Yann Rébillé
email: yann.rebille@univ-nantes.fr

Lionel Richefort
email: lionel.richefort@univ-nantes.fr

Document de Travail

Keywords: network, strategic complements, equilibrium, efficiency, social tragedy. JEL: A14, C72, D85

We model agents in a network game of strategic complements and negative externalities. Sufficient conditions for the existence of a unique Nash equilibrium and of a unique social optimum are established. Under these conditions, we find that players with more vulnerable locations in the network exert more effort at equilibrium, and that the most influential players should exert less effort at efficiency. We then find structural conditions under which each player exerts strictly more effort than her efficient level, whether the social optimum be interior or not.

Introduction

Although social dilemmas are situations in which individual and collective interests are in conflict, their outcome depends heavily on the nature of social (or geographic) interactions. By considering the case where a social dilemma is embedded in a directed and weighted network, this present paper engages the classic question of which social (or geographic) structures are conducives to a " social tragedy " and shows that social (or geographic) structure, pure strategy Nash equilibrium (henceforth PSNE) and social optimum (henceforth SO) are related in a mutually interesting way.

Here we consider a simultaneous-move game in which each player is more likely to exert an effort whenever her neighbors are exerting an effort, in other words, we assume that efforts are local complements. In addition, it is assumed that higher levels of effort by neighbors lower a player's payoff, i.e., we consider a game in which efforts generate local negative externalities. Our model is thus a network game of strategic complements and negative externalities. Examples of application include: advertising expenditures by firms that produce brand-differentiated substitute goods; investments in training, in infrastructure or in talents by teams in competition for a professional or sporting challenge; campaign commitments and spendings by candidates in an election; research and educational investments by universities and higher education institutions, etc.

The main result is to establish conditions on the structure of the network for a social tragedy, whether the SO be interior or not. First, we find sufficient conditions under which the PSNE exists and we show that in that case, it is unique and interior. These results can be seen as applications of [START_REF] Kennan | Uniqueness of Positive Fixed Points for Increasing Concave Functions on IR n : An Elementary Result[END_REF]'s uniqueness result to the case of games with local complementarities. Our contributions here are to highlight the role of local interactions between neighboring agents and to relate equilibrium behaviors to network centrality measures. Using a modified version of the Bonacich centrality measure, we show that players with more vulnerable locations in the network exert more effort at equilibrium.

Then, we find sufficient conditions under which the SO exists and is unique. Using another modified version of the Bonacich centrality measure, we prove that the efficient profile is a fixed point, we derive a structural condition for the SO to be interior and we show that the most influential players should exert less effort at efficiency. Our proofs are simple, intuitive and based on standard optimization techniques. Then, we show that there is always at least one player that exerts strictly more effort than her efficient level and we find structural conditions under which this is the case for each player. We call this situation a social tragedy. In general, such a situation appears whenever each player has a successor. But when the SO is interior, it appears if and only if each player has a successor or a predecessor.

The paper is built on the assumption that the most basic and common mechanism for a social tragedy is the bandwagon effect (or fashion effect): an agent will behave in a certain way because some or all of his neighbors also behave in that way. Hence, the effort level exerted by an agent depends positively on the effort levels exerted by some or all of his neighbors, which depend themselves positively on the effort levels exerted by some or all of their neighbors, and so on. The mechanism's similarity to the contagion of cheating behavior is not really appropriate; we do not consider a threshold above which effort would become illegal, i.e., there is no risk of getting caught, agents are not facing a lottery (see, e.g., [START_REF] Pascual-Ezama | Motivation, Money, Prestige and Cheats[END_REF]. Note also that we do not allow agents to forge alliance with other agents. In our game, all the players are " enemies " (direct or indirect).

The main motivation of the model is collective wasteful behavior towards goods such as power, prestige or status. The welfare losses caused by consumption of such goods have been established by [START_REF] Hirsch | The Social Limits to Growth[END_REF] and [START_REF] Frank | The Demand for Unobservable and Other Nonpositional Goods[END_REF] and is clearly demonstrated by increased investments made, in some occasions, by rival agents: from the early 1950s through the mid-1960s, the US brewing industry was involved in a " game of market power ", the advertising spending per barrel rising from $5.00 in 1950 to $8.10 in 1963 (Tremblay and Tremblay, 2007, p. 68); for several years now, US colleges and universities are clearly engaged in a " game of educational prestige ", seeking to attract the best students through increased spending or reduced price [START_REF] Winston | The Positional Arms Race in Higher Education[END_REF]. A structural analysis of collective wasteful behavior towards status goes back at least to Thorstein Veblen, who saw the consumption of some goods or services as " conspicuous expenditures " driven by " the stimulus of an invidious comparison which prompts us to outdo those with whom we are in the habit of classing ourselves " (Veblen, 1899, p. 103). Interestingly, Veblen emphasized the role of the hierarchic structure of social classes (directed network) on the diffusion of the pecuniary standard of living (wasteful behavior).

Model

There are n players and the set of players is N = {1, . . . , n}. Each player l ∈ N chooses simultaneously a level of absolute effort x l ∈ IR + . E.g., the players could be firms producing brand-differentiated substitute goods and x l could be the strategy of firm l's in advertising expenditures. We assume that the cost of a unit of x l is w l . Let x = (x 1 , . . . , x n ) denote an effort profile of all players.

Players are arranged in a network, which we represent as a weighted directed graph G which consists of a set of nodes (the players), a set of arcs (the directed links between players) and a mapping from the set of arcs to a set of positive weights (the intensities of the directed links). We will use kl to denote an arc directed from node k to node l. If kl is an arc, then we say that node l is a vulnerable neighbor of node k, or that node k is an influential neighbor of node l. A directed path in G is a sequence of distinct nodes connected by arcs corresponding to the order of the nodes in the sequence. The length of a directed path is its number of arcs. The weight of a directed path is the product of the weights of its arcs. To continue our example, the network could reflect the geographic structure of competition among firms. Two firms compete only when they are geographic neighbors. In that case, these two firms are linked, the direction and the intensity of the link reflecting the balance between their relative market powers (due, for instance, to the quality of the goods being produced).

The basic representation of G is given by its weighted n × n adjacency matrix Λ = [λ kl ] ∈ IR n×n + where λ kl > 0 if kl is an arc and λ kl = 0 otherwise (by convention λ kk = 0). Let ρ(Λ) denote the spectral radius of the network. For the rest of the paper, we require:

Assumption A0. ρ (Λ) < 1.
Since Λ ≥ 0, it is well-known that A0 holds if and only if (I -Λ) -1 exists and is nonnegative [START_REF] Berman | Nonnegative Matrices in the Mathematical Sciences[END_REF]. Therefore,

M = (I -Λ) -1 = ∞ i=0 Λ i , hence M ≥ I + Λ ≥ I.
The entry m kl counts the total weight of all directed paths in G starting at node k and ending at node l.

We suppose that the players are " enemies " and hence that network externalities are negative. Let e l denote player l's effective effort, defined as the difference between player l's absolute effort level with the absolute effort levels of her influential neighbors multiplied by the corresponding weights. Players value their effective effort according to a twice differentiable strictly concave value function f l (e l ) defined on IR with f ′ l > 0 and f ′′ l < 0 for all l. According to our assumptions stated above, the resulting effective effort is determined according to e l = x lk:k =l λ kl x k . E.g., f l (e l ) could be the value (measured in monetary units) to firm l of advertising expenditures x of all firms.

The payoff function of player l, defined for all effort profile x ≥ 0, is given by

U l (x) = f l x l - k:k =l λ kl x k -w l x l ,
and we note G(Λ, f, w) the simultaneous-move game with payoffs U l : IR n + → IR and strategy space x l ≥ 0 for each player l ∈ N, where f is the vector of value functions and w the vector of marginal costs. Since ∂U l /∂x k ≤ 0 for all k = l, this is a game of negative externalities, and since ∂ 2 U l /∂x k ∂x l ≥ 0 for all k = l, this is a game of strategic complements.

In our game, the effort space is the positive real line and the payoffs are strictly concave, so every mixed strategy is dominated by its average pure strategy which is available. Then, we study Nash equilibria in pure strategies.

Equilibrium Analysis

For the equilibrium analysis, we require:

Assumption A1. For all l ∈ N, f ′ l (0) > w l and w l > f ′ l (∞).
This assumption, called the boundary conditions, guarantees that each player will exert a positive and finite level of absolute effort at equilibrium. Let x * l = (f ′ l ) -1 (w l ) denote player l's uninfluenced equilibrium effort and for ease of exposition, we will write x * = (f ′ ) -1 (w). Under A1, x * l is positive and finite, i.e., a player with no influential neighbors still exerts a positive and finite equilibrium effort level. Hence, the associated reactions functions are linear in strategies:

∀l ∈ N, x l (x -l ) = x * l + k:k =l λ kl x k ,
where x -l is the strategy vector x with player l's strategy removed. This allows us to find PSNE by solving a linear problem and we look for a closedform solution.

