Simon Varaine

Ismaël Benslimane

Raul Magni-Berton

Paolo Crosetto

Attacking the Weak or the Strong? An Experiment on the Targets of Parochial Altruism

Keywords: Parochial altruism, Terrorism, Social comparison, Inequality, Ideology, Intergroup conflict

Studies on parochial altruism have insofar focused on the causes leading individuals to attack any out-group on the behalf of one's group. Yet, we have no clue to understand why parochial altruists target specific groups, such as big firms in some contexts and refugees in other contexts. The present paper introduces an experiment to analyse the conditions under which individuals costly attack strong versus weak out-groups. In our study, 300 participants played a repeated Inter-group Prisonner Dilemma (IPD) involving multiple groups and inter-group differences in resources. The results show that individuals have a basic preference for targeting strong out-groups, but that attacks decrease when the inequality in destructive capacity between groups is high. Besides, individuals target weak out-groups when they are threatening their in-group status. Decisions in the game correlate with participants' political ideology and social dominance orientation. Overall, the results give clues to understand historical variations in the targets of political violence.

Introduction

A large body of research has investigated parochial altruism -i.e. the coexistence of in-group contribution (altruism) and out-group aggression (parochialism) [START_REF] Rusch | The evolutionary interplay of intergroup conflict and altruism in humans: a review of parochial altruism theory and prospects for its extension[END_REF][START_REF] De Dreu | Parochial cooperation in humans: forms and functions of self-sacrifice in intergroup conflict[END_REF][START_REF] Choi | The coevolution of parochial altruism and war[END_REF][START_REF] Bernhard | Parochial altruism in humans[END_REF]. The concept of parochial altruism captures a recurring pattern of inter-group relationships: that is, the costly attack of members of an out-group on behalf of one's in-group. In particular, this concept is applied to modern terrorism: the case of suicide-bombers sacrificing themselves to hurt the rivals of their political, religious or ethnic group is an archetypical example of parochial altruism [START_REF] Ginges | What motivates participation in violent political action[END_REF][START_REF] Atran | Dangerous terrorists as devoted actors[END_REF], Sheikh et al., 2014). While suicide-bombers may seem extreme cases of irrational behaviour, experimental studies demonstrate that parochial altruism is a widespread tendency. Indeed, in various economic game settings such as the Inter-group Prisoners' Dilemma (IPD) [START_REF] Bornstein | The free-rider problem in intergroup conflicts over step-level and continuous public goods[END_REF], a high proportion of people choose at their own cost to inflict losses to an out-group to the benefit of their in-group [START_REF] Abbink | Parochial altruism in inter-group conflicts[END_REF][START_REF] De Dreu | Parochial cooperation in humans: forms and functions of self-sacrifice in intergroup conflict[END_REF][START_REF] Bernhard | Parochial altruism in humans[END_REF].

Most research on parochial altruism has focused on its underlying motivations and the conditions under which it emerges. According to some research, the combination of both altruism and parochialism may have been an evolutionary stable strategy in human evolution under conditions of competition on resources [START_REF] Choi | The coevolution of parochial altruism and war[END_REF]. Such motivation would have been embedded in humans across evolution through hormonal and neurologic mechanisms, shaping inter-group psychology [START_REF] De Dreu | The neuropeptide oxytocin regulates parochial altruism in intergroup conflict among humans[END_REF][START_REF] Rusch | The evolutionary interplay of intergroup conflict and altruism in humans: a review of parochial altruism theory and prospects for its extension[END_REF].

Most experimental studies on parochial altruism have investigated the motivation for the aggression of any out-group on behalf of the in-group. Yet, in the real world, there is a great diversity of potential targets for parochial altruists. This may be illustrated by the heterogeneous targets of terrorist groups: depending on time and space, terrorism has been used as a strategy against minority groups, political authorities, large firms. Understanding the conditions under which parochial altruists choose different targets may help to predict why certain groups receive more or less hostility in the real world. More specifically, answering this question may help understand the causes of the emergence and targeting of terrorist movements.

The literature gives limited clues to understand why parochial altruists target certain groups instead of others. On the one hand, macro-level studies reveal that terrorist attacks targeting distinct out-groups emerge in very different socio-economic contexts [START_REF] Kis-Katos | On the heterogeneity of terror[END_REF], Brockoff et al., 2016[START_REF] Varaine | Revisiting the Economics and Terrorism Nexus: Collective Deprivation, Ideology and Domestic Radicalization in the US (1948-2016)[END_REF]). Yet, macro analyses do not allow to identify the underlying mechanism. On the other hand, experimental studies analyse the causes of different forms of out-group hostility, such as prejudice against minority groups [START_REF] Krosch | Economic scarcity alters the perception of race[END_REF][START_REF] Outten | Feeling threatened about the future: Whites' emotional reactions to anticipated ethnic demographic changes[END_REF] or aggression against advantaged groups [START_REF] Halevy | Relative deprivation and intergroup competition[END_REF][START_REF] Abbink | Inequality and inter-group conflicts: experimental evidence[END_REF]). Yet, to our knowledge, there is no experimental study comparing the causes of these different forms of out-group hostility.

In this paper we distinguish two types of parochial altruism: parochial altruism directed against out-groups that have more vs. fewer resources than the in-group. We call the first type left-wing, and the second right-wing parochial altruism, based on the common definition of left-wing versus right-wing terrorism [START_REF] Kis-Katos | On the heterogeneity of terror[END_REF]. Indeed, a definitional feature of left-wing terrorist ideologies is the strive for equality -which may take the form of the preference for reducing the resources of the richest. Left-wing terrorism often aims at attacking dominant groups -e.g. large firms, political authorities. In contrast, a definitional feature of right-wing terrorist ideologies is their hostility toward minority groups, such as ethnic minorities and immigrants [START_REF] Kis-Katos | On the heterogeneity of terror[END_REF].

Based on the literature on social comparison, we posit that these types of parochial altruism result from two forms of inter-group comparison: envy and jealousy. According to the logic of envy, individuals should have a basic preference for targeting richer out-groups.

According to the logic of jealousy, individuals should tend to attack poorer out-group when they get close to the level of resources of the in-group. The effects of envy and jealousy should depend on the groups' destructive capacity: a higher destructive capacity of richer groups should limit the effect of envy while it should enhance the effect of jealousy.

To test our hypotheses, we introduce a novel economic experiment allowing to identify the targets of parochial altruism. Our game is an extended version of the Inter-group Prisonner Dilemma [START_REF] Bornstein | The free-rider problem in intergroup conflicts over step-level and continuous public goods[END_REF] with multiple groups and inequality in resources across groups. We exogenously manipulate the groups' variation in resources to test whether subjects target poorer out-groups when they get close to the level of resources of the in-group.

We also manipulate the destructive capacity of groups to test whether a higher destructive capacity of richer groups moderates the effects of envy and jealousy.

The hypotheses and experimental design were preregistered on Open Science Framework, and all details are available at http://osf.io/neaqu1 . Moreover, the paper includes not pre-registered analyses assessing the implications of our results to left-wing and rightwing terrorism, based on a post-experimental survey and on historical data on terrorist events.

Social Comparison and Parochial Altruism

In an environment with multiple groups, parochial altruists cannot attack all outgroups. This would be too costly and may dilute the effect of each attack. Moreover, this shooting-in-the-dark strategy may generate vindictive coalitions from the other groups against the in-group. For a given constant incentive to attack, parochial altruists have to choose which out-group(s) to target. The core of our argument is that the individual decision to target specific out-groups depends on the repartition of resources across the different groups [START_REF] Halevy | Relative deprivation and intergroup competition[END_REF][START_REF] Abbink | Inequality and inter-group conflicts: experimental evidence[END_REF].

Drawing from studies on social comparison, we focus on individual motivation to achieve the best relative position for their group compared to other groups. Many studies show that individuals evaluate their own payoff relatively to others rather than in absolute terms [START_REF] Solnick | Is more always better?: A survey on positional concerns[END_REF][START_REF] Boyce | Money and happiness: Rank of income, not income, affects life satisfaction[END_REF][START_REF] Card | Inequality at work: The effect of peer salaries on job satisfaction[END_REF]. For instance [START_REF] Solnick | Is more always better?: A survey on positional concerns[END_REF] find that most individuals preferred a lower absolute to a higher relative income. Similarly, [START_REF] Boyce | Money and happiness: Rank of income, not income, affects life satisfaction[END_REF] find that relative income predicted life satisfaction, while absolute income did not once controlling for the first. Neurologic evidence indicates that higher relative payments and lower relative losses compared to other individuals are related to the activation of the ventral striatum -indicating reward-related brain activity [START_REF] Dvash | The envious brain: the neural basis of social comparison[END_REF][START_REF] Luo | Social comparison in the brain: A coordinate-based meta-analysis of functional brain imaging studies on the downward and upward comparisons[END_REF]. Some evidence indicates that inter-group comparison elicit similar mechanisms to inter-individual comparison. For instance, a study based on football teams show that fans being presented with the failures of the rival team experienced higher ventral striatum activity [START_REF] Cikara | Us versus them: Social identity shapes neural responses to intergroup competition and harm[END_REF]. Individuals displaying high ventral striatum responses to the rivals' failures are also more willing to harm fans of the other team (see also [START_REF] Hein | Neural responses to ingroup and outgroup members' suffering predict individual differences in costly helping[END_REF].

Based on this reasoning, we distinguish two forms of inter-group comparison, i.e. envy and jealousy, that respectively motivate left-wing and right-wing parochial altruism.

Envy and Left-Wing Parochial Altruism

Envy occurs "when a person lacks another's superior quality, achievement, or possession and either desires it or wishes that the other lacked it" (Parrott & Smith, 1993: 906). Research demonstrates that upward comparison with people or groups with higher resources generate negative feelings that display neural signatures [START_REF] Takahashi | When your gain is my pain and your pain is my gain: neural correlates of envy and schadenfreude[END_REF][START_REF] Luo | Social comparison in the brain: A coordinate-based meta-analysis of functional brain imaging studies on the downward and upward comparisons[END_REF]. Envy in its malign form is related to "schadenfreude": gloating when the envied other falls from grace. Various studies show that individual display positive emotions and reward-related brain activity when the envied person or group experiences losses [START_REF] Takahashi | When your gain is my pain and your pain is my gain: neural correlates of envy and schadenfreude[END_REF][START_REF] Dvash | The envious brain: the neural basis of social comparison[END_REF]. In this line, economic experiments show that people are willing to pay to burn the money of richer individuals [START_REF] Zizzo | Are people willing to pay to reduce others' incomes?[END_REF][START_REF] Zizzo | Money burning and rank egalitarianism with random dictators[END_REF].

At the group level, [START_REF] Halevy | Relative deprivation and intergroup competition[END_REF] analyse the effect of between-group relative deprivation on parochial altruism in a version of Inter-group Prisoner Dilemma in which individuals can use money either to benefit the in-group or reduce the income of the out-group without benefits for the in-group (see [START_REF] Halevy | In-group love" and "out-group hate" as motives for individual participation in intergroup conflict: A new game paradigm[END_REF]. They find that players from groups that are relatively deprived are more likely to costly attack the out-group, thus reducing the ingroup absolute welfare but increasing its relative status.

