Camille Cornand
email: cornand@gate.cnrs.fr

Frank Heinemann
email: frank.heinemann@ww.tu-berlin.de

Gabriel Desgranges

John Duffy

Petra Geraats

Ed Hopkins

Paul Pezanis-Christou

Alexander Meyer-Gohde

Papers

Keywords: JEL Classification: C92, D82, D84 coordination games, strategic uncertainty, private information, public information

In games with strategic complementarities, public information about the state of the world has a larger impact on equilibrium actions than private information of the same precision, because public signals are more informative about the likely behavior of others. We present an experiment in which agents' optimal actions are a weighted average of the fundamental state and their expectations of other agents' actions. We measure the responses to public and private signals. We find that, on average, subjects put a larger weight on the public signal. In line with theoretical predictions, as the relative weight of the coordination component in a player's utility increases, players put more weight on the public signal when making their choices. However, the weight is smaller than in equilibrium, which indicates that subjects underestimate the information contained in public signals about other players' beliefs.

Measuring Agents' Reaction to Private and Public Information in Games with Strategic Complementarities *

-Introduction

Financial markets, macroeconomic environments, and network games are often characterized by strategic complementarities. In games with strategic complementarities, public and private information about an uncertain state of the world have distinct effects, because information that is shared by several agents is more informative about the likely beliefs and actions of others than private information. Since agents have an incentive to align their actions to the expected actions of others, equilibrium choices are affected more by public than by private signals of equal precision. This has been used to explain why markets 'overreact' to public signals, so that public information may raise price volatility on financial markets and cause negative welfare effects. The aim of this paper is to experimentally evaluate the weights that subjects attach to private and public signals in games of weak strategic complementarities 1 . We show that on average agents indeed attach a higher weight to public than to private information, but the asymmetry in weights is smaller than in equilibrium.

Our experiment builds upon [START_REF] Morris | Social Value of Public Information[END_REF], who present a generalized beauty contest game with weak strategic complementarities for analyzing the distinct effects of public and private information. In their game, an agent's action is a weighted average of a public and a private signal.

The equilibrium weight attached to the public signal is higher than its relative precision. This 'overreaction' [START_REF] Morris | Social Value of Public Information[END_REF], p. 1530) is due to the higher informational content of public signals regarding the likely beliefs of other agents. In this context, Morris and Shin show that raising the precision of public signals may reduce expected welfare. 2 Welfare effects of public and private information depend on two considerations requiring empirical investigation: the relative precision of the two signals and the relative weight that agents attach to these signals. These considerations interact: the higher the weight that agents attach to public signals, the more likely it is that an imprecise public signal will reduce welfare. In equilibrium, the relative weight is determined by the signals' relative precision and agents' payoff functions. This paper precisely aims at measuring and analyzing how much public signals are taken into account compared to private ones -in a game with weak strategic complementarities. We present an experiment on a game that is characterized by both fundamental and strategic uncertainty. As in [START_REF] Morris | Social Value of Public Information[END_REF], agents have to choose actions that are close to a fundamental state but also close to each other. We test predictions of this approach by implementing two-player versions of this game with varying weights on fundamental and strategic uncertainty. In line with theory, we find that:

(i) subjects put larger weights on the public signal and (ii) as the relative weight of the coordination component in a player's utility rises, players put more weight on the public signal. However, overall, the weights put on the public signal are smaller than theoretically predicted.

A cognitive hierarchy model with 2 levels of reasoning helps explaining the observation that subjects attach a lower weight on public signals than predicted by the equilibrium. Stated second order beliefs, however, indicate that subjects underestimate how informative the public signal is in assessing other players' expectations. This may be viewed as an alternative explanation of behavior. However, these non-Bayesian higher-order beliefs can explain only a part of the observed systematic deviation of behavior from equilibrium.

The next section presents the basic game. Section 3 sets the experimental design and motivates hypotheses. Section 4 states the experimental results and provides measures for the relative 1 [START_REF] Angeletos | Transparency of Information and Coordination in Economies with Investment Complementarities[END_REF] distinguish weak and strong strategic complementarities. Under weak complementarities, the game has a unique equilibrium, while strong strategic complementarities define a coordination game with multiple equilibria. 2 [START_REF] Svensson | Social Value of Public Information: Comment: Morris and Shin (2002) Is Actually Pro-Transparency, Not Con[END_REF] has argued that detrimental welfare effects of public information are unlikely, because they require that public information be of lower precision than private information, which conflicts with empirical findings. For a response, see [START_REF] Morris | Social Value of Public Information: Morris and Shin (2002) Is Actually Pro-Transparency, Not Con: Reply[END_REF]. For discussions about the welfare effects of public information in closely related frameworks, see [START_REF] Cornand | Optimal Degree of Public Information Dissemination[END_REF], [START_REF] Baeriswyl | Reducing overreaction to central banks' disclosures : theory and experiment[END_REF], [START_REF] Woodford | Central Bank Communication and Policy Effectiveness[END_REF], Angeletos and Pavan (2007 a,b), [START_REF] Hellwig | Heterogeneous Information and the Welfare Effects of Public Information Disclosures[END_REF], and [START_REF] Myatt | On the sources and value of information: public announcements and macroeconomic performance[END_REF] among others.

weights that subjects put on private and common signals, relating them to equilibrium predictions.

Section 5 presents two behavioral theories to explain the observed deviation of behavior from equilibrium. Finally, section 6 concludes.

-Two-player game with weak strategic complements

We describe a game in which agents have to choose actions that are close to a fundamental state but also close to what the others believe. The framework is a two-player version of [START_REF] Morris | Social Value of Public Information[END_REF] adapted for conducting an experiment. 3The fundamental state of nature is given by θ and has a uniform distribution on the reals.

Agent i chooses action ℜ ∈ i a
, and we write a for the action profile over all agents. The utility function for individual i has two loss components:

2 2 0 ) ( ) ( ) 1 ( ) , ( j i i i a a r a r U a u - - - - - ≡ θ θ , ( 1
)
where j is the partner of i and U 0 is a constant that cannot be affected by player i. The first quadratic term captures losses arising from the distance between an action and the underlying state θ. The second quadratic term describes losses from miscoordination between both players' actions. The optimal action of agent i is given by the first order condition from maximizing (1):

( ) ( )

) ( 1 j i i i a E r E r a + - = θ , (2)
where ) (⋅ i E is the posterior expectation conditional on x i and y and j a is the action of the other player j. One needs to calculate ( )

θ i E and ) ( j i a E . Because both signals have the same precision, ( ) = θ i E ( ) ( ). , 2 , y x x E y x y x E i j i i = + = θ
The optimal strategy of any agent is a linear combination of his signals: In equilibrium, actions are distorted away from the expected θ towards y. This mirrors the disproportionate impact of the public signal in coordinating agents' actions and emphasizes the role of public information as a focal point for private actions. Note that the equilibrium weight on the private signal * γ decreases in the relative importance of coordination measured by r. [START_REF] Morris | Social Value of Public Information[END_REF] concentrate on the welfare effects of public information and show that if welfare depends only on how close actions are to the fundamental, public signals of low precision may actually reduce ex-ante expected welfare, even though they improve agents' ability to predict the fundamental. Here, the negative welfare effect from distorting actions towards the public signal exceeds the positive welfare effect that arises from the additional information. In these environments, the economy is better off without a public signal. This argument has been used to question the merits of central bank transparency.

( )y x a j j γ γ - + = 1 . So, ( ) ( ) ( ) y y x x E y x a E a E i j i j j i γ γ - + = = 1 , , ) ( ( ) . 2 1 y x y i + + - = γ γ Inserting this in equation (2) yields ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] 2 1 1 1 1 , 1 , γ γ θ - + + - - = - + = r y r x y x E r y x a rE a i i i i i i i . Thus, in equilibrium 2 ) 1 ( 1 * * γ γ - - = r
Negative welfare effects of public signals require that agents put a sufficiently large weight on public signals. In beauty contest games like the one used by [START_REF] Morris | Social Value of Public Information[END_REF], real players' behavior has been found consistent with limited levels of reasoning [START_REF] Nagel | Unraveling in Guessing Games: An Experimental Study[END_REF]. As will be explained in Section 5 below, such behavior leads to weights on the public signal that are lower than in equilibrium. [START_REF] Cornand | Higher-Order Beliefs and the Social Value of Public Information[END_REF] show that negative welfare effects of public disclosures require that agents put weights on public signals that are at least as large as the weight associated with level-2 reasoning. Thus, it is an empirical question, whether real players put a weight on public signals that is sufficiently strong to cause negative welfare effects of public signals. 4 To 4 Other experiments concerning the welfare effects of public information have concentrated on games with multiple equilibria: [START_REF] Anctil | Information Transparency and Coordination Failure: Theory and Experiment[END_REF] demonstrate that private signals with high precision are not sufficient to achieve coordination on an efficient equilibrium. [START_REF] Heinemann | The Theory of Global Games on Test: Experimental Analysis of Coordination Games with Public and Private Information[END_REF] compare perfect public and noisy private signals in a speculative attack game and find small effects towards higher efficiency with perfect information. [START_REF] Cornand | Speculative Attack and Informational Structure: An Experimental Study[END_REF] shows that subjects put a larger weight on the public signal if they receive both a private and a public signal about the state of the economy. For other experiments dealing with public versus private information, see [START_REF] Forsythe | Asset Valuation in an Experimental Market[END_REF], [START_REF] Plott | Rational Expectations and the Aggregation of Diverse Information in Laboratory Security Markets[END_REF], [START_REF] Mckelvey | Elections with Limited Information: A Fulfilled Expectations Model Using Contemporaneous Poll and Endorsement Data as Information Sources[END_REF], [START_REF] Mckelvey | Public and Private Information: An Experimental Study of Information Pooling[END_REF] and [START_REF] Hanson | Correction to MacKelvey and Page, "Public and Private Information: An Experimental Study of Information Pooling[END_REF]. They investigate how individuals use public information to answer this question, we propose an experiment that measures the actual weights that subjects put on private versus public information.