For a weighted adjacency matrix Λ ∈ IR n×n + , the Bonacich centrality measure is given by

c (α, β, Λ, 1) = α (I -βΛ) -1 Λ1,
where 1 is the vector of ones and α, β ∈ IR are two scalars [START_REF] Bonacich | Power and Centrality: A Family of Measures[END_REF]. Under A0, we have the following power expansion:

c (α, β, Λ, 1) = α ∞ i=0 β i Λ i+1 1.
When β is positive, c(1, β, Λ, 1) denotes the expected number of influences directly or indirectly caused by a node. The Bonacich centrality measure has proved to be very useful to provide geometric intuitions of how equilibrium efforts are related to network position in a variety of network games [START_REF] Ballester | Who's Who in Networks. Wanted: The Key Player[END_REF][START_REF] Corbo | The Importance of Network Topology in Local Contribution Games[END_REF][START_REF] Ballester | Interactions with Hidden Complementarities[END_REF][START_REF] Ilkiliç | Networks of Common Property Resources[END_REF][START_REF] Allouch | On the Private Provision of Public Goods on Networks[END_REF]Rébillé and Richefort, 2012a;[START_REF] Bramoullé | Strategic Interactions and Networks[END_REF]. We introduce a modified version of the Bonacich centrality measure.

Definition 1. Let G(Λ, f, w) be a game. Under A0, the vector b -(Λ, x * ) = x * + c 1, 1, Λ T , x * = I -Λ T -1 x * is called the vulnerability measure.
Using the power expansion presented above, we observe that the vulnerability measure of a player l in the network is a sum of her uninfluenced equilibrium effort x * l with the total weight of all directed paths that end at her in the network, where a directed path that starts at player k = l is weighted by x * k , the uninfluenced equilibrium effort of the corresponding player. Hence, having many influential neighbors increases vulnerability, and if one player's influential neighbors themselves have many influential neighbors, vulnerability is increased, and so on.

Example 1 (Acyclic network). There are four firms (players) that produce brand-differentiated substitute goods. Each firm must choose her advertising expenditures. Firm 3 is a geographic neighbor of all the other firms, and all the other firms are only geographic neighbors to firm 3. Firms 1 and 2 produce goods of better quality than firm 3, who produces herself a good of better quality than firm 4. The geographic structure of competition among the four firms is given by the following weighted directed network: Since the network is acyclic, ρ(Λ) = 0 [START_REF] Nicholson | Matrices with Permanent Equal to One[END_REF]. Then, A0 is always satisfied. For all firm l ∈ N = {1, . . . , 4}, let

U l (x) = 1 -exp -x l - k:k =l λ kl x k - 1 2 x l
where x l is firm l's advertising expenditures and λ kl is the intensity of the influence from firm k to firm l. For ease of exposition, we set the intensity of each influence to 0.5 (i.e., λ 13 = λ 23 = λ 34 = 0.5). We get

b -(Λ, x * ) =     ln (2) ln (2) 2 ln (2) 2 ln (2)     ,
so firms 3 and 4 have the most vulnerable locations.

Letting x ≥ 0 denote a PSNE of G(Λ, f, w), we obtain the following result.

Proposition 1. Let G(Λ, f, w) be a game. Under A0 and A1, the PSNE exists, is unique and is interior. Moreover,

x = b -(Λ, x * ) = b -Λ, (f ′ ) -1 (w) ≥ x *
and for all l ∈ N,

U l (x) = f l (x * l ) -w l xl .
The interpretation is straightforward. When the boundary conditions are met and when the spectral radius of the network is sufficiently low, the PSNE exists, is unique, is interior and is such that players with a more vulnerable location in the network exert a higher level of absolute effort (if the spectral radius is too high, no PSNE exists; see, e.g., [START_REF] Ballester | Who's Who in Networks. Wanted: The Key Player[END_REF][START_REF] Corbo | The Importance of Network Topology in Local Contribution Games[END_REF]. When the network is undirected, the spectral condition can be interpreted as follows (see, e.g., Ballester and Calvó-Armengol, 2010): the game G(Λ, f, w) admits a unique PSNE whenever the network is not too much dense. When the network is directed, things are less clear, although A0 still carries details on the topological structure of the network (for a survey of results on the spectra of graphs, see [START_REF] Cvetkovic | The Largest Eigenvalue of a Graph: A Survey[END_REF], and for a survey of results on the spectra of directed graphs, see [START_REF] Brualdi | Spectra of Digraphs[END_REF]. We now turn to examine social optima.

Welfare Analysis

The social welfare or efficiency of a strategy profile in G(Λ, f, w) is defined as the sum of payoffs of all players. Let SW be the social welfare function defined for all x ≥ 0 by

SW (x) = l f l x l - k:k =l λ kl x k -w l x l .
An effort profile is said to be socially optimal or efficient if it maximizes the social welfare function. Thus, the efficient behavior of a player depends on how she is impacted by her predecessors' behavior and how she impacts her successors' behavior. Consequently, we need an opposite version of the vulnerability measure for getting a characterization of the efficient profile.

Definition 2. Let G(Λ, f, w) be a game. Under A0, the vector

b + (Λ, w) = w + c (1, 1, Λ, w) = (I -Λ) -1 w is called the influence measure.
Using the power expansion presented in the previous section, we observe that the influence measure of a player l in the network is a sum of her marginal cost of absolute effort w l with the total weight of all directed paths that start at her in the network, where a directed path that ends at player k = l is weighted by w k , the marginal cost of absolute effort of the corresponding player. Hence, having many vulnerable neighbors increases influence, and if one player's vulnerable neighbors themselves have many vulnerable neighbors, influence is increased, and so on.

Example 2 (Example 1 continued). We get b + (Λ, w) =     7/8 7/8 3/4 1/2     ,
so firms 1 and 2 are the most influential firms.

We prove that SW is strictly concave whenever A0 is met and reaches a maximum on a compact and non-empty set if A1 is satisfied. This allows us to derive an existence and uniqueness result for the SO. In addition, we show that the efficient effort profile is a fixed point when the SO is corner. Let D = {l : xl > 0}, D its complement and x ≥ 0 the SO of G(Λ, f, w). Theorem 1. Let G(Λ, f, w) be a game. Under A0 and A1, the SO exists, is unique and is a fixed point, i.e.,

xD = b -Λ D×D , (f ′ ) -1 b + (Λ D×D , φ (x D )) , where φ(x D ) = w D + Λ D× D f ′ (-Λ D× DT xD ).
This formula provides intuitions on how efficient behavior is related to centrality. When D = N, there are inactive players at efficiency and the efficient efforts of the active players can be computed by solving the fixed point equation. When D = N, the SO is interior. In that case, φ(x D ) = w D . The efficient efforts of the active players are constant and given by a combination of the vulnerability and the influence measures. In that case, the connection between PSNE and SO can be established. Then, we now look for structural conditions under which the SO is interior. We require:

Assumption A2. For all l ∈ N, f ′ l (0) > b + l (Λ, w) and w l > f ′ l (∞).
This assumption, called the modified boundary conditions, guarantees that each player should exert a positive and finite level of absolute effort at efficiency (note that since b + l (Λ, w) ≥ w l , A1 holds whenever A2 holds). Let x * l = (f ′ l ) -1 (b + l (Λ, w)) denote player l's uninfluenced efficient effort and for ease of exposition, we will write x * = (f ′ ) -1 (b + (Λ, w)). Under A2, x * l is positive and finite, i.e., a player with no influential neighbors still has a positive and finite efficient effort level. Hence, the first order conditions become linear in efforts:

∀l ∈ N, x l (x -l ) = x * l + k:k =l λ kl x k .
We obtain the following result.

Proposition 2. Let G(Λ, f, w) be a game. Under A0 and A2, the SO exists, is unique and is interior. Moreover,

x = b -(Λ, x * ) = b -Λ, (f ′ ) -1 b + (Λ, w) ≥ x * and for all l ∈ N, U l (x) = f l (x * l ) -w l xl .
The interpretation is as follows. When the modified boundary conditions are met and when the spectral radius of the network is sufficiently low, the SO exists, is unique, is interior and is such that players with a more influential location in the network should exert a lower level of absolute effort. Then, b -(Λ, x * ) can be interpreted as the social vulnerability measure: players with a more socially vulnerable location in the network should exert a higher level of absolute effort at efficiency. Interestingly, this result is in line with the closed-form expression obtained for the SO in games with substitutabilities (Rébillé and Richefort, 2012a and 2012b). Moreover, it holds even if we consider social welfare functions that are weighted sums of individual payoffs.

Remark 1 (Weighted social welfare functions). More generally, the social welfare function could be weighted, reflecting the interest of the social planner for the various players with respect to their location in the network. Let α = (α 1 , . . . , α n ) >> 0 be social weights and consider the social welfare function

l α l f l x l - k:k =l λ kl x k -w l x l .
Then, the SO is given by

x = b -Λ, (f ′ ) -1 1 α b + (Λ, αw) .
where (1/α) l = 1/α l and (αw) l = α l w l for all l, provided that A2 holds (that is, provided that α l f ′ l (0) > b + l (Λ, αw) and w l > f ′ l (∞) for all l).

Social Tragedy in Networks

From now on, we assume that Λ = 0. The fact that in network games (of strategic complements or strategic substitutes), the PSNE is always inefficient is not surprising because players do not take into account the (positive or negative) externalities which they generate on their successors. However, network games of strategic complements differed from network games of strategic substitutes in that the best response of each player is increasing in efforts of her direct predecessors. Then, when externalities are negative, we shall prove the following result.

Theorem 2. Let G(Λ, f, w) be a game. Under A0 and A1, each player exerts as much or more effort than her efficient level with at least one player exerting strictly more effort, i.e., x < x.

