Dynamic club formation with coordination

We present a dynamic model of club formation in a society of identicalpeople. Coalitions consisting of members of the same club can form for oneperiod and coalition members can jointly deviate. The dynamic process isdescribed by a Markov chain defined by myopic optimization on the part ofcoalitions. We define a Nash club equilibrium (NCE) as a strategy profilethat is immune to such coalitional deviations. For single-peakedpreferences, we show that, if one exists, the process will converge to a NCEprofile with probability one. NCE is unique up to a renaming of players andlocations. Further, NCE corresponds to strong Nash equilibrium in the clubformation game. Finally, we deal with the case where NCE fails to exist.When the population size is not an integer multiple of an optimal club size,there may be `left over' players who prevent the process from `settlingdown'. To treat this case, we define the concept of $k$ -remainder NCE, which requires that all but $k$ players are playing a Nash clubequilibrium, where $k$ is defined by the minimal number of left overplayers. We show that the process converges to an ergodic NCE, that is, aset of states consisting only of $k$ -remainder NCE and provide somecharacterization of the set of ergodic NCE.


(Communicated by Ahmet Alkan)
Abstract. We present a dynamic model of club formation in a society of identical people. Coalitions consisting of members of the same club can form for one period and coalition members can jointly deviate. The dynamic process is described by a Markov chain defined by myopic optimization on the part of coalitions. We define a Nash club equilibrium (NCE) as a strategy profile that is immune to such coalitional deviations. For single-peaked preferences, we show that, if one exists, the process will converge to a NCE profile with probability one. NCE is unique up to a renaming of players and locations. Further, NCE corresponds to strong Nash equilibrium in the club formation game. Finally, we deal with the case where NCE fails to exist. When the population size is not an integer multiple of an optimal club size, there may be 'left over' players who prevent the process from 'settling down'. To treat this case, we define the concept of k-remainder NCE, which requires that all but k players are playing a Nash club equilibrium, where k is defined by the minimal number of left over players. We show that the process converges to an ergodic NCE, that is, a set of states consisting only of k-remainder NCE and provide some characterization of the set of ergodic NCE.

