckardt (2007) point out the closeness of the OMC to thenotion of laboratory federalism. Both processes set out to reach a higherquality of policies through the diffusion of successful innovations. The OMCwas introduced to achieve convergence in sensitive policy areas where theEU does not hold binding power. While the benchmarking process is coor-dinated by the European Commission and Council, the OMC allows for ex-perimentation and diversity. For a discussion of the OMC see de la Porte andPochet (2001), Borras and Jacobsson (2004), Pochet (2005), Zeitlin (2005),and Daly (2007).

The OMC has so far been studied mainly from the perspective of po-litical science or at an institutional level. This paper makes a contributionfrom an economic theory perspective. More generally, our model intendsto capture the features of policy innovation and diffusion in a decentralizedfederal system. While incentives for policy innovation have been analyzed(see, e.g., Strumpf 2002, Kotsogiannis and Schwager 2007), no previous studytakes explicit account of the resulting dynamics. We use concepts and resultsfrom evolutionary game theory and stochastic learning in games to analyzethe iterative process in Figure 1 with an emphasis on mimicking best policychoices.

Laboratory Federalism and Evolutionary Learning 769

Specifically, we embed the OMC into a standard public-finance frame-workof redistribution from rich to poor in a multijurisdictional settingwith mobile welfare recipients. Governments are inequality-averse utilitari-ans with a preference for redistribution. This stylized application is in thecontext of social policy for which the OMC was originally intended. Decen-tralized redistribution is also widely analyzed in the literature on fiscal fed-eralism and counts among the fields where laboratory federalism could beparticularly effective (Oates 1999, section 5).

As with tax competition, mobility implies that decentralized choices onredistribution policies may have external effects on the policy effectivenessin other countries. In the standard game-theoretic treatment of decentraliza-tion, countries choose transfer policies as mutual best responses. By contrast,in laboratory federalism and under the OMC, governments are expected tocompare their performance and copy the policy that appears best, relative towhat other governments do. Our analysis compares the Nash equilibria of adecentralized redistribution game with the evolutionary equilibria that resultfrom relative payoff comparisons. As a first result, we show that the equilibriaassociated with the OMC are a strict subset of those in decentralized compe-tition. Hence, the OMC is generally compatible with a decentralized federalsystem. At the same time, the OMC favors coordination on a more narrowrange of equilibria that exclude some extreme outcomes that might ariseunder full decentralization.

To capture the iterative and experimental nature of laboratory federal-ism and the OMC, the second part of the paper takes a dynamic approach.Governments make repeated choices on their transfers to the poor. Thesechoices and their success in terms of welfare become public every period.While in the OMC this would happen through national reports on the com-monly agreed indicators, in our model a symmetric welfare functions allowsfor an unambiguous identification of best-performing policies. Mutual learn-ing is modeled as follows. Countries, in principle, mimic best-performingpolicies when they revise their strategies. By the open nature of the method,this policy guideline is not binding and countries can adopt other policiesbased on their own motivations, which may range from national politicalinterests to mistakes. This feature is introduced in our model through ran-dom experimentation. Innovations introduced by experimentation may befollowed by other countries, but only if they prove to be more successful thanexisting policies; otherwise, they will not persist. The process is iterated andcountries proceed asynchronously with the revision of their policies.

We show that the dynamic process of policy imitation and experimenta-tion leads to the emergence of evolutionary equilibria, as in the static anal-ysis. However, not all static equilibria are equally robust in the long run. Inthe long run, governments coordinate on “intermediate” levels of the policyinstrument, which are characterized by the fact that they can be sustainedby a simple majority of countries. As we will show, long-run equilibria arelikely to emerge through step-by-step coordination of an increasing number


of countries choosing higher subsidies. Disturbing these equilibria would re-quire the unlikely coordination of a majority of countries at once. Finally, wewill illustrate that long-run equilibria need not be efficient. The OMC mayresult in subsidy levels that are too high or too low compared to the one thatwould maximize joint welfare. Still, the kind of decentralization introducedby the OMC does not necessarily imply the erosion of the welfare state.

An extensive literature in political science studies the adoption ofpolicies in federations as the result of imitation. According to this lit-erature, the probability that a state adopts a particular policy is higherif neighboring states have already adopted it (Walker 1969; Shipan andVolden 2008). A trend of convergence of social policies and the absenceof a race to the bottom for OMC participants has recently been ob-served by Coelli, Lefebvre, and Pestieau (2010). These findings fit ourpredictions.

The literature on the OMC rarely mentions strategic interdependenciesamong Member States as a relevant factor for its performance. An excep-tion is Buchs (2008) who informally analyzes the OMC as a two-stage gamewhere governments first agree on objectives and then implement policiesto meet these objectives. De la Porte (2011) views the process of OMC in-stitutionalization as a contractual relation between the EU Commission andMember States which could be captured in a principal–agent model. Someauthors relate the OMC to yardstick competition (e.g., Pestieau 2006), sincethe incentives for policy imitation could emerge as a result of voters us-ing relative performance evaluations to draw inferences on the quality ofpoliticians. Models of yardstick competition tackle the problem of politi-cal agency (see, e.g., Belleflamme and Hindricks 2005). They focus on theinformational asymmetry between politicians and the voters in their juris-dictions and on the strategic use of information to influence voters’ deci-sions. Our paper does not deal with opportunistic, rent-seeking politicians.Rather, we view the OMC as trying to solve a different problem, that of pol-icy convergence when governments’ knowledge about appropriate policiesis limited.

Formally, our analysis is related to that of oligopolistic price competi-tion in markets for a homogeneous good with convex costs (Dastidar 1995).Our prediction corresponds to that in Alos-Ferrer, Ania, and Schenk-Hoppe(2000), who show that price imitation perturbed by occasional experimen-tation results in coordination on central prices. Our static concept of evo-lutionary equilibrium follows Schaffer (1988); our dynamic analysis followsstochastic learning models (Kandori, Mailath, and Rob 1993; Ellison 2000),surveyed in Alos-Ferrer and Schlag (2009).

This paper is organized as follows. Section 2 sets up a fiscal game of de-centralized redistribution. Section 3 derives Nash and evolutionary equilibria(static analysis). Section 4 analyzes the imitation dynamics with experimenta-tion. Section 5 discusses the assumptions and provides extensions. Section 6concludes. All proofs are relegated to the Appendix.


2. The Model

2.1. Mobility and Redistribution

There are n ≥ 2 identical jurisdictions, called countries, that form an inte-grated economic area with free mobility. In each country, there is a massof individuals owning immobile assets, which can be taxed by the govern-ment. This part of the population is normalized to size one and denotedthe rich with total wealth wR . Mobile members of the population, calledthe poor, do not own taxable assets, but dispose of mobile wealth wP . Weassume wR > wP ≥ 0 and that the wealth of rich and poor is the same acrosscountries.

Each country i = 1, 2, . . . , n decides on the level of a nonnegative lump-sum transfer, si , payable to each poor in the country and financed by a lump-sum tax, ti , on the rich. We denote �i the amount of mobile poor living incountry i and assume 0 ≤ �i ≤ ν, where ν ≥ n is the total size of poor popula-tion in the economic area, considered to be constant. Government budgetsare required to balance; that is, si · �i = ti for all i . With this redistributionscheme, consumption levels of the rich and poor who reside in country i aregiven by cR

i = wR − ti = wR − �i · si and cPi = wP + si , respectively.

We henceforth write s = (s1, . . . , sn) ∈ Sn for vectors of redistributivepolicies. We use notation s = (si | s−i ), where si is the subsidy chosen bycountry i and s−i is the vector of subsidies chosen by countries other thancountry i or any permutation thereof.

Individuals care only about their consumption. Thus, mobile individu-als establish their residence in one of the countries with the most gener-ous redistribution policy. Given i , let s i = max j �=i s j be the maximum sub-sidy chosen in all countries other than i . Given s = (si | s−i ), denote byMi (s−i ) = { j �= i | s j = s i } the set of countries offering the highest subsidywhen we exclude i and let mi (s−i ) = |Mi (s−i )| be its cardinality.

