L Denant-Boèmont
email: laurent.denant-boemont@univ-rennes1.fr

S Hammiche

Richard Arnott

Rachel Croson

Michael Pickhardt

Public Transit Capacity and User's Choice: An Experiment on Downs-Thomson Paradox

We study the Downs-Thomson paradox, a situation where an additional road capacity can cause an overall increase in transport generalized cost and therefore a decrease in welfare for transport users. To this end, we build an experiment based on a double market-entry game (DMEG) where users have to choose between road and public transit after that the operator has choosen public transit capacity. The optimal strategy for operator is to minimize capacity, and the equilibrium for users depend on the endogeneous public transit capacity compared to exogeneous road capacity. The most important result is that we observe the Downs-Thomson paradox empirically in the laboratory: An increase in road capacity causes shift from road to rail and, at the end, increases total travel costs. But the contrary is not true: A decrease in road capacity does not cause lower total travel costs, which is in contradiction with our theoretical model. Results also show that the capacity chosen by operator di¤ers from Nash prediction, levels being signi…cantly higher than those predicted by our model. Moreover, users coordinate remarkably well on Nash equilibrium entry rate while capacity has been chosen by operator.

Introduction

In transport economics, an increase in road capacities, by causing shifts from public transit to private transport, could lead to a new tra¢ c equilibrium where total transport costs are higher. Indeed, by implementing an additional road capacity (eg a new route or more generally a new transport alternative), road speed increase, attracting public transit users towards road transport. Then, as public transit tra¢ c decreases, there is a loss of revenue for public transit operator, who, in the absence of subsidies, might raise faire or cut service. This well-known phenomenon has been called "Downs-Thomson"paradox [START_REF] Downs | The Law of Peak-Hour Expressway Congestion[END_REF][START_REF] Thomson | Great Cities and their Tra¢ c[END_REF][START_REF] Mogridge | The Downs-Thomson Paradox and the Transportation Planning Process[END_REF]Mogridge, 1990a,b, Mogrigde, 1997 ;[START_REF] Arnott | The Economics of Tra¢ c Congestion[END_REF][START_REF] Jones | Options for reducing tra¢ c congestion: an overview[END_REF][START_REF] Litman | Generated tra¢ c and induced travel[END_REF]. Such a phenomenon is an important argument for people who are defending urban pricing for private cars and consequently investment in (collective) high-capacity systems, in order to increase journey speeds (See Webster, 1985).

Although the possibility of the Downs-Thomson (DT) Paradox seems to be well accepted, there is little evidence in the literature to indicate when it might occur. [START_REF] Holden | Wardrop's Third Principle: Urban Tra¢ c Congestion and Tra¢ c Policy[END_REF] suggests that it may occur "in a city like London, where a signi…cant fraction of peak tra¢ c is carried on an extensive rail network". [START_REF] Mogridge | The Downs-Thomson Paradox and the Transportation Planning Process[END_REF] suggest that it may occur by "allocating even more space to roads when roads are a less e¢ cient carrier of the ‡ow of tra¢ c". These statements are mainly qualitative, based on intuition and experience.

Economic experiments about congestion tend to be in a growing number [START_REF] Selten | Commuters Route Choice Behaviour[END_REF]Ziegelmeyer et al., 2007 ;Hartmann, 2006 ;Rapoport et al., 2005 ), mainly about road tra¢ c (departure time or route choice) but, to our knowledge, no experiment exists on the possible interactions beetween public transit and road tra¢ c, as this interaction could be for instance investigated in the Downs-Thomson Paradox.

The aim of this paper is to provide empirical support about this paradox. To this end, we build an economic experiment in which subjects have to choose between two markets, e. g road and public transit. Our theoretical framework is built upon a class of coordination games named Market Entry Game [START_REF] Selten | Equilibrium Point Selection in a Class of Market Entry Games[END_REF][START_REF] Gary-Bobo | On the existence of equilibrium points in a class of asymmetric market entry games[END_REF]. Basically, in a MEG, players have to choose to enter a market or not to enter: The payo¤ is a decreasing function of the number of entrants whereas option to stay out gives a constant payo¤. In such games, Nash equilibriums will give excess entry, implying a social dilemma.

Such games are a very simple and accurate stylisation of congestion process that frequently occur in the transport …eld, …rstly because such game implements a negative externality of entry decision on individuals'payo¤s that are to enter, which implies an e¢ ciency problem. Secondly, at a theoretical level, such games are charactherized by many equilibriums that imply high levels of coordination for players, such equilibriums being not welfare maximizing.

The main problem if we consider transport choice is that users can choose between alternatives (routes, modes, departure times, etc.). The idea is that all of these alternatives give to users a certain utility, this utility being potentially a¤ected by congestion, or, more generally, by the total number of users who choose this speci…c alternative. It is not the case in the usual MEG. As we are interested in choices between modes, we use a Double Market Entry Game framework in order to deal with such choice alternatives, where the user can choose between two options, both being a¤ected by tra¢ c level on each option. The second originality of our theoretical framework is to deal with crossed-externalities, e.g a situation where the choice of player i choosing X a¤ects the situation of player j, even if player j is to choose the Y option. Last but not least, the externality produced by one market is negative, whereas the other market gives a positive externality for individuals choosing it.

But such phenomenons of crossed-externalities are not relevant only for transport …eld, of course. For instance, recently, some politicals in France claim to decrease levels of taxes on oil in order to absorb the impact of exploding oil prices for households. The impact of that could be that, for residential heating especially, households are …rst incited to consume more oil and second, households are incited to di¤er decisions which could reduce oil consumption, as solar or geothermic solutions for heating. This will produce additional negative externalities (CO2 emissions and so on) and also reduce positive externalities linked to the choice of more ecological solutions. Many examples of such phenomenons could be observed in economics, and our theoretical model can be used for investigating situations that do not deal only with transport economics.

Our experimental game is a two-stage game where a …rst mover A (the operator) has to choose the public transit capacity. Then, given a road capacity that is …xed exogenously by the experimenter (the planner), subjects B (travellers) have to choose between road transport (option X) and public transit (option Y). As in a usual MEG, the payo¤ from the …rst market decreases with number of entrants and increases with exogenous capacity. For the second market (public transit) things are quite di¤erent: The payo¤ from the second market increases both with the number of entrants and with the capacity chosen by the operator, i.e. player A. Thus, entry generates negative externality on the road market, and, on the contrary, entry generates a positive externality on the public transit market. At the tra¢ c users Nash equilibrium, the entry rate on road increases with exogenous capacity of road and decreases with endogenous capacity of public transit. As the marginal revenue of public transit entry is less than the marginal revenue of road entry, the optimal strategy for player A is to …x the lowest possible level for public transit capacity; The Subgame Perfect Equilibrium corresponds to a situation where player A chooses the minimum capacity, implying too much entry on road and lack of e¢ ciency for the transport system (Transport cost is not minimized at the tra¢ c equilibrium). In such a game, an exogenous increase in road capacity generates shifts from public transit to road that leads to a new tra¢ c equilibrium where e¢ ciency level is lower.