Introduction.
We provide a dynamic model of club formation motivated by a model of an economy with local public goods. In interpretation, those individuals sharing the same location form a club in order share a facility or to provide a local public good for themselves exclusively. We assume that the public good is financed by a poll tax, or in other words, equal cost sharing on the part of the members of the club. An individual's utility depends on his consumption of a private good, the public good, and on the size of the club; that is, we consider anonymous crowding -people care about the number of members of their club, but not about their identities, and there may be congestion. Examples are the choice of a leisure club, 342 TONE ARNOLD AND MYRNA WOODERS hospital, or restaurant. Our model reduces to a congestion game, as in Rosenthal (1973) [26], where each player's strategy set is the set of all locations, or clubs (each club is identified with its location), and each player's payoff is a function of the number of players choosing the same strategy, i.e. location, as himself.
In the presence of congestion effects, an increase in the number of members of a club has two opposing effects on the members' utilities: On the one hand, cost shares are diminished; on the other hand, congestion may be exacerbated. Thus there is a trade-off between cost sharing and crowding effects. Note, however, as in the literature on local public good economies with anonymous crowding, crowding effects are not necessarily negative. For instance, there might be positive externalities in consumption, or fashion effects. Finally, crowding effects might be both positive and negative over different ranges of the club size. In any of these cases, an player's marginal utility from an increase in club size is increasing up to a point, the 'agent optimal' club size, and then decreasing. Such models now have a long history going back to Buchanan (1965) [7]. For our main results we assume that preferences are single-peaked over club sizes. For our dynamic model of club formation we assume that players who are members of the same club can communicate costlessly. While our results do not depend on this assumption, we view it as intuitive and modelling a situation where members of a club can relatively easily talk to each other. Indeed, one purpose of a club may be for the individual members of the club to be able to interact with each other and coordination of strategies is one form of interaction. A viable coalition consists of a subset of the members of a club who can improve upon a strategy profile for themselves by moving to other locations. We allow for coalitional deviations only by groups of players within a club. Our motivation is that members of the same club may communicate easily with each other; for example members of a fitness club might chat around the water fountain. A subgroup of the set of members of a club may form a coalition for one period, and deviate jointly to some other strategy, eg. join another club, move to an unoccupied location, or distribute themselves across different locations or different existing clubs. In the next period, new coalitions may be formed. In each period, each coalition's opportunity to move arises at random.
Our model is truly dynamic: At each time step, a probability distribution determines the state, i.e. the strategy vector, for the next period, where the transition probabilities are derived from the myopic best-reply rules together with the random opportunities of strategy revision. 1 We call a vector of strategy choices that is immune to improving deviations by coalitions contained in any club a Nash club equilibrium (NCE).
If the optimal coalition size, say s * -the club size that maximizes per capita utility of the club membership -is not an integer divisor of the population size then, provided that the population is larger than the optimal club size, a Nash club equilibrium may fail to exist because some players will be 'left over'. This problem often arises in the literature on club economies and more generally in cooperative games. To treat the problem, some authors consider approximate equilibria that allow for leftover players. 2 The idea is intuitively that 'most' players will be in a stable state and a few left-overs can be ignored. 3 This idea, however, has not been given a rigorous formulation, which is one of the main aims of this paper. For this purpose, we first introduce the concept of a k-remainder Nash club equilibrium, a strategy profile with the following property: k players can be removed from the population in such a way that the remaining n − k players are playing a NCE (on the reduced strategy space), and k is the minimal number of left over players. Note that a k-remainder equilibrium, for k = 0, is a NCE. We then introduce an ergodic NCE as a subset of k-remainder NCE with the property that the all states in that set can be reached by the dynamic process and once a state in the ergodic NCE set is reached, the process will never leave the set. We demonstrate: Existence of a k-remainder Nash club equilibrium and nonemptiness of the set of ergodic MCE, and Convergence of the dynamic process to ergodic NCE. Our approach emphasizes the mobility aspect of the model, and is based on the assumption of myopic optimization on the part of the players. For our model and results, myopic optimization seems particularly appropriate. Indeed, since the membership of a club cannot prevent others from entering, long-run optimization can serve no long run purpose.
The remainder of the paper is organized as follows. The subsection below provides some relationships to the literature. The next section presents the formal framework of our model, i.e. the stage game. Section 3 describes coalitional deviations and the myopic adaptation process on the part of the players and defines Nash club equilibrium (NCE). Section 4 defines the dynamic finite Markov chain, while sections 5 and 6 deal with existence and efficiency of NCE, respectively. Convergence of the Markov chain is analyzed in section 7. Finally, we deal with the problem that NCE might not exist. We introduce the concept of ergodic NCE in section 8 and prove convergence of our dynamic process to the set of ergodic NCE. The last section concludes.
1.1. Related literature. Much of our motivation for this paper comes from the literature on economies with local public goods and clubs (and jurisdictions, firms, coalitions, for example). The Tiebout Hypothesis, which dates from Tiebout (1956) [31], is the claim that when public goods are local -subject to congestion and exclusion -then 'large' economies are 'market-like'. Recall that the core of an economy consists of those states of the economy that are immune to coalitional deviations. The Tiebout Hypothesis was formulated by Wooders (1978Wooders ( ,1980 [32,33] as the hypothesis that, when public goods are local, then approximate cores are nonempty, equilibria or approximate equilibria exist, and, as the economy grows large, the approximations can become arbitrarily 'close'. Moreover, with appropriate notions of equilibrium, approximate cores converge to equilibrium outcomes; see Allouch and Wooders (2008) [2] and references therein. The fundamental difficulty is that when public goods are local, even if the economy is 'well-behaved', the core may be empty. 4 In cooperative games, which provide abstract models of economies, it becomes clear that the problem of the emptiness of the core in club economies is that there may be leftover players; in games with sufficiently many players if a relatively small set of players were removed from the player set the resulting game would have a non-empty core; see Shubik (1971) [28] for an early example and Kovalenkov and Wooders (2003) [20] where the results underscore the generality of the problem. The intuition for the current paper comes from consideration of club economies and the left-over problem in these situations.
A static version of our model fits into the class of congestion models of Rosenthal (1973) [26], who showed existence of a pure strategy Nash equilibrium. Our model is also a simple version of the local public good games analyzed by, among others, Weber (1997a, 1998) [16,19]. The existence of pure strategy Nash equilibria of such local public good games has been shown, for various specifications of the game, by several authors. While Konishi et al. (1997a) [16], and, for an even more general model including external effects of group formation on non-members Hollard (2000) [12], prove existence of equilibrium in the general case, Holzman and Law-Yone (1997) [13], Konishi et al. (1997b) [17], and Milchtaich (1996) [21] are concerned with the special case of congestion games, where each player's payoff is non-increasing in the number of players choosing the same strategy. The latter two articles also provide conditions for the existence of strong Nash equilibria. Conley and Konishi (2000) [8] consider a simplified model, as in the current paper, and a variant of the free mobility equilibrium; as other papers in the literature, they require that the economy be sufficiently large to obtain existence of equilibrium. In all these papers, in equilibrium the partition derived from the players' strategy choices is stable against unilateral deviations by individuals. Page and Wooders (2007) [24] also treats the same basic model as in the current paper using a concept of farsighted stability.
In contrast to the literature noted above, we model player mobility explicitly: We provide a dynamic model where the club formation game is played by myopically optimizing players, who can move between existing clubs (or if an empty location is available, create their own) at each step in time. Also, we take up an issue often addressed in the context of club formation, namely the possibility of coordinated action on the part of a group of players. For instance, if a club becomes too crowded, a subset of its members might decide to move to another club or to jointly open a new club, even if every single player would not want to do that if he were on his own. To ensure stability against such coordinated deviations, we analyze equilibria that are immune to joint deviations by groups of players.
The problems of emptiness of the core and nonexistence of Nash equilibrium in club economies are pervasive. Part of the beauty of two-sided matching modelssee Roth and Sotomayor (1990) [27] for a review -is that, under mild conditions, equilibria exist. Unfortunately, once one leaves the matching framework, even introducing same-gender matchings, the nice properties of the two-sided matching disappear. See for example Fagebaume, Gale and Sotomayor (2010) [11] and Demange, Gale and Sotomayor (1986) [10]. 5 Nevertheless, if the games have many players and effective coalitions are small, then approximate equilibria exist under mild conditions. These equilibria frequently involve a set of "left-over" players, players who cannot be fitted into any "optimal" coalition. Our work in this paper provides a formal foundation for this motivation in a simple setting.
2. The basic model. While our model will reduce to a congestion game, since our motivation comes from club economies and economies with local public goods, we begin with that framework. We consider a finite set N = {1, ..., n} of individuals, or players. Each player can choose a location from a finite set 6 G = {a, b, ..., m}. Individuals choosing the same location form a club in order to provide a public good for themselves exclusively, or to use a common facility and share costs equally. The cost of the facility is exogenously given. Since there is no danger of confusion, we will identify a club with the location at which it is formed, e.g. club a is the name of the club formed at location a ∈ G.
A player's strategy is his choice of a club. A strategy profile is thus a vector g = (g 1 , ..., g n ) ∈ G n , indicating a club (location) for each player. We consider only pure strategies. Note that a strategy profile induces a partition of players into clubs, a club structure. Since our work was motivated by the study of club economies (also known as Tiebout economies), we first motivate our work with the model as one of an economy with local public goods.
Each person's utility depends on the size of his club, that is, the number of players choosing the same club. Formally, this crowding effect is captured by a function h : G × N → R, where h(a, s) is the (dis)utility to a member of club a when the total number of members (himself included) is s. We design a non-cooperative game Γ = {N, G, (u i ) i∈N } where N is the set of players, G is the common strategy set -each player can choose any one of the possible locations -and u i : G n → R is player i's payoff function. Such a game is commonly called a congestion game.
For any given strategy profile g = (g 1 , ..., g n ), let n a (g) denote the number of players choosing strategy a ∈ G, and let c(s) denote the cost of providing the optimal amount of public good for s club members. The payoff to player i playing strategy g i = a in strategy profile g is then given by the indirect utility function: c(n a (g)) n a (g) + h(a, n a (g)), where v(a) denotes the player's utility derived from the local public good.
A Nash equilibrium of Γ is a strategy profile g with the property that, for each i ∈ N and all b ∈ G , it holds that for all i ∈ N and for all b ∈ G,where a is the strategy adopted by player i in strategy profile g.
the condition that every "balanced" collection of admissible coalitions contains a partition, there are few applications of their conditions. 6 If |G| is large relative to the set of players, this assumption is not at all restrictive. Further, we are able to analyse the interesting case of |G| being small, so our model is in fact richer than one with an unlimited set of locations. Proposition 1 (Rosenthal, 1973). The game Γadmits a Nash equilibrium in pure strategies.