-Experimental design

The experiment aims at measuring reactions to private and public information in the game introduced in the last section. In particular, we are interested in testing whether subjects put larger weights on public than on private signals, whether the observed weights are in line with equilibrium predictions, and how changes in r affect these weights.

18 sessions were run at the BETA (Bureau d'Economie Théorique et Appliquée) laboratory in Strasbourg (using software Regate [START_REF] Zeiliger | A Presentation of Regate, Internet based Software for Experimental Economics[END_REF]) between January 2008 and March 2010. Each session had 16 participants who were mainly students from Strasbourg University (most were students in economics, mathematics, biology and psychology). Subjects were seated in random order at PCs. Instructions were then read aloud and questions answered in private. Throughout the sessions, students were not allowed to communicate with one another and could not see each others' screens. Each subject could only participate in one session. Before starting the experiment, subjects were required to answer a few questions to ascertain their understanding of the rules. Instructions and questionnaires are given in Appendix A. The experiment started after all subjects had given the correct answers to these questions.

Each session consisted of four stages with a total of 50 periods. While Stages 1, 2, and 4 were the same in all sessions, Stage 3 contained three different treatments across sessions. In each period, subjects were paired up randomly. They did not know the identity of their partner and they knew that they would most likely not meet the same partner in the next period. For each pair of subjects, a fundamental state θ was drawn randomly using a uniform distribution from the interval [50,450]. . Signals were drawn independently from these intervals using a uniform distribution. 5 The random process was explained in the instructions.

In the first three stages, each subject had to decide on an action a i conditional on her signals. 6 Signals and actions were limited to one decimal point.

augment their original private information and whether, in doing so, a rational expectations equilibrium is attained. 5 Having a sufficiently large support of the prior distribution enables us to reduce the informational content conveyed by the prior mean. In addition, this reduces the set of signals, for which the conditional posterior distribution is skewed. 6 To limit potential losses that might arise from typing mistakes, we restricted choices and stated expectations to In the second stage ("r=1"), the payoff function was given by (

)

2 100 j i a a - -
, where a i denotes the partner's action. Here, subjects should coordinate their actions irrespective of the fundamental state: this is a pure coordination game from which subjects should learn that public information is more helpful in overcoming coordination problems than private information. Any strategy that maps the public signal into the reals is an equilibrium 8 , provided that both subjects coordinate on the same strategy. The public signal, however, provides a focal point, 9 so that we expected subjects to coordinate on actions y a i = .

Stage 3 contained three different treatments, each conducted in 6 of the sessions. In Treatment "r=0.5", the payoff function was

( ) ( ) 2 2 200 θ - - - - i j i a a a , while it was ( ) ( ) 2 2 3 400 θ - - - - i j i a a a in Treatment "r=0.25" and ( ) ( ) 2 2 3 400 θ - - - - i j i a a a
in Treatment "r=0.75". Contrary to Stages 1 and 2, the three treatments of Stage 3 exhibit both a fundamental and a coordination motive. They correspond to the game by [START_REF] Morris | Social Value of Public Information[END_REF]. Given that players choose linear combinations of both signals In Stage 4 we elicited higher-order beliefs. Subjects also received private and public signals.

But here, each subject was asked to state two expectations: one regarding the fundamental state and one regarding her partner's expectation of the fundamental state. The payoff function was 7 When signals are smaller than 60 or larger than 440, the posterior distribution of θ is skewed, because of the bounded support of θ. 8 In the limiting case where fundamental uncertainty disappears and subjects' payoffs depend only on the distance between their actions, equilibrium theory does not yield a unique prediction. Any coordinated strategy is an equilibrium. However, the limit of equilibria in games with a decreasing weight on fundamental uncertainty uniquely selects a strategy in which all agents follow the public signal and ignore all private information. 9 A formalization of focal points is provided by [START_REF] Alós-Ferrer | Hidden Symmetries and Focal Points[END_REF]. 10 Due to the bounded support of θ, the equilibrium deviates from this linear combination into the direction of the center of the support. To see this, imagine that player i receives signals x i = 40 and y = 50. From her private signal, she can deduce that θ = 50. The posterior distribution of the other player's signal is uniform in [40,60]. Since the other player should never choose an action below 50, the expected action by the other player is above 50. Therefore, player i should also choose an action above 50. Obviously, it is too demanding to assume that players update their beliefs correctly at the edges of the support. In the data, we find systematic effects for signals up to 60 and above 440. We checked that restricting data analysis to situations with 430 70 < < θ does not alter our results. , where e i (•) denotes the stated expectation of subject i. The last term of the payoff function incentivizes the statement of higher-order beliefs. To our knowledge, this is the first experiment with a direct elicitation of higher-order beliefs.

Since individual decisions are not independent in repeated coordination games, we conducted 18 sessions with at least 6 sessions per treatment. 11 For statistical tests, we use the average weight on the private signal within a session as an independent observation. Table 1 gives an overview over sessions, stages, and treatments. After each period, subjects were informed about the true state, their partner's decision and their payoff. Information about past periods from the same stage (including signals and own decisions) was displayed during the decision phase on the lower part of the screen. Subjects knew from the start that they were playing the aforementioned stages in this order. At the end of each session, the points earned were summed up and converted into Euros. 100 points were converted to 25 cents in sessions 1 -6 and 13 -18, and to 43 cents in sessions 7 -12. Payoffs ranged from € 12 to € 33. The average payoff was about € 22. 13 Sessions lasted for around 90 minutes.

Sessions

Our main hypotheses are derived from the theory predictions: Table 2 summarizes the equilibrium weights on the private signal * γ for interior states. The natural hypothesis arising from theory is that subjects attach weights to private signals that are distributed around the equilibrium weights. As explained above, we intend to test this hypothesis against the alternative arising from a cognitive hierarchy model which predicts higher weights on the private signal for r > 0. Two other testable hypotheses are consistent with equilibrium and with limited levels of reasoning: the weights put on the private signal are smaller than 0.5 (for r > 0) and decreasing in r. 11 We also conducted 6 control sessions to include treatments with only one signal in order to test the welfare reducing effect of private signals for r=1 directly by comparing payoffs. Control sessions elicited beliefs at the second stage before strategic games and with a slightly different representation of the payoff function. We only present standard sessions in the main text. The results of control sessions are reported in Appendix B. 12 In Session 13, Treatment "r=0.75" unintentionally stopped after 26 periods. 13 In all stages, it was possible to earn negative points. This actually occurred in about 3.3% of all decision situations in standard sessions. Realized losses were of a size that could be counterbalanced by positive payoffs within a few periods. In general, losses were covered by earnings from the next three periods or balanced by earnings of previous periods.

-Experimental results

We use the following structure in analyzing data. First, we analyze whether subjects' choices are between their two signals as predicted by theory. Here we also look at the proportion of choices that are closer to the public or private signal contingent on the stage or treatment. Then, we estimate the average weights that subjects attach to the private signal for each session, stage, and treatment. We investigate whether average weights on the private signal are positive, smaller than 0.5, and whether they deviate systematically from the equilibrium value. If subjects best respond to the observed actions of their respective partners, their strategies should converge over time towards the equilibrium. We test whether there is a time trend by comparing the data from the first and second halves of a stage.

Comparing observed weights from different treatments, we test comparative static predictions arising from theory.

Some considerations about rationality

Theory predicts that subjects choose actions that are weighted averages of their two signals. . We observe that the proportion of choices outside this interval also becomes large when the interval is small.

Furthermore, the smaller is 1 I , the larger is 2 I . Since both of these intervals have some appeal for reasonable choices, we check, how many choices were contained in the union of both sets, 2 1 3

I I I ∪ =
. Note that the size of interval 3 I is less sensitive to the randomly drawn signals and never gets smaller than 10, while the size of 1 I and 2 I may shrink to zero for appropriate signals.

Result 1: 98% of choices are contained in I 3 . The proportion of choices inside I 1 is 85%.

I I I ∪ = 97% 99% 99% 99% 98%
Table 3 -Crude classification of individual choices Table 3 displays the percentage of all choices within these intervals. Counting the number of choices that are closer to the common signal, closer to the private, or in the middle provides a crude first impression of whether subjects put a larger weight on the common signal and how dispersed the distribution of relative weights is. 14 As r is increased, the proportion of choices closer or equal to the common signal y increases.

Estimated weights on the private signal

We estimate the relative weights of the two signals on subjects' decisions by fitting linear regressions to data. Because the payoff function rewards coordination when r > 0, we may expect that the weights applied by different subjects within a session converge over time. Hence, individual weights are not independent. To be on the conservative side, our main statistical analyses are based on the average weights per session. We estimate these weights by joining the data from all subjects within a session.

The regression equation is

it t it t it u y x c y a + - + = - ) ( γ ,
where γ is the session-specific relative weight on private information. The constant c stands for a bias towards higher or lower numbers. Using the Bonferroni correction for multiple hypotheses, this constant turns out to be insignificant in all stages and treatments. Implicitly, our regression imposes a restriction that the absolute coefficients on private and common signals add up to 1. In regressions without this restriction, we checked that violations of this restriction are not significant either. 15 Table 4 displays average weights on the private signal for all sessions and stages and compares them with theoretical weights. For games that were played at least 10 rounds, we split the data from the first and second halves of the stage so that we could analyze whether there was some systematic change of behavior over time. Figure 1 presents the evolution of these weights over periods; this representation captures the differences across stages within each group of sessions and between 14 To break ties in counting: if both signals and the action coincide, we count the choice as "Middle". If

1 . 0 | | = -i x y
, and ] [ y a x a

i i i = =
, we count the choice as "equal to x i " ["equal to y"]. 15 Data and programs are available at http://www.macroeconomics.tu-berlin.de/menue/working_papers/. different treatments across sessions. Note that average weights on private signals stay above the equilibrium values (for r > 0) in all periods. Data indicate that the relative weight on the private signal is around 0.5 for "r=0" and tends to be decreasing in r. For r=1, r=0.5, and r=0.25, all session-specific weights are larger than in equilibrium. For r=0.75, there is one exception (Session 17, 2 nd half). Comparing first and second halves, there seems to be some trend towards equilibrium in stage "r=1", but not in treatments with 0 < r < 1.