Hence, we predict that, in general, parochial altruists have a basic preference to target groups with high resources to advance their relative status.

H1: the higher the resources of an out-group the higher the level of attacks.

Jealousy and Right-Wing Parochial Altruism

Jealousy is "the belief, suspicion or fear that what one desire to keep for one self is in danger of being diverted to a known or suspected rival" (Salovey & Rodin, 1991: 396). In the case of inter-group comparison, jealousy refers to the fear that an out-group, which was initially less well off, challenges and overtakes the in-group's status.

It has long been proposed that perceived threats to their status may lead the majority group to backlash against minority groups [START_REF] Blalock | Toward a theory of minority-group relations[END_REF][START_REF] Funke | Going to extremes: Politics after financial crises, 1870-2014[END_REF]. Experimental studies reveal that making a future shift in the demographic composition of the society salient increases prejudice against minority groups [START_REF] Outten | Feeling threatened about the future: Whites' emotional reactions to anticipated ethnic demographic changes[END_REF] and right-wing attitudes in general [START_REF] Craig | On the precipice of a "majority-minority" America: Perceived status threat from the racial demographic shift affects White Americans' political ideology[END_REF], due to the feeling that the in-group status is threatened.

Besides, some historical analyses suggest that increases in minority and immigrant populations are positively related to hate crimes and far-right offenses [START_REF] Green | Defended neighborhoods, integration, and racially motivated crime[END_REF][START_REF] Boutcher | Strain, ethnic competition, and power devaluation: white supremacist protest in the US, 1948-1997[END_REF]. In this line, minorities overtaking the position of the traditional majority group is a recurring theme in far-right ideologies [START_REF] Hewitt | Understanding terrorism in America: From the Klan to Al Qaeda[END_REF][START_REF] Mcveigh | The rise of the Ku Klux Klan: Right-wing movements and national politics[END_REF].

Hence, we posit that individuals are more likely to target poorer out-groups when they get close to the in-group level of resources.

H2: the closer the out-group is getting to the in-group the higher the level of attacks.

How Destructive Capacity Affects Envy and Jealousy

Some conditions may affect the logics of envy and jealousy. Regarding envy, if the advantage of the out-group is far too high, the effect of an attack may be too small to alter the relative position of the in-group. This may lead to a resignation effect and decrease the actual level of attacks. In an economic contest between two groups, [START_REF] Abbink | Inequality and inter-group conflicts: experimental evidence[END_REF] find that in a setting with increased inter-group inequality, the level of attacks from the disadvantaged group significantly decrease. Furthermore, if richer groups have higher destructive capacity, targeting a richer out-group may be dangerous because the latter may retaliate and cause great damages to the in-group in the future. Hence:

H1bis: the effect of envy (H1) is reduced if richer groups have higher destructive capacity

In contrast, we posit that the logic of jealousy should be especially strong when richer groups have a higher destructive capacity. Indeed, experimental studies show that fear of future attacks is a major motivation for first strikes against an out-group [START_REF] Böhm | What makes people go to war? Defensive intentions motivate retaliatory and preemptive intergroup aggression[END_REF][START_REF] De Dreu | The neuropeptide oxytocin regulates parochial altruism in intergroup conflict among humans[END_REF], more than nastiness and out-group anger [START_REF] Simunovic | Preemptive strike: An experimental study of fear-based aggression[END_REF][START_REF] Abbink | Trust on the brink of Armageddon: The first-strike game[END_REF]. Letting a group overtake one's rank is dangerous since the out-group will be able to destroy the in-group in the future. Hence, to secure its rank, the best option for a group is to attack groups that is getting close to his level of resources. Hence:

H2bis: the effect of jealousy (H2) is increased if richer groups have higher destructive capacity Implications for Left-Wing and Right-Wing Terrorism What are the implications of our hypotheses regarding the societal conditions generating left-wing and right-wing terrorism? Arguably, richer groups have a higher destructive capacity in societies with high economic inequality. Indeed, in unequal societies, economically powerful groups have a higher political influence on policies [START_REF] Gilens | Affluence and influence: Economic inequality and political power in America[END_REF].

Richer groups may have higher impact on governmental repression of left-wing insurgencies, or even fund far right movements or paramilitary groups in response to threats from left-wing movements [START_REF] Acemoglu | Economic origins of dictatorship and democracy[END_REF]. If this is true, H1-bis implies that attacks against richer groups should decrease when inequality is high. In contrast, H2-bis implies that attacks against poorer groups that get close to the in-group should increase when inequality is high.

As mentioned earlier, attacking stronger groups is a definitional feature of let-wing terrorism, while right-wing terrorism mostly targets weaker groups [START_REF] Kis-Katos | On the heterogeneity of terror[END_REF]. Hence, H1bis implies that left-wing terrorism decreases when inequality is high, while right-wing terrorism increases when inequality is high.

Method

Measuring Parochial Altruism: the Inter-Group Prisoners' Dilemma

Parochial altruism is commonly captured through the Inter-group Prisoner's Dilemma (IPD) paradigm [START_REF] Bornstein | The free-rider problem in intergroup conflicts over step-level and continuous public goods[END_REF], 2003[START_REF] Halevy | In-group love" and "out-group hate" as motives for individual participation in intergroup conflict: A new game paradigm[END_REF], 2010[START_REF] Böhm | What makes people go to war? Defensive intentions motivate retaliatory and preemptive intergroup aggression[END_REF]. In a standard version of this game, subjects are divided in 2 groups of 3 players. Each subject receives a fixed initial amount of resources in points 𝑝 ! that the subject either choose to keep (𝑘) or to contribute to attack the out-group (𝑎). 𝑝 ! = 𝑘 + 𝑎. For each point kept, the subject gains 1 point. For each point contributed to attack the out-group, each subject of the in-group gains 0.5 point and each subject of the out-group loses 0.5 point. Decisions are made simultaneously.

The payoff of a given subject is determined by the number of points 𝑝 he gained at the end of the round. It is a function of the number of points 𝑘 he kept, the number of points 𝑎 ! contributed in attacks by the subjects 𝑖 of the in-group and the number of points 𝑎 ! contributed in attacks by the subjects 𝑗 of the out-group:

𝑝 = 𝑘 + 0.5× 𝑎 ! ! !!! -0.5× 𝑎 ! ! !!!
This game consists of two prisoner's dilemmas embedded into each other, one at the individual level and the other at the group level. Table 1 shows the payoff matrix for a subject depending on the decisions of the other in-group subjects, assuming that all subjects of the out-group keep their points2 . It demonstrates that at the individual level, the Nash equilibrium is for each subject to keep his points. Indeed, a subject gains 0.5 more points when keeping a point rather than attacking.

[Insert Table 1 around here]

Table 2 shows the payoff matrix for a subject depending on the decisions of the ingroup and out-group subjects, assuming that subjects of the same group take the same decision. It demonstrates that at the group level, the dominant group strategy is to contribute all points to attack. Indeed, each subject gains 0.5 more points if all in-group subjects attack rather than keep. Yet, the dominant collective strategy is for all subjects to keep their points.

Indeed, if all subjects attack, no subject gains any points as the gains form the in-group's attacks are offset by the losses from the out-group's attacks.

[Insert Table 2 around here]

The game captures core features of parochial altruism. From a purely individual perspective, it is costly to engage in attacks on behalf of one's group. Yet, the group would be better off if all their members engaged in such attacks. However, at the aggregate social level, attacks have negative consequences.

Design of the Present Experiment

We propose a new version of the IPD. In our version, subjects play a repeated IPD game with a number 𝑇 of rounds [START_REF] Bornstein | The effect of repeated play in the IPG and IPD team games[END_REF][START_REF] Halevy | In-group love" and "out-group hate" in repeated interaction between groups[END_REF]. Subjects stay in the same group for the whole duration of the game. We keep unchanged the payoff structure and the dominant strategies of the game. Yet, we bring three new elements to test our hypotheses.

Firstly, we increase the number 𝑛 of groups. Each subject may now contribute points to attack different out-groups. When deciding to contribute, subjects have to choose which out-groups to attack. The consequences for the in-group stay the same.

Secondly, we introduce inequality across groups. This allows us to test the conditions under which subjects attack richer or poorer out-groups. Before playing the IPD game, subjects perform a word-creation task akin to a Scrabble game (Brüggemann et al., 2016).

Differences in skill generate considerable variance. Each subject starts the IPD game with resources in points 𝑝 ! proportional to the results of the word-creation game. The groups for the IPD are then created based on the subjects' performance during the task: the three best subjects form a group, the three-second best another, and so on. This generates endogenous inter-group inequality in the IPD. Moreover, inequalities are (loosely) meritocratic, inducing feelings of entitlement in the subjects.

To test H2, we exogenously manipulate inequality across the groups. To do so, at each round, each subject receives a random endowment in points 𝛼. This allows manipulating the variation in resources of groups to test H2, independently from the variation that is due to the attacks across groups. The random endowment 𝛼 is determined by three economic conditions.

Every five rounds, each group is randomly assigned to one of three conditions with equal probabilities: in the improving condition, subjects have an endowment 𝛼 of 6% of their resources in points 𝑝 ! by round; in the stable condition, subjects have an endowment 𝛼 of 2% of their resources in points 𝑝 ! by round; in the declining condition, subjects have a negative endowment 𝛼 of -2% of their resources in points 𝑝 ! by round. We choose to assign groups to a given economic conditions for five rounds rather than every round because pure random variations of endowment from one round to another would probably not affect the targeting of parochial altruists because subjects could not draw expectations from the present on the future resources of the out-group.

Hence, at the end of a given round, the number of points 𝑝 of a subject is a function of the number of points 𝑘 he kept, the number of points 𝑎 ! contributed in attacks by the subjects 𝑖 of the in-group, the number of points 𝑎 !→!"!!"#$% invested in attacks against the in-group by the subjects 𝑗 of the 𝑛 -1 out-groups, and the subject's random endowment 𝛼:

𝑝 = 𝑘 + 0.5× 𝑎 ! ! !!! -0.5× 𝑎 !→!"!!"#$% !× !!! !!! + 𝛼
The number of point 𝑝 of a subject at the end of each round determines the resources in points 𝑝 ! with which he will start the next round. This also allows inequality to endogenously vary during the course of the game.

The payoff of each subject is determined by the number of points 𝑝 he gained at the end of the last round of the game.