A subsidy vector s triggers migration flows resulting in a vector(�1(s), . . . , �n(s)) of mobile poor residing in each country. Any two coun-tries, i and j , that choose the same transfer level attract the same amount ofpoor; that is, si = s j implies that �i (s) = � j (s). The poor settle only in thecountries with highest subsidies, according to the following pattern:

�i (s) = �(si | s−i ) :=⎧⎨⎩

0 if si < s iν

1+mi (s−i )if si = s i

ν if si > s i .

(1)

Clearly, 0 ≤ �i (s) ≤ ν and∑n

i=1 �i (s) = ν. A distribution of the poor acrosscountries remains unchanged unless some si changes; migration then takesplace without restrictions or adjustment costs until Equation (1) holds againfor the new subsidies. Depending on s , the distribution may be imbalancedwith some countries hosting no poor if their subsidy levels are lower than


others’. Equal subsidies everywhere, however, imply an equal distribution ofthe poor across countries.

The assumption of mobile poor following high subsidies is in line withWildasin (1991, 1994), Cremer and Pestieau (2003), and others. Alterna-tively, we could choose to make the poor immobile, let rich individuals bemobile, and governments set taxes instead of subsidies. This would not affectthe essence of our analysis. Potentially more critical is the assumption thatall poor are mobile. The assumption of frictionless mobility is unrealistic; itmakes migration responses discontinuous, with slight changes in transferscausing complete reshufflings of the population. In Section 5.2, we checkthe robustness of our results to the introduction of some immobile poor.

2.2. Policy Objectives

Government objective functions play a crucial role in decentralized re-distribution games when population size is endogenous (Mansoorian andMyers 1997). We consider here utilitarian governments that evaluate indi-vidual utility from consumption with some function u(c), assumed to betwice continuously differentiable, strictly increasing, and strictly concave forall c ≥ 0. We further assume that u(wP ) ≥ 0 and that u(0) < u(wR) − ν ·u (wP + 1/ν · wR).

The government of country i evaluates policies by the sum of the utilitiesof its residents,

πi (s) = �i (s) · u(cPi ) + u(cR

i ) = �i (s) · u(wP + si ) + u(wR − �i (s) · si ). (2)

We know from (1) that �i (s) is invariant to permutations of other countries’subsidies. This allows to write payoffs as follows:

πi (s) = π(si | s−i ) = �(si | s−i ) · u(wP + si ) + u(wR − �(si | s−i ) · si ), (3)

where now π(si | s−i ) is the welfare level of any country choosing s = si whenall other countries choose subsidies according to the vector s−i or any per-mutation thereof.

If all governments set identical transfers, the poor distribute equallyacross countries with �i (s) = ν/n. In that case, the optimal subsidy is givenby

s0 := arg maxs∈S

{ν

n· u(wP + s) + u(wR − ν/n · s)

}= wR − wP

1 + ν/n. (4)

Clearly s0 < n/ν · wR , so that the rich is not completely expropriated. We re-fer to s0 as the efficient symmetric solution with egalitarian income distribution.If countries set subsidies higher [lower] than s0, we say that there is too much[too little] redistribution.

The objective function in (2) corresponds to generalized utilitarianism,one of several possible utilitarian welfare functions (Blackorby, Bossert, andDonaldson 2009). A potentially serious ethical flaw is that it gives rise to the


so called repugnant conclusion (ibid.) in allowing substitution of populationsize for quality of life: for every population of arbitrary well-offs, there ex-ists another suitably larger population of paupers that is strictly preferred bythis kind of utilitarians. Assuming u(0) sufficiently low guarantees that nogovernment will want to expropriate the rich and avoids this problem to acertain extent.

The set of Nash equilibria and our theoretical predictions for the OMCclearly depend on the choice of objective function. Still, our results are ro-bust well beyond the specific functional form in (2). Small generalizations al-lowing different utility functions for rich and poor or strictly convex costs ofsubsidizing the poor could be easily accounted for. More substantial changesin the social welfare function are innocuous as long as certain structural fea-tures are preserved. A detailed discussion can be found in Section 5.3. wherewe also study average utilitarianism, an obvious alternative to avoid the re-pugnant conclusion.

3. Static Analysis: Nash Equilibria Versus EvolutionarilyStable Strategies (ESS)

Our model defines a game where players (=countries) i = 1, . . . , n simul-taneously choose strategies from a common set of feasible subsidies S =[0, wR]. Migration flows in (1) determine payoffs πi : Sn → R, given by thesocial welfare function in (2), which can be written in symmetric formπi (s) = π(si | s−i ) as in (3). We now proceed to the analysis of this staticgame. In the usual treatment of decentralized competition, governmentsmaximize national welfare, given what other countries do, giving rise to aNash equilibrium. In contrast, with the OMC, countries are expected to com-pare the relative performance of their policy choices. Maximizing relativeperformance corresponds to choosing ESS. The next definition formally re-views these concepts for the symmetric case in a finite population.

DEFINITION 1: A strategy sN is played in a symmetric Nash equi-librium if π(sN | sN , sN , . . . , sN ) ≥ π(s | sN , sN , . . . , sN ) for all s ∈ S. Alterna-tively, a strategy s E ∈ S is said to be an ESS if π(s E | s, s E , . . . , s E ) ≥π(s | s E , s E , . . . , s E ) for all s ∈ S.We say that a Nash equilibrium or an ESS is strictif the corresponding inequality holds strictly for all s ′ �= s .

Whereas in a Nash equilibrium no player can strictly benefit from de-viating, in an evolutionarily stable profile no player can gain a strict relativeadvantage by deviating. To identify an ESS, one compares the simultaneouspayoffs to the deviator, choosing s , with the payoffs to the nondeviators, stay-ing with s E . In finite populations, the deviator has a nonnegligible impacton others. Thus, an ESS may fail to be a Nash equilibrium strategy: if the lossimposed on others is larger than the own loss incurred by deviating, it may


pay in relative terms to abandon a Nash equilibrium. Definition 1 uses thefinite-population concept of ESS proposed by Schaffer (1988); a detailed dis-cussion with a comparison to the concept for a continuum population canbe found in Vega-Redondo (1996, pp. 31–33).

The following family of auxiliary functions will be useful to characterizeequilibria:

f (k, s) = ν

k· u(wP + s) + u

(wR − ν

k· s

), (5)

where k ∈ {1, . . . , n}. The value of f (k, s) can be interpreted as the payoffto any of k countries equally sharing all the poor with subsidy s . In the Ap-pendix we show that the family of functions { f (k, s)}k=1,...,n in (5) looks asdepicted in Figure 2. Indeed, our analysis is based completely on the struc-ture depicted in Figure 2. Any social welfare function with such propertieswould generate the same results (see also Section 5.3.).

Note that f (k, 0) ≥ u(wR) for every k. The strict concavity of u(c) andwP < wR also imply that f (k, s) is strictly increasing at s = 0. Moreover, givenk, f (k, s) is strictly concave in s . Let

s∗(k) = arg maxs≥0

f (k, s).

The properties of f guarantee that s∗(k) is strictly positive for all k andsatisfies the first-order condition u′(wP + s∗(k)) = u′(wR − ν/k · s∗(k)). Thisyields

s∗(k) = wR − wP

1 + νk

. (6)

u(wR)

f(1, s)

f(2, s)

f(n, s)

s∗(1) s(n) s(1) s(2) s( n2) s(n − 2) s(n − 1) s(n)

Figure 2: Properties of auxiliary welfare functions f (k, s).


When k countries share the burden of redistribution, they maximize socialwelfare by choosing s = s∗(k) with an equal distribution of income. Here,s∗(k) is increasing in k and satisfies 0 < s∗(k) < k/ν · wR . That is, redistribu-tion is optimally more generous the more countries coordinate their policiesand the rich are never fully expropriated. Note that s∗(n) = s0, the efficientsymmetric solution defined in (4).

Given any k ∈ {1, . . . , n}, define s(k) as the strictly positive value of s thatsolves

f (k, s(k)) = u(wR),

and s(k) as the strictly positive value that solves

f (k, s(k)) = f (1, s(k)).

