Laurence Kessler
email: laurence.kessler@chru-strasbourg.fr

M Mansilla

Keywords:

Regards croisés sur la transition

Épidémiologie et particularité du diabète à l'adolescence confondus, s'est dégradé ces dernières années, avec une moyenne d'HbA 1c de 8,4 % en 2016-2018 contre 7,8 % en 2010-2012, principalement en raison d'une augmentation des taux d'HbA 1c à plus de 9 % chez les adolescents [START_REF] Foster | State of type 1 diabetes management and outcomes from the T1D Exchange in 2016-2018[END_REF]. L'effet cumulé des années passées en hyperglycémie explique la réduction d'espérance de vie des diabétiques de type 1 de 16 ans quand le diabète survient avant 10 ans, et de 9,4 ans lorsque celui-ci survient après l'âge de 25 ans [START_REF] Rawshani | Excess mortality and cardiovascular disease in young adults with type 1 diabetes in relation to age at onset: a nationwide, register-based cohort study[END_REF]. La prise en charge de l'adolescent diabétique représente donc une situation particulièrement difficile à la fois au plan métabolique, psychologique, social et institutionnel.

Modifications hormonales au cours de l'adolescence et impact sur le diabète

• Le passage de l'enfant vers l'âge adulte est marqué par deux processus physiologiques qui vont interférer avec le contrôle des glycémies. La puberté est la période du développement sexuel, elle a lieu vers l'âge de 11 ans chez les filles et, en moyenne, vers l'âge de 12 ans chez les garçons. Elle implique des changements physiques, la survenue des menstruations chez la fille. Le deuxième processus qui va fortement impacter le contrôle glycémique est la croissance staturo-pondérale qui va s'accélérer pendant cette période.

Épidémiologie du diabète de type 1 chez l'enfant et l'adolescent

• Le diabète de type 1 (DT1) touche environ 500 000 enfants dans le monde avec des incidences variables selon les pays [START_REF]Fédération Internationale du Diabète. Atlas du diabète de la FID[END_REF]. On estimait en 2005 que 15 000 enfants/an déclaraient un diabète en Europe, et que ceci touchait les enfants plutôt plus âgés (24 % de 0-4 ans, 35 % de 5-9 ans, et 41 % de 10-14 ans) [START_REF] Patterson | Incidence trends for childhood type 1 diabetes in Europe during 1989-2003 and predicted new cases 2005-20: a multicentre prospective registration study[END_REF]. En France, il n'existe pas de registre national, mais des données régionales. La prévalence du diabète chez les enfants de moins de 15 ans était estimée en 2013 à 1,24-1,55/1 000, soit entre 21 000 et 26 000 enfants et adolescents de moins de 15 ans en 2017 [START_REF] Piffaretti | Trends in childhood type 1 diabetes incidence in France, 2010-2015[END_REF].

• Outre ces chiffres, c'est la dynamique qui est frappante. Les différents registres européens rapportent une augmentation annuelle du diabète chez l'enfant de +3,9 % par an [intervalle de confiance à 95 %, IC 95 % : 3,6-4,2] [3]. La France suit la même croissance, avec une augmentation annuelle de plus de 4 % par an au cours des 15 dernières années. L'incidence des nouveaux cas de diabète chez les jeunes de moins de 20 ans était de 7,4/100 000 enfants/an en 1988 [START_REF] Charkaluk | Incidence data of childhood-onset type I diabetes in France during 1988-1997: the case for a shift toward younger age at onset[END_REF] contre actuellement 18/100 000 enfants/an en 2013-2015, ce qui correspond à plus de 3 000 enfants et jeunes nouvellement diabétiques tous les ans [START_REF] Foster | State of type 1 diabetes management and outcomes from the T1D Exchange in 2016-2018[END_REF]. Enfin, il est à présent bien connu que cet accroissement touche de plus en plus les plus jeunes enfants : augmentation annuelle de 5,4 % [IC 95 % : 4,8-6,1] pour les 0-4 ans, 4,3 % [IC 95 % : 3,8-4,8] pour les 5-9 ans, et 2,9 % [IC 95 % : 2,5-3,3] pour les 10-14 ans [START_REF] Patterson | Incidence trends for childhood type 1 diabetes in Europe during 1989-2003 and predicted new cases 2005-20: a multicentre prospective registration study[END_REF]. les projections pour 2020 estiment qu'il y aura 24 400 nouveaux cas de diabète par an chez les enfants, contre 15 000 nouveaux cas en 2005, avec un doublement de nouveaux cas chez les moins de 5 ans qui représenteront alors 29 % des enfants nouvellement diabétiques, les 5-9 ans et 10-14 ans représentant, respectivement, 37 % et 34 % des enfants présentant une découverte de diabète [START_REF] Patterson | Incidence trends for childhood type 1 diabetes in Europe during 1989-2003 and predicted new cases 2005-20: a multicentre prospective registration study[END_REF][START_REF] Charkaluk | Incidence data of childhood-onset type I diabetes in France during 1988-1997: the case for a shift toward younger age at onset[END_REF] [START_REF] Foster | State of type 1 diabetes management and outcomes from the T1D Exchange in 2016-2018[END_REF]. Ces jeunes auront à vivre avec la maladie de plus en plus longtemps lorsque leur diabète se sera déclaré tôt, avec un parcours plus ou moins facile et des périodes d'adhérence aux soins et aux conseils, personnelle et/ou de l'entourage variables, agrémentées d'épisodes d'équilibre glycémique fréquemment fluctuant.

• La période de l'adolescence est fréquemment marquée par un déséquilibre glycémique et des moyennes d'hémoglobine glyquée (HbA 1c ) passant de 8,1 % à l'âge de 5 ans à 9,3 % à l'âge de 15-18 ans, et redescendant à 8 % à l'âge de 28 ans. À noter que cet équilibre glycémique, tous patients • L'augmentation de la prévalence du diabète de type 1 à l'adolescence est associée à une découverte du diabète de plus en plus tôt chez les enfants de moins de 10 ans.

• L'espérance de vie des diabétiques de type 1 est réduite de 16 ans quand le diabète survient avant l'âge de 10 ans.

• La prise en charge de l'adolescent diabétique représente un nouveau défi en diabétologie afin de limiter les effets néfastes du cumul des années en hyperglycémie.

• Les différentes composantes de l'adolescence : changements métaboliques, psychologiques et de modes de vie, vont interférer avec la gestion du diabète.

• L'adolescent diabétique va devoir être accompagné spécifiquement pendant cette période de transition, et ce pendant plusieurs années.

• La transition de l'adolescent des équipes pédiatriques aux équipes adultes nécessite une collaboration étroite et une bonne connaissance des pratiques de chacun des professionnels impliqués.

• La maison des adolescents, lieu neutre sans affichage médical, peut constituer un soutien pour ces jeunes diabétiques.

Les points essentiels

Dossier thématique

Regards croisés sur la transition de l'adolescent en diabétologie

• Les résultats plus médiocres chez l'adolescent en termes d'équilibre glycémique comparativement aux autres classes d'âges peuvent s'expliquer également par les variations endocriniennes physiologiques qui ont des effets sur le métabolisme du glucose. Ainsi, durant la puberté, on observe une diminution de la sensibilité à l'insuline qui entraîne une difficulté à équilibrer le diabète, suivie d'une amélioration à l'âge adulte. Ces modifications sont étroitement liées au facteur de croissance insulin-like growth factor-1 (IGF1), dont l'augmentation durant la puberté s'accompagne d'une réduction de l'insulino-sensibilité [START_REF] Benyi | The physiology of childhood growth: hormonal regulation[END_REF]. Il en résulte à l'adolescence une augmentation de l'HbA 1c moyenne par rapport à la période pré-pubertaire.

