Etienne Parizet
email: etienne.parizet@insa-lyon.fr

Vincent Koehl

Influence of train colour on loudness judgments

Keywords: Classification : 66, 002 : Cb -Loudness, absolute threshold 66, 010 : Lj -Perceptual effects of sound

This study replicates an experiment described by Patsouras et al. [5] or Rader et al. [8] in which subjects had to evaluate the loudness of a passing-by train noise while watching pictures of the train, the colour of which being varied. In contrast to these studies, no influence of train colour on loudness evaluation could be found. This discrepancy might be due to cultural differences (French subjects vs. German or Japanese ones). But a more realistic reason can be a jury sampling effect, as this influence can exist for very few subjects (one among twenty-two in this experiment).

Introduction

Influence of visual inputs on sound quality has been noticed in different situations, e.g. soundscapes [START_REF] Viollon | Influence of visual setting on sound ratings in an urban environment[END_REF], concert-hall evaluation [START_REF] Fastl | Psycho-acoustics and sound quality[END_REF], judgements of the quality of loudspeakers [START_REF] Toole | Hearing is believing vs. believing is hearing: blind vs. sighted listening tests, and other interesting things[END_REF] or of the powerfulness of a car sound [START_REF] Ellermeier | Visual bias in subjective assessments of automotive sounds[END_REF]. Similar effects have also been reported for a more elementary psychoacoustic quantity as loudness. Fastl and colleagues asked subjects to evaluate the loudness of a passing-by ICE train sound [START_REF] Patsouras | Influences of color on the loudness judgement[END_REF][START_REF] Fastl | Audio-visual interactions in loudness evaluation[END_REF][START_REF] Fastl | Recent developments in sound quality evaluation[END_REF]. When listening to the sound, subjects could see a picture of the train, the colour of which being varied. It appeared that a red coloured train lead to a higher estimation of loudness than a green coloured one, though this effect was small. This study was conducted with German subjects and later confirmed with Japanese ones [START_REF] Rader | Crosscultural effects in audio-visual interactions[END_REF]. More recently, a similar effect was related for the sound of a sports car [START_REF] Menzel | Hellbrück: Influence of vehicle color on loudness judgments[END_REF].

Nevertheless, this colour effect was small and the experiments were conducted with a rather low number of subjects (e.g. 9 in [START_REF] Patsouras | Influences of color on the loudness judgement[END_REF]). Therefore, we intended to replicate the experiment involving ICE train stimuli with a larger number of subjects.

Procedure

Stimuli

Audio-visual conditions were similar to those used in the above mentioned experiments [5 , 6, 7, 8]. A passing of an ICE train was monophonically recorded at a distance from 25 m. Signal duration was 11 seconds. The level was adjusted to match three different values (the A-weighted sound pressure level being 82 dB, 78 dB and 74 dB). This level modification was realized using the output potentiometer of the soundcard of the computer (Gina24 from EchoAudio), the maximum output setting being selected for the maximum desired sound level. Sounds were diotically presented to listeners through electrostatic headphones (Stax SR-303) in a sound proof booth. During sound presentation, a fixed picture of the train could be seen by the listener. The train could have four colours (white, i.e. the original colour of the ICE train, red, blue and green). This picture was presented on a screen at approximately 2 meters in front of the subject. The picture was approximately 1.5 meters wide and 1 meter high.

Experiment

The three levels of the sound and four images lead to twelve different combinations. Each combination was randomly presented three times to the subject. Loudness of sound was evaluated using a "free magnitude estimation" procedure : the first stimulus had to be evaluated absolutely and each stimulus was considered as a standard for the evaluation of next one [START_REF] Stevens | Psychophysics : introduction to its perceptual, neural and social prospects[END_REF]. Subjects were asked to use positive numbers (integer or not) to give their answer. They typed this value using the keyboard of the computer running the experiment.

Subjects

22 normal-hearing subjects (19 men and 3 women) participated to this experiment. They were students [START_REF] Ellis | Color preferences according to gender and sexual orientation[END_REF] or staff from the laboratory [START_REF] Fastl | Recent developments in sound quality evaluation[END_REF]. Their age varied from 22 and 58 (average : 32.5).

Results and discussion

In a first step, the influence of the two factors was investigated using an analysis of variance. For each couple of factors, the geometric mean of the three evaluations given by a subject was computed. An ANOVA (repeated measures) was conducted using the set of data thus obtained. It appeared that the only significant factor was level (F(2; 42) = 14.9, p < 0.001). Colour was not significant (F(3; 63) = 1.75, p = 0.165), nor the interaction between factors (F(6; 126) = 1.3, p = 0.25).

In order to directly evaluate the influence of colour, the influence of level and the inter-individual variability was eliminated. For each subject and each sound level, the ratio between the geometric mean obtained for a given train colour and that one of the white train was computed (the actual colour of the ICE train being selected as a reference). For each colour, a set of 66 (3x22) values was thus available. The mean, median and interquartile range of these data are presented in figure 1. Differences between median values were not significant (Wilcoxon test, p = 0.01). Therefore, it seems that train colour did not influence the loudness of the sound. As this could be true for some subjects, ANOVAs were conducted using each individual's set of data (including the three repetitions of each couple of stimuli). It appeared that a significant influence of colour was found for one subject only (p = 0.01). For this subject, the sound from the red train was evaluated as a little less loud than the one from the white train (post-hoc comparison using Scheffe's correction, p = 0.017).

This result is quite different from the ones already published by German and Japanese colleagues.

Using the same procedure (with the exception that another set of stimuli was without any image of the train), these papers related significant influence of the train colour. One noticeable effect was that the red train appeared louder than the green one [START_REF] Ellermeier | Visual bias in subjective assessments of automotive sounds[END_REF]. This discrepancy could be due to cultural effects (French subjects vs. German or Japanese ones). Such a cultural effect has already been found in some other topics (e.g. booming noise of a car [START_REF] Shibuya | The difference in sensitivity to vehicle noise performance between European and japanses subjects[END_REF], or the negative or neutral connotation of the word "loudness" [START_REF] Kuwano | Emotional expression of noise: a cross-cultural study[END_REF]). But such differences were mainly found between Japanese and European subjects -the perception within Europe being homogeneous.