This result entails that when a PSNE exists, too much collective effort is always exerted at equilibrium, i.e., l xl < l xl , whether the SO be interior or not. There may exist, however, some players whose equilibrium effort level is equal to their efficient effort level. Next, we consider the situation in which each player exerts strictly more effort than her efficient level. Definition 3. Let G(Λ, f, w) be a game. Under A0 and A1, a situation in which

x << x is called a social tragedy.

A player who has no successors, i.e., a sink player, does not generate externalities. Such a player has the particularity that her uninfluenced efficient effort is equal to her uninfluenced equilibrium effort. This observation provides the intuition for our next result, that shows that network games without sink players always lead to a social tragedy, even if the SO is corner.

Corollary 1. Let G(Λ, f, w) be a game. Under A0 and A1, there exists a social tragedy whenever there are no sink players.

Typical examples of network games without sink players are games played in undirected networks (with uniform or non-uniform weights attached to the undirected links). The proof clearly shows, however, that the no sink property is only sufficient. In other words, there exists games with complementarities which lead to a social tragedy even if there are sink players. The following example illustrates this point.

Example 3 (Example 1 modified). Consider the same network structure as in example 1, but with the following modified payoffs. For all l ∈ N = {1, . . . , 4}, let

U l (x) = 1 -exp -x l - k:k =l λ kl x k - 3 5 x l .
Assumptions A0 and A1 are met while A2 is not (we still assume that λ 13 = λ 23 = λ 34 = 0.5). We get

x =     ln (5/3) ln (5/3) 2 ln (5/3) 2 ln (5/3)     and x =     0 0 ln (10/9) 1/2 ln (2/3) + 3/2 ln (5/3)     .
Firm 4 is a sink player, but the game still leads to a social tragedy, x << x.

A player who has neither successors nor predecessors, i.e., an isolated player, does not generate and is not impacted by externalities. Such a player has the particularity that her efficient effort is equal to her equilibrium effort. Clearly, the absence of isolated players is a necessary condition for the appearance of a social tragedy. Moreover, the SO is interior under A0 and A2.

In that case, the efficient profile of G(Λ, f, w) can be interpreted as a PSNE of another game with " modified " uninfluenced equilibrium efforts. That is, we have

x = b -(Λ, x * ) , where x * = (f ′ ) -1 b + (Λ, w) ≤ (f ′ ) -1 (w) = x * .
This last observation leads us directly to our next result, that shows that when the SO is interior, the absence of isolated players becomes a sufficient condition for the appearance of a social tragedy.

Theorem 3. Let G(Λ, f, w) be a game. Under A0 and A2, there exists a social tragedy if and only if there are no isolated players.

It is worth noting that under A0 and A2, the global over effort and the resulting loss in welfare can be computed, thanks to Proposition 2. The global over effort is given by

x -x = b + (Λ, x * -x * ) >> 0
and the loss in welfare is given by

l U l (x) -U l (x) ∈ 0, l w l (x l -xl ) .
These measures indicate the performance of the network and interestingly, they both depend on network centrality measures.

Then, a situation in which xl < xl for all l appears if and only if each player has at least one (influential or vulnerable) neighbor, provided that the SO is interior. The reason is that players, who are rational and selfish, are trapped in a continuous prisoner's dilemma. Many continuous prisoner's dilemmas can be found in the literature, and a large majority of them have been developed from an underlying social dilemma [START_REF] Eaton | The Elementary Economics of Social Dilemmas[END_REF][START_REF] Acharya | The Calculus of the Security Dilemma[END_REF]. Depending on the nature of the interactions among players, social dilemma games can be classified into four groups (i.e., games of strategic substitutes/complements and negative/positive externalities; see [START_REF] Eaton | The Elementary Economics of Social Dilemmas[END_REF].

A famous multi-player generalization of the prisoner's dilemma is the commons dilemma, which leads to what has become known as the " tragedy of the commons " [START_REF] Gordon | The Economic Theory of a Common-Property Resource: The Fishery[END_REF][START_REF] Hardin | The Tragedy of the Commons[END_REF]. Our results are closely related to such a situation, since the commons dilemma is generally represented as a game of negative externalities (see, e.g., [START_REF] Hardin | Collective Action as an Agreeable N-Prisoner's Dilemma[END_REF][START_REF] Cornes | On Commons and Tragedies[END_REF][START_REF] Gardner | Rules, Games, and Common Pool Resources[END_REF]. With the exception of [START_REF] Ilkiliç | Networks of Common Property Resources[END_REF], this literature typically assumes that players are arranged in a complete, undirected network (even if the assumption on the completeness of the network is generally not explicitly stated). Taking our previous example on firms producing brand-differentiated substitute goods, Corollary 2 and Theorem 3 may then be interpreted as a " tragedy of the commons in networks ", with x l representing the share of a common-pool resource consumed by firm l (individual consumptions are strategic complements when users imitate each other, and generate negative externalities when the consumption by one user reduces the quantity available for other users). The following example illustrates a social tragedy when the SO is interior.

Example 4 (Example 1 continued). Since f l (e l ) = 1 -exp(-e l ) and w l = 1/2 for all l, Assumption A2 is satisfied. So the PSNE and the SO are both unique and interior. We get Though this example focus on acyclic networks, it is worth noting that our results are not restricted to such network structures; they apply to each kind of network, i.e., with or withtout cycles, directed or undirected, weighted or unweighted, connected or disconnected, etc. Moreover, it is interesting to note that, whatever the network structure, a social tragedy always appears for people belonging to a " community ".

x =     ln (2) ln (2) 2 ln (2) 2 ln (2)     and x =     ln (8/
Corollary 2. Let G(Λ, f, w) be a game. Under A0 and A2, there exists a social tragedy for non-isolated players, i.e., xl < xl if and only if l is not isolated, otherwise xl = xl = x * l . In the economic theory of negative externalities, the focus is on how to share and reduce the cost due to the external effects [START_REF] Montgomery | Markets in Licenses and Efficient Pollution Control Programs[END_REF][START_REF] Baumol | The Theory of Environmental Policy, Second Edition[END_REF]. One well known way to perform this goal is to impose a tax mechanism which achieves the efficient profile (this is the so-called " Pigouvian " tax when negative externalities are pollution). The mechanism punishes players for their deviations from the efficient effort level and hence, players prefer to exert the efficient effort level. In networks, it appears that each player should pay a tax that is precisely equal to the difference between her influence measure and her marginal cost of absolute effort. Thus, the more influential a player, the higher her optimal tax. Remark 2 (Optimal tax). The SO of game G(Λ, f, w) may be restored through a tax plan on marginal costs. Assume A2 holds (with w). Following Proposition 2, we obtain

x (Λ, w) = b -Λ, (f ′ ) -1 b + (Λ, w) .
Let us introduce the tax rate as follows

τ = b + (Λ, w) -w = Λb + (Λ, w) ≥ 0.
Then, A2 holds (with w + τ ) and by Proposition 1, the PSNE of game G(Λ, f, w + τ ) is given by

x (Λ, w + τ ) = b -Λ, (f ′ ) -1 (w + τ ) = x (Λ, w) .
Therefore, the network matters to restore optimality in game G(Λ, f, w), i.e., the optimal tax plan involves different tax rate at each location throughout the network.

Finally, it is worth noting that our results hold even if we consider weighted social welfare functions.

Remark 3 (Weighted social welfare functions continued). Let us remind that a PSNE is an ordinal property. Thus, whatever the social weights α = (α 1 , . . . , α n ) >> 0 chosen to satisfy A2, the corresponding SO will also satisfy x(α) << x if and only if there are no isolated players, leading therefore to a social tragedy.

Discussion

This work is related to the branch of the literature on social dilemmas that study games of strategic complements and negative externalities. This line of research has been pioneered by Augustin Cournot (1838, ch. IX), who developed a model where two component goods are produced to form a composite good (e.g., copper and zinc to form brass). This model is referred by [START_REF] Eaton | The Elementary Economics of Social Dilemmas[END_REF] as the " Cournot model of price competition ". In a way, we extend this model by adding a (social or geographic) network structure between the players. We now discuss the main assumptions we use for its analysis.

In our game, we assume that efforts are unbounded. When efforts are bounded from above, there always exists a unique PSNE which can not be characterized through the Bonacich centrality measure [START_REF] Belhaj | Network Games under Strategic Complementarities[END_REF]. When efforts are unbounded, we show that there exists a unique PSNE whenever the spectral radius of the network is sufficiently low (A0). This condition is known to guarantee the uniqueness of PSNE in network games of strategic complements or strategic substitutes [START_REF] Ballester | Who's Who in Networks. Wanted: The Key Player[END_REF][START_REF] Corbo | The Importance of Network Topology in Local Contribution Games[END_REF][START_REF] Ballester | Interactions with Hidden Complementarities[END_REF]. Recent research on the uniqueness of PSNE in network games of strategic substitutes has moved beyond this spectral condition, showing that the lowest eigenvalue is key to PSNE outcomes when the network is undirected [START_REF] Bramoullé | Strategic Interactions and Networks[END_REF][START_REF] Rébillé | Equilibrium Existence and Uniqueness in Network Games with Additive Preferences[END_REF]. In network games of strategic complements, there is no equilibrium whenever A0 is not met [START_REF] Ballester | Who's Who in Networks. Wanted: The Key Player[END_REF][START_REF] Corbo | The Importance of Network Topology in Local Contribution Games[END_REF][START_REF] Ballester | Interactions with Hidden Complementarities[END_REF]. Interestingly, our paper is the first, to our knowledge, to highlight the role of A0 for the uniqueness and the characterization of SO in such games, and for the appearance of a social tragedy.