• Par ailleurs, l'adolescence est la période de la croissance osseuse accélérée en lien avec une modification de la pulsatilité de la sécrétion d'hormone de croissance (hGH), faite de pics plus fréquents et plus importants, notamment durant la phase nocturne, expliquant la fréquente hyperglycémie matinale observée chez l'adolescent diabétique [START_REF] Benyi | The physiology of childhood growth: hormonal regulation[END_REF]. Chez la fille, l'apparition de cycles ovulatoires et des menstruations va être responsable de variations de la glycémie, du fait de la variation cyclique des oestrogènes. Ce phénomène est illustré par les variations glycémiques et des besoins en insuline observées pendant les périodes de menstruation chez la jeune fille diabétique. L'adolescent est donc confronté à une transition métabolique source de déséquilibre glycémique. Il s'agit d'un processus dynamique qui va durer jusqu'à l'âge adulte, période durant laquelle l'adaptation du traitement insulinique sera souvent difficile. • À côté de ces grands changements hormonaux, l'adolescent diabétique entre aussi dans une période où le contrôle glycémique est rendu difficile de par l'effondrement progressif de la sécrétion insulinique résiduelle. En effet, lors de la découverte du DT1 durant l'enfance, la sécrétion insulinique endogène est attestée par un C-peptide sanguin résiduel, certes très abaissé -mais non nul -qui peut contribuer à un meilleur équilibre glycémique. Progressivement, à l'adolescence, puis chez le jeune sujet diabétique, le C-peptide « encore » positif devient C-peptide négatif, ce qui rend le sujet totalement dépendant de l'insuline exogène [START_REF] Vanbuecken | Residual C-peptide in type 1 diabetes: what do we really know?[END_REF].

• Chez la jeune fille diabétique, l'adolescence est également la période de la contraception, source de déséquilibre glycémique lors du recours à une pilule oestro-progestative minidosée, quoique mieux tolérée en raison de l'absence de spotting et du maintien des règles [START_REF]Gender differences in glucose homeostasis and diabetes[END_REF]. L'ensemble de ces modifications hormonales auront pour conséquence thérapeutique une multiplication des injections d'insuline ou le recours à la pompe à insuline, l'intensification de l'autosurveillance glycémique, et le contrôle des apports alimentaires et de l'activité physique. L'ensemble de ces éléments va représenter une contrainte maximale du traitement que l'adolescent diabétique aura des difficultés à accepter.

Le diabète chez l'adolescent : l'exacerbation des problématiques de l'adolescence

L'apprentissage de l'autonomie chez l'adolescent diabétique est particulièrement complexe. • L'adolescent se construit en opposition, essentiellement à sa famille, et le diabète va être embarqué dans cette opposition. Progressivement, la gestion de la maladie au quotidien, assurée par ses parents, va être transférée chez l'adolescent avec un partage de la responsabilité de la gestion de la maladie, source de conflits et de déséquilibre glycémique. Un relâchement trop rapide de l'autorité parentale dans le but d'acquérir une indépendance plus importante peut se manifester par une altération de l'observance thérapeutique. Au contraire, une supervision parentale du diabète trop présente en ce qui concerne la gestion de l'insulinothérapie, l'alimentation, les activités physiques, peut s'opposer à l'autonomisation de l'adolescent [START_REF] Aboab | Traverser l'adolescence avec un diabète[END_REF].

• La période de l'adolescence est une période d'expérimentation qui engendre parfois des conduites à risque. La manipulation des traitements insuliniques avec l'induction des hypoglycémies ou des épisodes de cétose peut être dangereuse. Les consommations aiguës d'alcool peuvent entraîner des hypoglycémies sévères. Les passages à l'acte facilités pendant l'adolescence peuvent conduire à des arrêts intempestifs des injections d'insuline ou à des injections factices amenant à des conduites parfois très suicidaires [START_REF] Moreau | Factitious self-manipulation of the external insulin pump in adolescents with Type 1 diabetes[END_REF][START_REF] Jaser | Psychological problems in adolescents with diabetes[END_REF]. • L'adolescent diabétique présente également une exacerbation des préoccupations de tout adolescent face à son avenir, à ses risques de transmission de la maladie. L'adolescent diabétique est préoccupé par le risque de transmission du diabète à sa descendance. Paradoxalement, l'adolescent diabétique vit dans le présent et n'arrive pas à se projeter, notamment en ce qui concerne les complications dégénératives de la maladie bien qu'il en ait une pleine connaissance. • L'adolescence est également la période des paradoxes avec chez l'adolescent la revendication d'une identité, mais également la difficulté d'assumer ses différences par rapport aux autres et ses modifications corporelles. Le diabète représente une différence fondamentale avec la difficulté chez certains adolescents de porter sur le corps des dispositifs externes (capteur de glucose, pompe à insuline) ramenant le jeune diabétique à la différence par rapport aux autres. La problématique de l'identité corporelle chez l'adolescent diabétique est particulièrement importante à un moment où le corps est en pleine transformation [START_REF] Deeb | Common issues seen in paediatric diabetes clinics, psychological formulations, and related approaches to management[END_REF].

• Enfin, le diabète peut modifier et altérer le cours de l'adolescence. Le diabète est en effet un facteur de risque de troubles psychiatriques [START_REF] Buchberger | Symptoms of depression and anxiety in youth with type 1 diabetes: A systematic review and meta-analysis[END_REF]. Il s'agit le plus souvent de dépression et d'anxiété généralisée. Les troubles du comportement alimentaire sont particulièrement fréquents chez la jeune fille, l'amenant à un sous-dosage en insuline pour des raisons pondérales. Il est important d'identifier ces états dépressifs qui peuvent amener à des conduites suicidaires. En effet, les troubles du comportement alimentaire semblent plus fréquents dans la

Difficultés de gestion du diabète à l'adolescence en lien avec les évolutions sociétales

• Pendant la période de l'adolescence, le/la jeune diabétique va devoir s'adapter en tant qu'adolescent aux changements de ce mode de vie, mais également intégrer la gestion de son diabète. Cette période sera très souvent la période du départ du milieu familial avec la vie en foyer ou en colocation. L'horaire des repas, de même que la composition, deviennent très variables. Ce changement de mode de vie est indissociable de l'autonomie que doit acquérir l'adolescent diabétique.

• La période de l'adolescence est également la période de l'orientation professionnelle. L'adolescent diabétique pourra alors être confronté à l'impossibilité d'accéder à certaines professions (pompier, maître-nageur,…), ce qui va engendrer chez le jeune un grand sentiment d'injustice, parfois une révolte. La réalisation du permis de conduire nécessitera une déclaration en préfecture de la maladie. Tous ces éléments sont vécus par l'adolescent diabétique comme un obstacle à son intégration dans la société [START_REF]Permis de conduire et diabète[END_REF].

Défaut de structure dédiée à la prise en charge de l'adolescent diabétique

Pour répondre à toutes les problématiques de la prise en charge de l'adolescent diabétique, il n'existe pas actuellement de structure, ni de personnel, dédiés à leur prise en charge dans des services accueillant les adolescents diabétiques de type 1 [START_REF] Spizzo-Guellati | Représentations de l'adolescent diabétique par le personnel soignant en service de diabétologie adulte et pédiatrique[END_REF]. D'un côté, les services de pédiatrie qui prennent en charge les enfants depuis plusieurs années, qui ont accompagné l'adolescent et ses parents à la découverte de la maladie, ont des difficultés à considérer l'adolescent diabétique comme un futur adulte. De l'autre, les services de diabétologie adulte orientés vers la prévention et la prise en charge des complications dégénératives peuvent renvoyer à l'adolescent diabétique une image très négative de l'évolution de la maladie qui, pour certains jeunes adolescents non encore adultes, peut être extrêmement traumatisante. Devant ce défaut de structure adaptée, les recommandations de prise en charge de l'adolescent diabétique préconisent une transition pour permettre un travail collaboratif entre les équipes de diabétologie pédiatrique et de diabétologie adulte pour permettre à l'adolescent diabétique d'être accompagné pendant cette période et éviter les ruptures de prise en charge [17,18]. Depuis une dizaine d'années, des structures se sont développées pour répondre à la prise en charge de l'adolescent. C'est ainsi qu'ont été mises en place les maisons des adolescents qui sont amenées à recevoir des adolescents présentant ou non des maladies chroniques. La maison des adolescents s'adressant au jeune, considéré comme un adolescent avant d'être un diabétique, va permettre dans certaines situations une aide à la prise en charge de ces patients [19].

Déclaration d'intérêt

Les auteures déclarent n'avoir aucun conflit d'intérêt en lien avec cet article.