At s(k), a country is indifferent between paying transfers s(k) to ν/k poorand not attracting any poor at all. At s(k), a country is indifferent betweenpaying transfers s(k) to ν/k poor and, at the same transfer, hosting all poor.In the Appendix we show that the s(k) and s(k) are ordered as shown inFigure 2. In particular,

s∗(1) < s(n) < s(1) < s(2) < . . . < s(n). (7)

This observation allows an easy characterization of Nash equilibria and ESS.Our first proposition describes the set of Nash equilibria, that is, the sub-sidy profiles that countries can decentrally coordinate on when taking intoaccount their strategic connectedness.

PROPOSITION 1: Under generalized utilitarianism the set of pure-strategy Nashequilibria is given by

�N = {s = (s, . . . , s) | s(n) ≤ s ≤ s(n)}.

As Figure 2 illustrates, attracting immigrants is beneficial to a country,but redistribution policy also entails costs. For low subsidy levels the benefitsexceed the costs. At the same time, more generous redistribution is possi-ble the more countries participate. Proposition 1 shows that, in equilibrium,there is a large multiplicity of subsidy levels countries could coordinate on.They range from s(n), which is the lowest subsidy at which a country prefersto cooperate with all others, to s(n), which is the highest possible subsidythat is sustainable when all countries cooperate. For subsidies below s(n),the cost associated to redistribution is still low enough that it pays for a coun-try to attract all migrants by unilaterally increasing its subsidy (see f (1, s) inFigure 2). Subsidies larger than s(n) would render redistribution unprof-itable (welfare would be higher without any redistribution) even if allcountries participated equally (see f (n, s) in Figure 2). Inside the interval


[s(n), s(n)] there is no incentive to deviate. Note that equilibrium trans-fers can be quite generous, including levels larger than the social optimums0 = s∗(n). This is partly a consequence of the welfare function attachingvalue to large population sizes.

The next proposition characterizes the set of ESS. In line with laboratoryfederalism and the workings of the OMC, where the performance of nationalpolicies is mutually compared, it focusses on subsidy levels that maximizerelative payoffs.

PROPOSITION 2: Under generalized utilitarianism the set of ESS is the interval

S E = [s(1), s(n − 1)].

From Proposition 2, the set of evolutionarily stable subsidies is a sub-set of those that can be sustained as a Nash equilibrium. In this respect, theOMC is in accordance with the outcome of a decentralized system. The set ofESS excludes, however, some subsidies in the lower and in the upper rangeof Nash equilibria. To see the difference, consider any s ∈ [s(n), s(1)) thatconstitutes a Nash equilibrium according to Proposition 1. A single devia-tion to an alternative s ′ such that s < s ′ ≤ s(1) would induce an additionalcost of redistribution that does not compensate the benefits of the additionalinflow of migrants. From an individual perspective, it is irrational to imple-ment such a generous social policy alone, instead of coordinating with allothers at s . However, the welfare loss imposed on other countries due to themigration outflow is even larger and the deviator gains a relative advantage.This makes coordination on s unstable. Similarly, any Nash equilibrium witha subsidy s ∈ (s(n − 1), s(n)] is fragile, since unilateral deviations to somelower s ′ would make s unsustainable for the remaining countries.

Remark 1: All ESS in S E are strict, except for the upper bound s(n − 1).Starting at the symmetric profile where all countries set s = s(n − 1), a de-viation to some s ′ < s(n − 1) would induce migration inflows to the n − 1countries still offering s(n − 1). Their welfare would equal u(wR), the samelevel as for the deviator to s ′. Hence, the deviator would not suffer a strictdisadvantage. This will make a subtle difference in the dynamic analysis.

4. Laboratory Federalism as an Imitative Process

Coming to what we consider is the spirit of laboratory federalism and theOMC, we now explicitly take a dynamic approach, allowing countries to ob-serve each others’ outcomes and make sequential decisions based on thisinformation.

The idea of the OMC, sketched in Figure 1 and applied to the redis-tribution game of Section 3, is modeled here as an evolutionary learning


process. It is beyond the scope of this paper to discuss how agreement oncommon objectives is reached and how performance is actually comparedacross EU Member States. For the purpose of this paper, a country’s socialwelfare given by expression (2) is taken as the indicator for good policy andattaining its highest possible level is the commonly agreed objective. Sinceour learning dynamics will not require that countries act strategically, it isnot essential to our analysis that they have full understanding of the objec-tive function, but they must be able to observe its realized values. We considergovernments that recurrently decide on their redistribution policies deter-mined by the subsidy levels (these correspond to their national strategies)and these policy choices and their success in terms of national social wel-fare become public (in the OMC this would happen through the nationalreports). Performance comparisons allow to identify best practice, that is,subsidies associated with the highest welfare levels currently observed. Thefact that all countries are identical permits unambiguous identification ofbest-performing policies, but in Section 5.4. we will discuss how our analysiscould be extended to the asymmetric case.

We consider countries mainly following the OMC principle of adoptingbest-performing policies whenever they are called to revise their nationalstrategies. However, this is not binding and countries may occasionally de-viate from the prescriptions of best performance and adopt other policiesbased on their own motivations, national political interests, or mistakes. Wemodel these political innovations as random experimentation. New strate-gies introduced by experimentation may be followed by other countries onlywhen they prove to be more successful than existing policies; otherwise theywill not persist. Our imitation dynamics intends to capture the iterative loopby which the diffusion of best practice takes place; experimentation capturesthe open nature of the process.

The analysis is applied to a discretized version of the model in Section 3.Specifically, countries choose subsidies from a finite set � ⊂ S. For simplicityof exposition, we assume that � contains the values s(k) for all k = 1, . . . , n.The state space of the process is �n with states at t = 1, 2, . . . given by the vec-tors of subsidies, s(t) = (s1(t), . . . , sn(t)) . Subsidies s(t) determine payoffsat t , π (s(t)) = (π1 (s(t)) , . . . , πn (s(t))) , where πi (s(t)) is social welfare ofcountry i in state s(t) as given by expression (2). At any t = 1, 2, . . . countriesobserve the vectors s(t) and π (s(t)). Given the current state s(t), define

B(s(t)) = {s ∈ � | s = si (t) for some i and πi (s(t)) ≥ π j (s(t)) for all j

},

the set of all subsidies that gave highest welfare at t . With an imitate-the-bestguideline, only subsidies in B(s(t)) should be adopted with positive proba-bility. This gives a clear recommendation if B(s(t)) is a singleton. Otherwise,it is unclear what other motives should determine policy choice. Define theset

B(s(t)) = B(s(t)) ∩ {maxj

s j (t)}


that singles out the highest subsidy in period t whenever this gave highestwelfare. Note that B(s(t)) may be empty when the highest subsidy was notbest.

“The iterative process:” Strategy revision is done according to the fol-lowing imitative rule. With probability 0 < λ < 1 a country is called to reviseits subsidy for the next period. A revising country chooses the recommen-dation contained in B(s(t)), whenever this set is not empty. If B(s(t)) = ∅,best-performing countries keep their strategies unchanged while all othercountries choose any of the subsidies in B(s(t)) with positive probability.With probability 1 − λ no revision opportunity arrives. To formalize the non-binding character of the OMC we allow countries to engage in experimentationas follows. At any t , each country has probability 1 − ε to follow the imitativerule just described and no experimentation takes place. With probability ε

the country gets an experimentation draw and may choose any subsidy s ∈ �

at random. Imitation and experimentation opportunities are drawn inde-pendently across countries and across periods. A country keeps its subsidyunchanged if no imitation or experimentation opportunity arrives at t .

Note that the learning rule entails a certain preference for transfer gen-erosity: countries grant high subsidies if they are sustainable. Such behaviorappears plausible when the OMC is applied in the context of social policy.The rule also incorporates some conservative upholding: countries do notincrease their subsidies if observed higher subsidies have proved to be un-sustainable. Note also that strategy revision displays some inertia and allowsfor asynchronous policy adoption due, for example, to restrictions in the ad-ministrative or political process. Finally, experimentation captures the soft-law feature of the OMC: countries can ignore the policy guidelines withoutpunishment. They could even implement tailor-made policies in pursuit ofnational interests. Experimentation also captures mistakes, innovation, ortrial and error.