Coordination of coalitions within clubs.
We now turn to coalitional deviations and Nash club equilibrium. In each period, the strategy choices of that period induce a partition of the set of agents into clubs. Given this partition, we assume that the only admissible coalitions consist of players within the same club. These coalitions last for one period. A coalition is thus a subset of the set of members of a club. Note that the set of admissible coalitions may change from period to period. If a coalition is formed, its members will jointly decide which location each of them will choose. That is, coalition members may jointly deviate to another location, or the members may distribute themselves across different locations. A coalition will form whenever it is in the best interest of every single coalition member to do so. That is, each member must strictly benefit from forming the coalition and, for each member, there must be no other coalition (within the same club) that yields a higher payoff to that player. Such coalitions will be called viable, which will be defined formally below. In the next period, all coalitions dissolve, and new ones can be formed. Formally: Given any strategy profile g, define the resulting partition of the player set by where N a (g) denotes the set of all players choosing location a under the strategy profile g. Definition 1. Given any strategy profile g, a coalition C is a nonempty subset of a club induced by g, i.e. C ⊂ N a (g) for any N a (g) ∈ N (g).

Myopic strategy choice.
For the remainder of the paper, we assume that all locations are identical. Thus the utility of membership in a club depends only on the number of members of the club. We can therefore express utility as a function of club size. Let a be the location of player i in strategy profile g. Define That is, the payoff to a player i who is a member of club a under any strategy profile g with n a (g) = s will realize the payoff u(s). Further we assume that preferences are single peaked. 7 This implies that, from the point of view of an individual player, there exists an optimal club size, which may be any number between 1 (singleton clubs) and n (a grand club). Denote this number by s * . 8 Note that our restriction that coalitions to be contained in clubs makes our result stronger; it would be easier if we allowed any subset of players to form a coalition. Also, we will require that if members of a coalition decide to move to other locations, then no member of the coalition can remain in his current location.
Finally, we use the term potential coalition to refer to any nonempty subset of the total player set. Given a club structure the only admissible coalitions are those that are subsets of the membership of some club.
Assumption. Preferences for clubs are single peaked over club sizes. That is, we assume that there exists an integer s * ∈ {1, ..., n} such that 1. for any clubs a, b with n a < n b ≤ s * we have u(n a ) < u(n b ), and 2. for any clubs a, b with n a > n b ≥ s * we have u(n a ) < u(n b ).
We now turn to the dynamic adaptation process. 9 Time is divided into discrete periods t = 0, 1, 2, .... In the initial period t = 0, we start with an arbitrary strategy profile g ∈ G n . 10 Each period, each player receives a payoff, determined by the strategy choices of all players in that period. This payoff depends on the club size.
In any period t, given any strategy profile g and the resulting club structure N (g), the adaptation process consists of the following steps.
1. For every club a induced by the strategy profile g, members of a can form coalitions. Given any strategy profile g, a coalition C is called viable if (i) it forms a subset of a given club, i.e. C ⊂ N a (g); (ii) there exists a strategy profile y = (y C , g− C ) such that y i = g i and u i (y) > u i (g) for all i ∈ C and (iii) for any i ∈ C, there is no other coalition C containing i and strategy profile x = (x C , g− C ) such that i gets a higher payoff in C than in C, that is, a payoff u i (x) > u i (y). 11 Note that there may be more than one viable coalition within each club and viable coalitions may overlap. Also, viable coalitions may consist of individual players. 2. In each period, one potential coalition (not necessarily admissible nor viable) 12 is picked at random and then gets the opportunity to revise its strategy. 3. If any coalition gets the opportunity to revise, it will do so if and only if the coalition is viable. A viable coalition, if picked, will revise its strategy so that the coalition members will receive their highest possible payoffs in the next period, i. e. they chose a best reply to the current strategy configuration. If there is more than one best reply, the coalition randomizes, placing strictly positive probability on each. 4. Coalitions that do not get the chance to move will remain unchanged.
Note that we do not explicitly model the formation of viable coalitions within clubs, nor procedures for arriving at joint strategy profiles. We assume that a coalition forms if it is in the interest of all its members, i.e. if they can coordinate their strategies in such a way that each member's payoff will be increased. Also note that in our model, the formation of both coalitions (for one period) and clubs is always reversible, although it may take some time.
The aim of our definition of a viable coalition is to require that when the members of a coalition move to other locations, they are 'doing their best' and that all the coalition members actually move. The following example illustrates viable coalitions and a subtlety of the definition. 13 : Example. There are 8 players, 3 locations, s * = 2. and the initial club sizes are 4, 2, 2 at locations a, b, c respectively. Clearly there exists profitable moves by members of club a. Let the members of a be i j, k, and . We claim that any singleton set {i} is a viable coalition. Player i could change his strategy to b and be better off. Clearly conditions (i) and (ii) of the definition of a viable coalition are satisfied. But is condition (iii)? Observe that if {i, j.k. } form a coalition, j moves to b and k moves to c then i is even better off, since he would then be in a club of size two. But i has not changed his location. Without the restriction that y i = g i there would not exist a viable coalition. It follows that a viable coalition could not have three members. A viable coalition, though, could have two members, say i and j who could change their locations to b and c respectively.
We define a Nash club equilibrium as a strategy profile that is stable against deviations by coalitions, that is, in a Nash club equilibrium, no viable coalition exists.