Antoine Mandel
email: antoine.mandel@univ-paris1.fr

Xavier Venel
email: xavier.venel@univ-paris1.fr

Dynamic competition over social networks

Keywords: Game theory, Social Network, Dynamic games, Targeting, Stochastic games

Introduction

" Delivering the right message to the right person at the right time" is a common dictum among influencers in politics, lobbying, and marketing. The large increase in information about individual characteristics and social interactions brought about by the development of internet and online social networks has generated tremendous interest, among both practitioners and scientists, on the problem of identifying appropriate targets to maximize influence. In marketing, targeted advertising on social networks has become a cornerstone of the industry in less than a decade. In politics, Barack Obama's campaign for the U.S. presidential election in 2008 has illustrated the tremendous potential for influence carried by social networks [START_REF] Cogburn | From networked nominee to networked nation: Examining the impact of web 2.0 and social media on political participation and civic engagement in the 2008 obama campaign[END_REF]. In the academic literature, identifying the key target in a network in order to gain control, influence, or market shares has become a central focus of computer science, economics, and operations research. Most of the existing literature examines optimal/efficient strategies for a single agent [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF][START_REF] Ballester | Who's who in networks. Wanted: The key player[END_REF]. More recently, the competitive nature of the targeting problem has been emphasized and game-theoretic contributions have started to analyze the behavior of players competing for prominence over a network [START_REF] Goyal | Competitive contagion in networks[END_REF][START_REF] Bimpikis | Competitive targeted advertising over networks[END_REF][START_REF] Grabisch | Strategic influence in social networks[END_REF].

We note, however, that a common feature of most of the existing approaches is their focus on identification of the key target, i.e., "the right person". The temporal dimension, "the right time", hasn't yet been approached in a competitive setting. Its role is nevertheless crucial in applications. In political campaigning, as emphasized by [START_REF] Granato | Political campaign advertising dynamics[END_REF], "the relation between voters and campaign strategists is dynamic and evolves until voters' views on a candidate crystallize." In marketing, an important element of a firm's strategy is the sequence in which a product is offered to potential buyers [START_REF] Hartline | Optimal marketing strategies over social networks[END_REF].

The main objective of this paper is to include this temporal dimension in the analysis of the competitive targeting problem on networks. Therefore, we place ourselves in a competitive setting where two players choose a sequence of targets in a social network in order to influence the average opinion that gets formed about a given topic, e.g., a political candidate, a product, or the truth value of a piece of information. The network is made up of non-strategic agents whose opinion is represented by a real number and who update their opinions using a weighted average of their neighbors' opinions (as in [START_REF] Golub | Naive learning in social networks and the wisdom of crowds[END_REF]. Strategic players choose a sequence of targets among non-strategic agents. Hence, they directly influence the opinion of the targeted agent and indirectly influence the whole network through the dynamics of opinion formation.

In this setting, we are concerned with two main issues: the existence of a uniform value in the underlying (infinite-horizon) game and the characterization of optimal strategies, i.e., the determination of an optimal sequence of targets. A priori, both issues can be sensitive, on the one hand to the influence potential of the two strategic players, which measures the extent to which strategic players are able to change the opinion of the targeted agent and which may differ between players, and on the other hand to the informational structure of the game, that is, what each player observes of his opponent's actions.

The existence of a uniform value is non-trivial in dynamic/stochastic games with compact state spaces. However, in the general case, it may not exist, as shown by [START_REF] Ziliotto | Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture max min = lim v n[END_REF]. The first part of this paper is devoted to this issue. We highlight the relation between the linear updating of the vector of opinions, using the network's matrix of weights that we consider here, and the conventional stochastic game obtained when weights are interpreted as transition probabilities. We show that if the matrix is primitive, strategic influence "contracts" the vector of opinions in the network. This suffices to show that there exists a uniform value and that this value is independent of the initial distribution of opinions. Furthermore, this result holds independently of the informational structure of the game.

As for the characterization of equilibrium strategies, i.e., the determination of an optimal sequence of targets, a key observation is that players must implement a trade-off between a forward-looking perspective, according to which they aim to maximize the future spread of their opinion in the network, and a backward-looking perspective, according to which they aim to counteract their opponent's previous actions. It is likely impossible to provide a closed-form analytical characterization of dynamic strategies implementing this trade-off in the general case. Nevertheless, we provide several series of results that highlight the relationship between network structure, the strength of the influence exerted by strategic players, and the type of strategies that ought to be used at equilibrium. We first show that, in highly symmetric networks, players must adopt a purely backward-looking perspective. They should focus on counteracting the influence of their opponent so as to keep the distribution of opinions in an acceptable state. Second, we show that in very hierarchical, hence asymmetric, networks, players must adopt a purely forward-looking perspective. Independently of the current distribution of opinions, they should target (sequences of) influential nodes in order to foster the forward diffusion of their opinion in the network. Finally, we show that, if the level of influence is small, the game can be approximated by a static game similar to the one considered in [START_REF] Grabisch | Strategic influence in social networks[END_REF]. Note in particular that, if the players have equal influence and the non-strategic agents are equally influenced, they should target the node with the largest eigenvector centrality (a fact that went unnoticed in [START_REF] Grabisch | Strategic influence in social networks[END_REF]). Our results also highlight the importance of the temporal structure of strategies. For certain networks, there exist pure equilibria with dynamic targeting strategies, whereas no pure equilibria exist when a fixed target must be chosen (as in [START_REF] Grabisch | Strategic influence in social networks[END_REF]. This stems from the fact that dynamic strategies are less easily counteracted than static ones.

The remainder of the paper is organized as follows. Section 2 briefly reviews the related literature. Section 3 presents a game-theoretic model of dynamic competitive targeting and proves the existence of the uniform value. Section 4 provides a partial characterization of equilibrium strategies. Section 5 compares dynamic and static strategies. Section 6 concludes. All proofs are given in the appendix.

Related literature

Since the seminal work in quantitative sociology of the 1970s (notably [START_REF] Bonacich | Factoring and weighting approaches to status scores and clique identification[END_REF][START_REF] Freeman | A set of measures of centrality based on betweenness[END_REF], the identification of central agents has been a key research question in the analysis of socio-economic networks. With the rise of digital social networks and the correlative flow of data, many applications have been identified in communication and marketing, and the problem has been approached from a more goal-oriented perspective at the interface between economics and computer science. The targeting problem, i.e., the choice of an optimal target in view of the diffusion of a product or a message, has been widely investigated from an algorithmic perspective in computer science, following on from the early contributions of [START_REF] Domingos | Mining the network value of customers[END_REF] and [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF][START_REF] Kempe | Influential nodes in a diffusion model for social networks[END_REF]. In economics, two important contributions are [START_REF] Ballester | Who's who in networks. Wanted: The key player[END_REF] and [START_REF] Banerjee | Diffusion of microfinance[END_REF]. The first characterizes a key target through an "inter-centrality measure" that takes into account both a player's centrality and his contribution to the centrality of others, while the second develops a model that discriminates between information passing and endorsement and derives a measure of "diffusion centrality" to characterize efficient targets.

Now, the bulk of the literature has approached the targeting problem as an optimization problem for a single influencing agent. A competitive perspective has been introduced in a few recent contributions that develop game-theoretic models in which agents choose, ex-ante, a targeting strategy to maximize their influence on the outcome of the ensuing opinion dynamics. The contributions of [START_REF] Carnes | Maximizing influence in a competitive social network: a follower's perspective[END_REF][START_REF] Bharathi | Competitive influence maximization in social networks[END_REF][START_REF] Goyal | Competitive contagion in networks[END_REF][START_REF] He | Price of anarchy for the n-player competitive cascade game with submodular activation functions[END_REF] offer an algorithmic perspective on the issue while [START_REF] Lever | Strategic Competitions over Networks[END_REF][START_REF] Dubey | Competing for customers in a social network[END_REF][START_REF] Bimpikis | Competitive targeted advertising over networks[END_REF][START_REF] Grabisch | Strategic influence in social networks[END_REF] focus on the characterization of optimal targets. In particular, [START_REF] Grabisch | Strategic influence in social networks[END_REF] is a direct predecessor of this work and puts forward "intermediacy centrality" as a minimax characterization of the key target which accounts for the fact that, in a competitive context, agents should focus on relative, rather than absolute, centrality. [START_REF] Dubey | Competing for customers in a social network[END_REF] considers a different model where strategic players can interact with several non-strategic agents at the same time: the players choose an expenditure for each nonstrategic agent at a given cost. Their model is also static in the sense that the expenditures of the players are fixed at the beginning of the game.

The importance of adopting a dynamic perspective when designing influence strategies is also clearly identified in the literature. In political science, [START_REF] Box-Steffensmeier | A dynamic model of campaign spending in congressional elections[END_REF] emphasizes the fact that "at different stages of the campaign process candidates have different goals, and their expenditures should have different effects on the final election outcome". [START_REF] Granato | Political campaign advertising dynamics[END_REF] provides a simulation model of the dynamics of an electoral campaign from this perspective, while [START_REF] Demange | A communication game on electoral platforms[END_REF]; [START_REF] Aragones | Electoral competition through issue selection[END_REF] and [START_REF] Amorós | Issue convergence or issue divergence in a political campaign?[END_REF] investigate in a game-theoretic setting how candidates should allocate their campaign time to the different issues of concern for the electorate.