The iterative OMC process defines a stationary Markov process, with λ

and ε determining the transition probabilities between any two states. We re-fer to the process with ε = 0 as the unperturbed imitation dynamics. The steadystates of this process are called absorbing. A common property of imitation,which also holds here, is that it leads to so-called monomorphic states, whereall countries choose the same subsidy. Formally,

M = {s ∈ �n | s = (s, . . . , s), s ∈ �}.

Imitation alone cannot introduce new strategies. Therefore, from any states(t) �∈ M , a unique subsidy is eventually singled out as the best-performingpolicy and adopted by all countries. Once a monomorphic state is reached,imitation cannot lead the system away.

LEMMA 1: M is the set of absorbing states of the unperturbed imitation dynamics.


With ε > 0, there is positive probability to exit every state. We refer tothe Markov process with experimentation as the perturbed imitation dynamics.This process is ergodic, since there is positive probability that any subset ofcountries experiment simultaneously with any subsidy, and aperiodic, sinceat any time there is positive probability that no country revises its subsidy dueto inertia and the process stays at the same state for one period. For every λ

and ε, this guarantees convergence to a unique invariant distribution, μλ,ε,which gives both, the average frequency with which the process visits eachstate along any sample path, and the probability that the system is at thatstate in the long run.

Following the evolutionary literature, our analysis focuses on the limitinvariant distribution, μ∗ = limε→0 μλ,ε. States with positive probability in μ∗

are called stochastically stable (or long-run equilibria). Only these states will beobserved in the long run as the probability of experimentation goes to zero.Distribution μ∗ exists and only gives positive probability to absorbing statesof the unperturbed dynamics, the monomorphic states in M from Lemma 1.A detailed introduction to stochastic stability and the characterization of thesupport of μ∗ can be found, for example, in Young (2001, chapter 3). Therest of this section provides an intuition of our main result in Proposition 4.Formal proofs using the radius and coradius technique by Ellison (2000) aregiven in the Appendix.

Define the set E as the subset of monomorphic states where all countrieschoose an ESS,

E = {s ∈ M | s(1) ≤ s ≤ s(n − 1)}.This essentially correspond to the interval S E from Proposition 2. By defini-tion of an ESS, single countries deviating from states in E suffer a relative dis-advantage. This disadvantage is strict everywhere except at the upper boundof the interval (see Remark 1). Our learning rule guarantees that no devi-ation from E would be followed by imitation. This also holds for s(n − 1),since in case of payoff ties only the highest subsidy is adopted.

All other states in M can be abandoned after a single deviation in favorof some state in E . To see this, suppose the process is at a monomorphicstate with s < s(1) and, at t , one of the countries deviates to s(1). The de-viator attains welfare exactly equal to u(wR), the same as nondeviators butwith a higher subsidy. Thus, B(s(t)) = {s(1)}, which will be followed at thenext revision opportunity. Analogously, suppose the process is at a monomor-phic state with s > s(n − 1). A deviator lowering its subsidy at t (say, to somes ′ ∈ S E ) triggers migration into the n − 1 countries still offering s , render-ing s unsustainable as welfare falls strictly below u(wR). In this case, B(s(t))is empty and B(s(t)) = {s}; if called to review its strategy again, the devia-tor would not change its subsidy, while all other countries would follow thedeviator at the next revision opportunity.

The previous arguments show that E cannot be abandoned after anysingle deviation, but it can be reached from everywhere else with a single


deviation to a subsidy in S E . This makes the process more likely to moveinto rather than out of E . Recall that S E is a subset of the Nash equilibriumstrategies. Our prediction is that the process eventually moves into E , thusconverging to a Nash equilibrium.

PROPOSITION 3: If s∗ is stochastically stable, then s∗ ∈ E ⊂ �N .

Once E has been reached, single deviations to a different subsidy can-not gain a relative advantage and will not be followed. Yet, it is possible toabandon states in E after the simultaneous deviation of several countries.We will argue that the number of simultaneous deviations needed to disturbstates in E increases as we move toward intermediate levels of the subsidy. Inour construction, it will be enough to argue with simultaneous deviations tothe same alternative subsidy: if a transition between two monomorphic statesis possible after simultaneous deviation to different subsidies, then the sametransition is possible with the same number of deviations to the same subsidy.Intuitively, this is because s(k) increases with k and, if a certain subsidy is sus-tainable with k deviators, then it would also be sustainable if more deviatorschose it.

To refine our prediction within the set E , define the collection of sets

E k = {s ∈ M | s(k − 1) ≤ s ≤ s(k)}, k = 2, . . . , n − 1.

States in E k are monomorphic states with subsidies in the interval[s(k − 1), s(k)] and we have that E = ⋃n−1

k=2 E k . It is convenient to use nota-tion s k = (s(k), . . . , s(k)). Notice E k ∩ E k+1 = {s k} for k = 2, . . . , n − 2. Of-ten these boundary states must be treated separately in our analysis. Also ob-serve that positive net welfare gains can be obtained if at least k countriescoordinate on a subsidy in [s(k − 1), s(k)]. Instead, if less than k countriescoordinate on s ∈ (s(k − 1), s(k)], welfare falls strictly below u(wR). There-fore, for the process to abandon states in E , enough countries must coordi-nate on a higher subsidy to make net welfare gains, or enough must chooselower subsidies to make the current one unsustainable. We now argue this inmore detail.

(1) Moving from lower to higher subsidies: Consider any pair of states s, s ′ ∈E k with s < s ′. Notice how s ′ can be reached from s if k countriessimultaneously deviate from s to s ′. The same is true if the processstarts at any s ∈ E k ′

with k ′ ≤ k. Consequently, the boundary state s k ,with the highest subsidy in the interval s(k), can be reached eitherdirectly from any monomorphic state with lower subsidy if k devia-tors choose s(k), or step-by-step after a sequence of transitions goingthrough states in the sets E k ′

with k ′ ≤ k. The latter chain of transi-tions is more likely to happen, since it requires less coordination ateach stage. The last transitions in this chain, that is, the final moveto some s ∈ E k \ {s k−1} from lower subsidies, cannot take place with


s(1) s(2) s(3) . . . s n2

− 1 s n2

. . . s(n − 3) s(n − 2) s(n − 1)

1 122 33n2

− 1 n2

− 1n2

Figure 3: Most likely transition paths. The case of n odd.

less than k simultaneous deviations. It follows that transitions upwardsrequire an increasing number of countries and become less likely ask increases.

(2) Moving from higher to lower subsidies: Since migration takes place onlyto countries with highest subsidies, deviating to a lower subsidy im-plies giving up its benefits and results in welfare u(wR). At the samewelfare level, higher subsidies are favored by imitation, so that devia-tions to lower subsidies can only be successful if nondeviators are leftwith strictly less than u(wR). Thus, we need at least n − k + 1 coun-tries simultaneously lowering their subsidies in order to move down-wards from states in E k \ {s k−1}. Abandoning the boundary state s k−1

in favor of a lower subsidy requires n − k + 2 deviations, since s(k −1) is still sustainable with k − 1 countries. Since the number of devi-ations needed for a downward transition is independent of the statewe intend to reach, there is no particular advantage of step-by-steptransitions in this case. In general, transitions downwards requireless simultaneous deviations and, thus, become more likely as k in-creases. Respectively, reaching a state s ∈ E k from any other states ′ ∈ E k ′

for k ′ ≥ k with higher subsidy requires more deviations thelower is k ′ and, thus, the closer s ′ is to s .

To summarize: from states in E k with low k the process is more likely tomove upwards; for high k the process is more likely to move downwards. Inintermediate states these probabilities are balanced and the process is likelyto stay.

Figure 3 illustrates the case of n odd. Our prediction in this case is theset E n

2 �. The lower and the upper bounds of the interval associated withthis set can be reached from outside the set with at most n

2� − 1 deviations.The most likely way to exit the set is downwards, which requires at least n

2�deviations. Within E n

2 � it is possible to move up and down with exactly n2�

deviations. A typical path is likely to move into E n2 � eventually and then stay

bouncing up and down in this set.Detailed counting gives a slight difference between the odd and even

cases. If n is even, the central state, sn2 , can be sustained by exactly half of the

countries. This state can be reached from both states with higher and withlower subsidies with at most n

2 simultaneous deviations to s( n2 ). Instead, it

would take at least s( n2 ) + 1 deviations to exit s

n2 in favor of any other subsidy.