Our experimental design consists in groups of 15 subjects with 1 subject A and 14 subjects B. At each period, subjects A and B play a two-stage game where subject A chooses …rst a capacity level for public transit. Being informed about A's choice, subjects B have to choose between two options, X (road) and Y (public transit). In each session, subjects play 2 treatments of 20 periods (within-subject design). For some sessions, subjects play …rst 20 periods of low capacity for road market and second 20 periods of high capacity for road market (condition ADD). In other sessions, subjects play …rst the high capacity condition during 20 periods and then play 20 periods in low capacity condition (condition DEL).

16 sessions (8 sessions in the ADD condition and 8 sessions in the DEL condition) of 15 subjects (namely 240 participants, students from various origins, economists, lawyers, psychologists, etc.) have been held in LABEX (LABoratory for EXperiments in economics and management), University of Rennes 1 from January to April 2008 under the ZTREE platform (see [START_REF] Fischbacher | z-Tree: Zurich Toolbox for Ready-made Economic Experiments[END_REF].

Our main results indicate that participants A reach remarkably well the entry rate on each market that is predicted by Pure Strategy Nash Equilibrium. It is far from being the case for participant A, who chooses average capacity level signi…cantly above the optimal level given by Nash equilibrium. Moreover, we observe Downs-Thomson Paradox in the laboratory, e.g. an increase in road capacity for a given group does increase total travel times and therefore is associated to lower e¢ ciency levels. But we do not observe the reverse phenomenon, that is a decrease in road capacity does not decrease signi…cantly total travel times.

In a …rst section, we will make some literature review concerning our initial question (Downs-Thomson Paradox). The second section is devoted to our theoretical model, which is a Double Market Entry Game both with a negative and a positive externality. Then, we present in a third section the experimental design, and parameters that had been implemented in the lab. The fourth section presents experimental results and the last one is to conclude.

Downs-Thomson Paradox and Mogridge' s conjecture

Martin J.H. Mogridge in his 1990's book "Travel in towns: Jam yesterday, jam today and jam tomorrow? " observes that, especially in London, increasing road capacity tends to increase tra¢ c congestion: all road investment in a congested urban area, will reduce the average speed of the transport system as a whole -for road and public transport (e;g. urban mass transit). He therefore conjectures that improving collective transport could increase welfare. In fact, Mogridge's analyze is a renewed version of the well-known Downs-Thomson paradox [START_REF] Downs | The Law of Peak-Hour Expressway Congestion[END_REF][START_REF] Thomson | Great Cities and their Tra¢ c[END_REF].

Arnott & Small (1994) made an extensive survey about the most famous transport paradoxes. The Downs-Thomson is a counterintuitive answer to a very simple question: If urban road capacity is increased, does this result in some improvement in tra¢ c speeds, or does it make congestion worse?

Shortly, the answer given by the Downs-Thomson Paradox [START_REF] Downs | The Law of Peak-Hour Expressway Congestion[END_REF][START_REF] Thomson | Great Cities and their Tra¢ c[END_REF][START_REF] Mogridge | Travel in Towns: jam yesterday, jam today and jam tomorrow? (or how to improve tra¢ c speeds in Central London)[END_REF][START_REF] Mogridge | The Downs-Thomson Paradox and the Transportation Planning Process[END_REF] is that an increase in road capacities, by causing shifts from public transit to private transport, could lead to a new tra¢ c equilibrium where total transport costs are higher. The consequence is then a decrease in welfare level for the conurbation.

The argument is that, by implementing an additional road capacity (eg a new route or more generally a new transport alternative), road speed increase, attracting public transit users towards road transport. The consequence is, as public transit tra¢ c decreases, there is a loss of revenue for public transit operator, who, in the absence of subsidies, might raise faire or cut service. As [START_REF] Mogridge | The Downs-Thomson Paradox and the Transportation Planning Process[END_REF] wrote :"This states that, in congested situation, the equilibrium travel costs will rise if road capacity is increased, if the cost of collective network rises as ‡ow falls. If follows that increasing road capacity in congested situations is counterproductive" 1 .

This paradox regained some interest with the analysis of Martin H. Mogridge in the 80s, because what was called "Mogridge conjecture" was an important argument for implementing urban pricing in London city 2 . The Mogridge's conjecture was that "in conditions of suppressed demand, the speed of the road network is determined by the speed of the high-capacity network (rail, bus, etc.)" [START_REF] Mogridge | The self-defeating nature of urban road capacity policy[END_REF][START_REF] Mogridge | Travel in Towns: jam yesterday, jam today and jam tomorrow? (or how to improve tra¢ c speeds in Central London)[END_REF].

More precisely, this paradox is the consequence of users equilibrium and of Wardrop's …rst principle (users reach an equilibrium where travel costs is equal for each alternative). Such a paradox is the consequence of induced/latent demand e¤ect [START_REF] Arnott | The Economics of Tra¢ c Congestion[END_REF]Abraham & Hunt, 2001 ;Nolan, 2001 ;A…meimounga et al., 2005).

To our knowledge, the only empirical evidence about Downs-Thomson had been given by [START_REF] Mogridge | Planning for optimum urban e¢ ciency; the relationship between congestion on the roads and public transport[END_REF][START_REF] Mogridge | The self-defeating nature of urban road capacity policy[END_REF]. Mogridge asserted that the decrease in speeds in London centre was the consequence of Downs-Thomson Paradox, e.g. that increase in road network capacity over time causes decline in tra¢ c for mass transit (train), the global consequence being a growth of travel times for both users (see table 1

below)

Insert table 1 1 Mogridge, 1997 p. 9. 2 « The …rst (paradox) is that in congested conditions, building more road capacity for cars makes both motorists and users of public transport worse o¤ . By encouraging a shift from public transport to cars it …lls up the new road space, makes public transport less frequent and more expensive, and results in a new equilibrium that is slower for all. The second paradox follows from the …rst: taxing the ine¢ cient road user (the motorist) and subsidising the e¢ cient (on buses and trains) will make all travellers better o¤ » The Times, March, 17th, 2000.

Mogridge conjectures then that a decrease in road capacities, or better, an increase in mass transit capacity, could shift road users to train and can therefore decrease total travel times for big conurbations. His argument was one important element in favour of London urban pricing that had been implemented some years after.

The problem of such empirical evidence is that it is almost impossible here to distinguish precisely the variables that could in ‡uence average travel time, because the ceteris paribus assumption is not respected. That is one major reason for using experimental economics in order to give empirical evidence to DT paradox, because in the lab, it is possible to isolate the impact of changing road capacity on users'choices and then on total travel times. The main question concerning implementation in the lab is about the accurate theoretical model to obtain experimental congestion, that is to say a divergence between private cost and social cost of transport as a consequence of a coordination problem between transport users.