The quasi-linear preferences we consider, although restrictive, help us focus on the effects of network structure at equilibrium and at efficiency. This class of preferences allows us to formulate the equilibrium and the efficient efforts in terms of the directed paths that end and start at each player in the network. These results are in line with previous PSNE characterizations in various network games using bilinear payoffs [START_REF] Ballester | Who's Who in Networks. Wanted: The Key Player[END_REF][START_REF] Corbo | The Importance of Network Topology in Local Contribution Games[END_REF] or quasi-linear payoffs (Ballester and Calvó-Armengol, 2010; [START_REF] Bramoullé | Strategic Interactions and Networks[END_REF]. In our paper, the relevance to consider quasi-linear preferences to study a game of strategic complements is that it allows us to compare PSNE to SO and to quantify this comparison (such a comparison is an open issue in network games of strategic substitutes because directed paths of even and odd length have opposing signs in the closed-form expressions of PSNE and SO; see Rébillé andRichefort, 2012a and[START_REF] Rébillé | Welfare Analysis in Games with Substitutabilities[END_REF]).

In our model, preferences are taken to be cardinal, thus admitting only an interpretation in terms of benefits and costs. A useful direction for further research would be to consider payoffs that increase as players improve their position in some ordinal ranking produced by the game, where orderings of the players could represent how well they have done in the game relative to their neighbors [START_REF] Brandt | Ranking Games[END_REF][START_REF] Hopkins | Status, Affluence, and Inequality: Rank-Based Comparisons in Games of Status[END_REF]. In this regard, a status loss function should be added to individual payoffs (see, e.g., [START_REF] Immorlica | Striving for Social Status[END_REF]). This will be the next step on our research agenda. conditions are sufficient and are the following:

U ′ l (x l , x -l ) = f ′ l x l - k:k =l λ kl x k -w l = 0, equivalently, x l - k:k =l λ kl x k = (f ′ l ) -1 (w l ) = x * l ,
and in matrix notation,

I -Λ T x = x * .
Now, I -Λ has a nonnegative inverse, thus the unique PSNE x is obtained by

x = I -Λ T -1 x * ≥ x * >> 0.
Proof of Theorem 1. (Existence). We shall build a sufficiently large box where the maximum is reached. Let x ≥ 0. We have, by increasingness of the f l 's,

SW (x) ≤ n l=1 f l (x l ) -w l x l .
Under A1, for all l, f ′ l (∞) < w l < f ′ l (0), so there exists x l > 0 such that f ′ l (x l ) < w l . Since f l is strictly concave and differentiable for all l, the convexity inequality entails

f l (x l ) ≤ f l (x l ) + f ′ l (x) (x l -x l )
and also

f l (0) < f l (x l ) -f ′ l (x) x l . Thus, SW (x) ≤ n l=1 f l (x l ) + f ′ l (x) (x l -x l ) -w l x l = n l=1 f l (x l ) -f ′ l (x) x l -(w l -f ′ l (x)) x l ≤ A -B n i=1 x l where A = n l=1 f l (x l ) -f ′ l (x)x l and B = min l w l -f ′ l (x) (> 0). So by the convexity inequality, A > n l=1 f l (0) = SW (0).
Hence, if n l=1 x l > C = (A -SW (0)) /B (> 0) then SW (x) < SW (0). So, being continuous, SW reaches a maximum on ∆ = {x ≥ 0 : n l=1 x l ≤ C} which is compact and non-empty (0 ∈ ∆), hence on IR n + since SW (x) < SW (0) whenever x / ∈ ∆. (Uniqueness). Let us prove that SW is strictly concave. We may ignore the linear part, we shall show that the following function F is strictly concave where

F (x) = l f l x l - k:k =l λ kl x k , x ≥ 0 .
Let x ′ , x ′′ ≥ 0 and θ ∈ (0, 1) with x ′ = x ′′ . Under A0, I -Λ is invertible so I -Λ T is invertible too. Now, there exists some l 0 such that

I -Λ T x ′ l 0 = I -Λ T x ′′ l 0 , that is, x ′ l 0 - k:k =l 0 λ kl 0 x ′ k = x ′′ l 0 - k:k =l 0 λ kl 0 x ′′ k .
By strict concavity of f l 0 and concavity of f l for l = l 0 it comes

f l 0 θ x ′ l 0 - k:k =l 0 λ kl 0 x ′ k + (1 -θ) x ′′ l 0 - k:k =l 0 λ kl 0 x ′′ k > θf l 0 x ′ l 0 - k:k =l 0 λ kl 0 x ′ k + (1 -θ) f l 0 x ′′ l 0 - k:k =l 0 λ kl 0 x ′′ k and f l θ x ′ l - k:k =l λ kl x ′ k + (1 -θ) x ′′ l - k:k =l λ kl x ′′ k ≥ θf l x ′ l - k:k =l λ kl x ′ k + (1 -θ)f l x ′′ l - k:k =l λ kl x ′′ k .
Summing these inequalities over i, we obtain

F (θx ′ + (1 -θ) x ′′ ) > θF (x ′ ) + (1 -θ) F (x ′′ )
and this establishes strong concavity of F , thus strong concavity of SW . Therefore SW 's maximum is unique.

(Characterization.) Let D = {l : xl > 0} and D its complement. For the inactive players, we have x D = 0. For the active players, the first order conditions of social welfare maximization are: ∀l ∈ D,

f ′ l xl - k:k =l,k∈D λ kl xk -w l - j:j =l,j∈D λ lj f ′ j xj - i:i =j,i∈D λ ij xi - g:g =l,g∈ D λ lg f ′ g 0 - h:h =g,h∈D λ hg xh = 0. Let e i = f ′ i (.
) for all i. We note Λ I×J the submatrix of Λ with rows in I ⊆ N and columns in J ⊆ N. In matrix notation, the first order conditions for the active players write

(I -Λ) D×D e D -Λ D× D e D -w D = 0,
where Λ D× D denotes the (possibly rectangular) submatrix of Λ consisting of rows with all the active players and columns with all the inactive players. We obtain

xD = b -Λ D×D , (f ′ ) -1 b + (Λ D×D , φ (x D )) , where φ(x D ) = w D + Λ D× D e D and e D = f ′ (-(Λ D× D) T xD ).
Proof of Proposition 2. Let us prove that the SO is interior. The first order conditions provide the following inequalities:

∀l, f ′ l xl - k:k =l λ kl xk - j:j =l λ lj f ′ j xj - i:i =j λ ij xi ≤ w l , that is, (I -Λ) f ′ I -Λ T x ≤ w, or, f ′ I -Λ T x ≤ (I -Λ) -1 w. By assumption, f ′ (0) >> (I -Λ) -1 w ≥ w, thus ((I -Λ) -1 w) l ∈ [w l , f ′ l (0)). Hence, x ≥ I -Λ T x ≥ (f ′ ) -1 (I -Λ) -1 w >> 0 .
Since the SO is interior, the first order conditions provide actually the following equalities, Put X = (f ′ ) -1 (w) -(f ′ ) -1 (I -Λ) -1 w. Then, the condition is equivalent to, ∀l, ∃k / I -Λ T -1 lk × X k > 0 or equivalently, ∀l, ∃k, ∃p ≥ 0 / Λ T p lk > 0 and (I -Λ) -1 w k > w k that is, ∀l, ∃k, ∃p ≥ 0, ∃q ≥ 1 / Λ T p lk > 0 and (Λ q w) k > 0 since (I -Λ) -1 -I = ∞ i=1 Λ i . By transposition, ∀l, ∃k, ∃p ≥ 0, ∃q ≥ 1 / (Λ p ) kl > 0 and (Λ q w) k > 0 or ∀l, ∃k, ∃p ≥ 0, ∃q ≥ 1 / (Λ p ) kl > 0 and (Λ q ) k. w > 0 or ∀l, ∃k, ∃p ≥ 0, ∃q ≥ 1 / (Λ p ) kl > 0 and (Λ q ) k. = 0 T since w >> 0. Assume l is not an isolated player. Then either l has an influential neighbor or a vulnerable neighbor. If l admits an influential neighbor l ′ , then take k = l ′ , p = 1 and q = 1 so (Λ 1 ) l ′ l = λ l ′ l > 0 and (Λ 1 ) l ′ . = 0 T . Otherwise, l admits a vulnerable neighbor l ′′ , then take k = l, p = 0 and q = 1 so (Λ 0 ) ll = 1 > 0 and (Λ 1 ) l. = 0 T since (Λ 1 ) ll ′′ = λ ll ′′ > 0.