The computer science literature documents empirically both the diffusion of influence in social networks such as Twitter and the dynamic behavior of agents in view of influence maximization. With respect to the former, [START_REF] Raghavan | Coupled hidden markov models for user activity in social networks[END_REF] shows that the predictive power of models of users' activity on Twitter is improved by taking into account the dynamics of social interactions. With respect to the latter, [START_REF] Lee | Discovering content-based behavioral roles in social networks[END_REF] provides empirical evidence for the evolution of behavior in online social networks, while [START_REF] Mekouar | Access and sharing contents through the social network: A pomdp approach[END_REF] highlights the importance of the dynamic updating of content in social networks in view of revenue maximization. More specifically related to the present paper, a number of contributions in the computer science literature have considered the problem of dynamic targeting in a social network for a single influencer. [START_REF] Golovin | Adaptive submodularity: Theory and applications in active learning and stochastic optimization[END_REF] considers the design of adaptive algorithms to address the targeting problem when influences spread randomly over the network. [START_REF] Zhuang | Influence maximization in dynamic social networks[END_REF] also puts forward an adaptive approach in a setting where the influencer periodically probes the network in order to adjust his influence strategy. [START_REF] Yadav | Healer: Pomdp planning for scheduling interventions among homeless youth[END_REF] approaches the same problem, but from the more formal perspective of a partially observable Markov decision process (POMDP). This is further developed by [START_REF] Yu | Finding spread blockers in dynamic networks[END_REF], which focuses on the identification of spread blockers in dynamic networks. [START_REF] Mekouar | Optimizing diffusion time of the content through the social networks: Stochastic learning game[END_REF] includes both a dynamic and a competitive perspective, since it focuses on the timely delivery of content from two competing sources.

From a more economic standpoint, [START_REF] Hartline | Optimal marketing strategies over social networks[END_REF] analyses the marketing strategy of a firm through a social network in a dynamic programming framework and emphasizes the complementarity between pricing and targeting: "In general it is advantageous to get influential buyers to buy the item early in the sequence; it even makes sense to offer such buyers smaller prices to get them to buy the item." There is also a wide literature on competition in advertising. [START_REF] Chakrabarti | An analysis of advertising wars[END_REF] provides a simple model of advertising wars. These have a long history, as documented in [START_REF] Beard | Comparative advertising wars: An historical analysis of their causes and consequences[END_REF]. [START_REF] Doraszelski | Advertising dynamics and competitive advantage[END_REF] models dynamic competition among firms through advertising and analyzes its impact on industrial dynamics. Finally, [START_REF] Pastine | Consumption externalities, coordination, and advertising[END_REF] analyzes the use of advertising by two competing firms as an expectation coordination device in the purchasing decision of consumers when there are consumption externalities. However, the present paper is to our knowledge the first to include both a dynamic and a competitive perspective on the problem of targeting in a social network. As such, it raises a number of theoretical issues. The first is the definition of an appropriate solution concept. We choose to focus on the meanaverage payoff and on the uniform approach [START_REF] Mertens | Stochastic games[END_REF], which seems to us well suited to extend the results of [START_REF] Grabisch | Strategic influence in social networks[END_REF] to a dynamical framework. However, the existence of the uniform value is a complicated question in our framework, where the set of states is compact. It has been solved positively in several classes of stochastic games (see [START_REF] Solan | Stochastic games[END_REF][START_REF] Jaśkiewicz | Zero-sum stochastic games[END_REF] or as an intermediate result for stochastic games with incomplete information [START_REF] Renault | The value of repeated games with an informed controller[END_REF][START_REF] Aumann | Repeated games with incomplete information[END_REF]. However, none of these results apply here. Moreover, it has been shown recently that, even under reasonable assumptions, the uniform value might not exist [START_REF] Ziliotto | Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture max min = lim v n[END_REF].

Model

Notation

We shall use the following notation throughout:

• In the vector space R K , where K ∈ N * , e i denotes the i th vector of the canonical basis and e ′ = (1, . . . , 1) = ∑ K i=1 e i . • Given K a finite set of cardinality K, we denote by ∆(K) the set of probability distributions over K. It can be viewed as a subset of R K : {a ∈ R K + , ∑ k∈K a k = 1}. We consider the . ∞ norm on ∆(K) defined by ∀a ∈ ∆(K) , a ∞ = max k∈K a k .

• Given two natural numbers i, j ∈ N, δ i,j denotes the Kronecker symbol such that δ i,j = 1 if i = j and zero otherwise.

• Given a matrix M of size K × K, M k denotes the k th row vector.

• Given two vectors x, y ∈ R K , x ⋅ y denotes their scalar product.

• Given a family of matrices

(M i ) i∈N , b ← i=a B i

Opinion game

We consider two strategic players who compete for influence on a social network through dynamic targeting strategies. The network is formed by K non-strategic agents whose social ties are represented by a row-stochastic matrix M of size K × K, accounting for the fact that the network is in general directed. Each of the non-strategic agents is characterized by his opinion on an issue of concern for the two strategic players, e.g., the relative quality of two products or the proximity relative to two opposing electoral platforms. This opinion is represented by a number in [-1, 1] and, by default, evolves according to the social influence exerted on the agent, i.e., each agent updates his opinion by linearly combining the opinions of his neighbors in proportions determined by the corresponding row of the social network matrix M. This boundedly rational model of opinion formation was introduced by DeGroot (1974) and has received wide attention in the economic literature as a model of opinion formation in social networks (see e.g. Golub and Jackson, 2010, and references therein). It is particularly well suited to contrasting the myopic behavior of a "crowd" with the strategic behavior of external influencers such as lobbies, advertisers, or political campaigners (see [START_REF] Bimpikis | Competitive targeted advertising over networks[END_REF][START_REF] Grabisch | Strategic influence in social networks[END_REF].

In our framework, the two strategic players are referred to as Player 1 and Player -1. They are characterized by their fixed opinions, 1 and -1, respectively, and their levels of influence, λ and µ, respectively. They aim to bring the average opinion in the network as close as possible to their own opinion, e.g., to trigger the purchase of their product or a vote for their political platform. In each period, they simultaneously choose a target among the non-strategic agents with a view to influencing his opinion. Targeted agents then update their opinions through a convex combination of the opinions of their non-strategic neighbors and of the strategic agent(s) who targeted them. A strategic player might only partially observe the decision of the other player.

Structure of the game

Formally, we define an n-stage game Γ n ∶= (n, K, M, x 1 , λ, µ, ν, C1 , C -1 , s 1 , s -1 ) through the following parameters: a non-negative integer n representing the length of the game, a finite set K representing nonstrategic agents, a row-stochastic matrix M representing the local interaction between the non-strategic agents, a vector x 1 in X ∶= [-1, 1] K representing the initial vector of opinions of the non-strategic agents, a vector ν in R K + representing the influenceability of the non-strategic agents, a positive real number λ representing the relative influence of Player 1, a positive real number µ representing the relative influence of Player -1, a finite set C 1 of signals and a function s 1 from K × K to C 1 representing the signaling function of Player 1, a finite set C -1 of signals and s -1 a function from K × K to C -1 representing the signaling function of Player -1. 1 The game Γ n is played in n periods between the two strategic players, Player 1 and Player -1, as follows:

• Initially, each non strategic agent, k ∈ K, holds an opinion x 1,k ∈ [-1, 1] given by the vector x 1 ∈ [-1, 1] K .

• In every period t ≥ 1:

-Player 1 chooses a target i t ∈ K and Player -1 chooses a target j t ∈ K.

-Each non-strategic agent updates his belief using a convex combination of the opinions of his non-strategic neighbors and of the strategic players who target him. For all k ∈ K,

x t+1,k = q k (x t , i t , j t ) ∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ M k ⋅ x t if ν k = 0, k ≠ i t and k ≠ j t , ν k M k ⋅ x t + λδ k,it -µδ k,jt ν k + λδ k,it + µδ k,jt otherwise (3.1)
This can be summarized in matrix notation as follows.