Definition 2.
A strategy profile g is a Nash club equilibrium (NCE) if there exists no viable coalition. A NCE club structure is the partition of the population induced by a NCE.
Remark. We could equivalently define a NCE as a strategy profile g satisfying the property that there is no club a ∈ G, no coalition C ⊂ N a (g), and no strategy profile y = (y C , g− C ) such that u i (y) > u i (g) for all i ∈ C. In other words, there is no coalition that would want to deviate if it were given the opportunity.
4. The dynamics. The myopic best-reply rules together with the stochastic opportunities for strategy revision on the part of coalitions define a Markov chain on the finite state space G n . A state of the system is a strategy profile, i.e. a strategy (choice of location) for each player. Note that a state of the system induces a partition of the set of players into clubs. The transition probabilities between states are determined by the best-reply rules and the fact that each coalition's opportunity to revise its strategy arises at random.
Observe that, once a NCE profile is reached, no player or coalition will switch clubs. An NCE profile is thus an absorbing state of the process, i.e. a state that cannot be left again once it has been entered. That is, once the process has reached a NCE, it will 'settle down' in that state forever. Conversely, any strategy profile that is not NCE cannot be an absorbing state, since at least one viable coalition will exist and will gain by deviating when it gets the chance to do so, which will happen with positive probability.

Observation.
A strategy profile is NCE if and only if it is an absorbing state of the Markov process.
The above observation, however, does not ensure convergence of the process to an absorbing state. Instead, the process may get trapped in a set of states, and perpetually oscillate between these states. We will show that this is not the case. First, however, we will deal with existence of NCE.
5. Existence of a Nash club equilibrium. The existence of NCE depends on the relationship between several parameters of the model. These are the size of the population n, the optimal club size s * , and the number of locations |G|. First note that if s * ≥ n, there is a unique NCE club structure, namely the grand coalition. This is unique up to a relabelling of locations. In what follows, we focus on the case of s * < n. Three cases have to be considered: 1. If |G| ≥ n/s * and n/s * is an integer, NCE exists. A NCE club structure consists of n/s * clubs of size s * . 2. If |G| > n/s * and n/s * is not an integer, a NCE might not exist.
Example. Let n = 10, |G| = 5, s * = 3, and assume u(1) < u (4). In this example, no NCE exists: If the players form three clubs of size 3, there will be one left over player who can gain by joining any of the three clubs. If there are two clubs of size 3 and one club of size 4, a coalition of any three members of the latter can gain by jointly deviating to an unoccupied location. (Note that if u(1) ≥ u(4), then a NCE does exist.) 3. If |G| < n/s * , a NCE exists. In a NCE club structure, all players are distributed as evenly as possible across all locations, and location size is given by integers n a (g * ) satisfying where n/|G| is the largest integer weakly smaller than n/|G| and n/|G| is the smallest integer weakly larger than n/|G|. Note that if n/|G| is an integer then n/|G| = n/|G . : Example 1. Existence of NCE with 'few' locations. Let n = 100, s * = 10, |G| = 7. A NCE club structure consists of five clubs of size 14 and two clubs of size 15. The optimal club size cannot be reached in NCE in this example. The reason is that, even though any coalition of ten players would prefer to jointly deviate to a new club, this is impossible because there are no unoccupied locations. Clearly a member of a 15 person club cannot benefit from moving to a 14 person club. 14 In case 2. above, NCE exists under the condition stated in the following Proposition.

Proposition 2.
Existence of NCE for the case |G| ≥ n s * and n = rs * + where r and are positive integers with < s * . Then a NCE exists if and only if u( ) ≥ u(s * +1). Moreover, if u( ) ≥ u(s * + 1) and g is a NCE, then the induced club structure will have r clubs of size s * and 1 club of size .
Proof. See appendix.
Note that, if a NCE exists, there are multiple NCE that differ only with respect to the names of players and locations. That is, the club structure induced by NCE is uniquely characterized by the number of clubs and the sizes of their memberships as follows: • If |G| ≥ n s * and s * divides n, then all nonempty clubs are of size s * . • If |G| > n s * but s * does not divide n, i.e. n = rs * + , 0 < < s * , and u( ) ≥ u(s * + 1), there are r clubs of size s * and one club of size .
• If |G| < n s * , players are distributed across clubs as evenly as possible, i.e. each club is either of size n G or of size n G . This gives rise to the following observation.
Observation. If a NCE exists, it will be unique up to a relabelling of players and locations.
6. Efficiency of Nash club equilibrium. We will next show that in our model a NCE is a strong Nash equilibrium profile, i.e. no group of players (not even from different clubs) could gain by jointly deviating.
Formally, a strategy profile g is a strong Nash equilibrium if for every coalition S ⊂ N and all strategy profiles y S = (y i : i ∈ S) for the members S, there does not exist a strategy profile (y S , g −S ) such that, for all i ∈ S such that It is obvious that every strong Nash equilibrium is a NCE. The converse is also true, as stated in the following proposition.