These impressions are widely confirmed by nonparametric tests, reported below. Specifically, we test whether estimated weights differ from 0.5, from the respective equilibrium values, and between first and second halves of a stage/treatment. We also test monotony of the weights in r.

Unless otherwise noted, all tests are based on counting session-specific averages as independent observations with a significance level of 5%. When we split data from the first and second halves of a stage, we apply tests to both halves separately.

Result 2: For r = 0, subjects put an equal weight on both signals consistent with Bayesian rationality.

In Stage "r=0", there is no significant difference between group specific estimated weights and 0.5.

Two-tailed Wilcoxon matched pairs signed rank tests yield p-values of 0.77. Since individual decisions are independent in this first stage, we also performed tests on estimated individual weights. 16The hypothesis that individual weights are distributed around 0.5 cannot be rejected either (the p-value is 0.70).

Our main result is that for r > 0, subjects put a lower weight on public signals than in equilibrium.

Result 3: For 0 < r < 1, subjects tend to put larger weights on public than on private signals, but the difference is smaller than theoretically predicted. There is no trend towards equilibrium.

Our prior expectation (arising from the widespread success of cognitive hierarchy models in beauty contest games) was that estimated group-specific weights on the private signal are smaller than 0.5 and larger than equilibrium weights. Therefore, we use one-sided tests on the null hypotheses that the median weight is equal to the respective equilibrium value against the alternative hypotheses that it is smaller than 0.5 or exceeds the equilibrium weight, respectively. One-tailed Wilcoxon matched pairs signed rank tests yield the p-values, reported in Table 5. While most tests reject the null hypotheses at 5%, we cannot reject that the median weight equals 0.5 for the first half of Treatment "r=0.25". Comparing the weights assigned to private signals in the first and second halves of each treatment, we find no significant differences. Using the two-tailed Wilcoxon matched pairs test, pvalues range between 15% and 85%. In particular, for Treatment "r=0.75", the p-value is 85%. Thus, we conclude that subjects tend to put larger weights on public than on private information17 , but the difference is smaller than theoretically predicted and there is no systematic trend over time.

Result 4: For r = 1, subjects assign larger weights to public than to private information: they lean towards coordinating on the public signal, but do not achieve full coordination during the course of the stage.

In Stage "r=1", all estimated weights are positive and below 0.5. Hence, the Wilcoxon matched pairs signed rank test yields p-values below 1% for the null hypotheses γ = 0 and γ = 0.5. Testing whether the weights assigned to private signals in the last 5 rounds were equal to those in the first half against the alternative of a systematic trend towards equilibrium (i.e. lower weights in the last rounds), the onetailed Wilcoxon matched pairs test yields p-values of 0.02%. Thus, we can reject the hypothesis of no trend towards equilibrium for this stage.

Result 5: Over all sessions, the weight assigned to the private signal tends to decrease in r as predicted by theory.

A one-tailed Wilcoxon matched pairs signed rank test finds that γ is getting significantly smaller with increasing r, if we compare "r=0" with "r=0.5" or "r=0.5" with "r=1". P-values are always below 4%.

Note that here the difference in r is 0.5. Comparing treatments where the difference in r is only 0.25, test results are mixed: comparing "r=0.5" with "r=0.25" or "r=0.5" with "r=0.75", one-tailed Mann-Whitney tests yields p-values below 3% except for "r=0.5" (first half) compared to "r=0.75" (second half), where p=6.6%. However, the one-tailed Wilcoxon matched pairs fails to reject the hypothesis of equal weights for Treatment "r=0.25" versus "r=0" and for Treatment "r=0.75" versus "r=1" (p>9%) except for Treatment "r=0.75" (second half) compared to "r=1" (second half), where p=3.1%. Figure 2 relates the average weight on the private signal for each treatment to theoretical predictions and also displays how the weight depends on r.

Alternative explanations for observed behavior

We know from [START_REF] Nagel | Unraveling in Guessing Games: An Experimental Study[END_REF], [START_REF] Kübler | Limited Depth of Reasoning and Failure of Cascade Formation in the Laboratory[END_REF] and many other experimental papers that the equilibrium may be a poor predictor for actual behavior when it requires infinite levels of reasoning. In their environments, cognitive-hierarchy models with limited levels of reasoning yield better predictions. 18 In this section, we study two models of bounded rationality: one is a model of a cognitive hierarchy (limited levels of reasoning) and the other one is based on non-Bayesian higherorder beliefs. Both models are able to explain weights on the private signal that exceed the equilibrium values. 19

Cognitive hierarchy

The model of cognitive hierarchies assumes that agents process information rationally, but fail to consider that others decide with the same degree of rationality. [START_REF] Nagel | Unraveling in Guessing Games: An Experimental Study[END_REF] and [START_REF] Stahl | Experimental Evidence on Players' Models of Other Players[END_REF] define level-0 types as subjects who choose an action randomly from a uniform distribution over all possible actions. For k > 0, a level-k type is playing best response to level-k -1. In our game, if the partner's action has a uniform distribution on the reals, expected losses from the coordination part of the payoff function are infinite for any action, so that any action is a best reply as long as r > 0.

However, expected losses resulting from deviations from the fundamental are minimized by choosing the expected state. Hence, we define the best response to a uniform distribution over all reals as 18 Shapiro, Shi, and Zillante (2009) analyze the predictive power of level-k reasoning in a game that combines features of [START_REF] Morris | Social Value of Public Information[END_REF] with [START_REF] Nagel | Unraveling in Guessing Games: An Experimental Study[END_REF]. They try to identify whether individual strategies are consistent with level-k reasoning. They argue that the predictive power of level-k reasoning is positively related to the strength of the coordination motive and to the symmetry of information. Their experiment however treats public and private signals in an asymmetric way by normalizing public signals. 19 Hypotheses that arise from combining non-Bayesian higher-order beliefs with limited levels of reasoning are derived and tested in Appendix D. , so that level-1 players place weight 0.5 on both signals. Note that ) (

1 θ i i E a =
is the best response of a player who ignores the strategic part of the payoff function. Starting from this definition, actions for higher levels of reasoning can be calculated as follows: suppose that a player j attaches weight k γ to her private signal. The best response to such behavior is

. ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( 1 y r x E r E r a rE E r a k j i k i j i i k i γ γ θ θ - + + - = + - = +
Since the expected private signal of the other player equals the expected state,

[ ] . ) 1 ( 2 ) 1 ( 2 ) 1 ( ) 1 ( ) ( ) 1 ( 1 y r r r x r r y r E r r a k k i k k i k k i       - + + - + + - = - + + - = + γ γ γ γ θ γ
Hence, the next level of reasoning is 2

) 1 ( 1 k k r r γ γ + - = +
. With higher levels, the weight on the private signal decreases from 0.5 towards the equilibrium value. As can be seen from Figure 2, the observed average weights are close to the weights of level-2 reasoning for all values of r. Comparing session-specific weights presented in Table 4 with weights from limited levels of reasoning displayed in Table 6, shows that in Treatments "r=0.5" and "r=0.25", average group weights are higher than the weights from level-2 reasoning with only one exception for r=0.5 (Session 5, first half). The hypothesis that average weights are equal to or below those from level-2 reasoning is rejected by one-tailed Wilcoxon matched pairs signed rank test at p-values below 5%. In Treatment "r=0.75", session-specific weights are distributed around the weight from level-2, and we cannot reject that the weights are equal to or below those from level-2 reasoning. In this treatment, one-sided tests reject level-3 reasoning (observed weights are not lower than 3 γ ), but two- sided tests do not (equality of observed weights with 3 γ cannot be rejected).

For r=1, second half, the hypothesis 2 γ γ ≤ cannot be rejected either. These inhomogeneous test results indicate that subjects might behave in accordance to higher levels of reasoning with increasing r.

As [START_REF] Cornand | Higher-Order Beliefs and the Social Value of Public Information[END_REF] show, the welfare-detrimental effects of public information require that agents put a weight on the private signal that is smaller than the one associated with level-2 reasoning. Since we cannot reject this for r ≥ 0.75, we cannot rule out that for a strong coordination motive (high value of r), public information may be detrimental to welfare, if welfare depends only on how close actions are to the fundamental as in [START_REF] Morris | Social Value of Public Information[END_REF].

Non-Bayesian higher-order beliefs

Weights on the private signal above equilibrium values can also be explained by systematic mistakes in the formation of higher-order beliefs. For using information rationally, a subject has to form conditional expectations about his partner's beliefs based on his own information. Theoretically, subjects should put a weight of 0.5 on the private signal when estimating θ and a weight of 0.25 in estimating the other's estimation of θ. This does not require any assumptions about others' behavior apart from Bayesian rationality. If subject j has rational first-order beliefs, then

x y E E E i i j i 75 . 0 25 . 0 5 . 0 ) ( 5 . 0 )) ( ( + = + = θ θ . More generally, if y x E j j ) 1 ( ) ( λ λ θ - + = , then y x y E E E i i j i ) 2 / 1 ( 5 . 0 ) 1 ( ) ( )) ( ( λ λ λ θ λ θ - + = - + = .
Previous studies have explored whether subjects update beliefs according to Bayes' rule, when information arrives sequentially. Various studies from the 1960s found that "[u]pon receipt of new information, subjects revise their posterior probability estimates in the same direction as the optimal model [Bayes' rule], but the revision is typically too small" (Slovic and Lichtenstein, 1971, 693).