Let A 1 (i, j) = λ ν k +λ+µδ j,k e i , A -1 (i, j) = µ ν k +λδ i,
k +µ e j and B(i, j) be the K × K matrix with coefficients

B(i, j) kl ∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ M kl if ν k = 0, k ≠ i and k ≠ j, ν k M kl ν k + λδ k,i + µδ k,j otherwise. (3.2)
Then, we obtain

x t+1 = B(i t , j t )x t + A 1 (i t , j t ) -A -1 (i t , j t ). (3.3)
A 1 (i t , j t ) is the influence of Player 1 in one stage and A -1 (i t , j t ) is the influence of Player -1, while B(i t , j t ) represents the remaining local interactions between non-strategic agents.

-Player 1 then receives the signal s 1 (i t , j t ), while Player -1 receives the signal s -1 (i t , j t ).

• If t < n, a new period begins. If t = n, the game terminates.

Remark 3.1 Given a sequence of n pairs of targets (i 1 , j 1 , . . . , i n , j n ), the vector of opinions of the non-strategic agents at stage t ≤ n can be written in matrix notation as

x t = t-1 ← m=1 B(i m , j m )x 1 + t-1 l=1 ⎛ ⎜ ⎝ t-1 ← m=l+1 B(i m , j m ) ⎞ ⎟ ⎠ (A 1 (i l , j l ) -A -1 (i l , j l )). (3.4)
The aim of Player 1 is to maximize the mean-average opinion across time and across non-strategic agents, while the aim of Player -1 is to minimize the mean-average opinion. More precisely, strategies and payoffs are defined as follows.

At stage t, Player 1 chooses a target, possibly using randomization, as a function of the information he learned in the previous stages. Hence, a strategy of Player

1 is a function σ ∶ ∪ +∞ m=1 H 1 m → ∆(K), with H 1 m ∶= (K × C 1 ) m-1
(the player observes his signals and his own sequence of actions). The corresponding set of strategies is denoted by Σ. Similarly, a strategy for player -

1 is a function τ ∶ ∪ +∞ m=1 H -1 m → ∆(K), with H -1 m ∶= (K × C -1 ) m-1
, and the corresponding set of strategies is denoted by T . The strategy of a player is said to be pure if it does not involve randomization, i.e., if for every history, the player targets a single node with probability 1. For convenience, we will refer to the choices of the players as actions. Moreover, we introduce the set of actions H a n = (K × K) n taken in the n-stage game by both players, while H a ∞ = (K × K) N is the set of infinite histories of actions.

Taken with the information structure, a triple (x 1 , σ, τ ) naturally induces a probability distribution on

H n ∶= (K × K × [-1, 1] K × C 1 × C -1
) n denoted by P x1,σ,τ . We denote by E x1,σ,τ the expectation under this probability distribution. When Player 1 follows the strategy σ and Player -1 follows the strategy τ , the average opinion across time and across non-strategic agents is therefore given by

γ n (x 1 , σ, τ ) = E x1,σ,τ 1 Kn n t=1 k∈K x t,k . (3.5)
The payoff of Player 1 is given by γ n (x 1 , σ, τ ) and the payoff of Player -1 by -γ n (x 1 , σ, τ ). Hence, Player 1 (resp. -1) aims to maximize (resp. minimize) the average opinion across time and across agents.

Interpretation of the parameters

Hence, we consider a zero-sum dynamic game with incomplete information. The parameters of the game define on the one hand an influence structure (M, ν, λ, µ) and on the other hand an information structure

(C 1 , C -1 , s 1 , s -1 ).
Within the influence structure, the matrix M determines the dynamics of opinions among non-strategic agents in the absence of external influence. It is row-stochastic to represent the fact that an agent forms his opinion using a weighted average of his neighbours' opinions. The vector ν measures the sensitivity of non-strategic agents to external influence: the lower the value of ν k , the more influenceable agent k. Finally, λ and µ measure the influence potential of the two strategic players. An interesting special case occurs when each non-strategic agent puts equal weight on his neighbours and the share of influence received from the network is proportional to the number of neighbors. In this case, ν k is the degree of agent k in the network and all non-zero entries of M k have value 1 ν k .

The information structure determines the type and extent of information a strategic agent receives about the actions of his opponent. Two cases will be of particular interest. First, the blind information structure, where s 1 and s -1 are constant on I × J. Then, the players have no information about their opponent's actions and only take into consideration their own history of play. Second, the complete information structure where s 1 = s -1 = Id I×J and both players are perfectly informed about the actions of their opponent. Note that, given a vector of initial opinions, the complete information structure is equivalent to a situation where the players perfectly observe the vector of opinions at the beginning of each period.

Value of the finite horizon game

For a given n, the game Γ n is in fact equivalent to a finite zero-sum game, i.e., a matrix game. It thus follows from the classical result of von [START_REF] Neumann | Zur theorie der gesellschaftsspiele[END_REF] that the game has a value v n (x 1 ). That is, Player 1 can choose a strategy that ensures that the time average of opinions is above v n (x 1 ) independently of the strategy of Player -1, and reciprocally, Player -1 can ensure that the average opinion is below v n (x 1 ) independently of the strategy of Player 1. The corresponding pair of strategies form a Nash equilibrium of the game. More formally, one has:

Proposition 3.2 The n-stage game Γ n (x 1 ) has a value: v n (x 1 ) ∶= max σ∈Σ min τ ∈T γ n (x 1 , σ, τ ) = min τ ∈T max σ∈Σ γ n (x 1 , σ, τ ).
In particular, there exists σ * , a strategy of Player 1, such that for every strategy τ of Player -1, we have

γ n (x 1 , σ * , τ ) ≥ v n (x 1 ).
Such a strategy will be said to be optimal. While following this strategy, Player 1 can guarantee the payoff v n (x 1 ) against any strategy played by Player -1. Similarly a strategy τ * of Player -1 is optimal if ∀σ ∈ Σ, γ n (x 1 , σ, τ * ) ≤ v n (x 1 ).

Note in particular that the value of the finite-horizon game depends on the vector of initial opinions because the strategic players can only exert a limited amount of influence in finite time.

Uniform value

Strategic agents ought to take into account the diffusion and mixing of opinions in the network, which might be arbitrarily long. In order to account for this process, we focus on games with an arbitrarily long horizon. The notion of value does not extend straightforwardly to a setting with an arbitrarily long horizon. As a matter of fact, many notions of value coexist in this context. However, the notion of uniform value [START_REF] Mertens | Stochastic games[END_REF] provides a well-suited solution concept in our setting. Informally, it implies that there exist strategies that can be used independently of the length of the game to guarantee a given value. Formally, the uniform value is defined as follows. We denote the sequence (Γ n ) n∈N by Γ, and by Γ(x 1 ) when we wish to emphasize the initial vector of opinions.

Definition 3.3 Let v be a real number.

• Player 1 can guarantee v in Γ(x 1 ) if, for all ε > 0, there exists a strategy σ * ∈ Σ of Player 1 such that lim inf

n inf τ ∈T γ n (x 1 , σ * , τ ) ≥ v -ε.
We say that such a strategy σ * guarantees vε in Γ(x 1 ).

• Player -1 can guarantee v in Γ(x 1 ) if, for all ε > 0, there exists a strategy τ * ∈ T of Player -1 such that lim sup

n sup σ∈Σ γ n (x 1 , σ, τ * ) ≤ v + ε.
We say that such a strategy τ * guarantees v + ε in Γ(x 1 ).

• If both players can guarantee v, then v is called the uniform value of the game Γ(x 1 ) and denoted by v ∞ (x 1 ). The preceding strategies are then said to be ε-optimal in the infinite game.

In our setting, the existence of the uniform value ensures that both players can guarantee that the mean-average opinion is above (resp. below) a fixed threshold. Moreover, the optimal strategy does not depend on the length of the game. This implies that a player can play almost optimally in any long enough game without knowing either the exact length or the strategy of his adversary beforehand. The existence of the uniform value v ∞ (x 1 ) also implies that the sequence (v n (x 1 )) n≥1 converges to v ∞ (x 1 ) [START_REF] Mertens | Stochastic games[END_REF].

In our framework where the state space is infinite, the existence of the uniform value is a complicated question. It was recently shown that, even under reasonable assumptions, the uniform value might not exist [START_REF] Ziliotto | Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture max min = lim v n[END_REF]. There are positive existence results for special classes of games (see [START_REF] Jaśkiewicz | Zero-sum stochastic games[END_REF], for recent survey), but they do not apply in our setting. The next section is thus devoted to the existence of the uniform value in Γ.