This is because higher subsidies cannot be profitably sustained with less thans( n

2 ) + 1 countries, while if only half of the countries lower their subsidies,s( n

2 ) will still be the highest, most profitable subsidy. Our main result, thus,is as follows:

PROPOSITION 4: The set of stochastically stable states is E n2 � if n is odd and

{s n2 } if n is even.

Proposition 4 shows that the most likely outcomes of the OMC lie in theintermediate range of the ESS. Low subsidies can be destabilized by a smallnumber of countries coordinating on a higher level; high subsidies becomeunsustainable when a small number of countries cut down their transfers.Only intermediate values that can be sustained by a simple majority of coun-tries are robust to small-group deviations.

Put more technically, along any sample path of the stochastic learningprocess, the fraction of periods in which intermediate subsidies are observedconverges to one as ε → 0: in the long run essentially only intermediate sub-sidies are chosen most of the time. Ellison (2000, Theorem 2) allows us to es-timate the expected waiting time until the process hits the prediction for thefirst time. As the set E of symmetric ESS profiles can be reached from every-where else with a single deviation, the expected waiting time to reach E is oforder ε−1 (the shortest possible). The central interval can be reached step-by-step with an increasing number of simultaneous deviations up to [n/2].Thus, the expected waiting time to reach E is of order ε−[n/2]. Hence, theprocess quickly leads to the set of ESS and stays there most of the time. Coor-dination closer to the central interval requires a long time that expands withthe number of countries.

5. Discussion and Extensions

Proposition 4 implies that in laboratory federalism governments coordinatetheir policies to a strict subset of what could emerge under full decentral-ization. Policies sustainable by a simple majority of countries are most likelyto emerge. This prediction and its underlying assumptions deserve furtherdiscussion.

5.1. Ef ciency

An important motivation for policy coordination is the avoidance ofexternality-induced efficiency failures of decentralization. It is therefore nat-ural to ask whether the learning process of the OMC indeed fosters ef-ficiency. Formally, we have to compare the symmetric efficient solution,s0 = ˜s∗(n) in (4), with the stochastically stable states characterized in Propo-sition 4. Whereas s∗(n) maximizes f (n, s), the values of s( n

2� − 1) and


Table 1: Ef ciency results

w R ν s 0 s(4) s(5)

2 12 0.43 0.46 0.552 21 0.30 0.28 0.344 27 0.75 0.58 0.72

s( n2�), relevant for E n

2 �, refer to properties of the functions f ( n−12 , s) and

f ( n2 , s). Technically, there is no reason why s0 should be related to E n

2 �.This is illustrated in

EXAMPLE 1: Suppose that u(x) = √x − 1. Take wP = 1 and assume that wR ≤

4. Then,

s(k) = 4(√

wR − 1) (

νk + √

wR)

(νk + 1

)2 .

For n = 9, E n2 � = [s(4), s(5)]. Table 1, which reports results for differ-

ent values of wR and ν, shows that the efficient outcome, s0, may be lowerthan s(4), higher than s(5), or it may be contained in the interval. The setof stochastically stable states may, thus, include the efficient outcome, but ingeneral there can be too much or too little redistribution.

5.2. Mobility of the Poor

While the assumption of perfectly mobile poor is extreme, it poses thestrongest demands on our model since it amplifies externality effects: if acomplete erosion of the welfare state does not occur here, it will not happenwith restricted mobility either. Yet, we check the robustness of our results tothe introduction of some immobility among the poor.

Suppose that each country has a fixed amount β of immobile poor, mea-sured relative to the size of the immobile rich. Symmetry is preserved withβ equal across countries. The number of poor in country i is now given by�i (s) := β + �i (s), with �i (s) determined as in (1). Governments still adhereto generalized utilitarianism (2), replacing �i (s) by �i (s). As in (5), a familyof auxiliary functions f (k, s) can be defined as follows:

f (k, s) =(β + ν

k

)· u(wP + s) + u

(wR −

(β + ν

k

)· s

)for k = 1, . . . , n. Functions f have the same properties as their f -counterparts in Section 3. In particular, they are strictly concave in s withmaximizer

s ∗(k) = wR − wP

1 + β + νk

,


which is increasing with k. To support their native poor, countries start redis-tribution even without any migrants. If �i (s) = β, the government in coun-try i maximizes

g(s) = β · u(wP + s) + u(wR − βs).

Observe that g(s) is strictly concave, u(wR) < g(0) < f (k, 0), and 0 <

g ′(0) < ∂ f (k, 0)/∂s for all k. In fact, g corresponds to the limit of f (k, s)when k → ∞ and, thus, possesses the same properties as the functions f inSection 3. The optimal transfer when only native poor reside in i is largerthan with mobile immigrants: arg max g(s) > s ∗(n).

In the analysis of equilibria, the function g assumes the role of the hor-izontal line at u(wR) in Figure 2, since g(s) is the welfare level of a countrywith lower subsidy than others. The f -curves would be replaced by their f -analogs. The points at which f (1, s) and f (n − 1, s) cross g(s) are now theboundaries of the ESS interval.1 All other elements of the analysis remainunchanged and our results, thus, qualitatively preserved.

5.3. Objective Function

The choice of a specific social welfare function casts doubt on the generalityof our results. However, our proofs only use the auxiliary functions f (k, s)illustrated in Figure 2. The properties of these functions that are essential toour results are the following:

(1) An inflow of migrants is beneficial to the country and the govern-ment is averse to extreme inequality; in particular, f (k, 0) > u(wR),so that without redistribution migrants raise welfare, and fs (k, 0) >

0 for any k, implying that introducing redistribution strictly in-creases welfare for any size of the migrant population.

(2) A coordinated increase of the subsidy for any k countries sharingall the poor has decreasing marginal net benefits; that is, the socialwelfare function f (k, s) is inversely U-shaped in s for any k, so thatmore redistribution becomes increasingly costly.

(3) Independently of how many countries are able to coordinate theirpolicy, increasing the fiscal burden on the rich eventually becomesless attractive than giving up redistribution entirely (and, thus, los-ing all poor); that is, the subsidy s(k) that makes the country indif-ferent between a small rich society and an unequal but larger pop-ulation exists for every k. This precludes that a social optimum ever

1 Similar as for f , values s(k) exist at which f (k, s(k)) = u(wR ). Since g(wR/β) <

u(wR ), there is an sg > s(n) such that g (sg ) = u(wR ) and g(s) > u(wR ) for all s < sg .Since f (k, 0) > g(0) and g (s(k)) > u(wR ) for all k, points sg (k) < s(k) exist at whichf (k, sg (k)) = g (sg (k)).


involves the complete expropriation of the rich for the sake of in-creasing population size.

Propositions 1 to 4 and their implications for laboratory federalism andthe OMC continue to hold for any government objective function satisfyingthese plausible properties. This considerably broadens the applicability ofour model. Nevertheless, violation of any of these might considerably affectresults. For illustration, let us consider the case of average (rather than gen-eralized) utilitarianism where government payoffs are given by the averageutility in the population:

πAUi (s) = 1

�i (s) + 1· [�i (s) · u(wP + si ) + u(wR − �i (s) · si )] . (8)

The crucial difference between (8) and (2) is the emphasis on the homo-geneity rather than on the size of the population. Whereas (2) gives positivevalue to migrant inflow, even at the expense of equality, with (8) small buthomogeneous and rich societies are preferred to larger but poorer ones.Graphically, the difference when Figure 2 is adapted to average utilitarian-ism is that the analogs of the f -functions lie entirely below the u(wR)-line(which still represents a society without poor).2

Although the symmetric efficient solution coincides for average and gen-eralized utilitarianism, the unique Nash equilibrium and ESS with (8) entailsi = 0 for all i . Since u(wR) exceeds any welfare level attainable with positivemigration, there is always an incentive to cut back transfers and discourageinflow migration. Clearly, imitative learning in the OMC cannot improve onthis inefficient zero-subsidy equilibrium. This example confirms that the re-sults of fiscal federalism with mobility are sensitive to changes in governmentobjectives (Mansoorian and Myers 1997). Nevertheless, the discussion in thissection shows that the class of objective functions supporting our main re-sults is much broader than specification (2) might suggest.