Recent studies explore whether subjects are Bayesian in updating self assessments. While some find significant deviations from Bayes' rule [START_REF] Eil | The Good News-Bad News Effect: Asymmetric Processing of Objective Information about Yourself[END_REF], others report that subjects update overconfident prior beliefs correctly [START_REF] Grossman | An Unlucky Feeling: Persistent Overestimation of Absolute Performance with Noisy Feedback[END_REF]. To our knowledge there is no previous experimental study exploring whether subjects use Bayes' rule in forming higher-order beliefs. [START_REF] Dale | Experiments on the Social Value of Public Information[END_REF] test some predictions of [START_REF] Morris | Social Value of Public Information[END_REF] comparing treatments with private and public signals of different precisions. The weights that subjects put on signals of low precision and on the prior (which also has a low precision for indicating the fundamental) are much higher than under Bayesian belief formation. Since in their experiment public signals always have a low precision, they find higher weights on the public signal than we do. They find weights that are distributed around the equilibrium values and negative welfare effects of public signals. 20 In our experiment, all signals have the same precision, so that we can detect mistakes in the formation of higher-order beliefs without this potentially confounding bias towards low quality signals.

Result 7: Subjects attach too high a weight to the private signal when predicting other subjects' beliefs about fundamentals.

Data from belief elicitation reveal that subjects attach (on average) a weight of 0.463 on the private signal when estimating θ and an average weight of 0.291 when estimating their partner's estimation of θ. Two-tailed Wilcoxon tests on estimated group-specific weights reject the hypotheses that the median weight in estimating θ equals 0.5 (p=0.005) in favor of lower weights on private signals, and reject that the median weight in estimating the other player's estimation equals 0.25 (p=0.038) in favor of higher weights on private signals. 229 out of 288 subjects attribute a weight higher than 0.25 to the private signal in this task. Although Bayesian rationality requires that the weight on the private signal in second-order beliefs be half of the weight in estimating θ, it is in fact 62.2% and thereby too high to be consistent with Bayesian rationality. This can explain a part of the differences between observed weights and equilibrium predictions in the other stages. As we elicited beliefs directly after Stage 3 (0 < r < 1), we may expect that higher-order beliefs in the later periods of Stage 3 are formed in about the same way as elicited beliefs. However, as we show in Appendix C, the deviations from Bayesian rationality in forming higher-order beliefs are too small to explain the whole difference between observations and equilibrium predictions in the second half of Stage 3.

It is surprising, though, that subjects underused private signals in forming their expectation of θ, especially in light of the results from Stage 1 ("r=0") that is equivalent to asking first-order expectations. We attribute this to an order effect: after 40 periods in Stages 2 and 3, in which the public signal was more important than the private, some subjects may have developed a routine in putting higher weights on public signals that was applied to both tasks in Stage 4. In Appendix C, we present the results of 6 control sessions, in which we elicited beliefs directly after Stage 1 and before all stages, in which common signals are theoretically more important than private ones. Here, we find that the weights on the private signal in first-order expectations are distributed around 0.5 (equality cannot be rejected) while the weights in second-order expectations are all above 0.3 and on average 83% of the weight in first-order expectations. Here, the violation of Bayesian rationality in forming higher-order beliefs is even more pronounced. These results indicate that subjects underestimate how informative public signals are for estimating other players' beliefs.

-Conclusion

Our experiment provides convincing data for the hypothesis that subjects attach larger weights to public than to private signals if they have incentives to coordinate their actions. The experiment also investigates whether players put higher weights on the public signal if the relative importance of the coordination component is increased. We find clear evidence in line with this qualitative prediction of equilibrium theory. The observed weights are, however, lower than equilibrium values. They are closer to the weights arising from level-2 reasoning of a cognitive hierarchy model, for which [START_REF] Cornand | Higher-Order Beliefs and the Social Value of Public Information[END_REF] show that public information cannot be welfare detrimental.

  weight attributed to the coordination component. The coordination component is reminiscent of Keynes' beauty contest. Agents face uncertainty concerning θ. They receive two kinds of signals that deviate from θ by some error terms with uniform distribution. All agents receive a public (common) signal and the noise of the public signal θ y are independent and their distribution is treated as exogenously given.

  weight of ½ in the Bayesian expectation of the fundamental, because the public signal conveys more information about the likely action of the other player than the private signal.

  action as close as possible to the fundamental state, independent from their partners' choices. This stage aimed at familiarizing subjects with the game form and controlling for their Bayesian rationality. Since both signals have the same precision, the rational choice are in the interval[60, 440].7

  for r = 1.

Figure 2 -

 2 Figure 2 -Weight on the private signal for different values of r

Table 1 -

 1 Composition of the different sessions

	Stage 1	Stage 2	Stage 3	Stage 4
			Treatment
	1 to 6		"r=0.5"
	"r=0"		(30 periods)	Belief Elicitation
	(5 periods)	"r=1"	Treatment	(5 periods)
	7 to 12	(10 periods)	"r=0.25"
			(30 periods)
			Treatment
	13 to 18		"r=0.75"
			(30 periods) 12

Table 2 -

 2 Equilibrium weight γ * as a function of r

Table 4 -

 4

	2nd half

Group specific weights on the private signal Figure 1 -Evolution of average weights on the private signal

Table 5 -

 5 P-values of one-tailed Wilcoxon matched pairs signed rank tests.

Table 6

 6 summarizes the resulting weights on the private signal for increasing levels of reasoning and the equilibrium weight on the private signal for different values of r. Comparing observed weights with these theoretical values allows testing the cognitive hierarchy model.

	Value of r	Level-1	Level-2 2 γ	Level-3 3 γ	Level-4 4 γ	… Equilibrium weight (infinite levels of reasoning)	γ	*
	0	0.50	0.50	0.50	0.50	… 0.50
	0.25	0.50	0.437	0.429	0.429	… 0.429
	0.5	0.50	0.375	0.344	0.336	… 0.333
	0.75	0.50	0.312	0.242	0.216	… 0.2
	1	0.50	0.25	0.125	0.062	… 0

*

γ

Table 6 -

 6 Theoretical weight on x i depending on r and on the level of reasoning

	Result 6: For interior values of r, most estimated group weights on private signals are close to or
	above the weights given by level-2 reasoning.

Compared to the model of[START_REF] Morris | Social Value of Public Information[END_REF], we required the number of players to be finite and we also changed the distributions of the signals from normal to uniform in order to have a simple distribution with bounded support for the experiment. In Morris and Shin, the constant U 0 adds the average losses from miscoordination, so that the coordination part becomes a zero-sum game and aggregate welfare depends only on the distance between actions and fundamental state.

. This restriction was not mentioned in the instructions. If a subject entered a number outside this interval, the screen displayed a message indicating that her choice is too far away from signals. Apparently, this restriction was not binding. We observed only one (out of 12,960) decision situations, where a subject actually chose an action at the border of this interval, and less than 50 decisions, where the difference between action and public signal exceeded 15. Some subjects actually reported typing mistakes.

Average weights per individual are estimated by a similar procedure as average weights per session, with a separate regression for each subject.

This result seems to be corroborated by subjects' written comments in the post-experimental questionnaire (this questionnaire is presented in Appendix A.3). 43% of subjects explicitly wrote (without being directly asked) that y is more informative than x i on the other participant's decision. The percentage of subjects reporting such statements is positively correlated to the value of r in stage 3.

In the light of our previous impression that the level of reasoning may be correlated with r, the high parameter of r=0.8, chosen by[START_REF] Dale | Experiments on the Social Value of Public Information[END_REF], may contribute to higher weights on public signals. They also inform subjects about the best response after each round which may enforce convergence to equilibrium.

for helpful comments, to Romain Zeiliger for programming Regate, to Kene Boun My for providing assistance, to the students for their participation in the experiment and to the Conseil Scientifique de l'Université de Strasbourg and the ANR-DFG joint grant for financial support.


Bernard Bonnard

Olivier Cots

Lionel Jassionnesse

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

Keywords: Conjugate and cut loci, optimal control, general ellipsoid, spin dynamics

We combine geometric and numerical techniques -the Hampath codeto compute conjugate and cut loci on Riemannian surfaces using three test bed examples: ellipsoids of revolution, general ellipsoids, and metrics with singularities on S 2 associated to spin dynamics.

Introduction

On a Riemannian manifold (M, g), the cut point along the geodesic γ emenating from q 0 is the first point where γ ceases to be minimizing, while the first conjugate point is where it ceases to be minimizing among the geodesics C 1 -close to γ. Considering all the geodesics starting from q 0 they will form respectively the cut locus C cut (q 0 ) and the conjugate locus C(q 0 ). The computations of the conjugate and cut loci on a Riemannian surface is an important problem in global geometry [START_REF] Berger | A panoramic view of Riemannian geometry[END_REF] and it can be extended to optimal control with many important applications [START_REF] Bonnard | Optimal Control with Applications in Space and Quantum Dynamics[END_REF]. Also convexity property of the injectivity domain of the exponential map is related to the continuity property of the Monge transport map T on the surfaces [START_REF] Figalli | Nearly round spheres look convex[END_REF]. The structure of the conjugate and cut loci surfaces diffeomorphic to S 2 was investigated in details Bernard Bonnard Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France, e-mail: bernard.bonnard@u-bourgogne.fr Olivier Cots INRIA, 2004 route des lucioles, F-06902 Sophia Antipolis, France, e-mail: olivier.cots@inria.fr Lionel Jassionnesse Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France, e-mail: lionel.jassionnesse@u-bourgogne.fr by Poincaré and Myers [START_REF] Myers | Connections between differential geometry and topology I. Simply connected surfaces[END_REF], [START_REF] Poincaré | Sur les lignes géodésiques des surfaces convexes[END_REF]. In the analytic case, the cut locus is a finite tree and the extremity of each branch is a cusp point. But the explicit computation of the number of branches and cusps points is a very complicated problem and only very recently was proved the four cusps Jacobi conjecture on ellipsoids [START_REF] Itoh | The cut loci and the conjugate loci on ellipsoids[END_REF], [START_REF] Sinclair | Jacobi's last geometric statement extends to a wider class of Liouville surfaces[END_REF].