Existence of the uniform value

A key assumption in models of opinion diffusion is the connectivity of the graph, which ensures that an opinion can propagate between any pair of agents. While studying the emergence of consensus, DeGroot (1974) stressed the fact that connectivity is not sufficient for the emergence of consensus. For example, if we consider a cycle of 2 agents {α, β}, where each agent is influenced only by the other agent, then the dynamic of opinions is a cycle of length 2: the agents exchange their opinions at every stage. The associated Markov chain is periodic. This implies in particular that there is no consensus if the agents disagree at the beginning. This leads [START_REF] Degroot | Reaching a consensus[END_REF] to use the following notion from the theory of Markov chains.

Definition 3.4 A positive matrix M of size K × K is primitive if there exists m ≥ 1 such that ∀k ∈ K, ∀l ∈ K, (M m ) kl > 0.
By extension, we will say that the network is primitive when the matrix M associated with the network is primitive. Informally, when the network is primitive, after m periods of diffusion, every non-strategic agent k ∈ K will be taking into account the opinion initially held by every agent: the network will be mixing opinions. A typical example of a network which is not primitive is a network in which every agent is listening to only one other agent. In this setting, the opinions will be circulating across the network but never mixing. On the contrary, a sufficient condition for the network to be primitive is that it should be strongly connected and that at least one non-strategic agent should put some strictly positive weight on his own opinion. This implies on the one hand that each agent eventually influences each other agent and on the other hand that at least one agent has some confidence in his own opinion.

In our setting, the primitiveness of the network ensures that the opinion game admits a uniform value and, moreover, that this value is independent of the initial distribution of opinions. In particular, one has: Theorem 3.5 Let Γ be an opinion game such that M is primitive. For every x 1 ∈ X, the opinion game Γ(x 1 ) has a uniform value. Moreover, there exists v ∈ R such that, for every

x 1 ∈ X, v ∞ (x 1 ) = v.
A key insight brought about by Theorem 3.5 is that influencers can design robust targeting strategies that are ǫ-optimal independently of the initial distribution of opinions in the network and independently of the length of the game (provided it is long enough). In other words, influence strategies can be determined on the sole basis of the structure of the network. Furthermore, these strategies can be designed a priori, before the player obtains information about the actions of his opponent. From the point of view of opinion dynamics, Theorem 3.5 implies the convergence of the average opinion in the network at equilibrium, but not the convergence of opinions towards a consensus. The latter result is standard in a setting where agents are non-strategic and thus contribute, not necessarily purposely, to a common objective (e.g., the discovery of the truth in [START_REF] Golub | Naive learning in social networks and the wisdom of crowds[END_REF] or distributed optimization in [START_REF] Nedic | Constrained consensus and optimization in multi-agent networks[END_REF]). In contrast, in our setting, strategic agents have a similar influence on the dynamics to stubborn agents in [START_REF] Yildiz | Binary opinion dynamics with stubborn agents[END_REF]: they prevent convergence to a consensus and imply that the asymptotic distribution of opinions is independent of the initial opinions of non-strategic agents (whence the independence of the value from initial opinions in Theorem 3.5). A major difference with [START_REF] Yildiz | Binary opinion dynamics with stubborn agents[END_REF], however, is that we also consider our "stubborn" agents to be strategic in the choice of their placement in the network. Hence the game-theoretic nature of our model.

The key step in the proof of Theorem 3.5 is to divide the game into blocks of sufficient time length to ensure that the transition function is contracting. Hence, our model shares some similarities with stochastic games with state-independent transitions [START_REF] Thuijsman | Optimality and equilibria in stochastic games[END_REF], where the transition does not depend on the state variable. Nevertheless, our assumption is weaker since it is only asymptotically true. From this first result, we deduce that the value in very long games is almost independent of the original vector of opinions, whence the existence of the uniform value.

Remark 3.6 Our proof fails if the network is not primitive, so the existence of the uniform value is open in the general case. It is interesting to comment on the non-primitive case and in particular why [START_REF] Grabisch | Strategic influence in social networks[END_REF] were able to assume only that the network is strongly connected, i.e., why they did not need the stronger notion that it is primitive. In [START_REF] Grabisch | Strategic influence in social networks[END_REF], the strategic agents cannot change their target during the game. Hence, even if the original network between non-strategic agents is periodic, the new network augmented by the strategic agents will be aperiodic: after some m ′ stages of the game, the opinion of every non-strategic agent k ∈ K is influenced at every stage by the opinion of the strategic players. In our model, the strategic agents may change their targets during the game and therefore adopt a cyclical behavior. In this case, it is possible for the opinion of a non-strategic agent at a given stage to be independent of the influence of the strategic agents. For example, in our previous example with two agents {α, β} who influence each other, this is the case if the two strategic players target α at even stages and β at odd stages: the opinion of β at an even stage is never influenced by the strategic agents. Under such strategies, the transition is not contracting and our proof does not hold.

Characterization of equilibrium strategies

In this section, we focus on the characterization of equilibrium strategies. As emphasized in Theorem 3.5, the uniform value of the game is independent of the initial distribution of opinions among non-strategic agents. Therefore, we will focus on the case where initial opinions are uniformly set to 0 unless otherwise specified. In particular, we let v n stand for v n (0), unless otherwise specified. For finite games, this assumption is not without loss of generality, but we will emphasize, where relevant, how our results can be extended to the case of arbitrary initial opinions.

The problem faced by the strategic players in a dynamic context involves a number of trade-offs. At each stage, a strategic agent must first decide whether to influence an easily influenceable agent, with low ν k , or a central agent, which might, however, have a high ν k and be harder to influence. Second, he must decide whether to confront the other influencer by choosing the same target or to shy away by choosing another target or playing a mixed strategy (e.g., if he has a much weaker influence potential than his opponent). Both trade-offs are already present in the static targeting problem where the influenced agent is fixed through time [START_REF] Bimpikis | Competitive targeted advertising over networks[END_REF][START_REF] Grabisch | Strategic influence in social networks[END_REF]. The specific issue in a dynamic context is the trade-off between adopting a forward-looking perspective, according to which players aim to maximize the future spread of their opinion in the network, and a backward-looking perspective, according to which they aim to counteract their opponent's previous actions. It is likely impossible to provide a closed-form analytic characterization of dynamic strategies implementing this trade-off in the general case. Nevertheless, we have identified three cases that illustrate the relation between network structure, level of influence, and equilibrium strategies in the opinion game. In highly symmetric networks, players must adopt a purely backward-looking perspective. They should only focus on counteracting the influence of their opponent so as to keep the current distribution of opinions in an "acceptable" state. In very hierarchical, hence asymmetric, networks, players must adopt a purely forward-looking perspective. Independently of the current distribution of opinions, they should target (sequences of) influential nodes in order to foster the diffusion of their influence in the network. Finally, if the level of influence of both players is small, the game is similar to the static game in [START_REF] Grabisch | Strategic influence in social networks[END_REF]; note in particular that, if the players have equal influence and the non-strategic agents are equally influenced, they should target the node with the largest eigenvector centrality.

Backward-looking strategies

A purely backward strategy consists in targeting the node that is most likely to be influenced by the action the opponent took during the preceding period. The implementation of such a strategy clearly requires complete information. A first class of network where this strategy can be effective is the class where each node influences a single node, that is, when there exists a permutation σ of K such that the adjacency matrix of the network is of the form M φ i,j = δ j,φ(i) . This strategy is also effective (like every other strategy) in the complete network (with equal weights).

In this section, we will focus on the broader class of networks with adjacency matrix of the form M φ,α i,j = αδ j,φ(i) + ( 1-α) K, where φ is a permutation of K and α ∈ [0, 1]. These are networks where each node puts a weight (1α) on the average opinion and a weight α on a specific node. This class notably contains the complete network with equal weights (when α = 0), the circle (when α = 1 and φ(i) = (i + 1) mod K), the network where all agents are stubborn (when α = 1 and φ(i) = i), and perturbations of these (when α < 1). Formally, we introduce the following assumption.

Assumption 4.1 The game Γ is such that:

• The network has an adjacency matrix of the form M φ,α i,j = αδ j,φ(i) + ( 1-α) K, where α ∈ [0, 1] and φ is a permutation of K.

• The players have equal influence (i.e., λ = µ) and there exists ν ∈ R + such that, for all k, k ′ ∈ K,

ν k = ν k ′ = ν ≤ λ.
One can then show that backward-looking strategies are optimal in networks satisfying Assumption 4.1.

Notice that, when α = 1, the matrix is not primitive, and this means that Theorem 3.5, which proves the existence of the uniform value, does not apply. Nevertheless, our construction of the optimal strategies will imply the existence of the uniform value.