(I -Λ) f ′ I -Λ T x = w, thus x = I -Λ T -1 (f ′ ) -1 (I -Λ) -1 w .
(Only if). Assume there exists some isolated player l. As part of a PSNE xl = x * l since l has no interaction with other players. And as part of SO xl = x * l since the social welfare function is separable with respect to x l . Thus

34 Figure 1 :

 341 Figure 1: An acyclic network with four players

Proof of Theorem 2 . 1 ≥

 21 At equilibrium, the FOCs are for all l, f ′ l xlk:k =l λ kl xk = w l , since the PSNE is interior. Under A0 and A1, the SO may be corner. Let D = {l : xl > 0}. For l ∈ D, the FOCs for the efficient profile aref ′ l xlk:k =l,k∈D λ kl xk -w lj:j =l λ lj f ′ j xji:i =j λ ij xi = 0,and since f ′ l > 0, we havef ′ l xlk:k =l,k∈D λ kl xk ≥ w l ,with a strict inequality if λ lj > 0 for some j, i.e., if l is not a sink player. Now, =l,k∈Dλ kl xk ≤ f ′ l xlk:k =l,k∈Dλ kl xk . The function f ′ l being decreasing, we have xlk:k =l,k∈D λ kl xk ≤ xlk:k =l,k∈D λ kl xk , that is, in matrix notation, I -Λ T D×D xD ≤ I -Λ T D×D xD . Then, since I -Λ T D×D has also a nonnegative inverse with I -Λ T D×D -I D , it comes by composition xD ≤ xD , and xl < xl whenever l is not a sink player. If D = N, then x < x since xN\D >> 0 = xN\D . If D = N, there is a player which is not a sink player (since Λ = 0), so x < x. Proof of Corollary 1. If there are no sink players, then xl < xl for all l (see the proof of Theorem 2). Hence, x << x, i.e., a social tragedy appears. Proof of Theorem 3. (If). According to Propositions 1 and 2, a social tragedy holds if 0 << I -Λ T -1 (f ′ ) -1 (w) -(f ′ ) -1 (I -Λ) -1 w .

  , xl = xl = x * l . Proof of Corollary 2. (If). Let I = {isolated players} and I c = {non-isolated players}. Consider the subgame G I c (Λ, f, w) played by non-isolated players in the subnetwork Λ I c ×I c with vertex I c . Then, A0 and A2 hold since Λ I c ×I c has positive inverse too, and b + (Λ I c ×I c , .) = (b + (Λ, .)) I c ×I c . By Theorem 3, a social tragedy appears with x(I c ) << x(I c ). Since isolated player do not intervene at PSNE nor at SO we have xI c = x(I c ) and xI c = x(I c ) hence xI c << xI c . And xI = xI = x * I , so a social tragedy for non-isolated players appears. (Only if). If l is isolated, then xl = xl = x * l (see Theorem 3).

Appendix

Proof of Proposition 1. Let l ∈ N and x ∈ IR n . Player l maximizes her payoff function without constraint given others efforts x -l . The first order


David Guerrero
email: david.guerrero@ifsttar.fr

Jean-Paul Rodrigue

The Waves of Containerization: Shifts in Global Maritime Transportation

Keywords: Maritime transportation, Ports, Containerization, Economic cycles

This paper provides evidence of the cyclic behavior of containerization through an analysis of the phases of a Kondratieff wave (K-wave) of global container ports development. The container, like any technical innovation, has a functional (within transport chains) and geographical diffusion potential where a phase of maturity is eventually reached. Evidence from the global container port system suggests five main successive waves of containerization with a shift of the momentum from advanced economies to developing economies, but also within specific regions. These waves are illustrative of major macroeconomic, technological and sometimes political shifts within the global economy. They do not explain the causes, but simply the consequences in the distribution in container traffic and growth (or decline). Yet, they provide strong evidence that containerization has a cyclic behavior and that inflection points are eventually reached, marking the end of the diffusion of containerization in a specific port or port range. Future expectations about the growth of containerization thus need to be assessed within an economic cycle perspective instead of the rather linear perspectives.

Introduction: the spatial and functional diffusion of the container Waves and Container Port Development

After more than half a century since its introduction, the container continues its spatial and functional diffusion within global transport systems. Diffusion can be investigated at the level of the container shipping network in terms of growth in the intensity and connectivity of ports, but such an approach would require a substantial dataset to review all the shipping services. A simpler approach is to look at the geographical growth structure of individual container ports. Containerization has diffused to an extensive array of locations and supports a wide variety of supply chains, from retail goods, parts and commodities. Such diffusion is far from uniform, on par with the changes in the commercial geography of the global economy. This is particularly the case relative to the export-oriented strategies of Asian countries that have rebalanced a global trade system that used to revolve around the economic triad (North America, Europe, and Japan). In light of these economic and technological changes, economic cycles are offering a relevant perspective to investigate the spatiotemporal evolution of containerization. Furthermore, there are three other reasons to consider and understand waves when analyzing containerization. The first is that containerization waves are indicative of global changes in a broader economic environment. The second reason is that the waves of containerization raise the general question of the circular relationship between maritime transport and economic development. The third relates to forecasting future containerized traffic, an exercise that commonly considers growth processes in a linear fashion while the extent and rate of the growth is nonlinear; cycles are a relevant perspective to articulate this nonlinearity. Waves (the term 'cycle' is used interchangeably) are amply covered in the business and economic literature but many of their aspects, such as duration and amplitude, are subject to debate. The duration of a wave is related to the process being looked at. One of the longest, the Kondratieff wave (often referred to as a K-wave), usually imply a time frame of 45 to 60 years and try to depict technological diffusion within economic systems [START_REF] Barnett | Kondratiev and the Dynamics of Economic Development[END_REF]. Kuznets waves (15 to 25 years; demographic changes), Juglar waves (7 to 11 years; major investments in fixed capital) and Kitchin cycles (3 to 5 years; manufacturing and inventory cycles) refer to events of shorter duration and have been evidenced [START_REF] Korotayev | A Spectral Analysis of World GDP Dynamics: Kondratieff Waves, Kuznets Swings, Juglar and Kitchin Cycles in Global Economic Development, and the 2008-2009 Economic Crisis[END_REF]. Changes are usually measured through growth rates of an economic activity, such as GDP, production, commodity prices or sales. A notable branch of investigation concerns product life cycles and the shifts in the geography of production as a product goes through distinct phases. These usually involve introduction (or adoption), growth (or acceleration), peak growth, maturity and eventual degeneration (e.g. [START_REF] Hayes | Link manufacturing process and product life cycles[END_REF]. There is also a seasonality in many commercial activities, particularly in retail, which can be considered a cycle occurring annually. Waves are also covered by the geographical literature on the spatial diffusion of innovations (e.g. Hägerstrand 1963, Alves and[START_REF] Alves | Diffusion Theory and Planning[END_REF][START_REF] Rogers | Diffusion of innovations[END_REF] with many processes following a diffusion process that can be contiguous or hierarchical.

The conceptual usage of waves within maritime and port studies is much more limited, but implicitly implied in port development models (e.g. [START_REF] Bird | The major seaports of the United Kingdom[END_REF][START_REF] Hayuth | Containerization and the load center concept[END_REF], in particular for the literature addressing the diffusion of containerization (e.g. [START_REF] Levinson | The Box: How the Shipping Container Made the World Smaller and the World Economy Bigger[END_REF][START_REF] Notteboom | The Future of Containerization: Perspectives from Maritime and Inland Freight Distribution[END_REF][START_REF] Rodrigue | Transportation and Spatial Cycles: Evidence from Maritime Systems[END_REF]). Yet, these approaches consider containerization as a whole and have not looked at what cycles imply at the individual port level and the fact that different ports have been part of different growth patterns and thus part of different economic cycles. While studies looking at the dynamics of individual port or port range are common, global investigations appear much less prevalent (e.g. [START_REF] Fremont | Global maritime networks: The case of Maersk[END_REF][START_REF] O'connor | Global City Regions and the Location of Logistics Activity[END_REF][START_REF] Ducruet | Maritime constellations: a complex network approach to shipping and ports[END_REF]. One reason that can be advocated for this shortcoming is that such an approach requires familiarity with global macroeconomic processes and international trade, areas where port studies have conventionally not placed much focus.

The wave approach can be applied to an individual container port, a port range, or to the whole global port system to better capture their temporal and spatial dynamics. The framework offered by Kondratieff waves (K-waves) appears suitable since its time frame matches the functional and spatial diffusion of containerization (45 to 60 years). Thus, it is argued that containerization is a K-wave process, which is obviously incomplete and characterized by different levels of maturity depending on the markets. It is also argued that within this K-wave, specific phases in container port development are taking place, such as introduction, acceleration, peak growth and maturity (Figure 1). Yet, what these phases imply in the geographical diffusion of port containerization is not clear and needs to be identified. For an individual port, a Kwave represents the full realization of its hinterland (gateway) and foreland (hub) potential in light of geographical and site characteristics. For a port range, a K-wave relates to the setting and interrelations of its ports to service its commercial hinterland, implying a rank/size outcome in the port structure with a few major gateways dominating. Often, there are few major drivers (hinterland or foreland related) behind each wave, which explain the evolution of a set of ports. However, it should be noted that two very different ports (i.e. hub and gateway) can belong to a same wave, even if their drivers are different. What matters here is the shape of their trajectories over time, namely when they adopt, develop, and reach a stage of maturity. A transition (also called medium wave) considers the changes taking place within a Kwave where the system is undertaking a shift from one phase to another; an inflection point taking place over a few years. For instance, during a transition a port may shift from peak growth to maturity due to changing market conditions, such as demand saturation in its hinterland. Transitions are therefore particularly useful at identifying the time frame and the conditions associated with this shift. Otherwise, there is no change in the growth conditions and patterns during the considered time frame and thus a port remains within its existing phase.

Methodology

The paper investigates economic cycles as components of the demand of containerized maritime transport over a K-wave, and this mainly at the global level. Within this temporal framework the paper tries to identify specific growth phases of the geographical diffusion of containerization that can be statistically evidenced. It will also look at recent transitions that can underline which segments of the container port system have achieved a phase of maturity.