Proposition 3.
A Nash club equilibrium of the game is a strong Nash equilibrium.
Proof. See appendix.
7. Convergence to a Nash club equilibrium. We will now show that if a NCE exists the adaptation process will converge to a NCE profile with probability one. We provide an algorithm describing a path of moves of viable coalitions that terminates in a NCE.
There is one situation that is slightly more delicate; this is the case where the NCE club size is s * (or s * and for some < s * ) and where u( ) < u(s * + 1) for some positive greater than one. For simplicity, to discuss this case, we suppose that the NCE club size is unique and equals s * = 4. Now suppose that |G| = 5 and the size of the population n is eight (n = 8). (Convergence would be quicker if there were fewer locations, subject still to the condition that |G| ≥ |N |/s * ). Let us also suppose that u(2) < u(3) < u(s * + 1). It is clear that a person in a one-person club will prefer to move to a club with s * + 1 members rather than to another club containing only one member. To illustrate the treatment of this situation, we describe a path by a series of lists where the k th number in a list represents the number of people in the k th club, k = a, ..., d. The subtlety is to first allow a viable coalition contained in one of the smallest clubs to move to a location containing s * members and then to allow a viable coalition contained in the club with s * + 1 members to move to the largest club with fewer than s * members to create a club with s * members. To take account of situations -such as that illustrated by the penultimate state 4, 3, 1, 0, 0during each 'loop' in the procedure, there is a positive probability that some viable coalition moves to a location with fewer than s * members.
The situation above is slightly different if we have u(2) < u(s * + 1) but u(3) > u(s * + 1). In this case, the following list illustrates a path terminating in a NCE. Proof . Suppose a NCE exists and let s * * denote a club size induced by NCE. Observe that if |G| ≥ n s * then, in the case that s * divides n, we have s * * = s * and otherwise s * * ∈ {s * , } where n = rs * + , r and are positive integers, < s * , and if |G| < n/s * then s * * ∈ { n |G| , n |G| }. We split the proof into three cases: Case (A) |G| ≥ n s * and s * divides n. Case (B) n = rs * + , r and are positive integers, < s * .
Case (C) |G| < n/s * and s * * ∈ { n |G| , n |G| }. Starting from a state that is not a NCE, we construct a path, i.e. a sequence of states with positive transition probabilities, that terminates in a NCE. This shows that any state that is not a NCE must be transient, which implies that it cannot be absorbing.

Case (A).
Consider a state g that is not NCE. Then there must be clubs of nonoptimal size. There are two mutually exclusive possibilities: either (a) there is a club a with n a (g) > s * or (b) all clubs are of size s * or less, with some clubs being strictly smaller than s * . Our approach is to first give coalitions contained in clubs with more than s * members an opportunity to move until there are no clubs of size greater than s * . We then consider viable coalitions smaller than s * and give all of them (one after the other) the opportunity to move, where their movement will not result in coalitions of size greater than s * . Finally, we deal with the situation where the only viable coalitions are those whose optimal move is to join a coalition with s * members. Here, as illustrated in the examples above, we mix the movement of singletons to clubs of size s * with movements of viable coalitions contained in clubs of size greater than s * to join smaller clubs until we arrive at an equilibrium.
Step 1. Suppose there is a club a with n a (g) > s * . This implies that there exists a location b with n b (g) < s * (including the case of n b = 0). Thus, any coalition C ⊂ N a (g) with s * − n b (g) players is viable since n b (g) + |C| = s * and s * maximizes per capita utility. Suppose one such coalition gets the chance to move, which happens with positive probability. Let g denote the state after the move of 352 TONE ARNOLD AND MYRNA WOODERS C. Note that |{d ∈ G : n g (d) = s * }| > |{d ∈ G : n g (d) = s * }|, i.e. the number of clubs of optimal size is increased. Repeat this argument until all club sizes are equal to or smaller than s * . If all clubs are now of size s * we are done. Otherwise, we continue by next treating cases where no clubs are larger than s * and some are smaller.
Let S * (g 1 ) denote the set of clubs of size s * in state g 1 at the conclusion of Step 1.
Step 2. We first need to consider the possibility that S * (g 1 ) = ∅. For this case letŜ(g 1 ) denote the set of clubs of maximal size, sayŝ. Observe thatŜ(g 1 ) = ∅ and, at this stage in our proof,ŝ < s * . (Equality will hold only if S * (g) = ∅.) In this case, at least one viable coalition's best move is to join a club of sizeŝ so as to form the largest possible club with no more than s * members. Suppose one such viable coalition C gets the chance to move. The movement of C to a club inŜ(g 1 ) will induce a new strategy profile g 2 with a (weakly) increased number of empty locations and with an increase in the size of the membership of at least one nonempty location. Repeating this argument as many times as possible will lead to a situation where eventually there are clubs of size s * and it is not possible for a new club of size s * to be formed by the movement of a viable coalition (with fewer than s * members) to a new location. Let g 3 denote the state at the end of Step 2.
Step 3. Suppose that there remain clubs containing fewer than s * members. There are two possible cases: : (3.1) A viable coalition will move to a location with fewer than s * members to create a club with h ≤ s * members. In this case it must hold that u(h) ≥ u(s * + 1). : (3.2) A viable coalition is a singleton and, if given the chance, will move to a club with s * members.
Step 3a. We next sequentially give all viable coalitions satisfying the conditions of (3.1) the opportunity to move. Let g 4 denote the resulting state. (If there are no viable coalitions satisfying the conditions of (3.1) then g 4 = g 3 ). Note that in the state g 4 , if there are any viable coalitions, they satisfy the conditions of (3.2).
Step 3b. Suppose in state g 4 there are viable coalitions satisfying the conditions of (3.2). (Note that for this to occur it must be the case that s * ≥ 3. Otherwise this procedure would terminate after Step 3a.) Let s 1 denote the size of one of the smallest clubs in the state g 4 and suppose without loss of generality that n a = s 1 . Note that in this case s 1 < s * − 1.
(*) Let one member of N a move to a club with s * members, say to N b . Now N b has s * + 1 members. Let g 5 denote the resulting state.
Next, let N c be the largest club with fewer than s * members in state g 4 . Then there is a viable coalition C contained in N b with |C| + n c = s * . Give such a coalition the opportunity to move. Note that this leaves the same number of clubs of size s * as in g 4 and with an increase in the size of at least one club containing fewer then s * members.
Return to Step 3a. and repeat the procedure until n a = 0. (Note that this is possible since there is at least one club with s * members.) This brings us to a state, say g 5 , with strictly more empty locations, at least one club with s * members, and an increase of at least 1 in the size of the smallest club. (**) Return to Step 3a and repeat the this procedure until it is no longer possible. We must then have reached a NCE.
Case B. Our procedure in this case is basically the same as for Case (A) except that at some point the largest club smaller than s * will contain members. Once such a club exists do not give possible viable coalitions contained in one such club an opportunity to move (except at the end of the procedure described for Case Abut then there will no longer be any viable coalitions).