5.4. Asymmetric Countries

Our model makes strong symmetry assumptions: preferences over consump-tion are common across individuals, the value of immobile assets for the richand the income of the mobile poor are the same across countries, migrationtakes place frictionless to the country with highest subsidies. However, EUMember States or jurisdictions in real-world federations are not identical.The OMC actually accounts for diversity among its participants, allowing for

2 For any si ≥ 0, a society that hosts any population �i of poor is inferior to one with ahomogeneous mass of rich individuals. To see this, let λi = �i

1+�iand use the strict mono-

tonicity and concavity of u to see that, for any �i > 0, πAUi < u(λi (wP + si ) + (1 − λi )(wR −

�i si )) = u( wR +�i wP1+�i

) < u(wR ).


differentiated targets (within the commonly agreed objectives) and multidi-mensional policy instruments (Cherchye, Moesen, and Puyenbroeck 2004;Coelli et al. 2010). While our analysis is greatly simplified by the choice ofa symmetric welfare function, it can be extended to asymmetric cases. Fol-lowing Dastidar (1995) and Tanaka (1999), we can explore the robustness ofour results to the introduction of differentiated welfare functions.3

Setting: Consider two types of countries, n1 with welfare function π (1)(s)and n2 with welfare function π (2)(s). Let n = n1 + n2 and assume nj ≥ 3, j =1, 2. This may correspond to a north–south, rich–poor, or large–small coun-try approach. Keeping all other elements of our model, function π ( j) givesrise to a family { f ( j)(k, s)}k=1,...,n for each group j , which we assume to havethe properties described in Section 5.3. For any k = k1 + k2, f ( j)(k, s) is thewelfare to any country in group j when k countries, k1 from group 1 and k2

from group 2, equally share the poor at subsidy s . The values of s j (n) ands j (k) can be defined as in Section 3 for each group j . In a Nash equilibriumno country can improve its welfare by deviating. A strategy is an ESS in thissetup if deviating cannot improve welfare relative to other countries in thesame group.

Static equilibria: While setting different subsidies may now be an equilib-rium, symmetric equilibria of the type derived in Section 3 may still exist. Inparticular, as in Proposition 1, symmetric profiles where all countries set asubsidy in the interval

SN = [max{s1(n), s2(n)}, min{s1(n), s2(n)}] (9)

constitute a Nash equilibrium, provided that this interval is nonempty (seeDastidar 1995, pp. 28–30). The logic of Proposition 2 can again be appliedto discard some of the subsidies in SN on the basis of relative performance.In particular, only symmetric profiles with subsidies in the interval

S E = [max{s1(1), s2(1)}, min{s1(n − 1), s2(n − 1)}] (10)

satisfy the ESS criterion. The interpretation of (9) and (10) is along the samelines as in Section 3. The difference for the ESS is that countries comparetheir performance only to that of other countries in the same group and notto all countries.

Dynamics: Consider the same imitation rule as in Section 4, but assumethat countries only imitate within their own group. To illustrate, take n1

and n2 to be even and assume that s1(k) < s2(k) < s1(k + 1) for all k =1, . . . , n − 1. This corresponds to the case where the two groups are not

3 Evolutionary game theory treats asymmetric contests as multipopulation games, whereplayers are assigned to different populations according to some “roles.” The analysis isoften done for symmetrized versions of the game, where ESS correspond to strict Nashequilibria. This led to weaker definitions of evolutionary stability (e.g., Hofbauer andSigmund 2003). So far, no standard treatment exists for asymmetric finite populations,and the few existing results lack robustness (Apesteguia et al. 2010).


too different but where countries in group 2 are richer or have a strongerpreference for redistribution. The logic of Proposition 4 still applies: first,monomorphic states in S E can be reached (but not abandoned) with achain of single deviations. Second, monomorphic states with s ≤ s1(k) for2 ≤ k ≤ n − 1 can be directly reached from absorbing states with lower sub-sidies after at most k = k1 + k2 simultaneous deviations, if ki ≥ 1 deviationsoccur in group i = 1, 2. This is because subsidies s ≤ s1(k) are sustainable inboth groups when k countries coordinate. Instead, monomorphic states withsubsidy s1(k) < s ≤ s2(k) can only be reached after at least k + 1 deviations.4

Finally, monomorphic states with s such that s1(k) < s ≤ s2(k) are easier todestabilize, since n − k deviations to a lower subsidy suffice to make s un-sustainable in group 1, whereas n − k + 1 downward deviations are neededif s ≤ s1(k). Altogether, the process is more likely to move to higher subsi-dies below s1(n/2) and to lower subsidies above s2(n/2). Whereas all stateswith s ∈ (s1(n/2), s2(n/2)] can be abandoned with n/2 deviations, the statewhere all countries coordinate on s1(n/2) can only be abandoned with atleast n

2 + 1 deviations. As in Proposition 4, countries are most likely to coor-dinate on intermediate subsidy levels, but now this will occur on the lowestlevel sustainable by simple majority.

6. Conclusions

In the spirit of laboratory federalism, the OMC, adopted as a mode of gov-ernance by the EU at the Lisbon Summit in 2000, can be understood as apolitical process of mutual learning of best practice among its participants.We model this as a stochastic learning dynamics of the type studied in evolu-tionary game theory and formalize this idea for a specific model of incomeredistribution in an integrated economic area with mobile transfer benefi-ciaries. Countries are able to observe the policies and their performanceelsewhere. Revision of national policy occurs when policies chosen by oth-ers have proven to be sustainable and successful. In addition, countries canexperiment with new policies. In our view, these features both capture theidea of learning from mutual experience and preserve the decentralized andopen nature of a federal system.

Our main observation is that the learning process strongly favors co-ordination on the subset of Nash equilibria that corresponds to ESS. Ina dynamic interpretation, countries coordinate over time on intermediate

4 Note that k countries are required to make s sustainable in group 2; these may involvesome countries from group 1 also experimenting with s . After these k deviations, subsidys will spread to all countries in group 2. As s > s1(k) cannot be sustained in group 1 afterk deviations, countries in group 1 will either stay or move back to some other subsidy. Thisleads to a non-monomorphic absorbing state with s adopted in group 2 only. At this point,our assumption that s2(k) < s1(k + 1) implies that a single additional deviation from acountry in group 1 to s would suffice for s to spread in group 1 as well.


transfer levels, that is, on redistribution policies that can be sustained by co-ordination of half of the Member States.

To our knowledge, this is the first paper providing a formal, game-theoretic analysis of the OMC or, more general, of the dynamic learningprocesses with fiscal federalism. Both opponents and advocates of the OMCwill, with good reason, argue that our stylized analysis ignores many of theOMC’s advantages (e.g., the higher degree of legitimacy), defines away anumber of problems (e.g., the definition and measurement of performanceindicators, communication procedures), and discusses the OMC in a setting(decentralized redistribution) to which it may not be best suited. Neverthe-less, our analysis entails important messages for policy makers and mecha-nism designers in the EU where welfare and redistribution policies are stillin the domain of national governments. The hope that the OMC will “re-calibrate” European welfare states (Ferrera, Hemerijck, and Rhodes 2000)seems justified. On a first pass, the OMC provides a successful way to at-tain policy coordination and to avoid extreme, undesirable outcomes. Thisis in line both with preliminary empirical evidence on the OMC (Coelli et al.2010) and with the general tenet of laboratory federalism: decentralized ex-perimentation and imitation converge toward good policies when there isincomplete knowledge about the social and economic environment.