The condition λ = µ allows the computation to be tractable. Unfortunately, when λ ≠ µ it becomes impossible to characterize optimal strategies even in our restricted class. Qualitatively, it is nevertheless clear that the player with a higher influence has a positive advantage, while the other player still has some non-negligible influence: for example, when there is only one node, his opinion converges to (λ-µ) (λ+µ).

Theorem 4.2 Suppose that Assumption 4.1 holds and that the players have complete information. Let (σ * , τ * ) be the pair of strategies defined as follows.

• In period 1, σ * (∅) is an arbitrary target in K and, for every t ≥ 1 and every history h t = (i 1 , j 1 , ..., i t , j t ), σ * (h t ) = φ -1 (j t ).

• In period 1, τ * (∅) is an arbitrary target in K and, for every t ≥ 1 and every history h t = (i 1 , j 1 , ..., i t , j t ),

τ * (h t ) = φ -1 (i t ).
Then, the strategy σ * (resp. τ * ) guarantees a value of 0 to Player 1 (resp. Player -1) in every n-stage game. Therefore, the value of any n-stage game Γ n is 0, as is the uniform value of Γ, and (σ * , τ * ) is, in each case, a pair of optimal strategies. Remark 4.3 The proof of Theorem 4.2 highlights the fact that the equilibrium strategies σ * and τ * can also be defined as the strategies for which, in every period, a player targets the non-strategic agent whose opinion is the most different from his.

Theorem 4.2 can be contrasted with the fact that there does not exist an equilibrium in pure strategies in the absence of information in networks satisfying Assumption 4.1 Proposition 4.4 Suppose Assumption 4.1 holds with K ≥ 3, α > 0, and n ≥ 3. If the players have blind information, there does not exist an equilibrium in pure strategies in the game Γ n .

The contrast between Theorem 4.2 and Proposition 4.4 highlights the role of information and dynamics in devising efficient targeting strategies. Indeed, in the class of networks satisfying Assumption 4.1 with α > 0, there is no equilibrium in pure strategies in the absence of information, whereas there exist equilibria with pure dynamic targeting strategies when information is complete. This contrast hinges on two facts. First, the structure of the network. In regular networks satisfying Assumption 4.1, the efficient strategy for players is to counteract the influence of their opponent. Second, the set of strategies available to the players. On the one hand, with complete information, players can counteract the influence of their opponent by implementing a purely backward-looking strategy. On the other hand, in the absence of information, players choosing a pure strategy ought to commit ex-ante to a given sequence of actions (see Section 4.2 below). His opponent can then gain a major advantage over the player by best-responding to this given sequence of targets. Hence, to prevent his influence from being too easily counteracted, the player must use a mixed strategy.

To further emphasize the role of network structure, one should note that the purely backward-looking strategies defined in Theorem 4.2 are not optimal in every type of network. Example 4.5 highlights this fact.

Example 4.5 We consider the set of non-strategic agents K = {α, β, γ} and the local interaction matrix M given by

α β γ α β γ ⎛ ⎜ ⎝ 1 2 1 2 0 0 1 2 1 2 1 2 1 2 0 ⎞ ⎟ ⎠
Informally, each non-strategic agent puts equal weight on his own opinion and that of his predecessor. We further assume that, for every k ∈ K, ν k = 0 and λ = µ = 1, and also that the initial vector of opinions is 0.

In this setting, the backward-looking strategy put forward in Theorem 4.2 is not optimal in the game of length 3. Indeed if Player -1 plays according to τ * , we consider the following strategy of Player 1:

• In stage 1, play uniformly over K,

• In the second stage, considering permutation symmetries, there are three types of vectors of opinions to take into account:

1. If x 2 = (1, -1, 0), i.e., Player -1 targeted the predecessor of the target of Player 1, then Player 1 follows σ * . This yields, in stage 3, the vector of opinions p 3 = (-1, 1, 1 2).

2. If x 2 = (-1, 1, 0), i.e., Player 1 targeted the predecessor of the target of Player -1, then Player 1 targets the agent with opinion 0. This yields the new vector of opinions (0, -1, 1).

3. If x 2 = (0, 0, 0), i.e., they chose the same targets, Player 1 then plays, in stage 3, a successor of his action in stage 2 such that x 3 is either (0, 0, 0) or (1, -1, 0).

We see that the payoff is positive in all cases and strictly positive in one of them, whence the mean average opinion is strictly positive. The strategy τ * is not optimal. One can check that the result holds true for any n-stage game with n ≥ 1, nevertheless, the computation shows that the short term advantage that is obtained by Player 1 at stage 2 decays over time. Hence, although the strategy is not optimal, it is ε-optimal in the infinite game. It is not clear whether this holds true in the general case.

Blind games and forward-looking strategies

In the absence of information, i.e., in the blind information game, the players cannot condition their actions on the history of the game. More precisely, a pure strategy of Player 1 in a blind game of length n is a sequence of actions (i 1 , ⋯, i n ) ∈ K n . Such a strategy guarantees a value v n for Player 1 if, for every strategy

(j 1 , ⋯, j n ) ∈ K n of Player -1, one has γ((i 1 , ⋯, i n ), (j 1 , ⋯, j n )) ≥ v n .
Hence, for an equilibrium in pure strategies to exist, Player 1 must determine a priori a sequence of actions that will guarantee him the value v n independently of the sequence of actions that his opponent chooses: he must be forward-looking. More precisely, a pair of strategies

(σ * , τ * ) = ((i * 1 , ⋯, i * n ), (j * 1 , ⋯, j * n )) ∈ K n × K n is an equilibrium in pure strategies of a blind game with value v n if one has ∀σ ∈ K n , ∀τ ∈ K n , γ n (σ * , τ ) ≥ v n ≥ γ n (σ, τ * ) (4.1)
Note that an equilibrium in pure strategies need not necessarily exist as Equation 4.1 is very demanding: players must be able to counteract a priori any sequence of actions chosen by their opponent. Yet, if such a pure equilibrium strategy exists, it will be very robust. It can, in fact, be used in any information structure. Indeed, the opponent cannot take advantage of any additional information as the original pure strategy is a best-response against any sequence of actions. In particular, given a pure strategy σ of Player 1 for the blind-information structure, let us denote by σ (C 1 ,C -1 ,s) the strategy in the game with information structure (C 1 , C -1 , s) that plays the sequence of actions σ independently of the signal received (and respectively τ (C 1 ,C -1 ,s) given a strategy τ of Player -1). Let us also denote by Γ b n the blind game associated with an arbitrary game Γ n by replacing its information structure by the blind information structure. Then one has: Proposition 4.6 Let Γ n be a game with information structure (C 1 , C -1 , s). Assume that (σ, τ ) is an equilibrium in pure strategies in the associated n-stage blind game Γ b n . Then, (σ

(C 1 ,C -1 ,s) , τ (C 1 ,C -1 ,s)
) is an equilibrium in the n-stage game Γ n . Moreover the two games have the same value2 .

In order to characterize these blind equilibrium strategies, it will prove useful to measure the total influence over time generated by the actions of the players in a given period. Given a history of actions h = ((i 1 , j 1 ), ⋯, (i n , j n )) ∈ H a n , this intertemporal influence can be measured as follows:

• The actions taken in stage n of the game only have an impact from stage n + 1 onward. Hence, they have no influence on the game of length n. In other words, the "influence potential" in period n is

y n = 0 ∈ R K .
• The actions taken in stage n -1 only have an impact on opinions in period n (there is no forward diffusion). Hence, the potential influence exerted through each target (not accounting for influenceability) is identical. Thus, the "influence potential" is distributed proportionally to y n-1 = 1 K e ′ . Accounting for the influenceability of the target, the actual impact on opinions of the actions of the players is given by y n-1 (A 1 (i n-1 , j n-1 ) -A -1 (i n-1 , j n-1 )).

• The actions taken by the players in stage n -2 of the game have an influence in periods n -1 and n.

The potential influence exerted directly on each of the non-strategic agents in period n-1 is identical, and is thus distributed according to 1 K e ′ . The potential influence exerted indirectly in period n depends on the forward diffusion of the target's opinion in the network and also the forward diffusion of the influence potential of agents in the next period. It is given by y n-1 B(i n-1 , j n-1 ). Overall, the influence potential in stage n -2 is distributed according to y n-2 = y n-1 B(i n-1 , j n-1 ) + 1 K e ′ , and the total intertemporal impact on opinions of the actions of players in period n -2 is given by

y n-2 (A 1 (i n-2 , j n-2 ) -A -1 (i n-2 , j n-2 )).
• More generally, the influence potential can be defined recursively as the sum of the instantaneous and forward influences. In particular, the distribution of influence potential in stage t is given by

y t = y t+1 B(i t+1 , j t+1 ) + 1 K e ′ . (4.2)
Accordingly, the intertemporal impact on opinions of the actions of players in period t is given by y t (A 1 (i t , j t ) -A -1 (i t , j t )).