Theoretical model: The Double Market Entry Game

Our theoretical model is an extension of Market Entry Game (Selten andGüth, 1982 and[START_REF] Gary-Bobo | On the existence of equilibrium points in a class of asymmetric market entry games[END_REF]). In our game, two types of agents, A and B have to make choices. The individual A is the public transit operator, and has to choose capacity level for market 2 (public transit).Then, individuals B (transport users) have to choose between market 1 (road transit) and market 2 (public transit). We will assume that market 1's capacity is given for transport users and that market 2's capacity is chosen by public transit operator. Our game is sequential, e. g. individual A chooses capacity of market 2, other paramaters being common knowledge. In the second step, users know capacities for each market, and choose to enter market 1 or market 2. Using backward induction, we will assume that, if public transit operator knows how players B choose between market 1 and market 2, he should choose the capacity level that maximizes his payo¤.

Users'tra¢ c equilibrium (one-shot game)

Players B have to choose simultaneously between the option X (Road or Market 1) and Y (Public Transit or Market 2). Di¤ering from the usual MEG, there is no outside option which gives a given payo¤ for sure (see above).

The individual payo¤ for a user choosing road transit is 3 :

i 1 = k 1 + r 1 (c 1 m 1 ) if i = X (1)
whereas the individual payo¤ of using public transit is:

i 2 = k 2 + r 2 (c 2 + m 2 ) if i = Y (2)
Where c 1 and c 2 are respectivly the capacities of road and public transit, m 1 and m 2 are respectivly the number of users choosing market 1 (road) and market 2 (public transit), r 1 , r 2 ; k 1 , k 2 are positive parameters with r 1 > r 2 and k 1 > k 2 : All these parameters are to be known by all participants B before they choose which market to enter 4 . Such information is also given to participant A (the operator for market 2).

Constraints are:

m 1 + m 2 = n (3)
or equivalently

m 2 = n m 1 (4)
and that

c 1 + c 2 < n ( 5
)
Where n is the …nite number of transport users.

In this model, using road transit is supposed to create a negative externality, as in the usual MEG, whereas using public transit is supposed to create a positive externality. This positive externality is linked to the fact that, if capacity is …xed, the increase in public transit enables to increase frequencies and then decrease time travel cost (see Arnott and Small, 1994 5 ).

Users equilibrium is reached when:

k 1 + r 1 (c 1 m 1 ) = k 2 + r 2 (c 2 + m 2 ) (6)
Then, combining equns. ( 8) and ( 10), and as r 1 + r 2 6 = 0;we have:

m 1 = k 1 k 2 r 2 (n + c 2 ) + c 1 r 1 r 1 r 2 (7)
And consequently

m 2 = 1 r 1 r 2 (k 2 k 1 + nr 1 c 1 r 1 + c 2 r 2 ) (8)
The entry rate (tra¢ c) on market 1 (market 2) is respectively increasing (decreasing) with c 1 ; k 1 ; r 1 and n, decreasing with c 2 ; k 2 and r 2 : Of course, such a theoretical prediction holds for a continuum of agents. In the experimental game, theoretical predictions will be made without this assumption (see below).

Optimal choice for public transit operator

We assume that the pro…t for public transit operator is simply the di¤erence between the number of public transit users and the capacity he chooses, e.g.:

= m 2 c 2 (9)
Thus, replacing m 2 with equ. (8), and using eqn [START_REF] Gary-Bobo | On the existence of equilibrium points in a class of asymmetric market entry games[END_REF], we obtain

= n k 1 k 2 r 2 (n + c 2 ) + c 1 r 1 r 1 r 2 c 2 = n c 2 k 1 r 1 r 2 + k 2 r 1 r 2 + n r 2 r 1 r 2 c 1 r 1 r 1 r 2 + c 2 r 2 r 1 r 2
If we assume that r2 r1 r2 < 1, the optimal solution is a corner solution where c 2 equals zero. Then, in the one-shot game, the theoretical prediction based on pure strategy Nash equilibrium is that the operator chooses the minimum capacity level and consequently, users coordinate to enter for some of them on "road market" and for the others on "public market", entry rates for road and public transit being given respectively by eqns [START_REF] Gary-Bobo | On the existence of equilibrium points in a class of asymmetric market entry games[END_REF] and [START_REF] Güth | An Experimental Analysis of Ultimatum Bargaining[END_REF]. If players know that the one-shot game is to be repeated a …nite number of times, then the equilibrium for the one-shot game should be implemented at each period6 .

Downs-Thomson (DT) Paradox

The Downs-Thomson Paradox is a situation where welfare is to be decreased when road (market 1) capacity increases, the other market (public transit) capacity remaining constant. This occurs because too many users shift from road market to public transit market.

The level of welfare in the Double Market Entry Game is simply the sum of all players payo¤s, that is:

W = (m 2 c 2 ) + m 1 (k 1 + r 1 (c 1 m 1 )) + m 2 (k 2 + r 2 (c 2 + m 2 )) (10)
Or, equivalently, given eqn ( 8)

W = (n m 1 c 2 ) + m 1 (k 1 + r 1 (c 1 m 1 )) + (n m 1 ) (k 2 + r 2 (c 2 + n m 1 )) (11)
Furthermore, we obtained previously the expression [START_REF] Gary-Bobo | On the existence of equilibrium points in a class of asymmetric market entry games[END_REF] on entry rate in market 1 (road), which could be replaced in the former equation giving welfare. After simpli…cation, it gives:

W = 1 r 1 r 2 k 2 k 1 + nr 1 c 1 r 1 c 2 r 1 + 2c 2 r 2 + n 2 r 1 r 2 nk 1 r 2 + nk 2 r 1 nc 1 r 1 r 2 + nc 2 r 1 r 2 (12
) Assume that capacity c 2 is to be constant, since we are for the moment not interested in the behavior of player A. Let call W HIGH the level of welfare obtained in a situation where road capacity is high, say c 1 and W LOW the level of welfare obtained in a situation where road capacity is low, say c 1 . We have then c 1 > c 1 :

If the DT Paradox is to occur, then an exogeneous increase in road capacity should decrease welfare level. If we compute the variation in welfare level from low capacity to high capacity, we have, after manipulation:

W = W HIGH W LOW = r 1 c 1 c 1 r 1 r 2 (nr 2 + 1) < 0 ( 13
)
This expression is negative, since all parameters are positive in the DMEG, i.e. increasing road capacity causes a decrease in welfare, which gives a theorem and its corollary.

Theorem 1 In the Double-Market Entry Game, an increase in market's 1 capacity, the other market capacity remaining constant, will decrease level of Welfare by shifting users from market 2 to market 1 (Downs-Thomson Paradox)

Corollary 2 A decrease in Market's 1 capacity will increase Welfare level.

Such a theoretical result is precisely what is to be tested in our experimental treatments.