Before analyzing the waves of containerization, the evolution of global container throughput and its concentration level is investigated. We apply the Gini coefficient, a widely used index that measures the deviation from a perfectly uniform distribution of container throughput between ports [START_REF] Kuby | Technological change and the concentration of the US general cargo port system: 1970-88[END_REF], Notteboom, 1997, 2010).

The identification of phases (waves) within the K-wave will rely on cluster analysis where a large sample of container ports is categorized according to their growth pattern, namely its time frame and scale. Hierarchical Cluster Analysis (HCA) is a statistical method commonly used for finding relatively homogeneous clusters of observations based on their measured attributes. From a dataset, HCA finds groups (clusters) that minimize their endogenous dissimilarity according to a set number of groups. Initially it places each observation in a separate cluster and then combines the clusters sequentially, reducing the number of clusters at each step until only one cluster is left. When there are N cases, this involves N-1 clustering steps. This hierarchical clustering process can be represented as a dendrogram where each step in the clustering process is illustrated by a join of the tree until only one branch is reached. It is for the analyst to decide on the relevant number of groups, but such a choice is made in light to have the minimum number of groups possible with an acceptable level of dissimilarity. The Ward clustering procedure takes into account the chi-square distances between the profiles and the associated observations [START_REF] Everitt | Cluster Analysis, 4th Edition[END_REF]. This way it provides a decomposition of inertia with respect to the nodes of a dendrogram. The total dissimilarity of the dataset is reduced by a minimum at each successive level of merging the observations. Cluster analysis has been used to identify the long waves of containerization between 1970 and 2010 in five-year increments. There are two reasons for using five year increments instead of annual data. The first and most important reason is that the approach on long waves focuses on long term trends of maritime transportation, rather than on short term traffic fluctuations (mainly linked to local specificities of ports, outside the scope of the study). Selecting five-year intervals is therefore an approach that captures long term trends while avoiding short term variations. The second reason is incomplete data over long time series covering a large number of ports. The HCA is particularly well adapted to the identification of long waves because it allows the distinction of homogeneous clusters of ports simultaneously based on the variation of their shares of global container throughput and on the shapes of their trajectories over time. A review of the literature underlines that this methodology appears to have had limited application to the dynamic analysis of cycles (e. g. [START_REF] Kuczynski | Have there been Differences between Growth Rates in Different Periods of the Development of the Capitalist World Economy since 1850? An Application of Cluster Analysis in Time Series Analysis[END_REF] and even fewer applications to port studies could be identified [START_REF] Guerrero | Les aires d'influence des ports de la France: entre réseau et gravitation[END_REF]. For instance, HCA has mainly been used in a static way, to classify ports by their function and performance levels (e.g. [START_REF] Tongzon | Systematizing international benchmarking for ports[END_REF]Ducruet, Koster, Ven der Beek, 2010). This methodology has also been applied to processes in other fields of geography. Two recent papers on the vegetative seasons of French wineyards [START_REF] Bonnefoy | Temporal and spatial analyses of temperature in a French wineproducing area: the Loire Valley[END_REF] and on the demographic trends of Romanian cities [START_REF] Camara | Demographic Evolution of the Small Towns in the North-East Development Region in the Post-Communist Period[END_REF] have used HCA in such a fashion. The HCA technique has also been used to categorize trends over time in statistics [START_REF] Maharaj | Cluster of time series[END_REF][START_REF] Liao | Clustering of time series data-a survey[END_REF]. According to Maharaj: "results show that it is possible for [time] series on different levels but having similar patterns to cluster together" (Maharaj, 2000, p. 312).

An analysis of Changes of Port Share (CPS) in global container throughput will be used to identify the nature and extent of transitions between long waves. Here, these transitions are called medium waves. The method used (CPS) is less sophisticated than HCA because it only takes into account the variation of the port shares of global container throughput between two time points. It allows to determine whether the trends identified for the long waves is continuing or if there is a divergence. For this purpose the focus is put on the 2000s decade, one that has seen fundamental changes in the structure of global shipping networks with the rapid rise of new ports, namely in China, but also of transshipment hubs at intermediary locations. This period has also seen since 2008 the most significant commercial shift since the beginning of containerization with a significant drop in traffic among the bulk of the world's major container ports, particularly in Europe and North America. The analysis CPS considers two periods, one acting as a reference and the other as the case to evaluate the difference from the reference period. If an observation has the same share of the total traffic in the case period compared with the reference period, then its shift share is close to zero. A negative CPS implies that an observation has less traffic than expected (if its share remained constant), while an increasing CPS (positive) implies the opposite. The analysis is very useful to evaluate the level of divergence from a reference distribution and to identify observations that are standing out.

Global shifts in production and transportation

The Dynamics of Containerization Containerization has been the most dynamic physical component of globalization, far exceeding the growth of the value of exports and the GDP (Figure 2). As globalization developed, each new individual, GDP or export unit was associated with a higher level of container flows. While up to 1980 the growth of container port throughput was on par with the growth of the value of exports and GDP, a divergence was noted afterwards; they were decoupling. Each unit of GDP was associated with a greater level of exports and even greater levels of container throughput. Nevertheless it should be noted that this decoupling between maritime container transportation and GDP in the early 1980s is not specific to the container market. The same trend has been observed, for instance, in the context of European short sea shipping [START_REF] Verny | The importance of decoupling between freight transport and economic growth[END_REF]. It was the outcome of industrial location changes and their impact on freight distribution. The irregular and limited growth of ton-kilometers in the late 1970s and early 1980s was mainly a result of an increase in distances travelled. The development of free trade agreements and the decline of tariffs have also contributed to the growth of trade in the last four decades, underlining a decoupling between GDP and exports. It is still too soon to measure the impacts of the financial crisis of 2008-2009 on free trade agreements, exports and container volumes. Protectionist attempts on trade remain so far marginal [START_REF] Kee | Is protectionism on the rise? Assessing national trade policies during the crisis of 2008[END_REF].

Containerization entered the acceleration phase of its diffusion cycle as a fundamental support of export oriented strategies pursued by Asian economies. Therefore, an array of growth factors is at play to explain the substantial growth of containerization and more interestingly the contribution of these factors in time varies. For instance, in the 1970s, growth involved the substitution of general cargo handling tools by the container. In the 1980s and 1990s the growth of international trade was a strong driver while in the 1990s and 2000s the development of intermediary hubs played a role.

While additional traffic resulting from economic growth is the most salient factor, imbalanced trade flows (empty containers) and the configuration of shipping networks relying on transshipment hubs (double counting of containers) have also contributed to additional containerized flows and port handlings. As economies of scale were applied to maritime shipping, transshipment became more salient. The quantity of containers being transshipped increased from roughly 11per cent of all cargo handled by container ports in 1980 to about 30 per cent in 2010. Also, Figure 2 brings forward questions about the additional growth and diffusion potential of containerization in light of receding economic opportunities (production and consumption) and the expected demographic shifts, namely the stabilization of the global population in the next decades as well as its ongoing aging [START_REF] Lutz | The end of world population growth[END_REF][START_REF] Lutz | The coming acceleration of global population ageing[END_REF]. For instance, if container throughput was to double from the 520 million TEU figures observed circa 2010 to 1 billion TEU, then what would be the level of related consumption and the physical capacity of transport infrastructures to handle such a level of material flows? Obviously, this question cannot be readily answered and on several occasions in the past such points were raised and turned out to be invalid. Still, the K-wave perspective underlines that maturity will eventually be achieved for global containerized freight distribution and that close attention must be paid to identify inflection points. Future expectations about the growth of containerization thus need to be assessed within an economic cycle perspective instead of the rather linear perspectives in which containerization is generally considered.

Identifying Port Containerization Phases

At the global level, the four phases of the product life cycle have been inferred to the development of containerization [START_REF] Notteboom | The Future of Containerization: Perspectives from Maritime and Inland Freight Distribution[END_REF][START_REF] Rua | Fixed Costs, Network Effects, and the International Diffusion of Containerization, FREIT: Forum for Research on Empirical International Trade[END_REF], which follows an S-shaped curve. During the adoption wave of the early 1960s, containerization was still an unproven technology with a few competing standards in terms of sizes and latching systems. The services offered were specific (point to point), mostly using converted cargo ships. Containerization demonstrated that it was achieving productivity gains since it involved a much more efficient form of port operations. During the early 1970s, containerization entered its acceleration wave and was recognized as a promising mean of transportation. New services and consequently new networks were being established, which multiplied its productivity, with growing volumes and the beginning of the application of economies of scale, both for modes and terminals. In the 1980s, pendulum services, which would become the standard network configuration for containerized maritime shipping, were being set across maritime ranges, particularly the Atlantic [START_REF] Slack | Across the Pond: Container Shipping on the North Atlantic in the Era of Globalisation[END_REF].

By the 1990s, containerization became the dominant support of global trade and entered its peak growth wave. Its diffusion was massive, particularly in industrializing economies such as China. Network development was facing growing complexities, which led to the setting of major transshipment hubs reconciling regional and global shipping networks. As economic cycles theory underlines, systems eventually reach a phase of maturity where growth is much less related to diffusion but with standard economic changes and the exploitation of remaining niches, both functional (e.g. the containerization of commodities) and locational (e.g. a new transshipment hub at a low deviation location). It remains uncertain to what extent this phase may have already been achieved in some parts of the world. For instance, the maturity of containerization in Japan appears well established with limited growth prospects and may also emerge in parts of North America and Europe. Yet, in regions such as Latin America, Africa, South Asia and the Middle East, containerization still has substantial potential for growth.