Finally, in our version of the game, subjects have a limit 𝛽 on the number of points 𝑎 they can contribute to attack out-groups at each round. 𝑎 ≤ 𝛽. To test H1-bis and H2-bis, we use two between-subjects treatments. In the absolute treatment, subjects face an absolute limit to the number of points 𝑎 they can contribute to attack. That is, 𝛽 is fixed. In the relative treatment, the limit 𝛽 is proportional to the subjects' resources in points 𝑝 ! at the beginning of each round. In the absolute treatment, a rich subject can attack as much as a poor subject 3 ; in the relative treatment, a rich subject can attack more than a poor subject. In both treatments, the rich have an advantage over the poor: the opportunity cost for attacking is lower for rich subjects because contributing a point is relatively less costly for a rich than a poor. Yet, the relative treatment enhances this advantage. In the relative treatment richer groups have more destructive capacity than poorer groups, while in the absolute treatment all groups have the same destructive capacity within each round. If H1-bis is correct, subjects should diminish their attacks against richer groups in the relative treatment. If H2-bis is correct, subjects should increase their attacks against close poorer groups in the relative treatment.

As shown in Table 2, the IPD game entails the possibility that subjects have negative payoffs. As it is impossible to withdraw money to subjects, negative payoffs are equivalent to zero payoffs, thus altering the structure of the game. The possibility of negative payoffs is even more likely in our version of the game because subjects may experience cumulative losses across rounds, and because the higher number of groups means that the in-group may be simultaneously attacked by multiple out-groups. The limit 𝛽 partly addresses this issue, since lowering the level of attacks decreases the probability that players have negative points.

However, it was impossible to avoid completely the possibility of negative payoffs. Hence, Half of the subjects played in the absolute treatment, with the limit 𝛽 fixed at 50 points; the other half in the relative treatments, the limit 𝛽 fixed at 10% of their points at the beginning of each round.

Exploratory Analyses

At the end of the IPD, subjects filled out a short questionnaire. It included sociodemographic questions -age, gender, level of education and professional status. We also included two questions about whether subjects found themselves and the other subjects cooperative during the game. This allows us to check whether attacking other groups is perceived as cooperative behaviour, in line with results from the literature on the IPD [START_REF] Halevy | In-group love" and "out-group hate" as motives for individual participation in intergroup conflict: A new game paradigm[END_REF][START_REF] Weisel | Ingroup love" and "outgroup hate" in intergroup conflict between natural groups[END_REF][START_REF] Halevy | In-group love" and "out-group hate" in repeated interaction between groups[END_REF]. Besides, we included a political and a psychometric measure to test the relationship between left-right parochial altruism and the subjects' left-right political attitudes. Subjects indicated their political self-placement on a scale from 1 (left) to 10 (right) and filled out the French short version of the Social Dominance Orientation scale in 8 items (Bizumic et al., 2008) -which captures individual's sense that some groups are inferiors and that some groups should dominate in society [START_REF] Sidanius | Social dominance: An intergroup theory of social hierarchy and oppression[END_REF].

Furthermore, we conduct not pre-registered analyses based on historical data to assess the implications of our experimental results to right-wing and left-wing terrorism. We look at the simple relationship between economic inequality and left-wing and right-wing terrorism in nine developed countries that experienced a significant number of terrorist events since 1970.

We measure inequality by the share of the pre-tax national wealth owned by the 1% richest [START_REF] Wid | World Wealth & Income Database[END_REF]. The number of terrorist events is based on the Global Terrorism Database (START, 2016). Left-wing and right-wing terrorist events are distinguished based on the political identity of the terrorist organisation following [START_REF] Kis-Katos | On the heterogeneity of terror[END_REF].

Results

Gameplay and Learning in the Game

Consistent with the IPD literature, and despite the Nash prediction of no attacks, subjects engage in a considerable level of attacks. On average, subjects contribute 390 points in attacks (sd = 181), which represent around 75% of their resources in points at the beginning of the game (mean = 522, sd = 156). A look at the payoffs confirms that subjects lose a significant amount of money because of their mutual attacks. The mean payment at the end of a session was 6.35 € (sd = 4.22). If no subject had ever attacked, the mean payment based on the observed resources at the beginning of the game would have been of 17.55 €. This means that subjects on average lost 65% of their payoff because of their mutual attacks. There was a difference in payoffs between the treatments: 5.28 € in the absolute treatment and 7.42 € in the relative treatment (Student's t = -4.5370, p < .001).

This level of attacks is inconsistent with Nash predictions. Nonetheless, subjects could attack in the first rounds but then learn in time to attack less. We see no evidence of learning.

There are no major differences in the level of attacks, i.e. the sum of the number of points contributed in attacks by the in-group subjects against a given out-group, between the initial and the final rounds, for neither treatment. We use negative binomial regressions to model the level of attacks 7 , controlling for the in-group level of resources, which mechanically affects the capacity to attack in the relative treatment. Figure 1 plots the marginal effect of such a regression for each round. We also test for a linear effect of rounds, controlling for in-group resources. Regression results show no significant effect of rounds on attacks in the absolute treatment (b = -.0408, p = .176) and a slight negative effect in the relative treatment (-.0538, p 7 We use negative binomial regressions in all subsequent analyses instead of linear regression (as pre-registered) because the level of attacks is over-dispersed (mean = 19.48, sd = 21.12). Note that in all of the subsequent analyses, we exclude observations relative to rounds in which one group or more is dead to alleviate potential biases in the comparisons. Nonetheless, including all observations in the analyses does not alter our conclusions.

The authors can send results upon request.

= .023). Overall, the level of attack is quite stable through the game. There is little convergence toward the dominant individual strategy.

[Insert Figure 1 around here]

Figure 2 plots the mean resources of the five playing groups, i.e. the sum of the resources in points of the subjects of each group, across the rounds of a session. We see that in both the relative and absolute treatments, the level of resources of all groups generally decreases during the course of the game because of mutual attacks. We see different patterns in the relative and absolute treatments. In the absolute treatment, rich groups lose relatively more resources than poor groups, leading to close levels of resources for all groups at the end of the last round. In contrast, in the relative treatment, the slopes are quite similar from the richer to the poorer groups, leading to a higher level of inter-group inequality in the last rounds.

[Insert Figure 2 around here]

Pre-Registered Analyses

Figure 3 displays the results relative to H1. Figure 3 plots the marginal effects of outgroup resources on the level of attacks based on negative binomial regressions, controlling for the in-group resources. We include a quadratic term to model the slope of the effect of the out-group resources. In line with H1, the higher the level of resources of the out-group the more it is attacked, both in the absolute and relative treatments. This finding suggests that the logic of envy is predominant, since subjects tend to attack richer out-groups more.

[Insert Figure 3 around here]

The relative treatment introduces an inequality in destructive capacities depending on the subjects' resources. According to H1-bis, the destructive capacity reduces the attacks against the richest. Results from figure 3 are consistent with this hypothesis. Indeed, in the absolute treatment, the relationship between the attacks and the out-group's resources follows an exponential slope: attacks are concentrated against the richest out-groups. In contrast, in the relative treatment, the slope is decreasing: a richer out-group is attacked more until it reaches a certain level of resources (around 2000 points). After that level, attacks stagnate.

This indicates that subjects refrain from attacking the richest out-groups when they have a higher destructive capacity, in line with H1-bis.

Figure 4 displays the results relative to H2. According to H2, the closer a poorer outgroup is getting to the in-group the higher its probability to be attacked. If H2 is true, poorer out-groups should be attacked more when they are in an improving economic condition -i.e.

they are getting closer to the in-group -while this should not be the case for richer out-group.

Figure 4 plots the marginal effect of the out-group economic conditions on the level of attacks, based on negative binomial regressions. We now split the sample depending on whether the target is a poorer out-group or a richer out-group.

[Insert Figure 4 around here]

Consistently with H2, Figure 4 shows that poorer out-groups are attacked more when their economic condition is improving. When analysing absolute and relative treatments together, compared to the declining condition, the stable condition has a positive significant effect (b = .125, p = .026), and the improving condition has a sizable significant effect (b = .415, p < .001) on the attacks received by the poorer out-group. However, Figure 4 shows that richer out-groups are also targeted more when their situation is improving. When analysing absolute and relative treatments together, compared to the declining condition, the stable condition has a positive but insignificant effect (b = .0779, p = .154), and the improving condition has a significant positive effect (b = .221, p < .001) on the attacks received by the poorer out-group. This does not generally support the logic of jealousy (H2): subjects attack more out-groups in improving economic condition but irrespectively of whether they are poorer or richer8 .

The absolute and relative treatments considerably influence the level of attacks against poorer and richer out-groups. As figure 4 shows, the richer out-groups in an improving condition are attacked more in the absolute treatment. In contrast, the poorer out-groups in improving condition are attacked more in the relative treatment. This means that when all groups have the same destructive capacity, subjects are envious: they tend to attack more richer out-groups that have prospects to become even richer. This effect is much weaker against poorer groups. This finding is consistent with H1bis. In contrast, in the relative treatment, the attacks are consistent with the logic of jealousy, as predicted by H2bis. The poorer out-groups with prospects to become richer are more strongly and significantly attacked than the others in the relative treatment, while this effect disappears for the richer out-groups. Overall, our results suggest that if the groups' destructive capacity depends on their resources, subjects attack more the poor who gets richer than the rich who gets richer.

Not Pre-Registered Analyses

In this section, we assess the implications of our results regarding left-wing and right wing terrorism. If we start from the realistic assumption that the destructive capacity of individuals is associated with their wealth in the real world, this capacity is equally distributed across individuals in equal societies, while it is unequally distributed in unequal societies.

Therefore, our results imply that left-wing terrorism targeting richer out-groups should decrease under high inequality, while right-wing terrorism targeting poorer out-groups should increase under high inequality. We here present favourable evidence of the correlation of our experimental measures of left-wing and right-wing parochial altruism with left-right political attitudes of subjects. We then show that the effect of inequality is observed within our experiment and provide evidence that it generalizes to historical data on terrorist events. [Insert Table 3 around here]

Results based on the post-experimental survey indicate that behaviours in the game correlate with subjects' attitudes. First, in line with previous study on parochial altruism, attacks were perceived as an altruistic behaviour. Model I shows that subjects who perceive themselves as cooperative during the game actually contribute more points in attacks. The effect of perceived cooperation on attacks is different for left-wing and right-wing parochial altruism. In model II, relative to attacks against richer out-groups, the coefficient of perceived cooperation only attains the 10% significance threshold. In contrast, in model IV, perceived cooperation is strongly related to attacks against poorer out-groups. By running a random slope model, we confirm that the effect of perceived cooperation on attacks is significantly higher for poorer than for richer out-groups (coefficient of the interaction between perceived cooperation and the out-group being poorer: b = .0739, p = .028).

Second, we test for the relationship between the targets of parochial altruism and the political attitudes of subjects. Regarding left-wing parochial altruism, model II shows that the more the subjects are at the right of the left-right scale the less they attack richer out-groups in the game. In contrast, model IV shows that subjects' left-right self-position has no significant effect on attacks against poorer groups. The Social Dominance Orientation9 has no effects on the attacks against richer groups, while it has a positive and significant effect on attacks against poorer groups. Overall, behaviors in the game correlate with political attitudes of subjects in the expected direction. Finally, the models in Table 1 include socio-demographic variables, i.e. gender, age, educational attainment and professional status. None of these variables are statistically related to any form of parochial altruism. Note however that selfemployed participants are significantly less prone to attack in general.