The aim of this article is to present techniques which led to the explicit computation of the cut and conjugate loci based on three examples, combining geometric techniques and numerical simulations using the Hampath code [START_REF] Caillau | Differential continuation for regular optimal control problems[END_REF]. Geometry is used in a first step to choose appropriate coordinates to analyze the metric and the geodesics flow: curvature computation and principal lines of curvature. Also the explicit computations will be refered to the micro-local complexity of the Hamiltonian flow. This is clear in the example of an ellipsoid of revolution: geodesics can be meridians, the equator and a family of geodesics such that representing the metric in the normal form g = dϕ 2 + m(ϕ)dθ 2 , θ increases or decreases monotonously while ϕ oscillates between ϕ -and ϕ + . The important task is to evaluate the first conjugate point t 1c which is solution to the Jacobi equation d 2 J dt + G(γ(t))J = 0, where G is the Gauss curvature and such that J(0) = J(t 1c ) = 0. Since in our case the usual Sturm theorem is not very helpful to estimate conjugate points, our approach is to estimate them in relation with the period mapping T of the ϕ-variable.

In the case of an ellipsoid of revolution it can be shown that conjugate and cut loci can be estimated with only the first and second order derivative of the period mapping [START_REF] Bonnard | Conjugate and cut loci of a two-sphere of revolution with application to optimal control[END_REF].

The Hampath code is useful to analyze the geodesics and to evaluate conjugate points and the conjugate locus, using Jacobi fields and continuation method. In particular the analysis of the case of revolution can be easily extended to a general ellipsoid.

The time optimal transfer of three linearly coupled spins with Ising coupling described in [START_REF] Yuan | Geometry, optimal control and quantum computing[END_REF] leads to study a one parameter Riemannian metric on S 2 with equatorial singularity and which is a deformation of the Grushin case g = dϕ 2 + tan 2 ϕdθ 2 . Again the analysis of the flow and conjugate points computation led to describe the conjugate and cut loci for various values of the parameter.

Riemannian metrics on surfaces of revolution

We briefly recall the general tools to handle the analysis of surfaces of revolution with applications to the ellipsoids [START_REF] Bonnard | Conjugate and cut loci of a two-sphere of revolution with application to optimal control[END_REF], [START_REF] Shiohama | The geometry of total curvature on complete open surfaces[END_REF].

Generalities

Taking a chart (U, q) the metric can be written in polar coordinates as

g = dϕ 2 + m(ϕ)dθ 2 .
One use Hamiltonian formalism on T * U, ∂ ∂ p is the vertical space, ∂ ∂ q is the horizontal space and α = pdq is the (horizontal) Liouville form. The associated Hamiltonian is

H = 1 2 p 2 ϕ + p 2 θ m(ϕ)
and we denote expt #-H the one-parameter group. Parameterizing by arc length amounts to fix the level set to H = 1/2. Extremal solutions of #-H are denoted γ : t → (q(t, q 0 , p 0 ), p(t, q 0 , p 0 )) and fixing q 0 it defines the exponential mapping exp q 0 : (t, p 0 ) → q(t, q 0 , p 0 ) = Π (expt #-H(q 0 , p 0 )) where Π : (q, p) → q is the standard projection. Extremals are solutions of the equations

dϕ dt = p ϕ , dθ dt = p θ m(ϕ) , dp ϕ dt = 1 2 p 2 θ m ′ (ϕ) m 2 (ϕ) , dp θ dt = 0.
Definition 1. The relation p θ = C t is called Clairaut relation on surfaces of revolution. We have two types of specific solutions: meridians for which p θ = 0 and θ (t) = θ 0 and parallels for which dϕ dt (0) = p ϕ (0) = 0 and ϕ(t) = ϕ(0).

To analyze the extremal behaviours, we fix H = 1 2 and we consider the mechanical system dϕ dt

2 +V (ϕ, p θ ) = 1 where V (ϕ, p θ ) = p 2 θ m(ϕ)
is the potential mapping depending upon the parameter p θ and parallels correspond to local extrema.

Assumptions 1

In the sequel we shall assume the following • (A1) ϕ = 0 is a parallel solution with a local minimum of the potential and the corrresponding parallel is called the equator. • (A2) The metric is reflectionally symmetric with respect to the equator: m(-ϕ) = m(ϕ).

Micro-local behaviors of the extremals

We describe a set a solution confined to the segment [-ϕ max , +ϕ max ] where ϕ max is the local maximum of V closest of 0. Such an extremal is such that ϕ oscillates periodically between -ϕ + and ϕ + . The dynamics is described by:

dϕ dt = ± 1 g , dθ dt = p θ m(ϕ)
,

where

g(ϕ, p θ ) = m(ϕ) m(ϕ) -p 2 θ
and for an increasing branch one can parameterize θ by ϕ and we get

dθ dϕ = g(ϕ, p θ )p θ m(ϕ) = f (ϕ, p θ ), where f (ϕ, p θ ) = p θ m(ϕ) m(ϕ) -p 2 θ .
The trajectory t → ϕ(t, p θ ) is periodic and one can assume ϕ(0) = 0. The period of oscillation T is given by

T = 4 ϕ + 0 g(ϕ, p θ )dϕ
and the first return to the equator is at time T /2 and the variation of θ at this time is given by

∆ θ = 2 ϕ + 0 f (ϕ, p θ )dϕ.
The mapping p θ ∈ I → T (p θ ) is called the period mapping and R : p θ → ∆ θ is called the first return mapping.

Definition 2. The extremal flow is called tame on I if the first return mapping R is such that R ′ < 0.

Proposition 1. For extremal curves with p θ ∈ I, in the tame case there exists no conjugate times on (0, T /2).

Proof. If R ′ < 0, the extremal curves initiating from the equator with p θ ∈ I are not intersecting before returning to the equator. As conjugate points are limits of intersecting extremals curves, conjugate points are not allowed before returning to the equator.

Assumptions 2 In the tame case we assume the following

(A3) At the equator the Gauss curvature G = -1 √ m(ϕ) ∂ 2 √ m(ϕ) ∂ ϕ 2
is positive and maximum.

Using Jacobi equation we deduce: Lemma 1. Under assumption (A3), the first conjugate point along the equator is at time π/ G(0) and realizes the minimum distance to the cut locus C cut (θ (0) = 0, ϕ(0) = 0). It is a cusp point of the conjugate locus.

Parameterization of the conjugate locus under assumptions (A1-2-3) for p θ ∈ I Fixing a reference extremal γ, Jacobi equation is the variational equation:

dδ z dt = ∂ #- H(γ(t)) ∂ z δ z, z = (q, p)
and a Jacobi field J(t) is a non trivial solution of Jacobi equation. According to standard theory on surfaces, if γ is parametrized by arc length, J 1 (t) = (δ q, δ p) denotes the crucial Jacobi field vertical at time t = 0, that is δ q(0) = 0 and such that p(0), δ p(0) = 0, that is restricting the Jacobi equation to H = 1 2 . Since J 1 (0) is vertical, α(J 1 (0)) = 0 and then α(J 1 (t)) = 0, and we have: Proposition 2. Conjugate points are given by the relation dΠ (J 1 (t)) = 0 and

dΠ (J 1 (t)) = ∂ ϕ(t, p θ ) ∂ p θ , ∂ θ (t, p θ ) ∂ p θ .
In particular we have at any time the collinearity condition:

p ϕ ∂ ϕ ∂ p θ + p θ ∂ θ ∂ p θ = 0.
The conjugate locus will be computed by continuation, starting from the cusp point at the equator. Let p θ ∈ I and t ∈ (T /2, T /2 + T /4). One has the formulae

θ (t, p θ ) = ∆ θ (p θ ) + t T /2 p θ m(ϕ) dt and on [T /2,t], dϕ dt < 0, ϕ < 0. Hence t T /2 p θ m(ϕ) dt = 0 ϕ(t,p θ ) f (ϕ, p θ )dϕ.
Using the collinearity condition, one has:

Lemma 2. For p θ ∈ I and conjugate times between (T /2, T /2+T /4) the conjugate locus is solution of

∂ θ (ϕ, p θ ) ∂ p θ = 0, (1)
where θ (ϕ, p θ ) = ∆ θ (p θ ) + 0 ϕ f (ϕ, p θ )dϕ. This gives a simple relation to compute the conjugate locus by continuation. One notes p θ → ϕ 1c (p θ ) the solution of eq. 1 initiating from the equator. Differentiating one has

∆ θ ′ + 0 ϕ ∂ f ∂ p θ dϕ = 0 at ϕ 1c (p θ ).
Differentiating again we obtain

∆ θ ′′ + 0 ϕ 1c ∂ 2 f ∂ p 2 θ dϕ - ∂ ϕ 1c ∂ p θ • ∂ f ∂ p θ = 0. One can easily check that ∂ f ∂ p θ > 0 and ∂ 2 f ∂ p 2 θ > 0. In particular ∂ ϕ 1c ∂ p θ = ∆ θ ′′ + 0 ϕ 1c ∂ 2 f ∂ p 2 θ dϕ ∂ f ∂ p θ -1
.

and one deduces the following.