Shifts in Traffic Concentration

The use of concentration measures, particularly the Gini coefficient, is prevalent in port studies, even if the coefficient is a technical measure that does not reveal the processes behind the changes in the level of concentration [START_REF] Notteboom | Traffic Inequality in Seaport Systems Revisited[END_REF]. Based on a sample of ports derived from Containerization International and having a traffic of more than 150,000 TEU in 2010, the number of such container ports grew from 57 in 1970 to 317 in 2010 (Figure 3). Compared to the growth in traffic the number of ports have followed an S-shaped curve as it has increased rapidly in the late 1970s and then slowed down in the 2000s. The number of ports appears to be a leading indicator for the growth of containerized traffic and the recent stabilization in the number of container ports may be indicative of a forthcoming stabilization of containerized traffic.

The concentration of cargo has also changed considerably between 1970 and 2010. The Gini coefficient reveals two distinct periods, mainly resulting from the strategies of global shipping alliances and terminal operators. The first ) is characterized by an uninterrupted trend of concentration of container throughput (from 0.57 to 0.70) with the emergence of mega hubs (e.g. Hong Kong, Singapore, Rotterdam, Hamburg). The second is characterized by a stabilization of the trend with the coefficient hovering around 0.70. This can be attributed to two main trends. The first being the growth of container ports in developing economies, particularly China, and the second is the setting of new transshipment hubs in locations that previously saw limited containerized trade (e.g. Dubai, Gioa Tauro, Algeciras, Panama). The trend of concentration from the 1970s to the 1990s is particularly evident at the top of the container port hierarchy (Figure 4). In 1970 7per cent of all ports (n=4) were handling 25 per cent of the total container throughput. In 1995, only 1 per cent of all ports (n=4) were handling the same traffic share while 7 per cent of all ports (n=28) where handling 50 per cent of the throughput. Since 1995, the trend, as evidenced by the Gini coefficient, has stabilized with around 21 per cent of all ports handling 75 per cent of the global throughput. Thus, between 1995 and 2010 there has been little change in the level of global traffic concentration, underlining that global container port system relies on different dynamics than concentration since the main hubs have already been established.

Waves of containerization Identifying Long Waves

The data for the analysis was taken from Containerization International (CI) figures and retained over five-year increments that include nine periods from 1970 up to 2010. The main selection criteria was that the port should have exceeded a traffic of 150,000 TEU for a year in the time sequence and that data was available for 2010. A total of 316 container ports were retained for the analysis. However, there were 61 missing observations from the CI database out of a total of 2844 (316 ports x 9 time periods), the majority of them for ports of less than 500,000 TEU. For these missing observations a linear interpolation was performed from data available for adjacent years (e.g. calculating missing traffic data for 1985 using available data for 1984 and 1986). This enabled us to keep 61 ports in the sample by inferring traffic with a good level of confidence. If there was more than one missing data in the time sequence, then the port was discarded.

The comparative size of ports requires caution as several ports can be considered more statistical agglomerations than functional entities. For instance, the port of Shenzhen in the Pearl River Delta is composed of several large port facilities (e.g. Yantian, Chiwan, Shekou) that act as distinct operational entities with each port often servicing different hinterlands. The same observation applies to Guangzhou and Shanghai that are multiport (terminal) entities. The agglomeration effect may hide different dynamics and even membership to different waves, but because the data is collected at the port authority level, these nuances cannot be identified and analyzed here.

Applying the HCA methodology to the container port dataset resulted in the selection of seven classes where each component has a similar temporal and growth behavior (Figure 5). This result can be justified by the shape of the bar chart, which reveals the structure of the port dataset. Two ports are considered similar when their growth pattern between 1970 and 2010 are alike. Inversely, two ports are considered dissimilar if their trajectories are not alike. When the increase in dissimilarity level between ports is strong (measured by the intra-class variance), a level where grouping groups of ports that are already homogeneous has been reached. This criterion is taken into account to decide when to stop aggregating groups of ports. In this case, adding more classes, such as going from seven to eight classes does not drop the level of dissimilarity in a significant manner (10.20 to 9.36), while using less than five classes substantially increases dissimilarity (18.95 to 30.96). While seven classes appear optimal, the dendrogram underlines their respective distinctiveness and the relevance of using sub-classes in two cases. Therefore, instead of having seven waves (A to G), five waves can be identified (A to E) with two waves having phases (B.1 and B.2 as well as D.1 and D.2), that are close but differ slightly on the times of adoption and acceleration. The earlier the class branches out on the dendrogram, the more the class is distinct and cohesive. By using HCA, the evolution of the global container port system can be divided into seven classes representing five long waves (Figure 6). Each wave represents a specific temporal growth pattern of containerization. It is worth underlining that the inclusion of a port to a specific class does not imply that the port was not handling container traffic in prior periods, but that traffic growth became significant in the wave the port belongs to. While at the aggregate level the growth of container traffic is high, each class depicts a different growth dynamic reflecting their specific development.

First Wave (A): The Pioneers of the Triad

The first containerized services were established by Sea-Land in the late 1950s and early 1960s (Figure 7). By the 1970s, regular trans-Atlantic (Northern range of Western Europe and American East Coast) and trans-Pacific services (Japan, Australia to a lesser extent and American West Coast) were established through ports that were the first to adopt containerization and are thus considered to be the pioneers (e.g. New York, Yokohama, Oakland and Hamburg). They are almost all part of the economic triad which led globalization; North America, Western Europe, Japan and Australia. These ports accounted for the dominant share (about 80 per cent) of the global throughput in 1970, but this share fell rapidly afterwards. Few of these ports kept their primacy, which is in part explained by the fact that the rationale behind their emergence played to a much lesser extent or that limited room was available for terminal expansion.

A port site could be suitable to a specific volume, but once this volume has been reached, there are limited opportunities for port expansion. For instance, the initial primacy of Oakland was overtaken by the ports of Los Angeles / Long Beach covering a wider regional market and having more room for expansion and better inland connections. The exception is Australia where Sydney and Brisbane remained dominant gateways. The first wave underlines that several pioneers were able to initially capture the opportunities of containerization, but due to market or technical reasons, were unable to keep this initial advantage often because of factors completely outside their control such as a shift in trade patterns. The second wave represents an expansion of containerization within the triad as well as with its regional trade partners. It takes place in two phases with the first (B.1) occurring in the mid-1970s and accounting for a larger share of the world's throughput (Figure 8). While many of these ports were operational in the early 1970s, their market share increased and several became the world's dominant container ports, until overtaken by Chinese ports in the 2000s. This shift took place not because of a lack of growth for wave B.1 ports, but because of differential growth rates. Salient examples include Rotterdam, Tokyo and Hong Kong.

Wave B.2 began in the late 1970s and mostly took place in ports adjacent to the triad, such as in the Caribbean, Latin America, the Mediterranean, and among emerging East Asian tigers (Thailand, Taiwan and Hong Kong). From a pattern dominated by the triad, containerization undertook the initial phase of its global diffusion by embedding itself in commercial relations through a substitution effect of conventional break bulk trades. More intermediary locations in the Middle East (and India to a lesser extent) were also involved since the growth of containerized shipping between Asia and Europe presented opportunities to add port calls along pendulum routes. Figure 8 underlines that this wave (and the first wave as well) is dominantly bound to the dynamics spearheaded by North America, Western Europe and Japan. The third wave concerns the largest number of ports and captures the massive diffusion of containerization, particularly the incorporation of East and Southeast Asia (without China) in global trade relations through the beginning of offshoring as well as the emergence of early transshipment hubs (Figure 9). As the number of container ports increased, the network strategy to serve them favored a shift from point-to-point services along pendulum routes to the usage of hubs-and-spoke services. Indeed, several ports that emerged during this wave became major transshipment hubs through their intermediary locations along major shipping corridors such as Singapore, Colombo, and Dubai. Many of these ports are small market oriented citystates located on the edge of much larger countries (e. g. Indonesia [Singapore], India [Colombo], Iran [Dubai]) not so open to global trade. During that period (mid-1980s) it was difficult to get transport documentation that encompassed all the steps through to the final destination country; so containers were consigned to these hub ports, at which local freight forwarders and banks with good contacts in the destination country took responsibility of the final leg of the journey. This wave is statistically the most distinctive since the growth takes place in a similar time frame (mid-1990s) and at a similar level. The fourth wave concerns two phases, D.1 and D.2, which are similar in their pattern and only different in their sequence (Figure 10). From the mid-1990s, the container became the standard mean for global freight distribution, particularly with the massive entry of Chinese ports in global shipping networks (D.1) and the emergence of post-panamax ships. Furthermore, this wave also represents the increasing regionalization of East Asian countries with the resulting trade growth as one of the drivers of the growth of East Asian ports. Several ports in this wave are new transshipment hubs that have been inserted to accommodate the growing network complexity and to better link regional ports to deep sea services (e.g. Salalah, Gioa Tauro, Colon, Freeport). New gateways are also emerging to accommodate growth in emerging economies (e.g. Vietnam, Mexico, India, Brazil). later stage of that wave (D.2) concerns China gateway ports that provide additional export capabilities (some spillover effect) for massive manufacturing clusters (e.g. Ningbo, Guangzhou). For the rest of the world, they mostly concern additional transshipment hubs being set by global terminal operators (e.g. Tanjung Pelepas). The cumulative waves are depicted in figure 12, which shows global traffic in 2010 and the wave (cluster class) to which each ports belongs. Table 1 provides a synthetic overview of the characteristics and drivers of each wave. The analysis underlines that each wave lasts about 8 to 10 years, which is similar to the observations made by Kondratieff about the phases within a K-wave. The diffusion pattern is hierarchical, starting at the then absolute centers of the global economy (the triad) and adopting a pattern incorporating new locations within their respective spheres of influence. It is thus highly reflective of the change in the hierarchy of trade relations that took place with globalization and containerization.