Results relative to the game variables first confirm previous results. An out-group is attacked more when it has high resources. An out-group is also attacked more when its resources are growing because of its economic condition, especially when the out-group is poorer than the in-group10 . Moreover, the results show that subjects tend to attack in retaliation of previous attacks by the out-group. Besides, subjects tend to attack more when members of their in-group attacked during the previous round.

We include a indicator of inequality across groups in the models. Inequality is measured, using a fractionalization index, as the sum of the squared share of resources respectively owned by the five groups 11 . Results from model I show that, overall, attacks decrease when inequality is high. Models II to V show that this effect of inequality is actually different when we distinguish between attacks against richer and poorer out-groups. Indeed, attacks against richer out-groups are lower when inequality is high while this is not the case against poorer out-groups. Results from model IV suggest that inequality even increase attacks against poorer out-groups, albeit the 5% conventional level of significance is not reached. Model III shows that inequality has a significant effect on the attacks against richer out-groups only in the relative treatment. Indeed, the main effect of inequality is no longer significant once we include an interaction term between inequality and the relative treatment.

In contrast, model V shows that there is no interaction effect between inequality and the relative treatment on the attacks against poorer out-groups.

Altogether our results suggest that inequality has different effects on left-wing and right-wing parochial altruism: when richer groups have a high destructive capacity, high inequality reduces left-wing parochial altruism, while it has no clear effect on right-wing parochial altruism. This dampening effect of inequality on left-wing parochial altruism is paradoxical since left-wing ideologies precisely strive against inequality. Yet, this effect generalize to historical cases of terrorist events.

[Insert Figure 5 around here]

To illustrate this, Figure 6 plots the number of left-wing and right-wing terrorist events in nine developed countries. Two main pieces of information in Figure 6 suggest our experimental findings have some generalizability to terrorist events. Firstly, in line with experimental results, we see that left-wing terrorism (in red) is more widespread that rightwing terrorism (in blue). Secondly, left-wing terrorism is more frequent when inequality is low, while there is no clear relationship in the case of right-wing terrorism, as in our experiment.

Conclusion

Research on parochial altruism has mostly focused on the causes leading individuals to attacks an out-group on the behalf of the in-group. Yet, we insofar had few clues to understand why, in a world with multiple groups, parochial altruism targets specific outgroups. The present study addresses this issue by introducing a new version of the Inter-group Prisoner Dilemma (IPD) with multiple groups, in which subjects have to choose which out-groups to attack. We distinguish between two types of parochial altruism: left-wing parochial altruism, directed against out-groups with more resources than the in-group, and right-wing parochial altruism, directed against out-groups with fewer resources than the in-group.

Results partly confirm previous results and partly introduce new evidence. We confirm that participants chose to spend substantial amounts of money to attack out-groups for the benefit of their group, thus reducing collective welfare [START_REF] Bornstein | Intergroup conflict: Individual, group, and collective interests[END_REF]. We also corroborate that attacks mainly emerge from "in-group love" rather than "out-group hate" [START_REF] Halevy | In-group love" and "out-group hate" as motives for individual participation in intergroup conflict: A new game paradigm[END_REF][START_REF] Halevy | In-group love" and "out-group hate" in repeated interaction between groups[END_REF][START_REF] Abbink | Inequality and inter-group conflicts: experimental evidence[END_REF]). New evidence is provided about the targets of parochial altruism: on average, subjects choose to attack the richest out-groups more.

However, when the destructive capacity of groups increases with their resources, rich groups are less attacked and the poor groups more, especially when they have good prospects to become rich.

Results based on the post-experimental survey indicate that the targets of parochial altruism correlate with subjects' political attitudes: left-wing parochial altruism correlates with subjects' left-wing political orientation and right-wing parochial altruism was higher among subjects high in Social Dominance Orientation. This confirms the usefulness of distinguishing between the two forms of parochial altruism that may have distinct psychologic and social antecedents. It also suggests that our experimental setting is an effective measure of the ideological orientation of individuals in inter-group interactions, and that the results may be extrapolated to inter-group conflicts.

Then, what are the implications of our findings regarding the targets of terrorism and political violence in general? Firstly, our results shed light on the linkages between inequality and the orientation of political violence. Indeed, a historical look reveals close relationships between ideological waves of political violence and the variation in inequality. For instance, high levels of inequality in the Europe of the 1930s coincided with a right-wing orientation of political violence, incarnated by the Nazi Party in Germany, and by movements such as the Young Patriots in France and the British Union of Fascists in the UK. In contrast, violent political movements in the 1970s, under unprecedented low levels of inequality, were mainly left-wing oriented, as illustrated by the Red Brigades in Italy, the Red Army Fraction in Germany or Action Directe in France. In the current context of increasing inequalities, rightwing violence seems again to gain prevalence over left-wing violence. Recent empirical studies on French radical movements [START_REF] Varaine | Bad times are not good times for revolutions: Collective deprivation and the mobilization level of F rench radical movements (1882-1980)[END_REF] and US terrorists [START_REF] Varaine | Revisiting the Economics and Terrorism Nexus: Collective Deprivation, Ideology and Domestic Radicalization in the US (1948-2016)[END_REF] statistically confirm this trend: the higher the increase in inequality the more right-wing the orientation of political violence.

The present experiment offers clues to understand this relationship. Our results suggest that inequalities may lead to a right-wing orientation of political violence. Not necessarily through increasing right-wing violence, but through decreasing the share of left-wing violence targeting strong groups. Indeed, we find that high inequality, through increasing the retaliation capacity of economically dominant groups, reduces the level of attacks targeting them. One could think this mechanism occur in real group interactions, in which economically powerful groups have a higher influence on policy responses [START_REF] Gilens | Affluence and influence: Economic inequality and political power in America[END_REF] and on direct repression of left-wing insurgencies [START_REF] Acemoglu | Economic origins of dictatorship and democracy[END_REF]. This mechanism of retaliation may reduce left-wing violence targeting economically dominant group, in parallel with a resignation mechanism, due to the feeling that inequalities are far too high to be reduced [START_REF] Abbink | Inequality and inter-group conflicts: experimental evidence[END_REF].

Furthermore, our experiment gives clues to understand the relationship between the economic mobility of social groups and the orientation of political violence. In this regard, various comparative studies find that increases in the demographic and economic status of minority groups are related to increases in right-wing violence [START_REF] Green | Defended neighborhoods, integration, and racially motivated crime[END_REF][START_REF] Boutcher | Strain, ethnic competition, and power devaluation: white supremacist protest in the US, 1948-1997[END_REF]. Our results suggest that a logic of jealousy may be at play here, leading members of strong groups to target weak groups with high prospects of prosperity, especially in contexts in which strong groups have a higher destructive capacity.

Finally, the data reveals an interesting difference between right-wing and left-wing parochial altruism. We find that subjects' perceived level of cooperation is related with attacks against poorer groups, while it has a limited effect on attacks against richer groups.

This result is close from the findings of [START_REF] Halevy | Relative deprivation and intergroup competition[END_REF] that, although attacks are generally based on altruistic motives [START_REF] Halevy | In-group love" and "out-group hate" as motives for individual participation in intergroup conflict: A new game paradigm[END_REF], attacks against relatively advantaged groups may be inspired by purely hateful motives. This is also congruent with recent evidence suggesting that the widespread individual tendency to punish non-cooperative players result from a basic human inequality aversion rather than from a desire of reciprocity [START_REF] Dawes | Egalitarian motives in humans[END_REF][START_REF] Raihani | Human punishment is motivated by inequity aversion, not a desire for reciprocity[END_REF]. An implication of this finding is that, individual identification with an in-group may not be a pre-requisite for all types of political violence, contrary to predictions of the identity fusion model of conflict [START_REF] Whitehouse | Brothers in Arms: Libyan revolutionaries bond like family[END_REF], see also [START_REF] Atran | Devoted actors sacrifice for close comrades and sacred cause[END_REF]. Although individual identification with an in-group may be crucial in motivating right-wing violence, left-wing violence might emerge solely from personal inequality aversion.

Tables Table 1: Note. We present individual payoffs assuming that each subject has one point to contribute (p 0 = 1) and that subjects of the out-group keep their points. Rounds in which at least one group was eliminated excluded from the analyses.

In this new game, you will be divided into 5 groups of participants depending on your ranking in number of points during the word creation game:

-The top three players will be in the yellow group.

-The next three in the red group.

-The next three in the purple group.

-The next three in the blue group.

-The last three players in the green group.

You will be part of the same group throughout the game.

In this new game, you have a number of individual points: your personal prize pool. At the beginning of the game, your prize pool contains 10 times the number of points you won during the word-creation game.

The game is composed of 15 successive rounds. In each round you must decide how many points you want to take from other groups to give them to your group.

Taking points from another group has a cost: to take 1.5 points from a group, you must spend 1 point of your personal prize pool. In that case :

-You take 1.5 points from the other group, that is 0.5 points per player from this group.

-You give 1.5 points to your group, that is 0.5 points per player in your group (including you).

[Relative treatment] { There is a limit to the number of points you can spend to take points from other groups: you can not spend more than 10% of your personal prize pool per turn. }

[Absolute treatment] { There is a limit to the number of points you can spend to take points from other groups: you can not spend more than 50 points of your personal prize pool per turn. }

[Screenshot Decision page 1]

Here is a capture of the game screen. Let's see an example. Say you want to take 45 points from the red group to give them to your group. Say you do not want to take points from other groups.

In practice, you have to click on each of the piles to select the amount you want to take from each group: zero for groups green, purple and yellow and 45 points for the red one.

As you can see, your decision is summarized below: to take 45 points from the red group and give them to your group, you spend 30 points of your personal prize pool. The 45 points you take from the red group will be evenly distributed in your group: 15 points for you and 30 for the other two players in your group.

Once you have indicated your choices, you must click on validate.

At the end of each round, your personal prize pool will increase or decrease depending on your choices and those of other players. But not only ... Chance will determine a part of your gains and losses.

In practice, every 5 rounds, the chance will decide the weather of your group for the next rounds. You will be randomly assigned to one the following weather conditions: -Good weather: your prize pool will increase by 6% at the end of each round.

-Average weather: your prize pool will increase by 2% at the end of each round.

-Bad weather: your prize pool will decrease by 2% at the end of each turn.

Each turn, the weather will affect your prize pool after the calculations related to your choices and those of the other players.

[Screenshot Result page]

Let's go back to the game screen. After validating your decision and once everyone has played, the results page is displayed. At the top of the screen, you still see information about the situation of your group and the other groups. This information has been updated based on the results of the tour that has just been played.