APPENDIX

Properties of the Payoff Function

We derive here the main properties of the functions { f (k, s)}k=1,...,n de-fined in (5). Recall that f is strictly concave in s with a maximum at s∗(k)given in (6). Given k, let s(k) be the strictly positive value of s such thatf (k, s(k)) = u(wR). The properties of f imply that f (k, s) > u(wR) for alls ∈ (0, s∗(k)]. By definition of s∗(k) and fs s < 0, f strictly decreases for alls > s∗(k). Moreover, for s = k

νwR we have that

f(

k,kν

wR

)= ν

ku

(wP + k

νwR

)+ u(0) < u(wR),

by the assumption that u(0) < u(wR) − νu(wP + wR/ν) and the fact thatνk u(wP + k

νwR) is strictly decreasing with k. This implies existence and

uniqueness of s(k) for all k. Moreover, s∗(k) < s(k) < kνwR . Finally, note that

s(k) satisfies

�(k, s) := kν

·[u(wR) − u

(wR − ν

ks)]

= u(wP + s).

Here, u(wP + s) is strictly increasing and strictly concave in s . Given k,�(k, 0) = 0 ≤ u(wP ) and �(k, s) is strictly increasing and strictly convexin s .


Given k, let s(k) be such that f (k, s(k)) = f (1, s(k)). To show existenceand uniqueness of s(k), define

�(k, s) := f (1, s) − f (k, s)

= ν

(1 − 1

k

)· u(wP + s) + u(wR − νs) − u

(wR − ν

ks)

.

Clearly, �(k, 0) ≥ 0. Moreover, strict concavity of u implies that

�(k, s) > ν

(1 − 1

k

) {u(wP + s) − s · u′(wR − ν · s)

}.

For s ≤ s∗(1) we have wP + s ≤ wR − ν · s , such that

�(k, s) > ν

(1 − 1

k

) {u(wP + s) − s · u′(wP + s)

}> ν

(1 − 1

k

)u(wP ) ≥ 0.

Hence, �(k, s) > 0 for s ∈ (0, s∗(1)]. That is, if s(k) exists, we must haves(k) > s∗(1). Furthermore, by u′′ < 0, for s ≥ s∗(1) we have that

�s (k, s) = ν

(1 − 1

k

)· u′(wP + s) − ν · u′(wR − νs) + ν

k· u′

(wR − ν

ks)

= ν

(1 − 1

k

)· [

u′(wP + s) − u′(wR − νs)]

+ ν

k·[u′

(wR − ν

ks)

− u′(wR − νs)]

< ν

(1 − 1

k

)· [

u′(wP + s) − u′(wR − νs)] ≤ 0.

Thus, �s (k, s) < 0 for all s ≥ s∗(1). Finally, note that for s = s(1), we havethat

�(k, s(1)) = f (1, s(1)) − f (k, s(1)) = u(wR) − f (k, s(1)) < 0,

since s∗(1) < s(1) < s(k) for k > 1. Existence and uniqueness of s(k) for allk > 1 follow and s∗(1) < s(k) < s(1) holds for all k.

Proof of Proposition 1: Step 1: All Nash equilibria are symmetric. Consider a non-symmetric profile (s1, . . . , sn) with si < s j for some i �= j . Country i attractsno poor and gets payoff u(wR). The payoff to all countries currently choosings = maxk=1,...,n sk is f (m, s), where m is the number of countries choosing s .For s > s(m), f (m, s) < u(wR) and we are not in equilibrium: any coun-try with s is strictly better off by lowering the subsidy. Similarly, for s ≤ s(m),f (m, s) ≥ u(wR) and country i can strictly increase payoffs by deviating fromsi to s , since s(m) < s(m + 1) implies f (m + 1, s) > u(wR).

Step 2: Characterization of Nash equilibria. In a symmetric profile s =(s, . . . , s), all countries get payoff f (n, s). For s < s(n), f (1, s) > f (n, s)


and any country that deviates to s ′ such that s < s ′ < s(n) gets payofff (1, s ′) > f (n, s), higher than at the symmetric profile. For s > s(n),f (n, s) < u(wR) and any country decreasing its subsidy strictly improves.Finally, whenever s(n) ≤ s ≤ s(n), we have f (n, s) ≥ f (1, s) and f (n, s) ≥u(wR). A subsidy decrease attains payoff u(wR) with no strict improvement.Notice that since s(n) > s∗(1), a subsidy increase to s ′ > s attracts all mi-grants, resulting in payoff f (1, s ′) < f (1, s) ≤ f (n, s), a strict welfare lossrelative to the symmetric profile. It follows that [s(n), s(n)] is the interval ofequilibrium subsidies. �

Proof of Proposition 2: At any symmetric profile s = (s, . . . , s), consider sin-gle deviations to some subsidy s ′ �= s . The relevant payoffs to characterizeESS are now those obtained after deviation. Suppose first s < s(1), then adeviation upwards with s < s ′ < s(1) attracts all migrants and gets f (1, s ′) >

u(wR), while all other get u(wR). Thus, s < s(1) cannot be ESS. Considernow s > s(n − 1), then a deviation downwards to any s ′ < s forces migrantsto leave, resulting in payoff u(wR) for the deviator, while nondeviators areleft with too much migration and get f (n − 1, s) < u(wR) by definition ofs(n − 1). Again, the deviator has a strict advantage and s > s(n − 1) cannotbe ESS. Finally, let s ∈ [s(1), s(n − 1)]. Deviations to s ′ < s ≤ s(n − 1) earnpayoff u(wR) while those countries sticking to s obtain f (n − 1, s) ≥ u(wR).Deviations to s ′ > s ≥ s(1) get f (1, s ′) < u(wR) while all others get u(wR).In this case, deviations have no strict relative advantage. It follows from (i)–(iii) that only s ∈ [s(1), s(n − 1)] are ESS. �

Proof of Lemma 1: At any s(t) �∈ M , B(s(t)) is either a singleton or empty.If empty, there is positive probability that all countries with subsidies not inB(s(t)) get a revision opportunity choosing some subsidy in B(s(t)), thus,reducing the number of different subsidies chosen. We can repeat this ar-gument until B(s(t)) is a singleton. When that is the case, there is positiveprobability that all countries simultaneously revise their subsidies, reachinga monomorphic state. At any s(t) = (s, . . . , s) ∈ M , B(s(t)) = {s} and imita-tion alone cannot move the process away from s(t). �

Proof of Proposition 3: Let the process start at s = (s, . . . , s) ∈ M in any periodt and call s ′ ∈ �n the resulting state when only one country deviates to s ′ ∈ �

at t + 1.

(1) Suppose s < s(1) and s ′ = s(1). The deviating country attracts allpoor and gets payoff u (wR) by definition of s(1). Nondeviatingcountries with lower subsidies also get u (wR). Only the country withhighest best-performing policy is imitated. Thus, B(s ′(t + 1)) = {s ′}.

(2) Suppose s > s(n − 1) and s ′ ∈ [s(1), s(n − 1)] ∩ �. The deviatingcountry with lower subsidy attracts no poor and gets u (wR). Theremaining n − 1 countries share all the poor at s > s(n − 1) with


payoff f (n − 1, s) < u (wR) by definition of s(n − 1). It follows thatB(s ′(t + 1)) = {s ′}.

(3) Suppose s(1) ≤ s ≤ s(n − 1). A country deviating to s ′ < s getsu (wR), while all other get payoff f (n − 1, s) ≥ u (wR) by defini-tion of s(n − 1) and offer maximum subsidy s . Alternatively, a de-viation to s ′ > s ≥ s(1) attracts all poor and attains payoff f (1, s ′) <

u (wR), while the nondeviating countries get u (wR). In both casesB(s ′(t + 1)) = {s} and the deviation is not followed.