Case C.
Recall that in this case all clubs in a NCE must have size n G or n G . Let S * denote the set of NCE club sizes and lets denote the maximal club size in S * and let s denote the minimal club size in S * Either s = s or s = s − 1.
Step 1. Suppose that g is not a NCE. Then there must exist at least one club, say N a , with n a > s and another club, say N b , with n b < s. Suppose, without any loss, that n b ≤ n c for all other clubs n c . Either n b < s * or n b ≥ s * . If n b < s * then there is a viable coalition contained in N a , say C, with |C| = s * − n b since C is viable. If n b ≥ s * then there is a viable coalition C ⊂ N a with |C| = 1. Let g 1 denote the strategy profile (y C , g) where y C is the strategy y i = b for the all i ∈ C}. Note that the difference between the largest club and the smallest club sizes has decreased by at least 1. Repeat Step 1 until the largest club has club size has size n G and the smallest has size n G and thus the resulting strategy profile is a NCE.

Ergodic Nash club equilibrium.
Note that under our assumptions if |G| ≤ n/s * a NCE always exists, where all players are distributed evenly across all clubs.
In this section, we focus on the case |G| > n/s * . When the optimal club size is such that n/s * / ∈ I, and |G| > n/s * , a NCE might fail to exist, as we demonstrated by an example in Section 5. Here is another example more convenient for the current purposes. In this game the optimal club size is s * = 2. But at most two clubs of size 2 can be formed. The left over player can then gain by joining any of the two clubs, since this increases their payoff from 1 to 1.66. However, in a club of size 3, any two players can gain by forming a coalition and deviating to an unoccupied location. Thus, no NCE exists.

TONE ARNOLD AND MYRNA WOODERS
The nonexistence of a NCE is due to an indivisibility of optimally sized clubs or, in other words, a 'non-balancedness' problem. 16 We now define a notion of NCE that takes this problem into account. This notion is a stepping-stone towards our concept of an ergodic club equilibrium. Definition 3. We define a strategy profile g as a k-remainder NCE if there exist k players, k ≥ 0, such that, if these players are removed from the population, the strategies of the remaining n − k players will form a NCE (on the reduced strategy set G n−k ) where k = n − s * n/s * .
In the example above, the strategy profiles g = (a, a, b, b, c) and g = (a, a, a, b, b) both form 1-remainder NCE: removing player 5 from g and player 3 from g yields a NCE in both cases. In contrast, the profiles (a, a, a, a, b) and (a, b, c, d, d) are not 1-remainder Nash club equilibria. Proposition 5 characterizes k-remainder NCE for those cases where k > 0.