In the middle of the screen, you now see a summary of the gains and losses of your group of the round that has just been played. In this example, you see on the left that the green group has taken 30 points from your group overall -that is 10 points per player in your group, including you. As for you, you have given 45 points to your group -15 points per player -by taking them from the red group. For that purpose, you spent 30 points of your personal prize pool. On the right, you see that another player in your group has also given 45 points to your group by taking them from another group. Then you see that the third player in your group has not brought points to your group by taking to other groups. Finally, on the far right, you see that the weather, which is of +2% for your group, has brought 30 points to your group. Finally, at the end of this round, your personal prize pool after these different calculations is 500 points. Indeed, you spent 30 points to take points from the red group. You have gained 15 points from this action and 15 points thanks to another player in your group. This compensates. In addition, you lost 10 points because of the green group and gained 10 points because of your weather. This also compensates. Therefore, you still have the same number of points.

At the end of the 15 rounds, you will see the number of points accumulated in your personal prize pool.

Thus, in this new part you put into play your gains of the word creation game.

  The screen is framed by the colour of your group. In this example, you are part of the blue group. At the top of the screen, you see information about the situation of your group and the other four groups. Atthe top left, you see a graph that shows the number of points of each group in previous rounds. At the top centre, you have the total number of points of each group in the round being played. At the top right, you have the weather: I'll explain what it means a little later.In the middle of the screen, you see piles of coins. The pile on the left is your personal prize pool. You see that here you have 500 points. The piles on the right each correspond to another group from which you can take points. It is at the level of these piles that you will have to indicate your decisions. You must choose how much you want to take from each of the other groups to give to your own group. To do this, you must click on each of the piles corresponding to the other groups and decide how many points you want to take. It is imperative to click on each pile, even when you decide to take away zero points, in order to validate your decision. You have a limited time to choose how many points you want to take from each of the other groups: you have 45 seconds to indicate your choices. After this time, if you have not indicated a decision for one or more groups, the computer will randomly decide how much you take from each of them. It is therefore important to validate your decisions quickly. To help you, you have a little extra-time in the first round.[Screenshot Decision page 2]

Table 3

 3 presents results from negative binomial regressions of subjects' attacks with subjects random effects. The models include at a first level independent variables related to the game, and at the second level subjects' characteristics -including attitudinal and sociodemographic variables. Model I explains parochial altruism in general, i.e. the level of attack against any out-group, while models II and III focus on left-wing parochial altruism, i.e. the level of attack against a richer out-group, and models IV and V focus on right-wing parochial altruism, i.e. the level of attack against a poorer out-group.

  payoffs of the IPD game, individual level

	Decisions of the two other in-group	My decision
	subjects	Keep Attack
	Both keep	1	0,5
	One attack, one keep	1,5	1
	Both attack	2	1,5

Table 2

 2

	: payoffs of the IPD game, group
	level
		In-group
	Out-group	Keep Attack
	Keep	1	1,5
	Attack	-0,5	0
	Note. We present individual payoffs
	assuming that each subject has one point to
	contribute (p 0 = 1) and that subjects of the
	same group take the same decision.

Table 3 :

 3 Standardised coefficients from mixed effect negative binomial regression analyses

		I	II	III	IV	V
	Dependent variables	Attacks
		against an	Attacks against a richer	Attacks against a
		out-group	out-group	poorer out-group
	Subject-level variables
	Perceived self cooperation	0.111***	0.0627+	0.0651*	0.135**	0.135**
		(0.0275)	(0.0341)	(0.0332)	(0.0423)	(0.0425)
	Left right self position	-0.0680*	-0.0945*	-0.0858*	-0.0306	-0.0264
		(0.0311)	(0.0378)	(0.0370)	(0.0482)	(0.0485)
	Social dominance orientation	0.0549+	-0.0115	-0.00811	0.106*	0.110*
		(0.0313)	(0.0382)	(0.0372)	(0.0485)	(0.0490)
	Socio-demographics (age, sex, education,
		Yes	Yes	Yes	Yes	Yes
	work)
	Subject*Round variables
	Out-group resources	0.291***	0.231***	0.237*** 0.418*** 0.423***
		(0.0186)	(0.0299)	(0.0298)	(0.0383)	(0.0385)
	Out-group variation due to economic	0.0778***	0.0514**	0.0524** 0.105*** 0.103***
	condition	(0.0153)	(0.0182)	(0.0182)	(0.0275)	(0.0276)
	Attacks by the subject against the out-	0.321***	0.216***	0.219*** 0.353*** 0.353***
	group at t-1	(0.0162)	(0.0215)	(0.0216)	(0.0243)	(0.0243)
	Attacks by the out-group against the in-	0.252***	0.244***	0.245*** 0.259*** 0.258***
	group at t-1	(0.0153)	(0.0196)	(0.0196)	(0.0248)	(0.0248)
	Attacks by other in-group subjects at t-1	0.261***	0.312***	0.298*** 0.237*** 0.236***
		(0.0260)	(0.0355)	(0.0355)	(0.0370)	(0.0371)
	Subject resources	-0.0418+	-0.00345	0.0104	-0.0443	-0.0418
		(0.0232)	(0.0426)	(0.0424)	(0.0380)	(0.0383)
	Inequality	-0.0653**	-0.159***	-0.0350	0.0582+	0.141+
		(0.0205)	(0.0283)	(0.0611)	(0.0329)	(0.0729)
	Relative treatment			-0.246***		-0.105

Note that the present paper focuses on the hypotheses

and

(about social comparison) of the pre-registered project. The hypothesis 1 (about the in-group situation) is explored in a separate study.

Attacks from subjects of the out-group symetrically reduce the payoffs presented in Table 1, which do not change the payoff structure. Hence, they do not alter the game strategies.

We chose this rule because it is implicitly applied in the standard IPD game, in which subjects may potentially be "over-attacked" by subjects of the out-group. Indeed, if subjects of the in-group do not contribute points to attack and if subjects of the out-group contribute more than 2/3 of their points to attacks, any additional points contributed by the out-group no longer makes any difference for the in-group -whose payoff is negative -while still giving points to the out-group.

Instructions and screenshots from the software are provided in the online appendix. The oTree package is available upon request.

The results of the word-creation task are described in a forthcoming paper.

Yet, supplementary analyses show that the effect on attacks of the economic condition of out-group resources is slightly smaller for richer than for poorer out-groups: there is a significant positive interaction between the improving condition, compared to the declining condition, and the out-group being poorer, compared to richer, on the level of attacks it receives (coefficient of the interaction: b =.194, p = .012).

The internal validity of the Social Dominance Orientation scale was satisfactory (Cronbach's 𝛼 = .73).

Further analyses confirm again that the coefficient of the variation of the out-group resources due to the economic is significantly higher when the out-group is poorer rather than richer.

The variable theoretically goes from 20% (perfect equality across the five groups) to 100% (perfect concentration of the resources in the richest group). Empirically, the variable goes from 20.0% to 34.2%.

Acknowledgments

We would like to thank Innovacs for funding this study. We also thank Anna Cortijos Bernabeu for her work on the text.

we decided that a subject who has negative points can no longer attack and no longer loses points from out-group attacks. Yet, subjects of the out-groups continue to gain points when attacking his in-group irrespectively of his number of points 4 . When all subjects of the ingroup attain zero, we consider the in-group "dead" and the out-groups can no longer attack it.

Procedure and Parameterization

The experiment took place at the GAEL experimental laboratory in Grenoble, France.

The experimental software was written in Python using the oTree platform (Chen et al., 2016). We recruited 300 subjects from the GAEL subject pool, made in roughly equal parts by students and subjects issued from the general population. We ran 20 sessions of 15 subjects, divided in 𝑛 = 5 groups in the IPD game. Upon entering the lab, subjects were randomly assigned to individual computers. Instructions were read aloud and presented on overhead and individual screens. Clarification questions were answered collectively 5 . Subjects received a 10€ show-up fee, which was not at stake during the game.

Subjects first took part to the word-creation task 6 . After completing the task, subjects were informed about their individual gains. Based on pre-tests, we estimated that subjects would start the IPD game with around 500 points on average. Subjects were then divided in groups labelled with colours -yellow, red, purple, blue and green -to be easily identified.

Subjects played the IPD game for 𝑇 = 15 rounds, putting at stake their individual gains from the first task. Hello and thank you for your participation in our study.

In order to preserve anonymity during the session and when analysing the data, we have assigned you a code. No identifying data will be collected. It will be impossible to link your decisions to your name. The data will only be used for academic research purposes, always respecting anonymity.

Any communication between participants is strictly prohibited, as is any comment on what is appropriate or inappropriate to do during the experiment. Please turn off your cell phones. We ask you to stay focused on your own computer screen. During the session, you will have to make simple decisions. Nevertheless, if you face any problem, please do not hesitate to consult us.

During the session, you will have to perform several tasks. Three phases will take place: First, you will participate in a word creation game, then you will engage in an economic group game, and finally you will be asked to fill out a questionnaire. A new phase will only begin when all participants have finished the previous phase. We will give you instructions before the beginning of each phase. For everyone's understanding, these instructions will be read aloud. The duration of this session will not exceed two hours.

You have 10 euros in cash for your participation. You will find this amount in your personal envelope. This sum is yours. In addition to this amount and depending on the choices you make, you can earn more money throughout the games. You will be informed of your final earnings at the end of the session.

Do you have any questions? [First task]

We will now begin the first phase: the word creation game.

(See instructions in Brüggemann et al., 2016)

[Inter-group prisoner dilemma]

You have now completed the word creation game. Therefore, we are going to move on to the economic group game.

At the end of the game, the number of points in your personal prize pool will be divided by ten, and this score will determine your final gains in Euros, at the conversion rate of 1 POINT = 0.25 €.

This final amount will be added to the 10 Euros in you envelope.

Do you have any questions?

[Questionnaire and end of experience] Now that you have completed the economic group game, we will ask you to complete a brief questionnaire.

Once this questionnaire is completed, we will call you to proceed to the payment.

We thank you for participating in this study.


D Tisseur

N Estre

L Tamagno

C Eleon

D Eck

E Payan

N Cherepy

Performance evaluation of several well-known and new scintillators for MeV X-ray imaging

Performance evaluation of several well-known and new scintillators for MeV X-ray imaging David Tisseur, Nicolas Estre, Léonie Tamagno, Cyrille Eleon, Daniel Eck, Emmanuel Payan, Nerine Cherepy Concerning imaging in MeV range for non-destructive testing (NDT), literature is less abundant [START_REF] Baker | Scintillator efficiency study with MeV xrays[END_REF]. In this paper, we present a study of scintillator imaging performance for a 9 MeV linear accelerator beam. We selected well-known phosphor screens, micro-structured, single crystal and transparent ceramic scintillators. These tests are intended to help define the desired scintillator properties, geometry and optical configuration to optimize our 9 MeV radiography system.