Recent transitions

The selection of the time frame for the analysis of Changes of Port Share (CPS) relates to the 2000 -2010 period, which on Figure 5 represents a critical juncture as the share of waves A, B and C was waning while wave D was fully asserting itself and wave E would be felt from 2005 onward. Containerization entered the inflection point with total traffic in the sample growing by a factor of 2.32 times over the decade (from 223 to 518 million TEUs), exceeding the combined absolute growth of all the previous decades.

The dataset is taken from Containerization International and includes annual port traffic from 2000 to 2010. Ports that had more than two missing yearly observations were removed from the dataset; a total of 264 ports were retained. 14 missing observations out of 2,904 (264 x 11) observations were estimated through linear interpolation using two adjacent years.

The analysis of Changes of Port Share (CPS) was divided over two periods, 2000-05 and 2005-10 to capture the shift in containerization that has occurred since then. The first period illustrates the fast growth where container ports in developing economies were incorporated in the global trade system (Figure 13) while the second period illustrates the impact of the financial crisis of 2008-09 on a global container transport system that was until then experiencing full-fledged and almost uninterrupted growth (Figure 14). It is important to underline that the results of the analysis of CPS must be interpreted in the global context of the container port system and not within respective regions (or port range).

A look at the ports that have experienced the highest gain and decline in share during the period is revealing as both the ports with the highest gain and decline are dominantly in Asia (Table 2). This underlines a growing divergence in regional containerization dynamics. A prime example of this trend is Hong Kong since the port dominated the Pearl River Delta [START_REF] Wang | The evolution of a regional container port system: the Pearl River Delta[END_REF]. In the 2000s it became the world's major port that has lost the most significant share of its traffic even as this traffic increased from 18.1 million TEU in 2000 to 23.7 million TEU in 2010. This underlines capacity limitations challenges in its ability to service its hinterland in light of the substantial growth of other ports in the Pearl River Delta (Shenzhen and Guangzhou). Med is taking place. The steep decline of the share of North American ports is marginally counterbalanced by the strengthening of niche gateways (Lazaro Cardenas, Savannah, Prince Rupert) and an ongoing rebalancing of transshipment in the Caribbean to the advantage of Panama and Cartagena. There is an overall advantage to South American ports linked to ongoing economic growth, their integration to the global economy, particularly to the trans-Pacific trade.

Conclusion

This paper provides evidence of the K-wave behavior of containerization through a decomposition of its phases. The container, like any technical innovation, has a market and diffusion potential where a phase of maturity is eventually reached. Evidence from the global container port system suggests five distinct waves of containerization with a shift in the momentum from advanced economies to developing economies, but also within advanced and developing economies. This pattern is reflective of the process of economic globalization from the initial dominance of the triad to the incorporation of export-oriented economies in the global trade system and the emergence of major transshipment hubs. These waves span about 8 to 10 years each, which matches the patterns initially identified by the work of Kondratieff.

Changes of Port Share (2005-10)

Transitions, particularly in light of the impacts of the financial crisis of 2008-09, are illustrative of an inflexion and a shift in the patterns of global container port growth.

The tendencies depicted by a shift-share analysis have shown that the core growth has focused mainly on coastal Chinese ports as well as on emerging transshipment hubs along the Asia-Mediterranean trade route. South Asia and Latin America have also fared comparatively well. In the same region, ports like Shanghai, Shenzhen and Ningbo are achieving peak growth while ports like Hong Kong, Singapore and Kaohsiung have achieved maturity. This is illustrative of the economic and manufacturing rebalancing that has taken place in the last decade.

While cyclical behavior has been identified within the global container port system, these waves are simply illustrative of major macroeconomic, technological and sometimes political shifts within the global economy. They do not explain the causes, but simply the consequences of such shifts in the distribution in traffic and growth (or decline). Yet, they provide strong evidence that containerization has a cyclic behavior and that inflection points are eventually reached, marking the end of the diffusion of containerization in a specific port or port range. Future expectations about the growth of containerization thus need to be assessed within an economic cycle perspective instead of the rather linear perspectives in which containerization is generally considered, particularly among commercial actors. This paper focused on the port level. It would be revealing if more detailed analysis could be performed at the port range level as well as if cycles can be observed within the configuration of shipping networks.

Figure 1 :

 1 Figure 1 : The K-wave, Phases of Containerization

Figure 2 :

 2 Figure 2 : Global Trade and Container Throughput (1970=100)

Figure 3 :

 3 Figure 3 : Concentration of the Global Container Port System, 1970-2010

Figure 4 :

 4 Figure 4 : Percentage of Ports Needed to Handle 25%, 50% and 75% of Total Container Throughput, 1970-2010

Figure 5 :

 5 Figure 5 : Hierarchical Cluster Analysis Parameters of Global Container Ports, 1970-2010

Figure 6 :

 6 Figure 6 : Evolution of Absolute and Relative Container Throughput by Hierarchical Cluster Analysis Class, 1970-2010 Figures 7 to 11 each depict the global container traffic at a representative year of the seven waves (each decade from 1970 to 2010). It is important to underline that the classification range of each wave is different to better depict the relative importance of container ports at specific points in time.

Figure 7 :

 7 Figure 7 : The First Wave of Containerization, 1970 -The Pioneers of the Triad Second Wave (B): Adoption in the Triad and its Periphery

Figure 8 :

 8 Figure 8 : The Second Wave of Containerization, 1980 -Adoption in the Triad and its Periphery Third Wave (C): Internationalization

Figure 9 :

 9 Figure 9 : The Third Wave of Containerization, 1990 -Internationalization Fourth Wave (D): Global Standard

Figure 10 :

 10 Figure 10 : The Fourth Wave of Containerization, 2000 -Global Standard Fifth Wave (E); Peak Growth The fifth wave concerns ports that have emerged recently (late 2000s) and is linked to peak growth in global container shipping. It is particularly one of niche ports filling a specific role, such as a new gateway to cope with congestion along a range (Yingkou or Taicang), or a new transshipment hub being inserted within maritime shipping networks (e.g. Tangier Med, Caucedo) (Figure 11). Prince Rupert in Canada is the only port of significance that has emerged on the North American west coast in recent

Figure 11 :

 11 Figure 11 : The Fifth Wave of Containerization, 2010 -Peak Growth

Figure 12 :

 12 Figure 12 : Waves of Containerization, 1970-2010

Figure 13 :

 13 Figure 13 : Changes of Port Share (CPS), 2000-05

Figure 14 :

 14 Figure 14 : Changes of Port Share (CPS) 2005-10

Table 1 :

 1 The Waves ofContainerization, 1970Containerization, -2010

		First Wave (A) Second Wave	Third Wave	Fourth Wave	Fifth Wave (E)
			(B.1 & B.2)	(C)	(D.1 & D.2)
	Period	1956 (1965) -	1970-1980 (B.1)	1980-1990	1995-(D.1)	2005 -
		1975	1975-1985 (B.2)		2000-(D.2)
	Overview	Pioneer ports	Expansion of the	Large diffusion	The container	Peak growth
		setting	triad and its	in new	becoming the	and the setting
		containerized	trade partners	international	standard	of niches.
		operations in	(Caribbean,	markets (Latin	transport
		the economic	Mediterranean,	America,	support of the
		triad (North America, Western	Asian Tigers).	Middle East / South Asia, Southeast	global economy. "The China Wave"
		Europe and		Asia).
		Japan).
	Driver	Early trade	Adoption of	Setting of	Expansion of	Spillover effect
		substitution.	containerization.	global supply	global supply	and new
				chains. Setting	chains. China	transshipment
				of	and	hubs.
				transshipment	transshipment
				hubs.	hubs.
	Representative	Antwerp, New	B.1: Rotterdam,	Singapore,	D.1 Shanghai,	Tangier Med,
	ports	York, Los	Tokyo, Hong	Colombo,	Shenzhen, Gioa	Caucedo.
		Angeles,	Kong	Busan, Dubai,	Tauro	Yingkou, Prince
		Oakland,	B.2: Kaohsiung,	Algeciras	D.2: Ningbo,	Rupert
		Nagoya	Jeddah,		Tanjung
			Kingston		Pelepas

Appendix. List of ports

First Wave (A) n=36

Antwerp, Los Angeles, New York-New Jersey, Bremen-Bremerhaven, Felixstowe, Nagoya, Zeebrugge, Le Havre, Oakland, Melbourne, Seattle, Sydney, Hampton Roads, Montreal, Auckland, Gothenburg, London, Miami, Liverpool, Fremantle, Teesport, Anchorage, Helsingborg, Esbjerg, Philadelphia, Lyttelton, Belfast, Grangemouth, Adelaide, Hull, Dunkirk, Yokkaichi, Boston, Cork, Cadiz, Waterford Second Wave (B.1) n=40