Experimental design

In this section, we present the di¤erent experimental treatments that have been implemented in the lab, and therefore, given the speci…c values we had for parameters, we give the experimental predictions implied by the theoretical model presented above.

Experimental treatments

In each session, a group of 15 subjects participate to the double market-entry game described above. At the beginning of the experiment, a randomly chosen subject plays role A and the others subjects play role B. Roles remain constant throughout the session (partners design, one subject A and 14 subjects B). Each participant plays 40 periods of a two-stage game. For each period, the two-step game is the following: In the …rst step, subject A choose a positive integer number concerning level for market 2, from 1 to 11. Then, participants B are informed about A's choice, and have subsequently to choose to enter market 1 or market 2. At the end of the period, all participants are informed about the current number of entrants on market 1 and market 2, about the payo¤s of each participant (payo¤s are symmetrical for participants B). Then, a new period begins.

In each session, participants plays two subsequent games for 20 periods. Instructions speci…ed that they will play two games of 20 periods subsequently. The only di¤erence between the two games is the capacity of market 1, which could be high or low (see table below about the parameters that have been used). For 20 periods, participants play …rst low capacity and then high capacity in some sessions (condition ADD) and, in other sessions, participants play …rst high capacity and then low capacity (condition DEL). There is no contextualization or framing in the instructions that do not talk about transport. Instructions refer to option X (market 1 or road) and option Y (market 2 or public transit) that has to be chosen by participants B. Participants B are aware about the value of c 2 chosen by participant A, which is not described to be an operator who has to choose a given capacity. At the end of the experiment, all participants answer to post-experimental surveys, one to elicit individual risk aversion 7 and the other to have some information about socioeconomic characteristics and to have some debrie…ng from subjects. The parameters that had been implemented in the experiment are the following:

Insert Table 2 about here In the experiment, participants payo¤s are given in points, and an exchange rate in Euros is announced for each point to be gained …nally at the end of the experiment. In order to avoid income e¤ect, and because losses were possible in a given round for any participants, 4 rounds among the 40 that have been played have been randomly chosen to determine the …nal payo¤, and we add as usual a participation fee.

Theoretical equilibriums

In this particular kind of game (multiple stage games with observed actions), the equilibria are obtained by implementing Subgame-Perfect Equilibrium (SPE) method. A strategy pro…le is a subgame-perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. As often, we use the common method of backward induction in order to determine subgame perfect equilibria. Given our particular parameters (see table 2), the SPE corresponds to a capacity choice of c 2 = 1 for player A and to an entry rate of 6 players B on market 1 (road), e.g. 8 players B on market 2 (public transit) when road capacity is equal to 3 (treatment LOW ). When road capacity is equal to 6 (treatment HIGH ), the optimal strategy remains to choose c 2 = 1 for player A, but the entry rates are changing, giving 10 players B on market 1 and consequently 4 players B on market 2. Of course, there is also di¤erent equilibria for subgame B depending on A's choice about capacity of market 2.

For instance, assume we have 7 players B on market 1 and 7 players B on market 2 after player A chose c 2 = 1. Each player B gains 2 points. This is not a Nash equilibrium since a player B could gain 0.25 point if she deviate and choose Y (2.25 points for option Y and 3 points for option X). At this point, we have 6 players B on m1 and 8 players B on m2. If an additional player B deviate to choose Y, he will gain 0.25, obtaining 2.5, but leaving 3 not to choose X. Then, for c 2 = 1, Nash Pure Strategy equilibria for subgame B imply 6 players B choosing X and 8 players B choosing Y.

For c 2 levels higher than 1, the number of entrants on m 1 decreases.

The following table summarizes theoretical predictions given the speci…c values of parameters chosen in the experimental design.

Insert table 3 about here

The increase in road capacity makes road choice more attractive, implying 4 additional users in HIGH treatment compared to the LOW treatment. The consequence is that payo¤s for all individuals decrease, and consequently the e¢ ciency level at the asymmetric Pure Nash Equilibria is lower in the HIGH treatment. As the maximum level of welfare should be obtained by choosing maximum capacity for public transit, with no user entering on road, the change in road capacity does not a¤ect e¢ ciency level.

Experimental results

Sessions have been held in LABEX, Rennes, from January to April 2008, under the ZTREE platform [START_REF] Fischbacher | z-Tree: Zurich Toolbox for Ready-made Economic Experiments[END_REF]. The average duration of a session was 1h30 for an average payo¤ of 15 euros for participants B, 20 euros for participant A. There had 16 sessions of 15 participants, ie 240 subjects. 8 sessions were made in the ADD condition and also 8 sessions were made in the DEL condition.

Capacity choices for public transit

Here we analyse subject A'choice about the capacity level for market 2 (public transit) and its determinants. Let remind that theoretical predictions given above indicate that public transit operator capacity choice should not be in ‡uenced at the end by the exogeneous road capacity. In all treatments, subject A should choose the minimum capacity in order to maximize her pro…t.

The main results concerning capacity choice by subject A are the following. First, there is no signi…cant di¤erence about the level of capacity chosen by A for public transit for treatments HIGH and LOW, as theoretical model suggests. Second, the average capacity chosen by A tend to be signi…cantly higher than levels to be predicted by theoretical model. But, it decreases with repetition, suggesting some kind of learning. Finally, it has to be noticed that there is considerable heterogeneity within and between the individual data.

Actually, average capacity chosen by subjects A is given in the following table. For ADD condition, the average capacity for market 2 is around 3.79 whereas it is 3.75 for DEL condition. The data analysis in the table below indicates that individual behaviours are quite heteregeneous, especially in the DEL condition for LOW treatment, where some subjects choose repeatedly the minimum capacity whereas other subjects tend to focalise on a "middle-point" level for capacity c 1 = 6 (HIGH treatment).

Insert table 4

The average capacity chosen in LOW treatments is not signi…cantly di¤erent from the average capacity chosen in HIGH treatments. A bilateral Mann-Whitney Wilcoxon rank-sum test about the equality of average capacity chosen in the ADD condition for LOW treatment to the average capacity chosen in the DEL condition for HIGH treatment could not reject the assumption (z = 0:630, p = 0:529). A similar test assuming the equality of capacity chosen for LOW treatment in the DEL condition to capacity chosen for HIGH treatment in the ADD condition gives comparable results (z = 0:421, p = 0:674).

If a within-subjects comparison is conducted, it is possible to observe that average capacity tend to be higher in the …rst treatment compared to the second treatment, as it is suggested by graphs below, especially for the DEL condition (see also table 4 for other evidence).

Insert …gure 1 about here Insert …gure 2 about here Such an intuition is con…rmed by non parametrical analysis (Wilcoxon matched pairs test). In ADD condition, there is a signi…cant di¤erence about capacity chosen by A depending on capacity level c1. Average capacity is higher in « LOW » treatment compared to « HIGH » treatment (Wilcoxon matched-paired sign rank, bilateral, z = 2:106, p = 0:0352**).