II. SCINTILLATORS PERFORMANCE EVALUATION SET-UP

Tests were performed with a 9-MeV Linear accelerator [START_REF] Estre | High-energy X-ray imaging applied to nondestructive characterization of large nuclear waste drums[END_REF] in a cell named KROTOS at CEA Cadarache [START_REF] Estre | Fast megavoltage X-rays radioscopy[END_REF]. The scintillators evaluated were: two GOS (Gd 2 O 2 S:Tb) phosphor powder screens (Medex from AST, Lanex Fast Back from Kodak) one micro-structured CsI(Tl) scintillator provided by Hamamatsu, and three transparent scintillators -BGO (Bi 4 Ge 3 O 12 ) from Saint Gobain, LYSO (Lu 1.9 Y 0.1 SiO 5 , Cerium content: 0.5mol%) from JT Crystal Technology and GLO (Gd 0. Laboratory [START_REF] Cherepy | Transparent Ceramic Scintillators for Gamma Spectroscopy and MeV Imaging[END_REF]. These scintillators present a wide range of density, thickness and conversion efficiency. Tables I andII summarize the physical and optical properties of the scintillators studied. Each scintillator screen is imaged by a low noise S-CMOS 2160x2560 pixels (ANDOR Zyla 5.5) camera through a 45° tilted mirror in a light-tight box. A lead block prevents damage to the camera (see Fig. 1). The average detector dose rate was 0.92 mGy/s. The light output measured for each scintillator results from its stopping power (primarily determined by density in the MeV pair-production regime), luminosity (a constant for each scintillator), and optical configuration (optical scattering, Considering the distance from the linear accelerator to the detector box of 3125 mm, source size blur was considered to be negligible.

III. RESULTS AND ANALYSIS

In order to compare the six different scintillators, LYSO was selected as the reference screen (best known light yield with ±10% uncertainty), and all as-measured light output values were normalized to LYSO, but are not normalized for thickness, density or any other physical property. Each MTF curve for each scintillator and optical MTF has been evaluated (see Fig. 2). In order to be independent of the optical MTF, scintillators MTF have been corrected from optical MTF. Spatial resolution is computed using [START_REF]Industrial computed radiography with storage phosphor imaging plates -Part 1: Classification of systems[END_REF] definition: Table III summarizes the results obtained. The highest light output was obtained with CsI. Thanks to CsI micro-columnar structure, spatial resolution (994 µm) is better than Medex (1259 µm) and LYSO (1037 µm). With the highest thickness (25 mm), LYSO scintillator has a light output comparable to Medex with a little better spatial resolution (1037 µm). The best spatial resolution is achieved with GLO (401 µm). The 1.58 mm thick GLO scintillator was 2 brighter than the 3 mm thick BGO crystal, both prepared with black backing to eliminate scattered and back-reflected scintillation light (see Fig. 3).

𝑆𝑝𝑎𝑡𝑖𝑎𝑙 𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = 1 2 * 𝑀𝑇𝐹 20%

Fig. 3. Measured scintillator light output vs spatial resolution

To analyze our results on scintillator performance, we have performed MCNP6 Monte Carlo simulation [START_REF] Goorley | Initial MCNP6 Release Overview -MCNP6 Version 1.0[END_REF] and compared the measurements to the expected results for transparent scintillators (BGO, GLO). Simulations are in accordance with measurement excepted for GLO. Work is under progress to understand this result.

IV. CONCLUSION AND PERSPECTIVES

This study measured imaging performance with a 9 MeV Bremsstrahlung source of a wide range of scintillators including the well-known BGO, CsI GOS and newer scintillators, LYSO, GLO. The completed final paper will present more details particularly in experimental data and simulation analysis.

  Abstract-Digital X-ray imaging systems for MeV range photon beams are based on a combination of a scintillator screen and either a camera or an amorphous silicon array. To limit dose rate on electronics and enhance imaging device lifetime, the scintillator screen is mirror-coupled to the camera. Performances of such devices are a compromise between exposure time and spatial resolution. These technical characteristics are especially scintillator dependent. In this paper, we present a performance evaluation of six different scintillators with a 9 MeV Bremsstrahlung X-ray source. The tested scintillators are composed of one micro-structured CsI(Tl) scintillator, two phosphor (GOS) screens and three transparent scintillators. These scintillators present a wide range of density, thickness and conversion efficiency. Each scintillator's performance is assessed based on the combination of light output (ADU number) and modulation transfer function (spatial resolution) obtained. The results are helpful to guide design and engineering of high energy imaging devices adapted to specific requirements. Index Terms -Radioscopy; High-Energy X-ray; Linac; Scintillator; X-ray imaging I. INTRODUCTION Many papers in the literature report performance of various scintillators for digital imaging in the range of classical X-ray tube energy [1][2][3] or synchrotron applications [4][5].

  3 Lu 1.6 Eu 0.1 O 3 ) from Lawrence Livermore National D. Tisseur, N. Estre, L. Tamagno, C. Eleon, D. Eck, E. Payan, are with CEA, DEN, Cadarache, Nuclear Measurement Laboratory, F-13108 Saint-Paul-lez-Durance, France. N. Cherepy is with the Lawrence Livermore National Laboratory, L168, Livermore, CA, 94550, USA.

Fig. 1 .

 1 Fig. 1. Experimental setup.

Fig. 2 .

 2 Fig. 2. MTF corresponding to the six scintillators and optical part.

TABLE I SCINTILLATOR

 I Light output was measured via mean ADU number in a 50 pixels x 50 pixels region of interest of the image with an exposure time of 1 s. To avoid photonic noise, 10 images are averaged. A 5 cm x 5 cm x 10 cm copper block placed in front of the scintillator was used for modulation transfer function (MTF) evaluation with classical edge method. Edge spread function (ESF) was fitted with a combination of 2 Gauss error functions. Line spread function was obtained from the analytical derivative of the ESF. Then the MTF was given by analytical LSF Fourier transform.

	reflective or black backing).
	PHYSICAL AND OPTICAL CONFIGURATION USED
	Scintillator	Density	Thickness	Configuration
		(g/cm 3 )
	Medex, GOS	7.32	780 µm	On 2 mm brass plate
	Lanex, GOS	7.32	290 µm	See [11]
	CsI	4.51	2 mm	On 2 mm Cu plate
	BGO	7.13	3 mm
	LYSO	7.25	25 mm	Blackened surfaces
	GLO	9.10	1.58 mm	other than exit surface
		TABLE II
	SURVEY OF CHARACTERISTICS OF SELECTED SCINTILLATORS
	Scintillator	Surface		Light yield	Optical
	name	density in	(photons/MeV)	index
		mg/cm 2
	Medex	347		65000
	Lanex Fast Back 134		65000
	CsI	902		54000
	BGO	899		8000-10000	2.15
	LYSO	18500		30000	1.82
	GLO	1438		55000	1.89


Kähler Manifolds

Y A Deng
email: deng@ihes.fr

SIMPSON CORRESPONDENCE FOR SEMISTABLE

Keywords: Mathematics Subject Classification. 53C07 semistable Higgs bundles, representation of the fundamental group, Simpson correspondence

In this note we provide an elementary proof of the Simpson correspondence between semistable Higgs bundles with vanishing Chern classes and representation of fundamental groups of Kähler manifolds.

Introduction

Recently, J. Cao [START_REF] Cao | Albanese maps of projective manifolds with nef anticanonical bundles[END_REF] proved a longstanding conjecture by Demailly-Peternell-Schneider: for any smooth projective manifold whose anticanonical bundle is nef, the Albanese map of X is locally isotrivial. A crucial step of his proof relies on an elegant criteria in [START_REF] Cao | Manifolds with nef anticanonical bundle[END_REF] for the local isotriviality of the fibration, which is based on deep results for the numerically flat vector bundles (see Definition 1.4 below) in [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF] and the Simpson correspondence in [START_REF]Higgs bundles and local systems[END_REF].

Theorem 0.1. Let E be a holomorphic vector bundle over a Kähler manifold X which is numerically flat. Then (i) [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF]Theorem 1.18]. E admits a filtration

{0} = E 0 E 1 • • • E p = E (0.1)
by vector subbundles such that the quotients E k /E k-1 are hermitian flat, that is, given by unitary representations π 1 (X) → U(r k ). In particular, E is semistable and all the Chern classes of E vanish.

(ii) [START_REF]Higgs bundles and local systems[END_REF]§3]. E has a holomorphic structure which is an extension of unitary flat bundles.

Theorem 0.1.(ii) is indeed a special case (i.e. the Higgs fields vanish) of the equivalence between the category of semistable Higgs bundles with vanishing Chern classes and the category of representations of the fundamental groups of Kähler manifolds established by Simpson [START_REF]Higgs bundles and local systems[END_REF]§3].

Theorem A ([Sim92, Corollary 3.10]). Let X be a compact Kähler manifold equipped with a smooth vector bundle V . Then the following statements are equivalent (1) (V, D) is a flat vector bundle over X, i.e. D 2 = 0.

(2) V can be equipped with a Higgs bundle structure (V, ∂, θ) which is semistable. Moreover, these equivalences are compatible with extensions in the following sense:

(i) let {0} = (V 0 , D 0 ) (V 1 , D 1 ) . . . (V m , D m ) = (V, D)
be the filtration of flat vector bundles such that D i := D ↾V i and induced graded terms (V i /V i-1 , ∇ i ) correspond to irreducible representations of the fundamental group π 1 (X). Then each V i is θ and ∂-invariant, and the induced Higgs bundle structures on the graded terms (V i /V i-1 , ∂i , τ i ) are stable with vanishing Chern classes. Moreover, (V i /V i-1 , ∂i , τ i ) is the (unique) Higgs bundle induced by

(V i /V i-1 , ∇ i ) from the Simpson correspondence. (ii) Let {0} = (V 0 , ∂0 , θ 0 ) (V 1 , ∂1 , θ 1 ) . . . (V m , ∂m , θ m ) = (V, ∂, θ)
be the filtration of sub Higgs bundles such that ∂i := ∂↾V i and θ i := θ ↾V i , and each induced graded terms (V i /V i-1 , ∂i , τ i ) is a stable Higgs bundle (the existence of such a filtration is proved by Simpson in [Sim92, Theorem 2] for projective manifolds and by Nie-Zhang [START_REF] Nie | A note on semistable Higgs bundles over compact Kähler manifolds[END_REF] for Kähler manifolds). Then each V i is a D-invariant subbundle and the induced flat bundle

(V i /V i-1 , ∇ i ) corresponds to irreducible representation of π 1 (X). Moreover, (V i /V i-1 , ∇ i ) is the (unique) flat bundle induced by (V i /V i-1 , ∂i , θ i ) from the Simpson correspondence.
In [START_REF]Higgs bundles and local systems[END_REF], Simpson introduced differential graded category [Sim92, §3], plus the formality isomorphism [Sim92, Lemma 2.2] to reduce the proof of Theorem A to his correspondence between polystable Higgs bundles with vanishing Chern classes and semisimple representations of fundamental groups of Kähler manifolds in [START_REF] Simpson | Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF]. While the correspondence for an extension of polystable Higgs bundles (i.e. m = 2 in Theorem A) was written down explicitly in [Sim92, §3, p. 37], the cases of successive extensions follow from the aforementioned differential graded categories.