Another possible reason for this discrepancy can be a sampling effect. Several limitations of our study can be pointed out:

-A few women (3) participated to the experiment. It has been shown that women and men perceive colours in a slightly different way [START_REF] Bimler | Quantifying variations in personal color spaces : are there sex differences in color vision ?[END_REF][START_REF] Pardo | An example of sex-linked color vision differences[END_REF], which is due to some physiological differences in the retina. But, in this experiment, train colours were so different that we think no gender effect could be expected. Some other gender differences are related to colour preferences [START_REF] Ellis | Color preferences according to gender and sexual orientation[END_REF]. But, as no preference was asked to subjects, this effect can also be neglected; -Some subjects were very used to sound experiments (7 of the 22). Such people may have paid their whole attention to the sound, without taking images into account. However, the subject who proved to be influenced by the train colour was one of these people (he was a staff member).

-Finally, the number of subjects could be increased.

Indeed, all these limitations also apply to previous studies, which concluded that the colour of the noise source influenced the evaluation of loudness. The number of subjects was 9 in [START_REF] Patsouras | Influences of color on the loudness judgement[END_REF], between 13 and 15 in [START_REF] Rader | Crosscultural effects in audio-visual interactions[END_REF], 16 in [START_REF] Menzel | Hellbrück: Influence of vehicle color on loudness judgments[END_REF]. All subjects were trained in psychoacoustic experiments in [START_REF] Patsouras | Influences of color on the loudness judgement[END_REF] -no information is provided in [START_REF] Rader | Crosscultural effects in audio-visual interactions[END_REF] and [START_REF] Menzel | Hellbrück: Influence of vehicle color on loudness judgments[END_REF]. In [START_REF] Menzel | Hellbrück: Influence of vehicle color on loudness judgments[END_REF], 4 women participated to the experiment, which is also true for one experiment related in [START_REF] Rader | Crosscultural effects in audio-visual interactions[END_REF], while the gender distribution is not mentioned in [START_REF] Patsouras | Influences of color on the loudness judgement[END_REF]. Therefore, it can be concluded that some more investigations are needed, using larger and better gender-balanced listener panels, before concluding that the colour of the sound source can influence the loudness of the sound.

On the other hand, the influence of loudness was quite close to what could be expected. Loudness values were averaged over the four train colours. The highest sound pressure level was used as a reference and the relation between loudness and the relative pressure of each stimulus can be seen in figure 2. The curve is a power-law adjusted to these data. The exponent of this power-law was best fitted to 0.52, which is quite close to the expected value (0.6 according to Stevens' power law [START_REF] Moore | An introduction to the psychology of hearing, 5 th edition[END_REF]).

Conclusion

In this experiment, no influence of train colour on loudness evaluations could be found. Such an effect is very subject-dependent and appeared for one subject only in the panel of 22 persons used in the experiment. We think that it can be neglected when averaging over a group of peoplethough audio-visual interactions certainly exist for more complex sound features. A limitation of this study is the rather low number of subjects, which could bias the results. This is also true for the referenced studies. Therefore, it appears that more investigations are needed to confirm any of these contradictory results. Moreover, the realism of the experiment could be improved, as the visual stimulus was just a picture and the sound was diotically presented, without any spatial effect. Using more realistic stimuli could modify these conclusions.

Figure 1 :

 1 Figure 1 : relative loudness (ref : white train). Closed circles : median values and interquartile ranges. Open squares : mean values in their 0.95 confidence intervals.

Figure 2 :

 2 Figure 2 : relative loudness of stimuli, represented as a function of the relative A-weighted sound pressure level. Median values are shown in their interquartile range.

Acknowledgments

The authors are grateful to Prof. Hugo Fastl who provided them the stimuli used in this experiment (pictures and sound of the ICE train).


Trung Quang

Laurent Le

Mathilde Bramerie

Michel Gay

Jean-Claude Joindot

Arthur Simon

O'hare

Trung Hoang

Jean-Louis Nguyen

Oudar

Quang Trung Le

Laurent Bramerie
email: laurent.bramerie@enssat.fr

Mathilde Gay

Michel Joindot

Jean-Claude Simon

Arthur O'hare

absorber in a recirculating loop experiment

Keywords:

come     All-optical phase-preserving amplitude regeneration of 28-Gbaud RZ-DQPSK signals with a microcavity saturable

Introduction

Phase-encoded modulation formats, such as return-to-zero differential quadrature phase-shift keying (RZ DQPSK), have recently attracted much attention for 100G Ethernet systems and this has led to significant research [1]. The reach of an optical phase-shift keying transmission is mainly limited by the accumulation of both linear and nonlinear phase noise. Linear phase noise results from amplified spontaneous emission (ASE) in optical amplifiers. Nonlinear phase noise (NLPN) results from intra-and interchannel nonlinearities, such as self phase modulation (SPM) and cross phase modulation (XPM) which convert amplitude noise to phase noise [2]. It is clear that by increasing the launched channel power, the impact of the linear phase noise can be reduced. However, in this case, the nonlinear phase noise (NLPN) becomes the major impairment factor.

In the literature, it is shown that phase-preserving amplitude regeneration would be an attractive solution to enhance the phase-encoded system performance [3]. By reducing the amplitude noise, which is the origin of NLPN, phase-preserving amplitude regenerators prevent the accumulation of NLPN during transmission. The cascadability of the phase-preserving amplitude regenerators based on four-wave mixing in fibre [START_REF] Matsumoto | Proc. ECOC'06[END_REF] and on nonlinear amplifier loop mirror [START_REF]Proc. ECOC'09[END_REF] have ben experimentally demonstrated. Both use short-pulse RZ DPSK signals at 10 Gbit/s.

Recently, we have shown that a novel microcavity saturable absorber allowing power stabilization (SA1) [START_REF] Nguyen | [END_REF] can provide phase-preserving amplitude noise reduction in a DPSK system [START_REF] Le | Proc. OFC'10[END_REF]. The cascadability of this device in a DPSK system has been demonstrated [START_REF] Le | Proc. ECOC'10[END_REF]. Compared to other phase-preserving regenerators, the microcavity saturable absorber provides the advantage of a simple, compact and WDM compatible solution (regeneration of several tens of WDM channel on less than 1 mm²) [START_REF] Bramerie | Proc. OFC'07, PDP1[END_REF].