In DEL condition, there is also a (weak) signi…cant di¤erence LOW and HIGH treatments (Wilcoxon, z = 1:820, p = 0:0687*), average capacity chosen by subject A being higher in HIGH treatment compared to LOW treatment.

Such a result is con…rmed if data are pooled together (sessions ADD and DEL) and if a Wilcoxon matched-pairs signed rank test is conducted. The assumption is that average capacity chosen by A during the …rst 20 periods is equal to the average capacity chosen by A during the last 20 periods. Such an assumption is strongly rejected (bilat. Wilcoxon, z = 2:795, p = 0:0052***), Another result is that entry rate on road tends to increase over time, indicating some learning process for participants B. Throughout the repetition of game, they clearly understood that they tend to underenter on road market, and they change gradually their choice to match Pure Strategy Nash equilibrium entry rate. But such a result is statistically signi…cant only for the HIGH treatments: Entry rate in ADD condition for the high treatment (part 2 of the experiment) is higher than the entry rate in DEL condition for the same treatment (9.43 compared to 8.63, bilateral Mann-Whitney, z = +2:298 ; p = :0027, signi…cant at the 1% level). But such a result has to be cautiously interpretated, because it could also be the consequence of participants heterogeneity.

In order to have more details about participant B behaviour, we conduct a parametrical analysis concerning variables that could in ‡uence B's choice. This parametric analysis explains factors that in ‡uence the probability p to enter on market 1 (road) for a participant B in period t (Panel Data analysis). Obviously, probability (1 p) will be the probability not to enter on market 1, and therefore to enter on market 2, since there is no outside option in our game. The results of this probit analysis are given in the following table.

Insert table 6 around here

The dependant variable is the probability to enter on market 1 in period t for a participant B. The explanatory variables are the following. c t 1 is the capacity of market 1 in period t, c t 2 is the capacity of market 2 chosen by participant A in period t, and c t 1 2 is the same lagged variable. m t 1 1 is the observed number of entrants in market 1 at the last period, RISK is the level of CRRA index for the participant8 , t 1 i is his payo¤ in the last period, CON D is a dummy variable concerning experimental condition (=1 if ADD and =0 if DEL) and t relates to the time variable.

When we analyse entry for B subjects, we …nd some intuitive results, that are in line with our theoretical predictions. Entry rate on a market depends positively on its capacity and negatively of the other market capacity. Moreover, previous choice of capacity for participant A in ‡uences entry choice for participant B: The more she chose in the previous period, the less he enters on road. An other interesting result concerns users' coordination during the experiment. The number of entrants in previous period increases the probability to enter in the current period (signi…cant at the 5% level for DEL condition and for the pooled data, but not for the ADD condition). Subjects B anticipate that high previous entry rate implies low entry rate for the current period and vice versa (they anticipate some cyclical oscillation around an average entry rate on road). There is also learning processes, since entry on road tends to increase with repetition (subjects understand that road choice is risky but no so much as anticipated (possible negative payo¤s)). As before, such a variable is signi…cant at the 5% level for DEL treatment and for pooled data, but not for ADD treatment. Then the above result is con…rmed by parametric analysis. Last but not least, the more participant risk averse, the less the probability to enter.

Payo¤s and Welfare

As road capacity grows, shifts from public transit to road choice may …rst decrease congestion risk and so increase payo¤s for those who choose road. All things being equal, higher capacity should increase group welfare. Of course, such assumption does not hold, since additional users on road may decrease the level for the positive externality of public transit. Moreover, if public transit operator reduce her capacity in order to decrease her operating costs, her pro…t may decrease also. At the end, these two negative e¤ects could decrease welfare level for the entire group.

Pro…t for Public Transit Operator

We have observed previously that increasing road capacity does not change signi…cantly capacity choice for subject A. But, it changes drastically the distribution of subjects B by encouraging them to choose road more frequently. The consequence is that, clearly, pro…t for A (i.e. M 2 c 2 ) decreases.

Watching the experimental data, the important result is that pro…t for B is higher in the LOW treatment compared to the HIGH treatment (see …gure 5).

Insert …gure 5

That is precisely one component of the Downs-Thomson paradox. Increasing road capacity will make public transit users shifting to road, which precisely decreases revenue for operator 9 . This empirical result is clearly established with non parametrical tests, for within and beetween-subject analysis. A Wilcoxon matched-paired sign rank test for each condition establishes a signi…cant di¤erence about A's pro…t between LOW and HIGH treatments (bilateral, z = 2:521, p = 0:0117, signi…cant at the 5% level). The positive value of z means that A's pro…t is signi…cantly higher in LOW treatment for both conditions. Such a result is con…rmed by between-subject analysis. A Mann-Whitney Rank Sign Test about the equality of A's pro…t for treatment LOW for ADD condition and for treatment HIGH for DEL condition is strongly rejected (z = 3:046, p = 0:0023, signi…cant at the 1% level). We …nd a comparable result by assuming the equality of average A's pro…t for treatment LOW for DEL condition and for treatment HIGH for ADD condition (z = 3:363, p = 0:0008, signi…cant at the 1% level).

Group Welfare

The following graphs show the evolution of total group payo¤ throughout the periods of the experiment. The total group payo¤ for a given group at period t is:

W t = t A + m t 1 t B1 + n m t 1 t B2 (14)
Average group payo¤ depending on period is given in …gure 6. In this …gure, the green dashed-line recalls maximum e¢ ciency level, while the red dashed lines recall e¢ ciency level predicted by Pure Asymmetric Nash equilibria.

Insert …gure 6

The average Group payo¤ for LOW treatment in ADD condition is 42.6 (indicated by the blue line), which is remarkably near Nash equilibrium e¢ ciency level. In the HIGH treatment, average group payo¤ is around 32, which is higher than Nash equilibrium level [START_REF] Selten | Equilibrium Point Selection in a Class of Market Entry Games[END_REF], but not signi…cantly. Clearly, welfare level decreases when road capacity is to be increased. A non parametrical test con…rms this intuition (Wilcoxon, p = 0.0078***) in ADD condition.

Insert …gure 7 about there In the DEL condition, the average group payo¤ is around 39.8 for HIGH treatment, to be compared to 38.7 in LOW treatment. Actually, there is no signi…cant di¤erence about welfare between the treatments. The decrease in road capacity does not succeed to increase signi…cantly payo¤s both for subjects B and subject A (theoretically, NE should give 28 points in HIGH treatment and then 40 points in the LOW treatment). This is because Group payo¤ in the …rst treatment (HIGH) is quite high compared to the Nash prediction (around 40 on average to be compared with 28 theoretically). This relatively high level of welfare results essentially from participant B's behaviour. As he is not extracting all the rent (he should gain 3 points in the HIGH treatment and he actually gains 1 point) by choosing high capacity levels, his payo¤ is very low. But these higher capacity levels do not give higher entry rate for market 2 (compared to Nash Perfect Subgame equilibrium), which is around 6. That means participants B who choose market 2 increases their payo¤ because capacity is higher for the market they choose.