The purpose of this note is to provide an elementary proof of Theorems A.(i) and A.(ii). Precisely speaking, we will construct the explicit equivalences in Theorem A. When m = 2, Simpson applied the Hodge decompositions for harmonic bundles in [START_REF]Higgs bundles and local systems[END_REF]§2] to build this concrete correspondence. In this note, we applied his ∂ ∂-lemma for harmonic bundles in [Sim92, §2] instead to deal with the general cases m > 2.

Technical Preliminaries

In this section we recall the definition of Higgs bundles, harmonic metrics for flat bundles, and the Simpson correspondence between polystable Higgs bundles and semisimple representations of fundamental groups. We refer the readers to [START_REF] Corlette | Flat G-bundles with canonical metrics[END_REF][START_REF] Simpson | Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF][START_REF]Higgs bundles and local systems[END_REF] for further details.

1.1. Higgs bundles. Definition 1.1. Let X be a n-dimensional Kähler manifold with a fixed Kähler metric ω. A Higgs bundle on X is a triple (V, ∂, θ), where V is a smooth vector bundle, ∂ is a (0, 1)-connection satisfying the integrability condition ∂2 = 0, and θ is a map

θ : V → V ⊗ A 1,0 (X, V ) such that ( ∂ + θ) 2 = 0. (1.1.2)
By the theorem of Koszul-Malgrange, ∂ gives rise to a holomorphic structure on V , and we denote by E the holomorphic vector bundle (V, ∂). Thus (1.1.2) is equivalent to that ∂(θ) = 0, and θ ∧ θ = 0.

Hence we can write abusively (E, θ) for the definition of Higgs bundle, where E is a holomorphic vector bundle, and θ :

E → E ⊗ Ω 1 X with θ ∧ θ = 0.
We say that a Higgs bundle (E, θ) is stable (resp. semistable) if for all θ-invariant torsion-free coherent subsheaves F E, say Higgs subsheaves of (F, θ), we have

µ ω (F ) := c 1 (det F ) • [ω] n-1 rank F < (resp. ) c 1 (det E) • [ω] n-1 rank E =: µ ω (E)
where det F = (∧ rank F F ) ⋆⋆ is the determinant bundle of F , and we say that µ ω (F ) is the slope of F with respect to ω. A Higgs bundle (E, θ) is polystable if it is a direct sum of stable Higgs bundles with the same slope.

For any Higgs bundle (E, θ) over a Kähler manifold X, if h is a metric on E, set D(h) to be its Chern connection with D(h) 0,1 = ∂. Consider furthermore the connection

D h = D(h) + θ + θ * h , where θ ⋆
h is the adjoint of θ with respect to h, and let

F h := D 2 h denote its curvature. Then the metric h is called Hermitian-Yang-Mills if ΛF h = µ ω (E) • 1.
1.2. Higher order Kähler identities for harmonic Bundles. Let (V, D) is a flat bundle equipped with a metric h. Decompose D = d ′ + d ′′ into connections of type (1, 0) and (0, 1) respectively. Let δ ′ and δ ′′ be the unique (1, 0) and (0, 1) connections respectively, such that the connections δ ′ + d ′′ and d ′ + δ ′′ preserve the metric h. Set

θ = d ′ -δ ′ 2 , ∂ = d ′′ + δ ′′ 2 , ∂ = d ′ + δ ′ 2 , (1.2.3) then we can decompose the connection D into D = ∂ + θ + ∂ + θ * h , here θ *
h is the adjoint of θ with respect to h, and it is easy to verify that ∂ + ∂ is also a metric connection. In general, the triple (V, ∂, θ) might not be a Higgs bundle.

However, since the hermitian metric h on V can be thought of as a map

Φ h : X → GL(n, C)/U(n),
by the series of the work of Siu-Sampson-Corlette-Deligne, when Φ h happens to be a harmonic map, (V, ∂, θ) is a Higgs bundle. Such a metric h on V is called harmonic metric, and we say that (V, D, h) is a harmonic bundle. Suppose that (V, D, h) is a harmonic bundle.The harmonic decomposition is defined by

D = D ′ + D ′′ , where D ′′ = ∂ + θ, D ′ = ∂ + θ * h . (1.2.4) Define the Laplacians ∆ = DD * + D * D ∆ ′′ = D ′′ (D ′′ ) * + (D ′′ ) * D ′′
and similarly ∆ ′ . Then by [START_REF]Higgs bundles and local systems[END_REF]§2] we have

∆ = 2∆ ′ = 2∆ ′′ , (1.2.5)
and thus the spaces of harmonic forms with coefficients in V are all the same, which are denoted by H • (V ). By the Hodge theory we have the following orthogonal decompositions of the space of V -valued forms with respect to the L 2 -inner product:

A p (V ) = H p (V ) ⊕ Im(D ′′ ) ⊕ Im (D ′′ ) * (1.2.6) = H p (V ) ⊕ Im(D) ⊕ Im(D * ) (1.2.7) Consequencely one has the following ∂ ∂-lemma for harmonic bundles in [Sim92, Lemma 2.1]. Lemma 1.2 (∂ ∂-lemma). If (V, D, h) is a harmonic bundle, then ker(D ′ ) ∩ ker(D ′′ ) ∩ Im(D ′′ ) + Im(D ′ ) = Im(D ′ D ′′ ). (1.2.8)
We will define the de Rham cohomology H i DR (X, V ) for the flat bundle (V, D). We identify V with the locally constant sheaf of flat sections of V . Consider the sheaves of C ∞ differential forms with coefficients in V :

V → A 0 (V ) D -→ A 1 (V ) D -→ • • • ,
which are fine, and thus the cohomology H i DR (X, V ) is naturally isomorphic to the cohomology of the complex of global sections

A • (V ), D = A 0 (V ) D -→ A 1 (V ) D -→ • • •
Let us finish this subsection by recalling the following Corlette-Simpson correspondence.

Theorem 1.3. Let (X, ω) be a compact Kähler manifold of dimension n.

(i) [START_REF] Corlette | Flat G-bundles with canonical metrics[END_REF][START_REF] Donaldson | Twisted harmonic maps and the self-duality equations[END_REF] A flat bundle V has a harmonic metric if and only if it arises from a semisimple representation of π 1 (X). (ii) [START_REF] Simpson | Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF] A Higgs bundle (E, θ) admits an Hermitian-Yang-Mills metric if and only if it is polystable. Such a metric is harmonic if and only if ch

1 (E) • {ω} n-1 = ch 2 (E) • {ω} n-2 = 0.
1.3. Numerically Flat Vector Bundle. Let E be a holomorphic vector bundle of rank r over a compact complex manifold X. We denote by P(E) the projectivized bundle of hyperplanes of E and by O P(E) (1) the tautological line bundle over P(E).

Recall the following definition in [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF].

Definition 1.4. Let X be a compact Kähler manifold.

(i) We say that a line bundle L is nef, if for any ε > 0, there exists a smooth hermitian metric h ε on L such that iΘ hε (L) -εω, where ω is a fixed Kähler metric on X. (ii) A holomorphic vector bundle E is said to be nef if O P(E) (1) is nef over P(E). (iii) We say that a holomorphic vector bundle E is numerically flat if both E and its dual E ⋆ is nef.

Proof of Theorem A

Proof Theorem A. (i) Let ρ : π 1 (X) → GL(n, C) be the representation of the fundamental group corresponding to the flat vector bundle (V, D). After taking some conjugation, one can put the representation in block upper triangular form

    ρ 1 * . . . * 0 ρ 2 . . . * . . . . . . . . . . . . 0 0 . . . ρ m    
such that for every i = 1, . . . , r, ρ i : π 1 (X) → GL(r i , C) is an irreducible representations. Thus there is a filtration of flat vector bundles

{0} = V 0 V 1 . . . V m = V such that V i corresponds to     ρ 1 * . . . * 0 ρ 2 . . . * . . . . . . . . . . . . 0 0 . . . ρ i    
In particular,

(i) each V i is invariant under the flat connection D, that is, D(V i ) ⊂ V i ⊗ A 1 (X).
(ii) The quotient connection D i on Q i := V i /V i-1 induced by D is also flat, which corresponds to the irreducible representation ρ i : π 1 (X) → GL(r i , C).

By Theorem 1.3, we can find a (unique) harmonic metric h i such that (Q i , D i , h i ) is an harmonic bundle. By (1.2.4) for each i = 1, . . . , m, there is a unique harmonic decomposition

D ′′ i = ∂i + θ i , D ′ i = ∂ i + θ * i , (2.9)
where θ * i is the adjoint of θ i with respect to h i . Moreover, Q i can be equipped with a Higgs bundle structure (Q i , ∂i , θ i ).

For simplicity we first consider the case that V is an extension of an irreducible representation by another one, that is, m = 2 and we have an exact sequence of flat vector bundles over X:

0 → Q 1 → V → Q 2 → 0,
and thus there is

η ∈ A 1 X, hom(Q 2 , Q 1 ) such that D is given by D = D 1 η 0 D 2 .
We denote by D 2,1 the induced flat connection on the bundle hom(Q 2 , Q 1 ) by D 1 and D 2 . By D 2 = 0, one has D 2,1 (η) = 0, and thus {η} ∈ H 1 DR X, hom(Q 2 , Q 1 ) . Claim 2.1. The cohomology class {η} ∈ H 1 DR X, hom(Q 2 , Q 1 ) characterizes the isomorphism class of (V, D) among all extensions of Q 1 by Q 2 .

Proof of Claim 2.1.For any η

′ ∈ A 1 X, hom(Q 2 , Q 1 ) such that η ′ ∈ {η}. Then η ′ = η + D 2,1 (a) for some a ∈ A 0 X, hom(Q 2 , Q 1 ) . We define a gauge transformation g ∈ Aut ∞ (V ) by (2.10) g = 1 -a 0 1 , Then g • D • g -1 = D 1 η ′ 0 D 2 =: D. (2.11)
Hence (V, D) and (V, D) are isomorphic flat bundles.

Since both

(Q 1 , D 1 , h 1 ) and (Q 2 , D 2 , h 2 ) are both harmonic bundles, so is hom(Q 2 , Q 1 ), D 2,1 , h 1 h * 2 . Set D ′
2,1 and D ′′ 2,1 to be the harmonic decomposition of D 2,1 as (1.2.4), and let ∆ 2,1 and ∆ ′′ 2,1 be the Laplacians of D 2,1 and D ′′ 2,1 respectively. By Claim 2.1 one can assume that η is the (unique) harmonic representation in its extension class {η} ∈

H 1 DR X, hom(Q 2 , Q 1 ) . Then ∆ ′′ 2,1 (η) = 1 2 ∆ 2,1 (η) = 0.
In particular

D ′′ 2,1 (η) = 0. (2.12)
Let η ′ and η ′′ to be the (1, 0) and (0, 1)-parts of η respectively. Set

∂ := ∂1 η ′′ 0 ∂2 and θ := θ 1 η ′ 0 θ 2 .
Then (2.12) is equivalent to ( ∂ + θ) 2 = 0, and by Definition 1.1 (V, ∂, θ) is a Higgs bundle over X. Moreover, it is compatible with the Higgs bundle structures (Q i , ∂i , θ i ).