In this paper, we assess for the first time the efficiency of all-optical phase-preserving amplitude regeneration in a DQPSK system. The SA1-based regenerator is concatenated in a recirculating loop at 28 Gbaud. Different regeneration spans are studied to find a better trade-off between the gain provided by regeneration and the device insertion loss.

Demonstration of SA1-regeneration effect

Before system experiments, we perform a first experiment demonstrating the effect of the regenerator. We emulate a link with a regenerator located where only linear phase noise is present (1 st transmission section). By strongly reducing the amplitude noise, the SA1 prevents the creation and accumulation of NLPN in the next transmission span (2 nd transmission section).

The experimental setup is presented in Fig. 1. The DQPSK signal, at 1549 nm, is generated by a nested Mach-Zehnder modulator (MZM) with 2 9 -1 bit length pseudorandom binary sequence (PRBS). A second MZM is sinusoidally modulated at 28 GHz to convert 28 Gbaud NRZ DQPSK to 50% RZ DQPSK. The 1 st transmission section is emulated by an OSNR degradation obtained by an ASE source followed by a 3-nm band-pass optical filter, centred at the working wavelength followed by an optical attenuator (ATT).

OMT5.pdf

OSA/OFC/NFOEC 2011

OMT5.pdf

The signal is then boosted to 15 dBm by EDFA1 before being sent to SA1 via an optical circulator. The 2 nd transmission section is composed of 100 km of non-zero dispersion-shifted fibre (NZ DSF) with chromatic dispersion of 4.5 ps/km/nm at 1550 nm, followed by a dispersion-compensating fibre (DCF). EDFA2 is used to increase the launched power up to 12.5 dBm (necessary to generate a high enough amount of NLPN in 100 km), a more realistic launch power is used in recirculating loop experiment with several amplification spans. EDFA3 compensates the residual loss. The preamplified DQPSK receiver consists of a fibre-based delay-line interferometer (DLI) and balanced detectors. The BERT is programmed to detect the correct incoming bit stream. Fig. 2 shows the measured BER versus the receiver input power. The first curve (diamonds) relates to our backto-back (B2B) reference with an OSNR of 32 dB. The square-mark curve refers to B2B with an OSNR of 20.7 dB. The BER in this case is limited by the linear phase noise induced by the ASE source. The triangle-mark curve refers to a transmission case with 12.5 dBm launched power and without SA1. A large power penalty is obtained and an error floor at BER of 2.10 -9 appears owing to linear and NLPN. When the SA1 based amplitude regenerator is used, a considerable improvement of the BER for the same fibre launch power is obtained (circle-mark curve). Indeed, the BER curve with regeneration is close to the one limited only by linear phase noise, and the error floor disappears from the measurable range of BER. The efficiency of the regenerator is thus demonstrated.

Recirculating loop experiment

In order to assess the regenerator cascadability and performance, it is concatenated in an experiment using a recirculating loop (Fig. 3). Tx and Rx are the transmitter and the receiver respectively, working at 1549 nm.

The amplification span is composed of 100 km of NZ-DSF, followed by a DCF. Fibre loss is compensated for by erbium amplification. A SA1-based phase-preserving amplitude regenerator is inserted in the transmission line every M amplification spans. In this work, because of the limited number of available amplifiers, we consider only M=1 and 2.

OMT5.pdf

OSA/OFC/NFOEC 2011 OMT5.pdf

Results

Fig. 4 shows the measured BER versus transmission distance with and w/o SA1 in the case of M=1. The fibre launched power is set at 6 dBm. Without SA1, the BER grows rapidly due to linear and NLPN accumulation. When the phase-preserving amplitude regenerator is used, the amplitude noise is reduced and the NLPN is partly removed. As a consequence, the transmission distance is enhanced thanks to the regenerator. A transmission distance of 1200 km is reached for a BER of 10 -4 compared to 800 km without regeneration.

In order to evaluate more accurately the distance improvement brought by SA1, we optimize the fibre launched power for both cases with and w/o it. BER is measured versus transmission distance for different launched powers, as depicted in Fig. 5, where the iso-BER curves associated with BER=10 -n , n=3, 4, 5 and 6 are drawn. For a given BER, the distance increases first with the launched power in the linear regime (OSNR increases with the launched power), reaches a maximum. Then the transmission distance decreases with the launched power because nonlinear effects become the main degradation source and degrade more and more severely the system performance. Let us consider the particular iso-BER curve for BER=10 -4 as depicted in Fig. 6 (dash line), the same iso-BER curve w/o regenerator is also plotted (solid line). Without regeneration, the maximum transmission distance corresponds to a launched power of 2 dBm. When the SA1-based regenerator is used, the largest transmission distance is multiplied by a factor of 1.2 and occurs at 5 dBm instead of 2 dBm. This reveals a significant improvement of system tolerance to the nonlinear effect. In the linear transmission regime, at low launch power, the regenerator does not bring any improvement as it preserves the signal phase. Moreover, in this case, because of its high insertion loss, the SA1 slightly degrades the OSNR.

Let us now consider the case of M=2 where the number of amplification spans is doubled. Consequently, for the same transmission distance, the number of concatenated regenerators is reduced by a factor of 2. Fig. 7 shows the iso-BER curves for the cases with and w/o regeneration loop. Whereas in the reference case, we remain almost unchanged, the performance of the regenerated case is better (distance improvement factor of 1.3). This can be explained by the fact that by reducing the number of regenerators, a better trade-off between amplitude regeneration effect and the device insertion loss can be found. Experiment will be completed by using a path switchable loop in order to define precisely the best regeneration spacing.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Experimental setup.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Recirculating loop setup.

Fig. 5 :Fig. 6 :Fig. 7 :

 567 Fig. 5: BER (log scale) with concatenated SA1 versus launch power and transmission distance

Conclusions

We have reported for the first time the efficiency of phase-preserving amplitude regeneration using a microcavity saturable absorber in an RZ DQPSK transmission system at 28 Gbaud. The tolerance to the nonlinear effect of the regenerated system is enhanced, and the transmission distance is improved thanks to the regenerator. A distance improvement ratio of 1.3 is obtained with regeneration every 200 km.