The last result is about the interaction beetween participants A and participants B. It is possible to see that participants B coordinate quite well on subgame Nash equilibrium after participant A chose market 2 capacity, as it can be viewed in …gure 8 below.

Insert …gure 8 about here The …gure gives the empirical average entry rate compared to theoretical entry rate given by Pure Strategy Nash equilibrium for the subgame of participants B 10 . The …rst observation is that empirical entry rate is close to theoretical one, whatever exogeneous capacity for market 1 (low or high), e.g indicating a high level of coordination for participants B. Another observation is that average observed entry rate tends to be higher compared to theoretical one in the LOW treatment, which is not the case in HIGH treatment. Such a result is similar to what was already observed by Camerer & Lovallo (1999) in the usual MEG, who observe that there is overentry when capacity is low and underentry when capacity of the market is high.

Concluding comments

The aim of this paper was to give empirical, and more precisely, experimental evidence to a wellknown phenomenon discussed since years in transport economics, the Downs-Thomson Paradox. To this end we build a theoretical model inspired on the usual Market Entry Game and based, …rstly, on an individual choice between two markets, and, secondly, on the fact each market generates either a negative externality or a positive one. The third originality in our model is to implement an endogenous choice of capacity for one of these two markets before individuals (users) have to enter, the other market capacity being exogeneous. The theoretical equilibrium (pure strategy Nash) is such that, given the particular subgame for users, the operator should choose the minimum capacity for his market, and for this level, given a certain …xed capacity for the other market, users allocate themself between the two markets. An exogeneous shock concerning the capacity of the …rst market should give an incentive for users to switch from one market to the other, the optimal strategy of the operator being the same: When exogeneous capacity increases, users ‡ew from market 2 to market 1 and vice versa. The consequences of such an exogeneous shock are that, …rst, on operator's revenue -it decreases when exogeneous capacity increases -and second on total welfare for the group -it decreases when exogeneous capacity increases since total users cost increases. That is, our simple theoretical model predicts Downs Thomson Paradox. To test it, we implement experimental sessions in which some subjects are …rst confronted to a low capacity level and second to a high capacity level (ADD condition) and for other subjects, it is the contrary (DEL condition). The …rst condition corresponds precisely to Downs-Thomson situation, whereas the reverse condition is closer to Mogridge conjecture.

The results we obtained are the following. Firstly, we observe that Downs-Thomson Paradox is produced in the laboratory, which gives empirical support for such a phenomenon. Secondly, we nevertheless do not observe that decreasing market capacity exogeneously does produce some improvement in welfare level.

The second result is that, for a given vector of capacity levels, users coordinate remarkably well on average around the entry rate predicted by pure strategy Nash equilibrium for the users' subgame. This result is in line with former results obtained in experimental studies for Market Entry Games, but we have to notice that our game is almost more complex for subjects, which precisely make coordination problem more di¢ cult to solve for them.

The third important result is that operator does not choose the minimum level of capacity, far from it, and that his choice responds very little to exogeneous road capacity. Such a result is obviously linked to the repeated partners design we implement in the lab, since it enables both reciprocity e¤ect and reputation e¤ect that might increase the level of capacity he chooses.

In terms of transport policies, what our results indicate is that, …rst, as was saying Roy Kienitz (executive director of the Surface Transportation Policy Project, cited by [START_REF] Litman | Generated tra¢ c and induced travel[END_REF]: "Widening roads to ease congestion is like trying to cure obesity by loosening your belt". It might be useless to solve road congestion by increasing road capacity, at least in the short term. Secondly, policy makers should be cautious about the temptation to solve congestion problems by thinking that a decrease in road capacity (or an increase in public transit capacity, the road capacity being constant) might reduce total transport costs.

Figure 1 .Figure 2 . 22 Figure 3 .Figure 4 . 23 Figure 5 .Figure 6 . 24 Figure 7 .Figure 8 .

 12223423562478 Figure 1. Average capacity chosen by B in each period (ADD condition) ADD condition

Table 1 .

 1 Average speed for private car and rail in London centre

	Source : Mogridge (1997)

Table 2 .

 2 Experimental

	sessions

Table 3 .

 3 Theoretical predictions (asymmetric Pure Nash Equilibria) treatment c 2 m 1 m 2

					i A	i B if i =	i B if i =	W N ash W M ax
						X	Y
	c 1 = 3	1	6	8	7	3	2.25	43	90.5
	(LOW)
	c 1 = 6	1	10	4	3	2	1.25	28	90.5
	(HIGH)

Table 4 .

 4 Capacity chosen by subject A per group capacity c 2 (mean and s.d., individual data)

	cond. group	LOW	HIGH
	ADD	1	2.55 (3.33)	1.9 (2.29)
		2	3.35 (2.21)	3.6 (2.70)
		3	5.15 (0.86)	4.65 (0.59)
		4	4.1 (2.31)	3.6 (2.56)
		5	4.5 (2.65)	2.7 (2.11)
		6	3.05 (2.28)	3.25 (2.53)
		9	5.75 (1.37)	3.95 (1.64)
		10	5.1 (2.31)	3.45 (2.74)
	average (s.d) 4.19 (2.47)	3.39 (2.33)
	DEL	7	1 (0)	1.5 (2.01)
		8	1 (0)	2.75 (3.23)
		11	3.9 (2.75)	4.25 (2.67)
		12	1.35 (0.81)	3.65 (3.08)
		13	6 (1.38)	5.9 (3.09)
		14	6.3 (0.57)	5.7 (2.15)
		15	1.8 (0.83)	5.8 (2.02)
		16	3.75 (1.89)	5.4 (2.74)
	average (s.d) 3.14 (2.44)	4.39 (3.02)

Table 5 .

 5 Panel censored Tobit regression on individual choice for c 2

	variable

2

+ 4 (CON D) + 5 (RISK) + 6 t + 0

The market 1 technology is similar to the payo¤ for entering on market in a usual Market Entry Game (see[START_REF] Erev | Coordination, "Magic,"and Reinforcement Learning in a Market Entry Game[END_REF].

Our game di¤ers from Rapoport et al., 2000 Two Market Entry Game (TMEG) in one major respect: Entering …rst or second market implies a decrease in individual payo¤s for both markets in TMEG, whereas in our DMEG case, more entrants on market 2 increase individual payo¤ for each entrant.

"If more people take the train, then trains run more frequently, saving people some waiting time at the station", p.4 . Such a property is a technological property of all types of mass transit, as it was shown by[START_REF] Mohring | Optimization and scale economies in urban bus transportation[END_REF]

Nevertheless, it is fair to remark that in the repeated game, such a prediction does not hold necessarily (seeGaechter & Falk, 2002, p5). One argument is that if players are from di¤erent "types", the game is of incomplete information which might rise possibility for playing cooperative behaviour.[START_REF] Kreps | Rational Cooperation in the Finitely Repeated Prisoners'Dilemma[END_REF] show that, even if there is a small probability that the adversary is, e.g., a "tit for tat" player, cooperative play can be supported until the …nal period.