Proposition 3. If ∆ θ ′′ > 0 on I, then ∂ ϕ 1c ∂ p θ = 0 and the curve p θ → (ϕ 1c (p θ ), θ 1c (p θ )
) is a curve defined for p θ ∈ I and with no loop in the plane (ϕ, θ ). In particular it is without cusp point.

To simplify the computations we use the following lemma:

Lemma 3. We have the relation R ′ (p θ ) = T ′ (p θ ) 2p θ .

Ellipsoids of revolution

The ellipsoid of revolution is generated by the curve

y = sin ϕ, z = ε cos ϕ
where 0 < ε < 1 corresponds to the oblate (flattened) case while ε > 1 is the prolate (elongated) case. The restriction of the Euclidian metric is

g = F 1 (ϕ)dϕ 2 + F 2 (ϕ)dθ 2
where F 1 = cos 2 ϕ + ε 2 sin 2 ϕ, F 2 = sin 2 ϕ. The metric can be written in the normal form setting: dϕ = F 1/2 1 (ϕ)dϕ. Observe that ϕ oscillates periodically and θ is monotonous. Hence the period mapping can be computed in the (ψ, θ )-coordinate, ψ = π/2 -ϕ and ψ = 0 is the equator. The Hamiltonian is

H = 1 2 p 2 ϕ F 1 (ϕ) + p 2 θ F 2 (ϕ)
and with H = 1/2, one gets

dψ dt = (cos 2 ψ -p 2 θ ) 1/2 cos ψ(sin 2 ψ + ε 2 cos 2 ψ) 1/2 . Denoting 1 -p 2 θ = sin 2 ψ 1 and making the rescaling Y = sin ψ 1 Z, where Y = sin ψ, one gets (ε 2 + Z 2 sin 2 ψ 2 1 (1 -ε 2 )) 1/2 (1 -Z 2 ) 1/2 dZ = dt.
Hence the formulae for the period mapping is

T 4 = 1 0 (ε 2 + Z 2 sin 2 ψ 1 (1 -ε 2 )) 1/2 (1 -Z 2 ) 1/2 dZ
which corresponds to an elliptic integral. The discussion is the following.

Oblate Case

In this case the Gauss curvature is increasing from the north pole to the equator and the problem is tame if the period mapping is such that

T ′ (p θ ) < 0 < T ′′ (p θ )
for each admissible p θ > 0. The cut point of q(0) = (ϕ(0), 0) is given by t 0 (p θ ) = T (p θ )/2 and corresponds to the intersection of the two extremal curves associated to φ(0), andφ(0). The cut locus C cut (q(0)) for not a pole is a segment of the antipodal parallel. If q(0) is not a pole nor on the equator, the distance to the cut locus is the half-period of the extremal starting from ϕ(0) with φ(0) = 0 and the injectivity radius is realized for ϕ(0) = π/2 on the equator, and is given by π/ G( π 2 ) where G is the Gauss curvature. The conjugate locus C(q(0)) of a point not a pole has exactly four cusps, two on the antipodal parallel which are the extremities of the cut locus segment and two on the antipodal meridian.

Prolate Case

In this case, the Gauss curvature is decreasing from the north pole to the equator and the first return mapping to the equator is an increasing function of p θ ≥ 0. Given q(0) = (ϕ(0), 0) not on a meridian circle, the cut point t 0 (p θ ) is given by solving θ (t 0 , p θ ) = π and corresponds to the intersection of the two extremal curves symmetric with respect to the meridian associated respectively to p θ and -p θ . The cut locus of a point not a pole is a segment of the antipodal meridian. The conjugate locus C(q(0)) of a point not a pole has exactly four cusps, two on the antipodal meridian which are the extremities of the cut locus and two on the antipodal parallel.

Conclusion

To resume both cases are distinguished by the monotonicity property of the Gauss curvature or equivalently of the first return mapping. The cut loci are computed using the symmetric property of the extremal curves: in the oblate case, the symmetry of the metric with respect to the equator and in the prolate case the symmetry of the metric with respect to the meridian. Additionaly to this discrete symmetry, the symmetry of revolution ensures the existence of an additionnal one-dimensional group of symmetry which gives according to Noether theorem the first integral p θ linear with respect to the adjoint vector and corresponds to a Clairaut metric.

General Ellipsoids

We shall extend the result on ellipsoids of revolution to general ellipsoids. Roughly spoken, the general case will mixed the oblate and the prolate case, which will be easily seen in the classification of the extremal flow.

Geometric Properties [7]

A general ellipsoid E is defined by the equation

x 2 1 a 1 + x 2 2 a 2 + x 2 3 a 3 = 1, a 1 > a 2 > a 3 > 0
and we use the branched double covering parameterization of

E (θ 1 , θ 2 ) ∈ T 2 = S 1 × S 1 → E: x 1 = √ a 1 cos θ 1 (1 -β ) cos 2 θ 2 + sin 2 θ 2 x 2 = √ a 2 sin θ 1 sin θ 2 x 3 = √ a 3 cos θ 2 β cos 2 θ 1 + sin 2 θ 1
where β = (a 2a 3 )/(a 1a 3 ) ∈ (0, 1) and the (θ 1 , θ 2 )-coordinates are related to the elliptic coordinates (λ 1 , λ 2 ) by

λ 1 = a 1 sin 2 θ 1 + a 2 cos 2 θ 1 λ 2 = a 2 cos 2 θ 2 + a 3 sin 2 θ 2 .
In the coordinates (θ 1 , θ 2 ) the restriction of the euclidian metric on R 3 takes the form

g = (λ 1 -λ 2 ) λ 1 λ 1 -a 3 dθ 1 2 + λ 2 a 1 -λ 2 dθ 2 2 .
The metric has three main discrete symmetries defined for i = 1, 2 by the following changes

θ i -→ π -θ i , θ i -→ -θ i .
The associated Hamiltonian is

2H = 1 λ 1 -λ 2 λ 1 -a 3 λ 1 p 2 θ 1 + a 1 -λ 2 λ 2 p 2 θ 2
and an additional first integral quadratic in (p θ 1 , p θ 2 ) is given by

F = 1 λ 1 -λ 2 λ 1 -a 3 λ 1 (a 2 -λ 2 )p 2 θ 1 - a 1 -λ 2 λ 2 (λ 1 -a 2 )p 2 θ 2 .
We have the following correspondences with the ellipsoid of revolution.

• In the limit case a 1 = a 2 > a 3 (oblate case), the metric g reduces to the form presented in 2.2 where (ϕ, θ ) ≡ (θ 1 , θ 2 ) and we have F ≥ 0. • In the limit case a 1 > a 2 = a 3 (prolate case), the metric g reduces to the form presented in 2.2 where (ϕ, θ ) ≡ (θ 2 , θ 1 ) and we have F ≤ 0.

According to Liouville theory [START_REF] Bolsinov | Integrable geodesic flows on two-dimensional surfaces[END_REF], the metric can be written in the normal form

g = (F 1 (u 1 ) + F 2 (u 2 )) du 2 1 + du 2 2 ,
where u 1 , u 2 are defined by the quadratures

du 1 = λ 1 λ 1 -a 3 dθ 1 , du 2 = λ 2 a 1 -λ 2 dθ 2
and see [START_REF] Klingenberg | Riemannian geometry[END_REF] for the relation with elliptic coordinates. The third fondamental form is given for x 3 = 0 by

III(dx 1 , dx 2 ) = x 1 x 2 x 3 a 1 a 2 a 3 1 a 3 - 1 a 1 dx 2 1 - x 1 x 2 x 3 a 1 a 2 a 3 1 a 2 - 1 a 3 dx 2 2 + x 3 a 3 1 a 1 - 1 a 2 x 3 a 3 2 - 1 a 2 - 1 a 3 x 1 a 1 2 - 1 a 3 - 1 a 2 x 2 a 2 2 dx 1 dx 2
and we get the four umbilical points

± √ a 1 1 -β , 0, ± √ a 3 β
which are {(0, 0), (0, π), (π, 0), (π, π)} in (θ 1 , θ 2 )-coordinates. Besides in elliptic coordinates the lines λ i = Cst are the curvature lines. Finally, taking the Liouville normal form, the Gauss curvature is given by

K(u 1 , u 2 ) = F ′ 1 (u 1 ) 2 + F ′ 2 (u 2 ) 2 2 (F 1 (u 1 ) + F 2 (u 2 )) 3 - F ′′ (u 1 ) + F ′′ (u 2 ) 2 (F 1 (u 1 ) + F 2 (u 2 )) 2
or similarly, in the elliptic coordinates, one has

K(λ 1 , λ 2 ) = a 1 a 2 a 3 λ 2 1 λ 2 2 , (λ 1 , λ 2 ) ∈ [a 2 , a 1 ] × [a 3 , a 2 ].
We give in Fig. 1 the Gauss curvature in the (θ 1 , θ 2 )-coordinates restricted to

(θ 1 , θ 2 ) ∈ [0, π/2] × [0, π/2
] by symmetry. The maximum of K is at the intersection of the longest and the middle ellipses while the minimum is at the intersection of the shortest and the middle ellipses.

Fig. 1 Gauss curvature in (θ 1 , θ 2 )-coordinates for (θ 1 , θ 2 ) ∈ [0, π/2] × [0, π/2].
The a 1 /(a 2 a 3 ) is obtained at (0, π/2) and the minimum a 3 /(a 2 a 1 ) is at (π/2, 0).

Geodesic Flow

Parameterizing by arc length H = 1 2 and setting F = c, the extremal equations are described by

ε 1 √ λ 1 dθ 1 √ λ 1 -a 3 √ λ 1 -a 2 + c = ε 2 √ λ 2 dθ 2 √ a 1 -λ 2 √ a 2 -c -λ 2 and dt = ε 1 √ λ 1 √ λ 1 -a 2 + c √ λ 1 -a 3 dθ 1 + ε 2 √ λ 2 √ a 2 -c -λ 2 √ a 1 -λ 2 dθ 2
where

ε i = ±1 is the sign of dθ i /dt, i = 1, 2. The value c of F varies between -(a 1 -a 2
) and (a 2a 3 ) and the behavior of the extremals depends on the sign of c.