Olivier Ley

Sadco Mar

Large time behavior of systems of Hamilton-Jacobi equations

Time-dependent systems of HJ equations

     ∂u i ∂t + F i (x, Du i ) + m j=1 d ij (x)u j = f i (x) T N × (0, +∞) u i (x, 0) = u 0,i (x)
T N for i = 1, . . . , m.

➪ Periodic setting ➪ Linear coupling ➪ More precise assumptions later Aim : Study the behavior of u(x, t) = (u 1 (x, t), . . . , u m (x, t)) when t → +∞

The scalar periodic case

F (x, p) ≥ F (x, 0) = 0 f (x) ≥ 0, A = {x ∈ T N : f (x) = 0} = ∅ regularity : |F (x, p) -F (y , p)| ≤ ω((1 + |p|)|x -y |)
Then, for every u 0 ∈ Lip(T N ), there exists

(c, v ) ∈ R × Lip(T N ) such that u(x, t) -ct → v (x) as t → +∞ uniformly in T N v is solution of F (x, Dv ) = f + c in T N c is the ergodic constant, unique, c = -min T N f = lim t→+∞ -u(x,t) t = 0
.

Goal

Is this theorem true in the case of systems ?

The Aubry set A := {x ∈ T N : f (x) = 0} plays a particular role. What is the equivalent for systems ? Dynamics : Ẋ (s) = b(X (s), α(s)) s ≥ 0 X (0) = x α(s) control, takes its value in a compact space K .

Value function :

V (x, t) = inf α(•) { t 0 f (X (s))ds + u 0 (X (t))} Then V is the unique viscosity solution of    ∂u ∂t + sup α∈K {-b(x, α) • Du} = f (x) u(x, 0) = u 0 (x)
Optimal trajectories are attracted by A = argmin f and V (x, t) t ∼ t→+∞ -c = min f . Control for systems : piecewise deterministic trajectories with random jumps (1)

Dynamics : Ẋ (s) = b ν(t) (X (s), α(s)) s ≥ 0 X (0) = x solution : (X (s), ν(s)) with ν(s) a Markov process with values in {1, 2, . . . , m}
Transition probabilities :

P (ν(t + h) = j | ν(t) = i, X (t) = x) = γ ij (x)h + o(h) for j = i.
Value function :

V i (x, t) = inf α(•) E x,i { t 0 f ν(s) (X (s))ds + u 0,ν(t) (X (t))}
Then V = (V 1 , ..., V m ) is the unique viscosity solution of the system      ∂u i ∂t + sup α∈K {-b i (x, α) • Du i } + m j=1 γ ij (x)(u i -u j ) = f i (x) u i (x, 0) = u 0,i (x)
for i = 1, . . . , m.

For instance Fleming-Zhang 98

m j=1 γ ij (x)(u i -u j ) = m j=1 d ij (x)u j with d ii = j =i γ ij ≥ 0 and d ij = -γ ij ≤ 0 for i = j

Assumptions on the Hamiltonian and initial conditions

The same as in Namah-Roquejoffre Theorem.

For all i = 1, ...m :

Periodicity : F i (•, p), f i , u 0,i are 1-periodic continuous convexity and coercivity of F i (x, •)

F i (x, p) ≥ F i (x, 0) = 0 f i (x) ≥ 0 regularity : |F i (x, p) -F i (y , p)| ≤ ω((1 + |p|)|x -y |) SADCO Mar 2011 1. Scalar case 2. Control 3. Result 4. Proof
Assumptions on the coupling matrix

D(x) = (d ij (x)) 1≤i,j≤m
For all x ∈ T N :

d ii ≥ 0, d ij ≤ 0 for j = i, m j=1 d ij ≥ 0.
➪ D is a M-matrix Classical assumptions to have a monotone system ⇒ maximum principle for the evolution problem d ij are periodic in x D(x) has non zero coefficients or : is an irreducible matrix in T N equivalent definitions : (i) ∀I {1, ..., m}, ∃i ∈ I and j / ∈ I s.t. d ij = 0 (ii) ∀i, j ∈ {1, ..., m}, there exists a "chain" i = i 0 , i 1 , ..., i n = j s.t. d i l-1 i l = 0. ➪ means that the coupling is nontrivial Introduction of a "Aubry set" and assumptions

Let F = 1≤i≤m argmin f i D = 1≤i≤m {x ∈ T N : m j=1 d ij (x) =0} We define A = F ∩ D.

Assume that

A is non empty all the f i 's have the same minimum f (➪ to simplify we take f = 0) D = T N (➪ to simplify) With these assumptions, since f i ≥ 0,

A = F = {x ∈ T N : m i=1 f i (x) = 0}

A result

Theorem. Under the previous assumptions, for every u 0 = (u 0,1 , ..., u 0,m ) ∈ Lip(T N ), there exists

((c 1 , ..., c m ), (v 1 , ..., v m )) ∈ R m × Lip(T N ) such that u(x, t) -ct → v (x) as t → +∞ uniformly in T N
v is solution of the stationary system Comparison result for the stationary system

F i (x, Dv i ) + m j=1 d ij (x)v j = f i + c i in T N , i = 1, . . . , m c is in the kernel of D(x) so c = (c 1 , ..., c 1 ), with c 1 = -f = lim t→+∞ -u i (x,t) t = 0 for all i. v i = v j on A.
F i (x, Dv i ) + m j=1 d ij (x)v j = f i in T N , i = 1, . . . , m
Theorem. Let u be a bounded subsolution and v a bounded supersolution.

(Classical case) If, for all i,

m j=1 d ij > 0 (D = ∅) then u ≤ v on T N . (Degenerate case) If m j=1 u j ≤ m j=1 v j on A then u ≤ v ➪ A may be empty Therefore ( j d ij )M -δd ik (x) ≤ (µ -1)f i ≤ 0 contradiction
λv λ i + F i (x, Dv λ i ) + m j=1 d ij (x)v λ j = f i in T N , i = 1, . . .

, m

Theorem. There exists a unique solution which is Lipschitz continuous with constant L independent of λ.