As many experimenters, we use the Holt-Laury procedure to elicit the level of Constant Risk Relative Aversion (CRRA) index. For more details, seeHolt & Laury, 2002.

Actually the number of safe lotteries chosen by the participant in our expost survey, this survey being based onHolt & Laury, 2002.

[START_REF] Arnott | Properties of dynamic equilibrium involving Bottlenecks, including a paradox and metering[END_REF] wrote p.148 "If road capacity is now expanded, users will shift to the road until it is as congested as before. If the railway has to balance its budget, the loss of revenue will force it to increase fares and cut service (...)".

0 The dashed line has to be interpreted cautiously, since for some capacity levels of market

2, there is more than one theoretical entry rate predicted by Pure Strategy Nash equilibrium. We took the "average" theoretical entry rate in the …gure for convenience.

suggesting clearly that average capacity chosen in the …rst 20 periods is signi…cantly higher than the average capacity chosen in the last 20 periods. This temporal trend is con…rmed by parametrical analysis. We conduct a Panel Regression about individual choices for market 2's capacity (left censored, Tobit Analysis, Random E¤ects GLS). Results are given in the following table.

Insert table 5

The above regression indicates that market 1 capacity plays an ambiguous role: The coe¢ cient is positive and signi…cant when data are pooled, but becomes non signi…cant when capacity choice is analyzed for each condition. Given the limited number of data, we have to be cautious with econometric results. Nevertheless, some variables play a key role, as risk aversion level and time (period). There is a clear trend to have a decrease in capacity level chosen by A from period to period, as non parametrical analysis suggested above. Moreover, the more risk averse participant A is, the lower capacity she chooses.

Entry on markets

Entry rate on markets is obviously related to capacity on each market. The theoretical model predicts that when road capacity is LOW, entry rate on road should decrease whereas higher levels of capacity for public transit decrease it. Actually, the important result is that entry on market 1 is higher when capacity for this market is higher. Figures 3 and4 give the average entry rate on market 1 (road) respectivly for ADD condition and for DEL condition.

Insert …gure 3 about here Insert …gure 4

Figures above indicate clearly that entry rate increases when road capacity is to be increased. Such empirical result is statistically signi…cant, as non parametrical tests show. In ADD condition the average entry rate is 6.07 in the LOW treatment compared to an average entry rate of 9.43 in the HIGH treatment (road capacity is doubled). A within subject Wilcoxon test is signi…cant at the 5% level (z = 2:521 ; p = :0117). That is also the case in DEL condition, where average entry rate tends to be higher in HIGH treatment (the average entry rate is around 6.41 in LOW treatment compared to 8.63 in HIGH treatement, the Wilcoxon matched paired sign-rank test indicates z = 2:524 ; p = :0116).

Such an empirical evidence is also clearly demonstrated when we implement between subjects comparison. If the entry rate of the LOW treatment in the ADD condition is compared to the entry rate in the HIGH treatment for the DEL condition (these two treatments constitute both the …rst part of the experiment for these participants), the entry rate are respectivly 6.07 and 8.63, which is signi…cant at the 1% level (bilateral Mann-Whitney test, z = 3:366, p = :0008). The same result is obtained by comparing average entry rate in each group for the HIGH treatment in the ADD condition and average entry rate per group for the LOW treatment in the DEL condition (such treatments constitute the …nal part of the experiment). A bilateral Mann-Whitney indicates clearly that entry rate when (road) capacity is high is statistically higher than entry rate when capacity is low (z = +3:363 ; p = :0008, signi…cant at the 1% level).


B Kibler

C Finot

Impact of structural irregularities on high bit-rate pulse compression techniques in photonic crystal fibre

The impact of structural irregularities on high bit-rate pulse compression techniques is evaluated in photonic crystal fibre. Specifically, we report more robust pulse compression to longitudinal fluctuations in the normal dispersion regime. We identify the physical limits of these pulse compression techniques in the presence of dispersion fluctuations and we confirm that state-of-the-art fabrication tolerances are sufficient for future experimental applications.

The advent of optical microstructured fibres was one of the most revolutionary progresses over the past decade in the field of nonlinear waveguide optics [START_REF] St | Photonic-Crystal Fibers[END_REF]. Indeed, photonic crystal fibres (PCF) offer a significant degree of freedom for the control of chromatic dispersion, such as exhibiting two zero dispersion wavelengths. Moreover, solid-core PCFs have also enabled the significant nonlinearity required for supercontinuum generation [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. Thanks to the growing control of the manufacturing processes [START_REF] St | Photonic-Crystal Fibers[END_REF], the use of such microstructures in the area of telecom applications now becomes conceivable. If a low normal or anomalous dispersion is needed at these infrared wavelengths, it is typically convenient to work in the vicinity of the second zero dispersion wavelength (IR-ZDW) of such holey fibres. But in this case, the dispersion properties become increasingly dependent on the geometrical fluctuations of the microstructure. In this letter, we determine what impact these changes may have on two specific high bit-rate nonlinear applications at 40 GHz, namely pulse compression techniques by multiple four-wave mixing (MFWM) in the anomalous dispersion regime or by conversion into a parabolic pulses in the normal dispersion regime with subsequent linear compression [START_REF] Pitois | Generation of 20-GHz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers[END_REF]. We confirm that state-of-the-art fabrication tolerances are compatible with future experimental applications.

To examine the consequences of the microstructure longitudinal fluctuations, we consider the typical highly nonlinear PCF structure shown in Fig. 1(a) with a hexagonal photonic crystal cladding surrounding a central solid core defect. We assume a hole diameter d = 1.1 m and pitch Λ = 1.2 m corresponding to a IR-ZDW at 1564.4 nm. Standard beam propagation method simulations were used to calculate the variation in dispersion and effective area with frequency [4]. From a recent review on PCF [START_REF] St | Photonic-Crystal Fibers[END_REF], it is now realistic to manufacture the microstructure in air-glass PCF to accuracies of 10 nm on the scale of 1 m. However, such fluctuations of structure parameters such as core diameter, hole diameter or pitch, are sufficient to induce significant variations of dispersion, nonlinearity and birefringence characteristics, which are highly detrimental to nonlinear processes [START_REF] Karlsson | Four-wave mixing in fibers with randomly varying zero-dispersion wavelength[END_REF][START_REF] Kibler | Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers[END_REF][START_REF] Farahmand | Parametric amplification in presence of dispersion fluctuations[END_REF]. In the following analysis, we focus on large dispersion variations induced by longitudinal fluctuations of the pitch Λ while the diameter-to-pitch ratio d / Λ remains constant. Indeed, Fig. 1(b) outlines the significant dependence of the dispersion on pitch variations about 1-2%. Note that pitch fluctuations of ±1% lead to variations as high as ±16 nm in the position of the IR-ZDW, equivalent to a ±110% fluctuation of the GVD value at 1550 nm. Such extreme dependences have already been reported experimentally [START_REF] Joly | Competition between spectral splitting and Raman frequency shift in negativedispersion slope photonic crystal fiber[END_REF].