• If 0 < c < a 2a 3 , then θ 1 (t) increases or decreases monotonously and θ 2 (t) oscillates between ν 2 (c) and πν 2 (c), where ν 2 (c) is defined by

sin ν 2 (c) = c a 2 -a 3 , 0 < ν 2 (c) < π 2 .
These trajectories do not cross transversely the segments θ 2 = 0 and θ 2 = π which degenerate into two poles in the oblate case. Here the longest ellipse θ 2 = π/2 plays the role of the equator from the oblate case.

• If -(a 1a 2 ) < c < 0, then θ 2 (t) increases or decreases monotonously and θ 1 (t) oscillates periodically between ν 1 (c) and πν 1 (c) where ν 1 (c) is defined by

sin ν 1 (c) = -c a 1 -a 2 , 0 < ν 1 (c) < π 2 .
These trajectories do not cross transversely the segments θ 1 = 0 and θ 1 = π which degenerate into two poles in the prolate case. Here the longest ellipse plays the role of the meridian circle from the prolate case.

• The separating case c = 0 is the level set containing the umbilical points.

Arc length geodesic curves γ(t) = (θ 1 (t), θ 2 (t)) can be seen as a function of the parameter but we use the one from [START_REF] Itoh | The cut loci and the conjugate loci on ellipsoids[END_REF], introducing the unit tangent vector:

ν(η) = cos ηe 1 + sin ηe 2
where (e 1 , e 2 ) is the orthonormal basis

e 1 = (λ 1 -λ 2 )λ 1 λ 1 -a 3 -1/2 ∂ ∂ θ 1 , e 2 = (λ 1 -λ 2 )λ 2 a 1 -λ 2 -1/2 ∂ ∂ θ 2 .

Results on the Conjugate and Cut Loci

According to [START_REF] Itoh | The cut loci and the conjugate loci on ellipsoids[END_REF] we have the following which generalized the case of an ellipsoid of revolution.

Proposition 4. The cut locus of a non-umbilical point is a subarc of the curvature lines through its antipodal point and the conjugate locus has exactly four cusps.

The Analysis

Fixing the initial point to (θ 1 (0), θ 2 (0)), the relation between η and c is given by:

c(η) = a 2 -λ 2 (θ 2 (0)) cos 2 η -λ 1 (θ 1 (0)) -a 2 sin 2 η
and let η 0 be the unique η such that c(η 0 ) = 0, 0 ≤ η 0 ≤ π 2 .

• The case c > 0, cf. Fig. 2. We use the parameterization (θ 1 , θ 2 ) with θ 1 ∈ T 1 and 0 ≤ θ 2 ≤ π.

-For η ∈ (0, η 0 ) ∪ (π -η 0 , π) the value of θ 2 along the geodesic increases until it reaches a maximum θ + 2 and then it decreases. The cut time t 0 (η) is the second positive time such that θ 2 takes the value πs 2 .

-For η = 2π -η, the value of θ 2 along the geodesic decreases until it reaches a minimum θ - 2 and then it increases. The cut time t 0 (η) is the first positive time such that θ 2 takes the value πs 2 .

-Besides, we have t 0 (η) = t 0 (η) and γ η (t 0 (η)) = γ η (t 0 (η)).

Fig. 2 Trajectories, cut and conjugate points in the case c > 0. The trajectory with θ2 (0) > 0 corresponds to η ∈ (0, η 0 ) while the other corresponds to η = 2π -η. The two conjugate points are plotted in red and come after the cut point in black. The period T of the θ 2 -variable is equal for each trajectory and is represented with the half-period.

0 pi/2 pi 3pi/2 2pi 0 pi/2 pi ν 2 π-ν 2 π-θ 2 (0) θ 2 (0) T T/2 c > 0, η ∈ [0, η 0 ) θ 1 θ 1 (0) θ 2 • The case c ≤ 0, cf. Fig. 3. We use the parameterization (θ 1 , θ 2 ) with 0 ≤ θ 1 ≤ π and θ 2 ∈ T 1 .
-For η ∈ (η 0 , πη 0 ), θ 2 increases monotonously and let t 0 (η) be the first positive time t such that θ 2 takes the value s 2 + π. The cut time is given by t 0 (η). -For η = 2π -η, θ 2 decreases monotonously and let t 0 (η) be the first positive time t such that θ 2 takes the value s 2 -π. The cut time is given by t 0 (η). -Besides, we have t 0 (η) = t 0 (η) and θ 1,η (t 0 (η)) = θ 1,η (t 0 (η)).

Fig. 3 Trajectories, cut and conjugate points in the case c < 0. The trajectory with θ2 (0) > 0 corresponds to η ∈ (0, η 0 ) while the other corresponds to η = 2π -η. The two conjugate points are plotted in red and come after the cut point in black. The periods of the θ 2 -variable are not equal for each trajectory and are represented with the half-period.

-2pi -pi 0 pi 2pi 3pi 0 pi/2 pi ν 1 π-ν 1 θ 1 (0) T T/2 π+θ 2 (0) c < 0, η ∈ (η 0 , π/2] θ 2 θ 2 (0) -π+θ 2 (0) T/2 T θ

Numerical Computation of Conjugate and Cut Loci

We fix the parameters of the ellipsoid a 1 > a 2 > a 3 > 0 such that a 1a 2 = a 2a 3 to avoid any additionnal symmetry. We take (a 1 , a 2 , a 3 ) = (1.0, 0.8, 0.5) for the following computations. The initial point is (θ 1 (0), θ 2 (0)) = (π/3, 2π/5) which gives generic results. Indeed, if the initial point is on θ 2 = 0 or π, c(η) ≤ 0 and there are only oblate-like extremals. Similarly, if θ 1 (0) = 0 or π, there are only prolate-like extremals. We present Figs. 4-7 the conjugate and cut loci.

One represents first the conjugate and cut loci on Fig. 4 using the double covering parameterization. We can distinguish the four cusps of the conjugate locus. Two of them on the horizontal line θ 2 = πθ 2 (0) coming from the oblate-like extremals and the two others on the vertical line θ 1 = πθ 1 (0) stemming from the prolatelike extremals. The cut locus is split into three parts, two on the line θ 2 = πθ 2 (0) and one on

θ 2 = π + θ 2 (0). -pi -pi/2 0 pi/2 pi 3pi/2 -pi -pi/2 0 pi/2 pi 3pi/2 2pi θ 1 (0) π-θ 1 (0) θ 2 (0) π-θ 2 (0) π+θ 2 (0) -π+θ 2 (0) θ 1 θ 2
Fig. 4 Conjugate (in red dots) and cut (in black dots) loci, using the double covering parameterization. The initial point is marked with a blue dot.

One has some trajectories for various η ∈ [0, 2π) with the cut and conjugate loci on Fig. 5 using again the double covering. One should notice that in red are plotted the four trajectories passing through the umbilical points with the parameterization (θ 1 , θ 2 ) with θ 1 ∈ T 1 and 0 ≤ θ 2 ≤ π. However, one could have used the second parameterization. These trajectories separate the oblate-like (c > 0) trajectories from the prolate-like (c < 0) ones. About the extremities of the three parts of the cut locus, two of them are coming from the trajectories γ 0 and γ π for which the conjugate point and the cut point coincide. The two others extremities are located in the intersection of γ η 0 and γ 2π-η 0 and the intersection of γ π-η 0 and γ π+η 0 , where c(η 0 ) = 0. The conjugate locus is revealed by the trajectories as we can see on the right subplot.

The conjugate and cut loci are given on the left subplot of Fig. 6 using the parameterization (θ 1 , θ 2 ) ∈ [0, 2π) × [0, π] and on the right subplot of Fig. 6

in (x 1 , x 2 , x 3 )- -pi -pi/2 0 pi/2 pi 3pi/2 -pi -pi/2 0 pi/2 pi 3pi/2 2pi θ 1 (0) π-θ 1 (0) θ 2 (0) π-θ 2 (0) π+θ 2 (0) -π+θ 2 (0) θ 1 θ 2 pi pi/2 θ 1 θ 2
Fig. 5 (top) Various trajectories γ η , η ∈ [0, 2π), with the conjugate (in red dots) and cut (in black dots) loci, using the double covering parameterization. (bottom) Zoom of the right-hand-side of the conjugate locus revealed by the trajectories. (both) For η = 0 or π, the trajectories are in black. They are tangential to the cut locus as we can see on the zoom. In blue are plotted the trajectories for η ∈ (0, π) and in magenta for η = 2π -η. The four trajectories in red such that c(η) = 0 pass through an umbilical point. They separate oblate-like behaviour from prolate-like one. The two intersections of these trajectories are junction of parts of the cut locus coming from oblate-like extremals and the cut locus coming from prolate-like extremals.

coordinates. To put together the four branches of the conjugate locus and the three segments of the cut locus, we first restrict θ 1 and θ 2 to [0, 2π) by modulo and then use a point reflection across (π, π) for points such that θ 2 ∈ [π, 2π) for prolate-like extremals, as illustrated on the left subplot. The cut locus is a subarc of the curvature lines through its antipodal point, as expected, which extremities are cusps of the conjugate locus. The four branches of the conjugate locus does not have any symmetry in the generic case. However, if the initial point is on the longest and/or shortest ellipse, the conjugate locus has cusps on this ellipse and is symmetric with respect to it. In other words, if