Up to extract, as λ → 0,

λv λ → -(c 1 , ..., c m ) = -(c 1 , ..., c 1 ) ∈ ker D v λ -v λ (x * ) → v ∈ Lip(T N )
and (c, v ) is solution of

F i (x, Dv i ) + m j=1 d ij (x)v j = f i + c 1 in T N , i = 1, . . . , m ➪ In fact -c 1 = min f i = f = 0 here
≥ |u 0 | ∞ s.t. v (x) -C ≤ u(x, t) ≤ v (x) + C The function u ε (x, t) = u(x, t ε ) is solution to the system      ε ∂u i ∂t + F i (x, Du i ) + m j=1 d ij (x)u j = f i (x)
T N × (0, +∞) u i (x, 0) = u 0,i (x) T N for i = 1, . . . , m.

Then, by stability, the half-relaxed limits u(x) = lim sup ε→0 * u ε (x, t) and u(x) = lim inf ε→0 * u ε (x, t)

are respectively sub and supersolutions to the stationary system.

  ∂u ∂t + F (x, Du) = f (x) T N × (0, +∞) u(x, 0) = u 0 (x) T N A lot of works : Lions 82, Fathi 98, Namah-Roquejoffre 99, Barles-Souganidis 00, Davini-Siconolfi 06, Ishii-Mitake 06,07,... Namah-Roquejoffre Thm for ∂u ∂t + F (x, Du) = f (x) Theorem. [Namah-Roquejoffre 99] Periodicity : F (•, p), f , u 0 are 1-periodic continuous convexity and coercivity of F (x, •)

A

  Lemma on the coupling matrix Lemma. Suppose that : D(x) is an irreducible M-matrix on T N m j=1 d ij = 0 (i.e., D = T N ) Then, for all x ∈ T N , : D(x) is degenerate of rank m -1 the kernel of D(x) is spanned with (1, ..., 1) there exists a positive function Λ :T N → R m such that D(x) T Λ(x) = 0 (i.e., m i=1 Λ i (x)d ij = 0) [Proof : Perron-Froebenius+continuous dependence]For simplicity, we suppose that Λ(x) = (1, ..., 1)

  Proof of the convergence theorem u(x, t)ct → v (x) as t → +∞ (1)Recall that -c 1 = -min f i = 0Comparison for the evolution problem : ∃C

  convergence theorem u(x, t)ct → v (x) as t → +∞ (2)Summing the evolution equations for 1 ≤ i ≤ m,

(Classical case) there exists a unique viscosity solution to the stationary system.

(Degenerate case) For all g continuous on A with compatibility conditions (see Fathi-Siconolfi 05, Ishii-Mitake 07), there exists a unique viscosity v solution such that v = g on A.

➪ A is a uniqueness set

≥0 (convexity and max.point)


Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions A recursive structure leading to an expression similar to the previous one holds in general for stationnary repeated games. Let us describe this representation for games with incomplete information.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions A recursive structure leading to an expression similar to the previous one holds in general for stationnary repeated games. Let us describe this representation for games with incomplete information. M is a product space K × L, π is a product probability p ⊗ q with p ∈ ∆(K ), q ∈ ∆(L) and in addition a 1 = k and b 1 = ℓ.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions A recursive structure leading to an expression similar to the previous one holds in general for stationnary repeated games. Let us describe this representation for games with incomplete information. M is a product space K × L, π is a product probability p ⊗ q with p ∈ ∆(K ), q ∈ ∆(L) and in addition a 1 = k and b 1 = ℓ. Given the parameter m = (k, ℓ), each player knows his own component and holds a prior on the other player's component.

Introduction: Shapley

Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions A recursive structure leading to an expression similar to the previous one holds in general for stationnary repeated games. Let us describe this representation for games with incomplete information. M is a product space K × L, π is a product probability p ⊗ q with p ∈ ∆(K ), q ∈ ∆(L) and in addition a 1 = k and b 1 = ℓ. Given the parameter m = (k, ℓ), each player knows his own component and holds a prior on the other player's component. From stage 1 on, the parameter is fixed and the information of the players after stage n is a n+1 = b n+1 = {i n , j n }.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions An auxiliary stochastic game Γ ′ is as follows: the "state space" M ′ is ∆(K ) × ∆(L) and is interpreted as the space of beliefs on the true parameter.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions An auxiliary stochastic game Γ ′ is as follows: the "state space" M ′ is ∆(K ) × ∆(L) and is interpreted as the space of beliefs on the true parameter. X = ∆(I ) K and Y = ∆(J) L are the type-dependent mixed action sets of the players; g is extended on X × Y × M ′ by g (p, q, x, y ) = k,ℓ p k q ℓ g (k, ℓ, x k , y ℓ ).

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions An auxiliary stochastic game Γ ′ is as follows: the "state space" M ′ is ∆(K ) × ∆(L) and is interpreted as the space of beliefs on the true parameter. X = ∆(I ) K and Y = ∆(J) L are the type-dependent mixed action sets of the players; g is extended on X × Y × M ′ by g (p, q, x, y ) = k,ℓ p k q ℓ g (k, ℓ, x k , y ℓ ). Given (p, q, x, y ), let x(i) = k x k i p k be the (total) probability of action i and p(i) be the conditional probability on K given the action i, explicitly p k (i) = p k x k i x(i) (and similarly for y and q).

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The resulting form of the Shapley operator is:

T (λ, f )(p, q) = sup x∈X inf y ∈Y {λ k,ℓ p k q ℓ g (k, ℓ, x k , y ℓ ) +(1 -λ) i,j
x(i)y (j)f (p(i), q(j))} (1) These equations are due to Aumann and Maschler (1966) and Mertens and Zamir (1971).

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

Contents

The classical fixed point formula for the discounted value is

v λ = T[λ, v λ ] (2)
Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The classical fixed point formula for the discounted value is

v λ = T[λ, v λ ] (2)
and the recursive formula for the n stage value is obtained similarly

v n = T[ 1 n , v n-1 ] (3)
with obviously v 0 = 0.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider now an arbitrarly evaluation probability µ on I N ⋆ . The total payoff is t µ m g m .