The nonlinear coefficient γ is found to be less sensitive (δ γ < 1%) and its variations will be neglected (γ 63 W -1 km -1 for the considered range of wavelengths). Therefore, we investigate the impact of such dispersion fluctuations on two pulse compression applications. In both cases, an initial sinusoidal signal obtained from the beating of two continuous laser diodes is compressed in order to generate picosecond pulse trains at 40

GHz (see Fig. 2).

The first method (Fig. 2(a)) is based on a MFWM process taking place in the anomalous dispersion regime and simultaneoulsy leading to a strong spectral broadening and a temporal compression of the sinusoidal signal [START_REF] Fatome | 20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave mixing in optical fibers[END_REF]. Consequently, we consider the central wavelength of the laser diodes fixed to 1550 nm where the corresponding GVD is D = 9.92 ps/km . nm. Generalized nonlinear Schrödinger equation (GNLSE) simulations including the global dispersion characteristics [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF] were used to determine the corresponding output pulses. From [START_REF] Fatome | 20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave mixing in optical fibers[END_REF], the optimum input average power and fibre length, for a maximum compression and minimum chirp, were respectively found to P opt = 5.1 mW and L opt = 3.5 km. Our numerical simulations confirm these optimal values in the absence of dispersion fluctuations and Fig. 2(a) shows that the initial sinusoidal train is efficiently converted into a 40-GHz nearly Gaussian pulses train with a FWHM temporal width of 4.2 ps. In order to characterize the output pulse shape, we computed the misfit parameter M opt between the pulse intensity profile and a Gaussian fit of the same energy [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device[END_REF]. To approach a realistic description of random fluctuations of the pitch parameter and subsequent dispersion fluctuations, we consider the randomly varying part of the pitch as a real, Gaussian, stationary stochastic process with zero mean [START_REF] Karlsson | Four-wave mixing in fibers with randomly varying zero-dispersion wavelength[END_REF]. For each simulated step of propagation, the corresponding GVD curve is then determined from the random value of the pitch. Varying the standard deviation σ and the correlation length L C (fluctuation period) of the distribution, we have drawn maps in Fig. 3(a) and (b), illustrating variations of the output pulse temporal width T OUT (FWHM) in percentage and the normalized misfit parameter (M/M opt ). Here, the largest standard deviation of the pitch taken into account is 0.55% which corresponds to a maximal variation of the pitch around 2% (~4σ) and the largest correlation length is the PCF length. Each point of these maps has been calculated from an ensemble of 50 simulations. From Fig. 3(a), we globally note that a standard deviation of Λ below 0.2% leads to an acceptable fluctuation of T OUT inferior to 10%. In the case of higher standard deviations, the group-velocity dispersion may become normal and lead to local temporal broadening of the pulses. This then explains large variations up to 100% of T OUT (not shown by the color scale used) and over 40×M opt in the normalized misfit parameter (see Fig. 3(b)). For long-scale fluctuations and high standard deviations, these results show how this nonlinear pulse compression technique is extremely sensitive to local GVD variation (in particular its sign) during propagation. However, we generally observe from Fig. 3(a-b) that when L C 0, the effect of the fluctuations disappears whatever the standard deviation. Short-scale fluctuations of the pitch inferior to L PCF /40 (corresponding to γP 0 L C << 1) have poor impact, in agreement with previous results demonstrated in the context of parametric amplification [START_REF] Farahmand | Parametric amplification in presence of dispersion fluctuations[END_REF].

The second method (Fig. 2(b)) of 40-GHz pulse train generation relies on the progressive reshaping of the initial sinusoidal beat-signal into a train of linearly chirped quasi-parabolic pulses in the normal dispersion regime [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device[END_REF]. After linear compression of the parabolic pulses into a standard optical fibre with opposite dispersion (SMF-28), a train of wellseparated picosecond pulses is achieved [START_REF] Pitois | Generation of 20-GHz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers[END_REF]. Here, we consider the central wavelength of the laser diodes fixed to 1580 nm, where the GVD is D = -10.55 ps/km . nm. The optimum input average power and fibre length [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device[END_REF], for a quasi-linear chirp across the pulses and minimum overlapping between consecutive pulses, were respectively found to P opt = 7 mW and L opt = 1.5 km. The anomalous dispersive compression fibre is a SMF with the following fixed parameters: D SMF = 17.8 ps/km . nm, dispersion slope S SMF = 0.057 ps/km . nm 2 , nonlinear coefficient γ SMF = 1.2 W -1 km -1 and optimum length L SMF = 1.25 km.

The ouput pulses have a nearly Gaussian intensity profile with a 7.6 ps (FWHM) temporal width and the new optimal computed misfit parameter is M opt . Following the previous modeling of random fluctuations, Fig. In conclusion, these results demonstrate that longitudinal fluctuations of microstructure parameters in PCF are not detrimental for high bit-rate pulse compression techniques thanks to best current fabrication tolerances around 1% (σ~0.25%) on the scale of 1 µm.

Moreover, we have identified more robustness in the normal dispersion regime. We expect that these required tolerances encourage the use of kilometer-long PCF for future high bit-rate nonlinear experimental applications.

  3(c) and (d) show variations of T OUT in percentage and the normalized misfit parameter (M/M opt ) obtained for this pulse compression technique. From Fig.3(c), we observe that a standard deviation of Λ inferior to 0.2% only corresponds to a fluctuation of T OUT below 5%. For higher standard deviation, the groupvelocity dispersion can become anomalous and now leads to local temporal compression of the pulses in PCF, however variations of T OUT remain inferior to 15% and those of the normalized misfit parameter do not exceed 20×M opt (see Fig.3(d)). Indeed, the first propagation stage in PCF broadens the pulses and decreases the peak power before linear compression. The impact of local GVD fluctuations is then weaker than in the nonlinear compression technique. This explains the "smoothing" of fluctuations maps. As the previous technique, we also observe from Fig.3(c-d) that when L C 0, the effect of the fluctuations vanishes.

Figure Captions: Fig. 1 (

 1 Figure Captions:

Fig. 2

 2 Fig. 2Setup diagram with the input sinusoidal signal at 40 GHz, (a) for MFWM

Fig. 3

 3 Fig. 3Numerical results showing variations of (a,c) the output temporal width T OUT (in

Acknowledgements: This research was supported by the Agence Nationale de la Recherche (SUPERCODE project, ANR-06-BLAN-0401).