Both in (i) and (ii) there is positive probability that all countries revisetheir subsidies and choose s ′ at the end of t + 1. This shows that from anystate s �∈ E a state in E can be reached after a single deviation. By (iii) statesin E cannot be abandoned after a single deviation. In the terminology ofEllison (2000, Theorem 1), (i) and (ii) imply that the coradius of E is oneand (iii) implies that its radius is strictly greater than one. It follows that thestochastically stable states must be contained in E . �

Proof of Proposition 4: States in E cannot be abandoned with a single devi-ation, but it may be possible to abandon these states with more than onesimultaneous deviation. In a first step we show that it is sufficient to con-sider simultaneous deviations to the same alternative subsidy. To see this,take any two states s, s ′ ∈ E and suppose that the process moves from s tos ′ after multiple deviations with at least two different subsidies s ′ and s ′′.This transition requires that the process visits some intermediate state ofthe form s = (s, . . . , s, s ′, . . . , s ′, s ′′, . . . , s ′′). Reaching s ′ eventually from srequires s ′ ∈ B(s) so that s ′ survives revision. Two cases are possible. Eithers ′ = max{s, s ′, s ′′} and B(s) = {s ′}, or else B(s) is empty. In both cases thenumber of countries choosing s ′ after further revision is at least as high asin s . If B(s) = {s ′}, this is because enough deviators had chosen s ′ so as tosustain this subsidy profitably. Since s(k) is increasing with k, as more coun-tries choose s ′ welfare cannot fall bellow u(wR) and s ′ will still be the uniquebest-performing policy. If, instead, B(s) is empty, countries currently choos-ing s ′ continue to do so after revision and all other revising countries chooses ′ with positive probability, resulting in a state analogous to s where we canrepeat these arguments as long as some countries still choose s ′′. In any case,the state resulting after revision from s could have been reached directly withall deviating and revising countries choosing s ′ in the first place.

Henceforth, we focus on simultaneous deviations from states in E withat least two countries choosing the same alternative subsidy. The rest of theproof uses the radius and coradius technique in Ellison (2000). We let theprocess start at some s = (s, . . . , s) ∈ E k in any t and denote s ′ ∈ �n the re-sulting state after deviation.

(1) Suppose s ′ has k ′ countries choosing s ′ > s with s(k ′ − 1) < s ′ ≤˜s(k ′) and k ′ ≥ k. Deviating countries get payoff f (k ′, s ′) ≥ u (wR)


by definition of s(k ′) and offer the highest subsidy. All other coun-tries get u (wR). Thus, B(s ′(t + 1)) = {s ′}.

(2) Take s �= s(k − 1) and suppose s ′ has n − k + 1 countries choos-ing s ′ < s . Deviating countries get u (wR) and the remaining k − 1get f (k − 1, s) < u (wR). Thus, B(s ′(t + 1)) = ∅ and B(s ′(t + 1)) ={s ′}. The same holds with n − k + 2 deviations for s = s(k − 1).

In (i) and (ii) there is positive probability that all countries revise theirsubsidies and choose s ′ at the end of t + 1, reaching state s ′.

(1) If k < k ′ countries coordinate on s ′ > s with s ′ ∈ (s(k ′ − 1), s(k ′)

],

their payoff is f (k, s ′) < u(wR), implying B(s ′(t + 1)) = {s} andB(s ′(t + 1)) = ∅. Nondeviators keep their strategies if they arecalled to revise. Deviators go back to s at the first revision oppor-tunity. Analogously, if k < n − k + 1 countries choose s ′ < s and s �=s(k − 1), deviators get u(wR) and nondeviators offer higher subsidyand get f (n − k, s) ≥ u(wR), since n − k ≥ k. Thus, B(s ′(t + 1)) ={s}. The latter also holds if k < n − k + 2 and s = s(k − 1).

In (iii) the process will not exit state s . Denote R(s) the radius of s , whichcorresponds to the minimum number of simultaneous deviations neededto exit the basin of attraction of s . Since s(k ′) increases with k ′, it followsfrom (i) and (iii) that the minimum number of deviations needed to exits ∈ E k \ s k upwards is k ′ = k for s < s ′ ≤ s(k) and k + 1 for s k . From (ii) and(iii) at least n − k + 1 deviations are needed to exit states in E k \ s k−1 down-wards and n − k + 2 for s k−1. It follows that R(s k) = min{k + 1, n − k + 1} forall k. Furthermore, for any s = (s, . . . , s) with s ∈ (s(k − 1), s(k)), R(s) =min{k, n − k + 1}. These together imply that

(1) for any s k = (s(k), . . . , s(k)), k = 1, . . . , n − 1, we have

R(s k) ={

k + 1 if k ≤ n/2n − k + 1 if k > n/2;

(A1)

(2) for any s = (s, . . . , s) with s(k − 1) < s < s(k), k = 2, . . . , n − 1, wehave

R(s) ={

k if k ≤ n+12

n − k + 1 if k > n+12 .

(A2)

Observe in (A1) and (A2) that R(s) is highest for intermediate val-ues of k. More precisely, the maximum radius is n

2 + 1 attained at sn2 ,

if n is even, and n+12 attained at states in E

n+12 , if n is odd. This gives

the cost of leaving central states. We now focus on the cost of reachingthem. Denote CR∗(s) the so-called modified coradius of s , which gives thehighest minimum of cost of reaching s from any other state discounting


the costs of leaving potential intermediate absorbing states along apath.

The case of even n. From (i) and (iii) we have that at least n/2 simultane-ous deviations are needed to reach s

n2 directly from states s = (s, . . . , s) with

s < s(n/2). Alternatively, from s ∈ E k \ {s k} with k ≤ n/2, we can constructa path of the form (s1, s2, . . . , sT ) from s1 = s to sT = s

n2 going through ab-

sorbing states s k, s k+1, . . . , sn2 −1 consecutively, requiring less than n/2 devi-

ations at each step along the path. Denote c(st , st+1) the minimum num-ber of simultaneous deviations needed for a transition from st to st+1. ThenC∗(s, s

n2 ) = ∑T−1

t=1 c(st , st+1) − ∑T−1t=2 R(st ) = k ≤ n/2 is the so-called modi-

fied cost of the transition from s to sn2 , which gives the minimum cost of

setting the transition in motion. There is no other path leading from s to sn2

requiring less deviations at each step and C∗(s, sn2 ) is highest from states

s ∈ En2 \ {s n

2 }. On the other hand, by (ii) and (iii), we know that reach-ing s

n2 from above, that is, from states s ∈ E k \ {s k−1} with k > n/2, can be

done directly with n − k + 1 deviations, or with a path through absorbingstates s k−1, s k−2, . . . , s

n2 +1 at modified cost also equal to n − k + 1 (step-by-

step transitions do not reduce costs in this case). This cost is highest for s ∈E

n2 +1 \ {s n

2 }. It follows that CR∗(sn2 ) = maxs C∗(s, s

n2 ) = n

2 < n2 + 1 = R(s

n2 ).

By Ellison (2000, Theorem 2), sn2 is the only stochastically stable state in this

case.

The case of odd n. We now focus on the set En+1

2 = E n2 �. We know from

(A1) and (A2) that R(s) = n+12 for all s ∈ E n

2 �. Let us first focus on state sn−1

2

in this set. We can follow the same arguments used for sn2 in the even case

to see that C∗(s, sn−1

2 ) = k for all s ∈ E k \ {s k} with k ≤ n−12 and C∗(s, s

n−12 ) =

n − k + 1 for all s ∈ E k \ {s k−1} with k > n+12 . This implies

C∗(s, sn−1

2 ) ≤ n − 12

<n + 1

2= R(s

n−12 ) for all s �∈ E n

2 �. (A3)

Finally, for any pair s, s ′ ∈ E n2 �, we have c(s, s ′) = n+1

2 . Take for examples ′ > s ; if the process starts at s , we need at least n+1

2 countries coordinatingto sustain s ′; if the process starts at s ′, we need at least n+1

2 countries loweringtheir subsidies to render s ′ unsustainable for nondeviators. Furthermore, itis not possible to reduce the costs of a transition between s and s ′ by a chainof transitions going through states out of the set E n

2 �; for s ∈ E n2 � we have

c(s ′′, s) ≥ n+32 for all s ′′ > s( n+1

2 ) and c(s ′′, s) = n+12 for all s ′′ < s( n−1

2 ). It

follows that C∗(s, s ′) = n+12 for all s, s ′ ∈ E n

2 �. It follows that

CR∗(s) = maxs ′

C∗(s ′, s) = n + 12

= R(s) for all s ∈ E n2 �. (A4)

We can now use Ellison (2000, Theorem 3). First, (A3) implies that for allstates s �∈ E n

2 � we must have μ∗(s) = 0. Thus, the set of stochastically stable


states, which always exist, must be contained in E n2 �. Let s ∈ E n

2 � be suchthat μ∗(s) > 0. Then (A4) implies that μ∗(s ′) > 0 for all s ′ ∈ E n

2 �. �

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