Proposition 5.
Assume that |G| > n/s * and n = rs * + for some positive integers r and < s * . Also assume that u( ) < u(s * + 1). Then: (i) Any strategy profile g with an induced partition of the set of players into r clubs with no fewer than s * members and with the remaining clubs of size less than or equal to is a k-remainder NCE for k = . 17 (ii) Any strategy profile g with an induced partition of the set of players into fewer than r clubs of size greater than or equal to s * is not a k-remainder equilibrium.
Proof. Let g be a strategy profile satisfying the conditions of part (i) of the Proposition. Since there are r clubs each containing at least s * members we can remove players from these clubs so that there are only s * players remaining in each of these clubs. Also, remove all players from the clubs of size less than s * . The number of players removed is equal to k and the strategy choices of the remaining players constitute a NCE, since each nonempty club now contains s * players.
The proof of part (ii) of the Proposition follows from the observation that it is impossible to remove only agents and have all nonempty clubs of size s * (so that the outcome, restricted to the remaining agents, would be a NCE).
Obviously, if preferences are single peaked, k-remainder NCE always exist. The special case of k = 0 corresponds to the definition of NCE.
For k > 0, a k-remainder NCE is not an absorbing state since it is not a NCE, i.e. there are coalitions that will switch locations when they get the opportunity to adjust their strategies. In the example, for instance, in the strategy profile g * =  (a, a, b, b, c), player 5 would switch to either a or b, and in state g = (a, a, a, b, b), a coalition of players 1 and 2 (or 1 and 3, or 2 and 3) would switch to an unoccupied location. Our main result is to show that, in the long run, only k-remainder NCE will be observed. To this end we need the definition of an ergodic set. 16 The indivisibility problem is that the optimal club is indivisible. This would be solved if there were constant per capita benefits to club formation -in which case clubs containing more than one member would be redundant. An alternative approach, following Wooders (1978) [32], would be to allow a range of optimal club sizes containing two relatively prime integers, for example,s * and s * + 1. Then, since any sufficiently large population size n can be written as the sum of nonnegative integer multiples of s * and s * + 1, for all sufficiently large populations, an NCE would exist. The non-balancedness problem is simply the emptiness of the core. 17 Note that the number of clubs of sizes less than s * may be limited by the number of locations. Indeed, if there is only r + 1 locations there can be at most one club of size less than s * . Definition 4. An ergodic set E ⊂ G N is a set of states such that: (1) each state in E can be reached from every other state in E in a finite number of steps and (2) once the set E is reached, it cannot be left again, i. e. the probability of the system's going from some state g ∈ E to some other state g ∈ E is equal to zero.
Note that an absorbing state is the same as a singleton ergodic set. Also, if E is an ergodic set then there is no proper subset of E satisfying the above conditions, that is, ergodic sets are minimal. An ergodic NCE is then defined as follows.
Definition 5. Given any stage game Γ with optimal club size s * , an ergodic Nash club equilibrium is a set of states M ⊂ G N with the following properties: 1. For k = n − s * n/s * , every state g ∈ M is a k-remainder NCE, and 2. M is an ergodic set.
Obviously, an ergodic NCE is a subset of the set of all k-remainder NCE. Also, for the case of k = 0, a NCE is an ergodic NCE. Proposition 6. Let n = rs * + k where 0 < k < s * and let h ∈ G n be element of an ergodic NCE. Then (i) There is no club a in the induced club structure with n a (g) > s * + k/r and thus; (ii) There are r clubs in the induced club structure with size greater than or equal to s * .
Proof. We first show that any state that contains a club of size larger than s * + k/r will never be visited again once it has been left. Start from a state g with n a (g) > s * + k/r for some club a. According to the first step in the procedure for our prior convergence result eventually, a state will be reached where there are r clubs of size s * and one club of size k. Any such state is a k-remainder NCE. Call this state g . Now, if g were in ergodic NCE set, there would be a path from g to g. We will show that this is not the case. The idea is very simple; in any state g no viable coalition will move to an 'overly crowded' club. In state g , without loss of generality suppose the clubs a 1 , a 2 , ..., a r are of size s * and club a r+1 is of size k (with all other locations, if any, empty). Then, there are exactly k viable coalitions, namely the singleton subsets of club a r+1 . Giving these viable coalitions the chance to move, we will reach a state where either the a (r+1) th location is empty or the players still at the a (r+1) th location can not gain by moving to another location because the utilities of those players still at location a r+1 is at least as large as the utility of players in that locations a 1 , a 2 , ..., a r if one more player were added to each of these locations. Note that the clubs at locations a 1 , a 2 , ..., a r must be nearly equal in size -that is, for any a r , a r , r = r , r , r ∈ {1, ...r} is holds that n a r (g ) − n ar (g ) ≤ 1; otherwise one of the viable coalitions must have made a mistake in not moving to one of its most preferred club; mistakes are not allowed. Thus, after the move of any viable coalition each club will have no more than s * + k/r members.
Let n ar+1 denote the number of players (if any) remaining at location a r+1 . Then the only viable coalitions are of size s * − n ar+1 or of size s * (if there exists an empty location). These coalitions can form at any location with more than s * members and the first one that gets the opportunity will move to the location a r+1 or some empty location to bring the number of players at that location up to s * . Note that any move by any such viable coalition will induce club sizes of no more than s * 356 TONE ARNOLD AND MYRNA WOODERS + k/r . After the move of one such coalition, we are again in a case where one coalition, say a 1 , has fewer than s * players, say n a1 ,and the only viable coalitions are of size s * − n a1 (or s * if there is an empty location). One such coalition, say at location a 2 can move to to a r+1 or an empty location, leaving a 2 with fewer than s * members and so on. We conclude that state g can never be reached again once it has been left. This contradicts g being part of an ergodic set.
To conclude the proof, suppose that the process has reached a state g where there are r clubs of size s * and one club of size k. To reach a state g where there are fewer than r clubs of sizes greater than or equal to s * , at some point in the process some coalition would have to move from a club at least s * members and, by its movement, create a club with fewer than s * members; this also would be a mistake as it could move so as to form a club with s * members.
Before presenting our next proposition we provide examples of two possible 'boundary' scenarios. They are boundary scenarios because each involves only a selected set of moves by viable coalitions and because in the set of ergodic NCE equilibrium, the process may go from one scenario to the other (depending on the number of locations) but does not leave the set of states illustrated in the two scenarios.
: Example 3. Let s * = 4, n = 15 and the number of locations be 6. To ensure that a NCE does not exist, we assume that u(3) < u(s * + 1).
Base case 1. As illustrated, the process can return to the initial state. Each state reached is a k-remainder NCE. The above depicted coalitional moves each leave the smallest number of empty locations. First, the 1-player viable coalitions were given the opportunity to move and then the largest viable coalitions were given an opportunity to move but, when possible and consistent with the rules of the process, all chose to move to nonempty locations. The number of empty locations can be as large as k, as next illustrated.
Base case 2. While we can see that it is possible to go from Base case 1 to Base Case 2, it is not possible to leave these to Cases.
The next example illustrates a case where there is never an empty location and also that not all k-remainder equilibria are ergodic NCE.