We prove the theorem when m = 2.

For general m 2, we will prove the theorem by inductions. Set

∇ j := D 1 ⊕ • • •⊕ D j to be the flat connection on Q 1 ⊕ • • • ⊕ Q j , and
∇ ′ j := D ′ 1 ⊕ D ′ 2 ⊕ • • • ⊕ D ′ j , ∇ ′′ j := D ′′ 1 ⊕ D ′′ 2 ⊕ • • • ⊕ D ′′ j . Then ∇ i = ∇ ′ i + ∇ ′′ i is the harmonic decomposition defined in (2.9). Assume that (2.13) ∇j =    D 1 B j . . . 0 D j    , is the flat connection on Q 1 ⊕• • •⊕Q j defining V j . Here B j ∈ A 1 X, End(Q 1 ⊕. . .⊕Q j ) which is strictly upper-triangle such that ∇ j (B j ) + B j ∧ B j = 0 (2.14) by ∇2 j = 0.
Here we write abusively ∇ j the induced flat connection of End(Q

1 ⊕• • •⊕Q j ) by (Q 1 ⊕ • • • ⊕ Q j , ∇ j ) Claim 2.2. Assume that we can find B j-1 ∈ A 1 X, End(Q 1 ⊕ . . . ⊕ Q j-1 ) which is strictly upper-triangle such that (i) for ∇j-1 defined in (2.13), the pair (Q 1 ⊕ . . . ⊕ Q j-1 , ∇j-1 ) defines V j-1 . (ii) ∇ ′′ j-1 (B j-1 ) + B j-1 ∧ B j-1 = 0, or equivalently ∇ ′ j-1 (B j-1 ) = 0. Then so is true for j. Proof of Claim 2.2.Since V j is an extension of V j-1 by Q j 0 → V j-1 → V j → Q j → 0, we denote by β ∈ H 1 DR X, hom(Q j , V j-1 ) the extension class. Choose any representa- tive A ∈ β, then (Q 1 ⊕ . . . ⊕ Q j , ∇j ) defining V j can be written as (2.15) ∇j =      D 1 B j-1 a 1 . . . . . . 0 D j-1 a j-1 0 . . . 0 D j      where A = a 1 ⊕ • • • ⊕ a j-1 with a i ∈ A 1 X, hom(Q j , Q i ) . Then by ∇2 j = 0, one has (2.16) ∇j-1 • A + A • D j = 0.
In particular, D j,j-1 (a j-1 ) = 0, where D j,i the connection on hom(Q j , Q i ) induced by D j and D i . Since hom(Q j , Q i ), D j,i , h i h * j is also a harmonic bundle, we set D ′ j,i and D ′′ j,i to be the harmonic decomposition of D j,i as (1.2.4). By (1.2.7) there exists

c j-1 ∈ A 0 X, hom(Q j , Q j-1 ) ⊂ A 0 X, hom(Q j , V j-1 )
such that ∆ j,j-1 (a j-1 + D j,j-1 c j-1 ) = 0, (2.17) where ∆ j,i (resp. ∆ ′ j,i ) is the Laplacian of D j,i (resp. D ′ j,i ). Denote by ∇j,j-1 the induced flat connection on hom(Q j , V j-1 ) by the connections D j and ∇j-1 , then

A 1 := A + ∇j,j-1 (c j-1 ) = a 1 ⊕ • • • ⊕ a j-1 + ( ∇j-1 • c j-1 -c j-1 • D j )
belongs to the same extension class as A 1 . If we write

A 1 = a ′ 1 ⊕ • • • ⊕ a ′ j-1 with a ′ i ∈ A 1 X, hom(Q j , Q i ) , then a ′ j-1 = a j-1 +D j,j-1 (c j-1 )
. By (2.17), one has ∆ ′ j,j-1 (a ′ j-1 ) = 1 2 ∆ j,j-1 (a ′ j-1 ) = 0, and thus D ′ j,j-1 (a ′ j-1 ) = 0. This gives us hints that we can use some ad hoc methods to find the proper A.

Assume now for some

A = a 1 ⊕ • • • ⊕ a j-1 ∈ β such that D ′ j,i (a i ) = 0 for all i = k + 1, . . . , j -1. By (2.16) we have D j,k (a k ) + j-1 i=k+1 b ki a i = 0, here b ki is the projection of B j-1 ∈ A 1 X, End(Q 1 ⊕ . . . ⊕ Q j-1 ) to the component A 1 X, hom(Q i , Q k ) . By the assumption that ∇ ′ j-1 (B j-1 ) = 0, we have D ′ i,k (b ki ) = 0. Hence 0 = D ′ j,k D j,k (a k ) + D ′ j,k ( j-1 i=k+1 b ki a i ) = D ′ j,k D j,k (a k ) + j-1 i=k+1 D ′ i,k (b ki )a i -b ki D ′ j,i (a i ) = D ′ j,k D ′′ j,k (a k ) = -D ′′ j,k D ′ j,k (a k ) Applying Lemma 1.2 to D ′ j,k (a k ), there exists c k ∈ A 0 X, hom(Q j , Q k ) such that D ′ j,k (a k ) = -D ′ j,k D ′′ j,k (c k ) = -D ′ j,k D j,k (c k ). (2.18) Set à : = A + ∇j,j-1 (c k ) = A + ( ∇j-1 • c k -c k • D j ) = a ′ 1 ⊕ . . . ⊕ a ′ k-1 ⊕ a k + D j,k (c k ) ⊕ a k+1 ⊕ . . . ⊕ a j-1
. which belongs to the extension class as A. In other words, the components of à in A 1 X, hom(Q j , Q i ) for i = k + 1, . . . , j -1 are the same as those of A, and the component of

à in A 1 X, hom(Q j , Q k ) are replaced by a k + D j,k (c k ), such that D ′ j,k a k + D j,k (c k ) = 0 by (2.18
). Thus by the induction on k we can choose A ∈ β properly such that D ′ j,k (a k ) = 0 for all k = 1, . . . , j-1. This is equivalent to ∇ ′ j (B j ) = 0. The claim is thus proved.

By Claim 2.2 we conclude that there exists

η ∈ A 1 X, End(Q 1 ⊕ . . . ⊕ Q m ) which is strictly upper-triangle, such that (Q 1 ⊕ . . . ⊕ Q m , ∇ m + η) defines the flat bundle V , and ∇ ′ m (η) = 0 ⇔ D ′ j,k (η kj ) = 0 ∀ 1 k < j m. (2.19)
Here we denote by η kj the component of

η in A 1 X, hom(Q j , Q k ) . Hence ∇ ′′ m (η) + η ∧ η = 0. (2.20) Set ∂ :=    ∂1 η ′′ . . . 0 ∂m    and θ =    θ 1 η ′ . . . 0 θ m    .
Here η ′ and η ′′ are the (1, 0) and (0, 1)-parts of η respectively, and D ′′ i = ∂i + θ i is defined as (2.9). Then (2.20) is equivalent to ( ∂ + θ) 2 = 0, and thus that (V, ∂, θ) is a Higgs bundle over X. In this setting, for each 1 i m, (V i , ∂↾V i , θ ↾V i ) is a Higgs subbundle of E, and the induced Higgs bundle structure on the graded term

Q i := V i /V i-1 coincides with (Q i , ∂i , θ i ).
(ii) The proof of Theorem A.(ii) proceeds along the same lines as Theorem A.(i). We will only sketch a proof for Theorem 0.1.(ii), i.e. the case that θ = 0. Let us start with a holomorphic vector bundle (V, ∂). Suppose that it admits a filtration

{0} = (V 0 , ∂0 ) (V 1 , ∂1 ) • • • (V m , ∂m ) = (V, ∂)
of holomorphic vector bundles such that for each i = 1, . . . , m, the graded term

Q i := (V i /V i-1 , D ′′ i ) is hermitian flat, i.e.
it can be equipped with a hermitian metric h i so that the Chern connection D i is flat. Set D ′ i := D i -D ′′ i , which is a (1, 0)-connection. Set ∇ j := D 1 ⊕ • • • ⊕ D j to be the hermitian flat connection on Q 1 ⊕ • • • ⊕ Q j , and

∇ ′ j := D ′ 1 ⊕ • • • ⊕ D ′ j , ∇ ′′ j := D ′′ 1 ⊕ • • • ⊕ D ′′ j .
Then (∇ ′ m ) 2 = (∇ ′′ m ) 2 = 0. A similar proof as Claim 2.2 shows the following result. Claim 2.3. There exists η ∈ A 0,1 X, End(Q 1 ⊕ . . . ⊕ Q m ) such that the complex structure of (V, ∂) is given by m is the (1, 0) component of D, the underlying holomorphic structure of (V, D) coincides with (V, ∂). It follows from our construction that, (V i , ∂i ) is a D-invariant subbundle for each i = 1, . . . , m, with D i the induced flat bundle structure on the graded terms.

∂ =    D ′′ 1 η . . .
Remark 2.4. (i) Note that by the proof of Theorem A.(i), the category of successive of unitary extensions of fundamental groups is larger than that of semistable vector bundles with vanishing Chern classes. In fact, even if all the graded terms are unitary representations of π 1 , the corresponding semistable Higgs bundles might have non-vanishing (nilpotent) Higgs fields.

(ii) The condition of numerical flatness in Theorem 0.1.(ii) is necessary. Indeed, in [START_REF] Biswas | Non-flat extension of flat vector bundles[END_REF] Biswas-Heu constructed an example of an extension of flat vector bundles, which does not admit any holomorphic connection.

  Let us denoteD := ∂ + ∇ ′ m , which is a flat connection for D 2 = ( ∂ + ∇ ′ m ) 2 = (∇ ′ m + ∇ ′′ m + η) 2 = ∇ ′ m (η) + ∂2 = 0. Namely, (V, D) is a flat bundle. Since ∇ ′

Acknowledgements. The first version of this note was written in my Ph.D thesis under the supervision of Professor Jean-Pierre Demailly, to whom I would like express my warmest gratitude. I thank Professors Junyan Cao and Xiaokui Yang for their interests on this work.


Reference Method and Reference Material

  Are Necessary Tools to Reveal the Variability of Cystatin C Assays Anne-Sophie Bargnoux, Nils Kuster, Vincent Delatour, Pierre Delanaye, Ana González-Antuña, Jean-Paul Cristol, Laurence Piéroni, Etienne Cavalier To cite this version: Anne-Sophie Bargnoux, Nils Kuster, Vincent Delatour, Pierre Delanaye, Ana González-Antuña, et al.. Reference Method and Reference Material Are Necessary Tools to Reveal the Variability of Cystatin C Assays. Archives of Pathology and Laboratory Medicine, 2016, 140 (2), pp.117 -118. 10.5858/arpa.2015-0198-LE. hal-01803383