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider now an arbitrarly evaluation probability µ on

I N ⋆ . The total payoff is t µ m g m . Then µ induces a partition Π of [0, 1] with t 0 = 0, t k = k m=1 µ m .
Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider now an arbitrarly evaluation probability µ on I N ⋆ . The total payoff is t µ m g m . Then µ induces a partition Π of [0, 1] with t 0 = 0, t k = k m=1 µ m . The repeated game is naturally represented as a game played between times 0 and 1, where the actions are constant on each subinterval (t k-1 , t k ): its length µ k is the weight of stage k in the original game. Let v Π be its value.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider now an arbitrarly evaluation probability µ on I N ⋆ . The total payoff is t µ m g m . Then µ induces a partition Π of [0, 1] with t 0 = 0, t k = k m=1 µ m . The repeated game is naturally represented as a game played between times 0 and 1, where the actions are constant on each subinterval (t k-1 , t k ): its length µ k is the weight of stage k in the original game. Let v Π be its value. The recursive equation is

v Π = val{t 1 g 1 + (1 -t 1 )Ev Πt 1 }
where Π t 1 is the normalization on [0, 1] of the trace of the partition Π on the interval [t 1 , 1].

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Define now V Π (t k ) as the value of the game starting at time t k with evaluation m µ m+k g m . One obtains the alternative recursive formula

V Π (t k ) = val{(t k+1 -t k )g k+1 + EV Π (t k+1 )} (4)
Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Define now V Π (t k ) as the value of the game starting at time t k with evaluation m µ m+k g m . One obtains the alternative recursive formula

V Π (t k ) = val{(t k+1 -t k )g k+1 + EV Π (t k+1 )} (4)
By taking the linear extension we define this way for every finite partition Π, a function Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

V Π (t) on [0, 1]. Lemma Assume µ(n) decreasing. Then V Π is C -Lipschitz in t,
We consider now the asymptotic behavior of v n as n goes to ∞, or of v λ as λ goes to 0, or more generally of V Π as the mesh µ(1) goes to 0.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

We consider now the asymptotic behavior of v n as n goes to ∞, or of v λ as λ goes to 0, or more generally of V Π as the mesh µ(1) goes to 0. 1) Concerning games with incomplete information on one side, the first results proving the existence of lim n→∞ v n and lim λ→0 v λ are due to Aumann and Maschler (1966), including in addition an identification of the limit as Cav ∆(K ) u.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

We consider now the asymptotic behavior of v n as n goes to ∞, or of v λ as λ goes to 0, or more generally of V Π as the mesh µ(1) goes to 0. 1) Concerning games with incomplete information on one side, the first results proving the existence of lim n→∞ v n and lim λ→0 v λ are due to Aumann and Maschler (1966), including in addition an identification of the limit as Cav ∆(K ) u.

Here u(p) = val ∆(I )×∆(J) k p k g (k, x, y ) is the value of the one shot non revealing game, where the informed player does not use his information and Cav C is the concavification operator: given φ, a real bounded function defined on a convex set C , Cav C (φ) is the smallest function greater than φ and concave, on C .

Introduction: Shapley

Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Extensions of these results to games with lack of information on both sides were achieved by Mertens and Zamir (1971). In addition they identified the limit as the only solution of the system of implicit functional equations with unknown φ:

φ(p, q) = Cav p∈∆(K ) min{φ, u}(p, q), (5)
φ(p, q) = Vex q∈∆(L) max{φ, u}(p, q) (6)

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Extensions of these results to games with lack of information on both sides were achieved by Mertens and Zamir (1971). In addition they identified the limit as the only solution of the system of implicit functional equations with unknown φ:

φ(p, q) = Cav p∈∆(K ) min{φ, u}(p, q), (5)
φ(p, q) = Vex q∈∆(L) max{φ, u}(p, q)

Here again u stands for the value of the non revealing game: u(p, q) = val X ×Y k,ℓ p k q ℓ g (k, ℓ, x, y ) and we will write MZ for the corresponding operator

φ = MZ(u). (7)
Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions 2) As for stochastic games, the existence of lim λ→0 v λ in the finite case (Ω, I , J finite) is due to Bewley and Kohlberg (1976) using algebraic arguments:

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions 2) As for stochastic games, the existence of lim λ→0 v λ in the finite case (Ω, I , J finite) is due to Bewley and Kohlberg (1976) using algebraic arguments: the Shapley equation can be written as a finite set of polynomial equalities and inequalities involving {x k λ , y k λ , v λ (k), λ} thus it defines a semi-algebraic set in some euclidean space R N , hence by projection v λ has an expansion in Puiseux series.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions 2) As for stochastic games, the existence of lim λ→0 v λ in the finite case (Ω, I , J finite) is due to Bewley and Kohlberg (1976) using algebraic arguments: the Shapley equation can be written as a finite set of polynomial equalities and inequalities involving {x k λ , y k λ , v λ (k), λ} thus it defines a semi-algebraic set in some euclidean space R N , hence by projection v λ has an expansion in Puiseux series. The existence of lim n→∞ v n is obtained by an algebraic comparison argument, Bewley and Kohlberg (1976).

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Starting with Rosenberg and Sorin (2001) several asymptotic results have been obtained, based on the Shapley operator: continuous absorbing and recursive games, games with incomplete information on both sides, absorbing games with incomplete information on one side, Rosenberg (2000).

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Starting with Rosenberg and Sorin (2001) several asymptotic results have been obtained, based on the Shapley operator: continuous absorbing and recursive games, games with incomplete information on both sides, absorbing games with incomplete information on one side, Rosenberg (2000). We describe here an approach that was initially introduced by Laraki (2002) for the discounted case.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Let t, p and q such that X(t, p, q, W ) is non-revealing and W (•, p, q)φ(•) admits a global maximum at t. Adding s -(•t) 2 to φ(s) if necessary, we can assume that this global maximum is strict.