Proposition 7.
The adaptation process will converge to an ergodic Nash club equilibrium with probability one as time goes to infinity, no matter where the process starts. Moreover, there exists an ergodic club equilibrium and induced club structure with r clubs of size s * and one of size k.
This implies that, once the process has reached a k-remainder NCE, only kremainder NCE will be observed forever after. Proof. The theory of finite Markov chains states that the process will reach an ergodic set with probability one as time goes to infinity. 18 Given this, the proof proceeds in three steps. First, we show that, from every state that is not a kremainder NCE, there is a path terminating in a k-remainder NCE. Second, we show that once a k-remainder NCE is reached, any other state that can be reached from that state will also be a k-remainder NCE. Third, we show that if g 1 and g 2 are the set, then each of these states can be reached from the other. The first two steps together imply that any state that is not a k-remainder NCE cannot be part of an ergodic set. This in turn implies that any ergodic set contains only states that are k-remainder NCE, and the theory of finite Markov chains ensures that this set will be reached with probability one as time goes to infinity. Let n = rs * + k.
Step 1. The first step of the proof simply follows the procedure in our prior convergence result. Eventually, a state will be reached where there are r clubs of size s * and one club of size k. Any such state is a k-remainder NCE and an ergodic NCE since it can be reached from any other state.
Step 2. Suppose the process has reached a state g with r clubs of size s * and k or fewer clubs of size less than or equal to k. In state g, without loss of generality suppose the clubs a 1 , a 2 , ..., a r are of size s * and club a r+1 is of size k (with all other locations, if any, empty). Then, there are exactly k viable coalitions, namely the singleton subsets of club a r+1 . Give these one-player viable coalitions the chance to move, one after another, until we will reach a state where either the a (r+1) th location is empty or the players still at the a (r+1) th location can not gain by moving to another location because the utilities of those players remaining at location a r+1 is at least as large as the utility of players in that locations a 1 , a 2 , ..., a r if one more player were added to each of these locations (as illustrated in Example 4 above). Note that all states in the process leading to this state are k-remainder NCE. Let the state be g 1 .
Now consider a club a with n a (g 1 ) > s * . Note that, because there are sufficient locations, in state g , any viable coalition is of size s * or less: in any club of size greater than s * a viable coalition contained in that club could move to another location of size less than s * (and possibly zero) to create a club of size s * . Any such move by a viable coalition will lead to another state g which is a k-remainder NCE. Allow all such viable coalitions to move and let the resulting state be g 2 . Again, all states in the process leading to g 2 are k-remainder NCE.
Movement of 'small' viable coalitions to increase the size clubs to be greater than or equal to s * and the movement of viable coalitions to clubs to create clubs of size s * cover all possibilities and, starting from a k-remainder NCE with k-clubs will always lead to another k-remainder equilibrium.
Finally, we need to show that any k-remainder equilibrium g that is in the set of ergodic NCE can be reached from any other k-remainder equilibrium g . But we have shown above that a state with r clubs of size s * and one club of size k can be reached from any state, and in particular from any k-remainder equilibrium that is in the set of ergodic NCE. Let one such state be g * . Now starting with g it holds that g * can be reached through the process and from g * the state g can be reached, which concludes the proof. 9. Concluding discussion. This paper provides a game theoretic model of club formation where player mobility is explicitly modelled by a dynamic process. In each period, the members of any given club may form coalitions, and then choose locations by a myopic best reply rule. We define a Nash club equilibrium (NCE) as a strategy configuration from which no coalition wants to deviate. If a NCE exists, the its induced club structure is unique up to a relabelling of players, and the NCE state is efficient in the sense of strong Nash equilibrium. Further, we show that, over time, our dynamic process defined by the myopic best-reply rules on the part of the coalitions converges to a NCE.
To broaden our existence results in an intuitive way, we define an ergodic Nash club equilibrium as a set of club profiles each of which constitutes a k-remainder NCE, where all but k players are in NCE states, and k is the minimal number of left over players. We show that an ergodic Nash club equilibrium exists and that, as time tends towards infinity, the process will converge to an ergodic Nash club equilibrium with probability one.
Our results are novel and interesting in several ways. First, on the one hand, even though strong Nash equilibrium takes into account deviations by all kinds of coalitions, whereas we allow only for deviations by subsets of the set of members of a club, our concept of NCE is shown to correspond to strong Nash equilibrium, in the context of our model. On the other hand, a NCE (and therefore a strong Nash equilibrium) may fail to exist. But, since non-existence is merely due to problems of 'numbers mismatching', this problem can be solved by our notion of a k-remainder Nash club equilibrium, which always exists.
Further, our equilibrium notions can be related to concepts of cooperative game theory, as employed in the literature on coalition formation in hedonic games, e.g. Bogomolnaia and Jackson (2002) [6]. Their concept of Nash stability corresponds to a Nash equilibrium of our club formation game, i.e. a strategy profile (player partition) that is immune to deviations by single players. Bogomolnaia and Jackson prove existence of a Nash stable partition of the population for the case of hedonic preferences, where a player's utility depends only on the members of a club but not on the location itself. The result on the existence of Nash equilibrium in the case of location dependent preferences can be seen as a corollary: If individuals are identical, and their preferences depend only on the club size, then a Nash equilibrium, and therefore a Nash stable partition, exists.
Moreover, the cooperative notion of a core stable partition corresponds to a strong Nash equilibrium of our game: No group of players can gain by deviating. Since our concept of NCE is weaker than that of a strong Nash equilibrium, the set of strong Nash equilibria (and thus the set of core stable partitions) is a subset of the set of NCE. Thus, we have the following relationship between cooperative and non-cooperative concepts: Strong Nash equilibrium (core stability) =⇒ NCE =⇒ Nash equilibrium (Nash stability). Second, our model is dynamic. In each period, new coalitions may form, and new locations can be chosen. This reflects player mobility, or the agents' "voting with their feet". Finally, even though players are myopic instead of farsighted (as in Page and Wooders (2007) [24]), equilibrium club profiles will be reached in the long run. Thus, being myopic may "help" a population to reach a desirable outcome.
It is easy to see that, if the number of locations is sufficiently large, then replicating both the player set and the set of locations will lead to a situation where the percentage of left-over players becomes small. Were transfer payments allowed, each agent in an optimal club could be charged a small fee (in the nature of unemployment insurance) and the totality of these fees could compensate the left over players, which would yield outcomes satisfying another notion of approximate stability. Similar ideas have appeared in game theory (c.f. Shubik, 1971[28]; Kovalenkov and Wooders 2003[20] among others) and in club theory (c.f, Wooders, 1980[33]; Allouch and Wooders 2008[2], among others). We do not pursue this further here.
There are a number of interesting ways in which this investigation could be continued; most important, perhaps, is extension of the results to multiple types of players where there are complementarities between types of players and optimal coalitions may mix types. Research on this situation is in progress.