θ 2 (0) = π/2 then t 1 (η) = t 1 (2π -η) and if θ 1 (0) = π/2 then t 1 (η) = t 1 (π -η),
where t 1 is the first conjugate time. Finally, the cut time t 0 , the first conjugate time t 1 and the half-period T /2 of the oscillating variable is plotted Fig. 7. We can first notice that in the generic case, the half-period is not equal to the cut time, even for oblate-like trajctories. This is still true for an initial point on θ 1 = 0 or π. The period is discontinuous when c(η) = 0 since the oscillating variable changes. The only one relevant symmetry is on the period mapping. Indeed, for η such that c(η) > 0, T (η) = T (2π -η). The symmetry on t 1 being obvious. 4 Dynamics of spin particles

The problem fully described in [START_REF] Yuan | Geometry, optimal control and quantum computing[END_REF] arises in the case of a spin chain of three linearly coupled spins with Ising coupling and the problem is to transfer the state from the first spin to the third spin in minimum time. Using appropriate coordinates the dynamics takes the form:

d dt   r 1 r 2 r 3   =   0 -cos θ (t) 0 cos θ (t) 0 -k sin θ (t) 0 k sin θ (t) 0     r 1 r 2 r 3  
where k = 1 corresponds to equal coupling and the transfer is from (1, 0, 0) to (0, 0, 1). Setting u 3 =cos θ , u 1 = -k sin θ , the dynamics is:

ṙ1 = u 3 r 2 , ṙ2 = -u 3 r 1 + u 1 r 3 , ṙ3 = -u 1 r 2 .
The minimum time problem is associated to the problem of transfering the system from r 0 = (1, 0, 0) to r(T ) = (0, 0, 1) and minimizing the functional:

T 0 I 1 u 2 1 + I 3 u 2 3 dt -→ min .
We introduce the metric:

g = I 1 u 2 1 + I 3 u 2 2 = I 3 dr 2 1 + I 1 I -1 3 dr 2 3 r 2 2
and this defines an almost Riemannian metric on the sphere S 2 :

g = dr 2 1 + k 2 dr 2 3 r 2 2 , k 2 = I 1 I 3 .
Lemma 4. In the spherical coordinates r 2 = cos ϕ, r 1 = sin ϕ cos θ , r 3 = sin ϕ sin θ the metric g takes the form g = (cos 2 θ + k 2 sin 2 θ )dϕ 2 + (k 2 -1) sin 2ϕ sin θ cos θ dϕdθ + sin 2 ϕ(sin 2 θ + k 2 cos 2 θ )dθ 2 cos 2 ϕ , while the associated Hamiltonian function is given by

H = 1 4k 2 sin 2 ϕ p 2 ϕ sin 2 ϕ(sin 2 θ + k 2 cos 2 θ ) + p 2 θ cos 2 ϕ(cos 2 θ + k 2 sin 2 θ ) - 2 (k 2 -1)p ϕ p θ sin ϕ cos ϕ sin θ cos θ .
We deduce the following

Lemma 5. For k = 1 H = 1 4 p 2 ϕ + p 2 θ cot 2 ϕ ,
and it corresponds to the so-called Grushin case g = dϕ 2 + tan 2 ϕdθ 2 . The Grushin case is analyzed in details in [START_REF] Bonnard | Conjugate and cut loci of a two-sphere of revolution with application to optimal control[END_REF]. We just remind that for each trajectory, θ increases or decreases monotonously while ϕ oscillates between ϕ -and ϕ + .

Moreover, we have Lemma 6. The family of metrics g depending upon the parameter k have a fixed singularity on the equator ϕ = π/2 and a discrete symmetry group defined by:

• H(ϕ, p ϕ ) = H(π -ϕ, -p ϕ ) (reflexion with respect to the equator);

• H(θ , p θ ) = H(-θ , -p θ ) (reflexion with respect to the meridian).

Numerical Computation of Conjugate and Cut Loci

In what follows the conjugate and cut loci are computed for the fixed initial conditions: ϕ(0) = π/2, θ (0) = 0, and are represented via the deformation of the parameter k starting from k = 1. There are two different cases to be analyzed: k > 1 and k < 1. Starting from the axis of symmetry, the Hamiltonian reduces to H = p 2 ϕ /4, and restricting the extremals to H = 1, we can parameterize the geodesics by p ϕ = ±2, p θ ∈ R. By symmetry we can fix p ϕ = -2 and consider p θ ≥ 0. For any k, the conjugate locus has a contact of order two at the initial point, as p θ → ∞.

• k ≥ 1. We study the deformation of the conjugate and cut loci for k ≥ 1 in Figs. 8-11. The key point is: when k > 1, θ is not monotonous for all the trajectories. This is true even for small k, like k = 1.01, taking p θ = 0.1 and t f > 14. We denote t 1 (p θ , k) the first conjugate time and q 1 (p θ , k) = (θ , ϕ) |t=t 1 (p θ ,k) the associated conjugate point. In Fig. 8, we represent the map k ∈ [1, 1.5] → q 1 (k) for p θ fixed to 10 -4 . The value 1.5 is heuristically chosen to simplify the analysis. We can notice that θ (t 1 (k)) only takes approximately the values 0 and π and so it is on the same meridian as the initial point. It switches three times at

1 < k 1 < k 2 < k 3 < 1.5, with k 2 -k 1 = k 3 -k 2 .
We then restrict the study of the conjugate locus to k ≤ k 3 to simplify.

Fig. 8 The first conjugate point with respect to k, for p θ fixed to 10 The deformation of the conjugate locus can be explained analysing the behaviors of the trajectories. We describe four types of trajectories in (θ , ϕ)-coordinates (see Fig. 10), limiting the study to k ≤ k 3 to simplify and p θ ≥ 0 by symmetry. These trajectories clarify the evolution of the conjugate locus.

-The first type occuring for any k s.t. 1 ≤ k ≤ k 3 , is represented in the top left subplot of Fig. 10. Its characteristic is that the θ -variable is monotonous non-decreasing on [0,t 1 ].

The three others trajectories do not have a monotonous θ -variable on [0,t 1 ]. We denote t the first time when the trajectory leaves the domain 0 ≤ θ ≤ π.

-The second type (top right) existing for k 1 ≤ k ≤ k 3 has no intersection on [0,t] and is such that θ (t) = 0.

The two last types of extremals have a self-intersection in the state-space in [0,t].

-The third kind of trajectories (bottom left) is such that θ (t) = 0 and is present for k ≤ k ≤ k 3 . -The last one (bottom right) is only appearing for k 2 ≤ k ≤ k 3 and has θ (t) = π. • k ≤ 1. The deformation of the conjugate locus in the case k < 1 is easier to interpret. We give on Fig. 11 the conjugate locus for k ∈ {0.8, 0.5, 0.2, 0.1} with 15 chosen trajectories. The key point is the non-monotony of the θ -variable for k < 1. We give a preliminary result about the cut loci to conclude these numerical computations. We denote p θ > 0 → ∆ θ k (p θ ) ∈ (0, π) the variation of θ at the first return to the equator (or first return mapping) as in §2.1. The previous numerical simulations suggest that ∆ θ k is well defined for k ∈ [0, k 3 ]. The figure 12 suggests that for any k, the first return mapping is monotonous non-increasing and surjective. As a consequence, for a fixed k and starting from ϕ(0) = π/2, θ (0) = 0, if there is no intersection between trajectories before the first return to the equator, then the cut locus is the equator minus the initial point. The figure 11 implies that there is no intersection before the first return to the equator for k < 1. Similar computations for k ∈ [1, k 3 ] suggest the same.

Fig. 6

 6 Fig. 6 Conjugate (in red dots) and cut (in black dots) loci, using the parameterization (θ 1 , θ 2 ) ∈ [0, 2π) × [0, π] for the left subplot and in (x 1 , x 2 , x 3 )-coordinates for the right subplot. The initial point is marked with a blue dot. The two vectors crossing (π, π) illustrate the point reflection used to put together the different parts of the conjugate and cut loci.

Fig. 7

 7 Fig.7The cut time t 0 , the first conjugate time t 1 and the halfperiod T /2 of the oscillating variable, with respect to the parameter η ∈ [0, 2π].

Fig. 9

 9 Fig. 9 The deformation of one branch (p ϕ = -2 and p θ ≥ 0) of the conjugate locus w.r.t. the parameter k ∈ [1, k 3 ]. (top) k = 1.0, 1.05. (left) k = 1.1, 1.2. (right) k = 1.3, 1.4.

Fig. 10

 10 Fig. 10 The four types of trajectories which clarify the evolution of the conjugate locus. (top-left) existing for 1 ≤ k ≤ k 3 . (top-right) present for k 1 ≤ k ≤ k 3 . (bottom-left) found for k ≤ k ≤ k 3 . (bottom-right) only for k 2 ≤ k ≤ k 3 .

Fig. 11

 11 Fig. 11 Conjugate locus with 15 trajectories for k = 0.8, 0.5, 0.2, 0.1 from top left-hand to bottom right-hand.

Fig. 12

 12 Fig. 12 First return mapping for different values of the parameter k ∈ [0.1, 50]. In red is plotted the curve for k = 1.

  -4 . In red is plotted θ (t 1 (p θ , k)) while we have in blue ϕ(t 1 (p θ , k)).We can see in Fig.9, three subplots which represent the deformation of one branch (p ϕ = -2 and p θ ≥ 0) of the conjugate locus resp. for k in[1, k 1 ], [k 1 , k 2 ] and [k 2 , k 3 ]. For any k ∈ [1, k 3 ],the branch is located in the half-plane θ ≥ 0. If we denote k 1 < k < k 2 , the parameter value s.t. ϕ(t 1 (k)) = π/2, then k ≈ 1.155 and the branch form a loop for k ≤ k ≤ k 3 .

		pi
	The
	θ -variable takes the values	pi/2
	0 and π. The values k 1 , k 2 ,
	k 3 are approximately and
	respectively 1.061, 1.250,
	1.429.	0
		1 k1	k2	k3