Let W ϕ(n) converge to W and define θ(n) ∈ {0, . . . , ϕ(n) -1} such that θ(n) ϕ(n) is a global maximum of W ϕ(n) (•, p, q) -φ(•). Then θ(n) ϕ(n) → t. One has W ϕ(n) θ(n) ϕ(n) , p, q = max x min y [ 1 ϕ(n)
g (x, y , p, q) + i,j

x(i)y (j)W ϕ(n) ( θ(n) + 1 ϕ(n) , p(i), q(j))]

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Let x n be optimal for the maximizer and y ∈ Y be any non-revealing strategy of player J.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Let x n be optimal for the maximizer and y ∈ Y be any non-revealing strategy of player J.

W ϕ(n) θ(n) ϕ(n) , p, q ≤ 1 ϕ(n)
g (x n , y , p, q) + i

x n (i)W ϕ(n) θ(n) + 1 ϕ(n) , p n (i), q

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Let x n be optimal for the maximizer and y ∈ Y be any non-revealing strategy of player J.

W ϕ(n) θ(n) ϕ(n) , p, q ≤ 1 ϕ(n)
g (x n , y , p, q)

+ i x n (i)W ϕ(n) θ(n) + 1 ϕ(n) , p n (i), q
By concavity of W ϕ(n) with respect to p i∈I

x n (i)W ϕ(n) θ(n) + 1 ϕ(n) , p n (i), q ≤ W ϕ(n) θ(n) + 1 ϕ(n) , p, q .

  of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation

  where C is a bound on the payoffs.

	Introduction: Shapley Introduction: Shapley
	Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general repeated games
	Extensions of the Shapley operator : general evaluation Extensions of the Shapley operator : general evaluation
	Asymptotic analysis: the main results
	Asymptotic analysis -the discounted case: games with incomplete information
	Asymptotic analysis -the continuous approach: games with incomplete information
	Asymptotic analysis -the continuous approach: extensions
	Contents
	Introduction: Shapley
	Extensions of the Shapley operator : general repeated games
	Extensions of the Shapley operator : general evaluation
	Asymptotic analysis: the main results
	Asymptotic analysis -the discounted case: games with
	incomplete information
	Asymptotic analysis -the continuous approach: games with
	incomplete information
	Asymptotic analysis -the continuous approach: extensions
	Sylvain Sorin	A Continuous Time Approach for the Asymptotic Value in

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

Contents

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The analysis is inspired from Larki (2001). Recall that the recursive equation is a fixed point operator:

x(i)y (j)v λ (p(i), q(j))} = v λ (p, q) (8)

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The analysis is inspired from Larki (2001). Recall that the recursive equation is a fixed point operator:

x(i)y (j)v λ (p(i), q(j))} = v λ (p, q) (8)

Remark that the family of functions {v λ (p, q)} is uniformly Lipschitz, hence relatively compact. To prove convergence it is enough to show that there is only one accumulation point.

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The analysis is inspired from Larki (2001). Recall that the recursive equation is a fixed point operator:

Remark that the family of functions {v λ (p, q)} is uniformly Lipschitz, hence relatively compact. To prove convergence it is enough to show that there is only one accumulation point. Note first that any accumulation point w satisfies

i.e. is a fixed point of the projective operator T (0, .). Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Assume now that w 1 and w 2 , w 1 ≥ w 2 are two different accumulation points and let (p 0 , q 0 ) an extreme point of the (convex hull of) the set where the difference w 1w 2 is maximal.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Assume now that w 1 and w 2 , w 1 ≥ w 2 are two different accumulation points and let (p 0 , q 0 ) an extreme point of the (convex hull of) the set where the difference w 1w 2 is maximal. Using (9) and the fact that all functions involved are saddle, this implies that the set X(0, w 1 )(p 0 , q 0 ) of profile of mixed actions x ∈ X optimal in T(0, w 1 ) is included in the set NR X (p 0 ) of non revealing actions at p 0 (meaning that for any move i having positive probability p(i) = p 0 ).

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider a sequence v λn converging to w 1 and let x n be optimal for T(λ n , v λn )(p 0 , q 0 ). Jensen's inequality leads to Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider a sequence v λn converging to w 1 and let x n be optimal for T(λ n , v λn )(p 0 , q 0 ). Jensen's inequality leads to

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider a sequence v λn converging to w 1 and let x n be optimal for T(λ n , v λn )(p 0 , q 0 ). Jensen's inequality leads to

Let x be an accumulation point of the sequence {x n }, one obtains as λ n goes to 0:

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider a sequence v λn converging to w 1 and let x n be optimal for T(λ n , v λn )(p 0 , q 0 ). Jensen's inequality leads to

Let x be an accumulation point of the sequence {x n }, one obtains as λ n goes to 0:

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Consider a sequence v λn converging to w 1 and let x n be optimal for T(λ n , v λn )(p 0 , q 0 ). Jensen's inequality leads to

Let x be an accumulation point of the sequence {x n }, one obtains as λ n goes to 0:

The dual property implies convergence. Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions For simplicity of notations, we consider the sequence of uniform partitions.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions For simplicity of notations, we consider the sequence of uniform partitions. For each integer n, let W n (1, p, q) := 0 and for m = 0, ..., n -1 define W n ( m n , p, q, ω) inductively as follows.

W n ( m n , p, q) = max

Extend W n (•, p, q) to [0, 1] by linear interpolation and consequently:

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Moreover if W is an accumulation point of the equi-continuous family {W n } then for all (t, p, q):

x(i)y (j)W (t, p(i), q(j))   Let X(t, p, q, W ) ⊆ ∆(I ) K be the set of strategies for player I that are optimal for the above game.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The variational Inequalities

Theorem For any accumulation point W of the family {W n }, all (p, q) ∈ ∆(K ) × ∆(L) and all C 1 test function φ : [0, 1] → R : (P1) If, for some t ∈ [0, 1), X(t, p, q, W ) is non-revealing and W (•, p, q)φ(•) has a global maximum at t, then u(p, q) + φ ′ (t) ≥ 0. (P2) If, for some t ∈ [0, 1), Y(t, p, q, W ) is non-revealing and W (•, p, q)φ(•) has a global minimum at t then u(p, q) + φ ′ (t) ≤ 0.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Let t, p and q such that X(t, p, q, W ) is non-revealing and W (•, p, q)φ(•) admits a global maximum at t.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Let t, p and q such that X(t, p, q, W ) is non-revealing and W (•, p, q)φ(•) admits a global maximum at t. Adding s -(•t) 2 to φ(s) if necessary, we can assume that this global maximum is strict.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Let t, p and q such that X(t, p, q, W ) is non-revealing and W (•, p, q)φ(•) admits a global maximum at t. Adding s -(•t) 2 to φ(s) if necessary, we can assume that this global maximum is strict.

Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Hence: Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions proof Hence:

Assume {x n } converges to some x (hence non-revealing).

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Passing to the limit:

Since this inequality holds true for every y , taking the maximum with respect to x yields: u(p, q) + φ ′ (t) ≥ 0 .

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

The comparison principle

Theorem Let W 1 and W 2 be two fixed points of Φ in G and suppose that:

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

Theorem Let W 1 and W 2 be two fixed points of Φ in G and suppose that:

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

Theorem Let W 1 and W 2 be two fixed points of Φ in G and suppose that:

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

Theorem Let W 1 and W 2 be two fixed points of Φ in G and suppose that:

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

We argue by contradiction, assuming that max

Then, for ε > 0 sufficiently small,

(10) Moreover δ(ε) → δ as ε → 0.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

We claim that there is (t ε , s ε , p ε , q ε ), point of maximum above, such that X (t ε , p ε , q ε , W 1 ) is non-revealing for player I and Y (s ε , p ε , q ε , W 2 ) is non-revealing for player J.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

We claim that there is (t ε , s ε , p ε , q ε ), point of maximum above, such that X (t ε , p ε , q ε , W 1 ) is non-revealing for player I and Y (s ε , p ε , q ε , W 2 ) is non-revealing for player J. Finally we note that t ε < 1 and s ε < 1 for ε sufficiently small, because δ(ε) > 0 and W 1 (1, p, q) ≤ W 2 (1, p, q) for any (p, q) by P3.

Sylvain Sorin

A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

Since the map t → W 1 (t, p ε , q ε ) -(t-sε) 2 2ε has a global maximum at t ε and since X (t ε , p ε , q ε , W 1 ) is non-revealing for player I, condition P1 implies that

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

Since the map t → W 1 (t, p ε , q ε ) -(t-sε) 2 2ε has a global maximum at t ε and since X (t ε , p ε , q ε , W 1 ) is non-revealing for player I, condition P1 implies that

In the same way, since the map s → W 2 (s, p ε , q ε ) + (tε-s) 2

2ε

εs has a global minimum at s ε and since Y (s ε , p ε , q ε , W 2 ) is non-revealing for player J, we have by condition P2 that

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The comparison principle

Since the map t → W 1 (t, p ε , q ε ) -(t-sε) 2 2ε has a global maximum at t ε and since X (t ε , p ε , q ε , W 1 ) is non-revealing for player I, condition P1 implies that

In the same way, since the map s → W 2 (s, p ε , q ε ) + (tε-s) 2

2ε

εs has a global minimum at s ε and since Y (s ε , p ε , q ε , W 2 ) is non-revealing for player J, we have by condition P2 that u(p ε , q ε ) + t εs ε ε + ε ≤ 0 .

This latter inequality contradicts (11).

Sylvain Sorin A Continuous Time Approach for the Asymptotic Value in Introduction: Shapley Extensions of the Shapley operator : general repeated games Extensions of the Shapley operator : general evaluation Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions Asymptotic analysis: the main results Asymptotic analysis -the discounted case: games with incomplete information Asymptotic analysis -the continuous approach: games with incomplete information Asymptotic analysis -the continuous approach: extensions

The same tols extend to the study of absorbing games and can be applied to the "splitting game". Sketch of the approach : The family of value functions is relatively compact Consider two accumulation points w 1 and w 2 and a point (t, ω) where the difference w 1w 2 is maximal. Deduce a variational inequality at (t, ω) for any majorant of w 1 and a dual property Prove a comparison principle.


Assessing precision of lasso conditional logistic estimates in the case-crossover design

Marta Avalos

International Biometric Society Channel network IBS Channel Network, 3rd Conference, Bordeaux, France, 11 -13 April 2011

ASSESSING PRECISION OF LASSO CONDITIONAL LOGISTIC ESTIMATES IN THE CASE-CROSSOVER

DESIGN

Avalos, Marta

INSERM U897 and University Bordeaux Segalen, Bordeaux, France

The case-crossover design introduced by Maclure ( 1991) is an observational epidemiological study for analyzing the effects of transient exposures on the risk of acuteonset events. This design is based on exclusively selecting subjects that have experienced the event under investigation. The aim is to identify the short-term, transient triggers of event, rather than to identify who is at highest risk of disease. The association between event onset and risk factors is then estimated by comparing exposure during the period of time just prior to the event onset (case period) to the same subject's exposure during one or more control periods. The standard tool for this analysis is conditional logistic regression (Breslow and Day, 1980;Maclure, 1991).

Until present, variable selection issues have not received much attention when modeling the risk of an event of interest under the case-crossover design. On the one hand, time-invariant potential confounders are controlled by design. On the other hand, studies in the literature using the case-crossover design are often concerned by few main risk factors. Nevertheless, advances in technology for data collection lead to complex situations such as those involving many predictors. This may also be the case in studies based on the casecrossover design.

The lasso procedure (Tibshirani, 1996), which relies on an L1 penalty, has been widely used for accommodating high-dimensional predictors. This approach has also been adapted to conditional logistic regression in the context of matched case-control studies (Avalos, 2009). However, a drawback of LASSO regression is that no measure of uncertainty is provided.

In this work we study by simulations the behavior of nonparametric bootstrap percentile confidence intervals in assessing significance of lasso point estimates. We also illustrate the use of this method with some examples under the case-crossover design. Volume 1: The analysis of case-control studies. IARC Scientific Publications, Lyon. 3. Maclure, M. (1991). The case-crossover design: a method for studying transient effects on the risk of acute event. Am J Epidemiol. 133, 144-153. 4. Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Royal. Statist. Soc B. 58,

  1. Avalos M. (2009) Model selection via the lasso in conditional logistic regression. In: Conference abstracts. Second International Biometric Society Channel Network Conference, Belgium (2009). 2. Breslow, N. E. and Day, N. E. (1980). Statistical methods in cancer research.