• Overall, the intertemporal impact on opinions of a history of actions h = ((i 1 , j 1 ), ⋯, (i n , j n )) is given by

← γ n (h) = n-1 t=1 y t (A 1 (i t , j t ) -A -1 (i t , j t )). (4.
3)

The intertemporal impact on opinions defined above does coincide with the mean belief across time and across agents, i.e., with the payoff. Namely, one has: Proposition 4.7 Assume x 1 = 0. Let n ≥ 1 and h ∈ H a n be a history of actions. Then,

γ n (h) = ← γ n (h).
Remark 4.8 The case of arbitrary initial opinions x 1 can be accounted for by altering the definition of the influence potential y n in period n.

One can then build on Proposition 4.7 to give a characterization of pure equilibrium strategies in the blind game with equal influence, provided that such strategies exist. Indeed, in the case of equal influence, the game is zero-sum and symmetric, so its value is zero. In this setting, at an equilibrium in pure strategy, a player ought to target, in each period, the node where his intertemporal impact is maximal. Otherwise, his opponent would gain a strictly positive profit by targeting the maximal impact node when the player doesn't, and mimicking his strategy otherwise. In particular, one has: Proposition 4.9 Assuming that a blind game Γ b n admits a pure Nash equilibrium (σ, τ ), then for every t ∈ {1, ..., n -1},

σ n ∈ k ∀j ∈ K, y n,k λ + ν k ≥ y n,j λ + ν j , and
τ n ∈ k ∀j ∈ K, y n,k λ + ν k ≥ y n,j λ + ν j .
Hence, in an equilibrium in pure strategy of the blind game with equal influence, a player ought to adopt a purely forward-looking strategy: in each period he ought to maximize the forward propagation of his influence in the network. Proposition 4.9 also shows why there might not exist equilibria in pure strategies in the blind game. The identity of the node in which the influence potential is maximum depends, in general, on the history of play, whereas in the blind game a player commits ex-ante to a fixed sequence of targets when he chooses a pure strategy.

Nevertheless, when one can decouple the identification of the maximal influence node and the history of play, necessary conditions for the existence of a blind equilibrium in pure strategies can be put forward. One has the following proposition: Proposition 4.10 Assume that there exists a sequence of nodes (i 1 , ⋯, i n ) ∈ K n such that, for all t = 1, ⋯, n -1 and all histories h ∈ H a ∞ , one has

i t = argmax k∈K y t,k λ + ν k . (4.4)
Then, (i 1 , ⋯, i n-1 ) is an equilibrium strategy for both players in the blind game of length n with equal influence λ.

Remark 4.11 If there exists ν such that, for all k ∈ K, ν k = ν, Equation 4.4 can be replaced by

i t = argmax k∈K y t,k . (4.5)
To ease the interpretation of our results, we shall focus on this particular case in the examples given in the remainder of this section.

Proposition 4.10 can then be used to characterize some networks for which there exist blind equilibria in pure strategies. To this end, one has to relate the influence potential y t defined above to the network characteristics. With this in mind, we introduce the following notions. Definition 4.12 Let a network with adjacency matrix M be given.

• A path of length n ∈ N is a sequence of n nodes p = (i 1 , ⋯i n ) ∈ K n such that, for all t < n, m it,it+1 > 0.

The weight of such a path is defined, with a slight abuse of notation, as m p = ∏ n-1 t=1 m (it,it+1) . We denote by P n (M ) the set of paths of length n in the network M , and by P n k (M ) the set of paths of length n whose end node is k.

• A path constraint on P n (M ) is an n dimensional vector C = (C 1 , ⋯, C n ) ∈ (2 C ) n inducing con-
straints on admissible paths. Informally, we restrict to paths that are not going through C t in stage t. Formally, a path

(i 1 , ⋯i n ) satisfies the constraint C ∈ (2 C ) n if, for all t < n, i t ∈ C t (if C t = ∅,
there is actually no constraint on i t .). We denote by P C n (M ) the set of paths of length n in the network M satisfying the constraint C, i.e.,

P n,C (M ) = {(i 1 , ⋯i n ) ∈ P n (M ) ∀ t ≤ n, i t ∈ C t }.
In the following, we focus on path constraints C = (C 1 , ⋯, C n ) such that, for all t ≤ n, C t has at most 2 elements. Accordingly we denote by C the set of subsets of K with at most 2 elements.

• The n-centrality of a node k ∈ K, denoted by ζ n i (M ) ∈ R + , is the sum of the weights of those paths of length less than n ∈ N leading to k, i.e., A notable class of networks for which there exist blind equilibrium in pure strategies are those where the sequence of nodes with maximal t-centrality is not altered if the network is perturbed by a Cconstraint, i.e., if at most two nodes are discarded every period. We introduce the following assumption: Assumption 4.13 The game Γ n is such that:

ζ n k (M ) = ∑ n t=2 ∑ p∈P t k (M) m p . In vector form, one has ζ n (M ) = e ′ ∑ n-1 t=1 M t . • Given a constraint C ∈ C n , the C-constrained n-centrality of a node i ∈ K, denoted by ζ n,C i (M ) ∈ R + ,
• The strategic players have equal influence, i.e., λ = µ.

• For all t < n, there exists i

* n-t ∈ K such that, for all C ∈ C n-t-1 , i * n-t = argmax k∈K ζ n,c k (M ) λ + ν k .
Remark 4.14 Similarly to Remark 4.11, in the case where there exists ν > 0 such that ν k = ν for all k ∈ K, the condition in the second part of Assumption 4.13 reduces to i

* n-t = argmax k∈K ζ n,c k (M ) independently of λ ∈ R + .
One then has: Proposition 4.15 Suppose Assumption 4.13 holds in the game Γ n . Then there exists a blind equilibrium in the game Γ n in which the equilibrium strategy of both players is given by the sequence of actions (i * 1 , ⋯, i * n ) satisfying Assumption 4.13 (and i * n is chosen arbitrarily).

Hence, under Assumption 4.13, in period t, a player ought to target the node that has maximal n-t-1 centrality, i.e., the node for which the sum of paths of length less than nt -1 is maximal. The player discards previous actions and current beliefs and aims at maximizing his influence over the course of the remaining periods. The earlier in the game it is, i.e., the smaller t is, the longer the horizon of the player, i.e., the longer the paths of the network that he takes into consideration.

In order to satisfy Assumption 4.13 and to admit a blind equilibrium in pure strategies, a network must channel the flow of influence towards specific nodes, even when the structure is perturbed by the deletion of certain nodes. Examples 4.16 and 4.18 put forward two classes of networks which satisfy the assumption in the benchmark case where non-strategic agents are equally influenceable (i.e., there exists ν > 0 such that, for all k ∈ K, ν k = ν). A basic example is the star network (Example 4.16) in which a blind equilibrium strategy is to target the center of the star. More broadly tree-like structures are natural candidates for Assumption 4.13, as they strongly constrain the flow of influence. Example 4.18 illustrates this point by defining a family of trees satisfying the assumption.

Example 4.16 The (connected) star is a network with K nodes with an adjacency matrix such that

M 1 = 1 K e ′ and, for i > 1, M i = e 1 .
The node 1 is then the center of the star. As soon as K ≥ 3, it is straightforward to check that, for t ≥ 2, the center of the star has maximal C-constrained t-centrality, independently of the constraint C. Remark 4.17 Example 4.16 mainly emphasizes the role of the topology of the network in the determination of optimal targets. The weights of links can also play a fundamental role. To illustrate this point, consider a star with two centers with unequal incoming weights. More precisely, consider the network with K + 1 nodes and adjacency matrix M such that M 1 = M 2 = 1 K e ′ (nodes 1 and 2 are the two centers of the star) and, for all i > 2, M i = (1ǫ)e 1 + ǫe 2 . It is straightforward to check that, for ǫ small enough, node 1 has maximal C-constrained t-centrality, independently of the constraint C and for all t ≥ 2. Thus, although nodes 1 and 2 are perfectly symmetric from a topological point of view, node 1 is the only relevant target in the game because of its quantitative characteristics.

Example 4.18 We consider rooted trees in which the root node is denoted by 1, the descendants of each node are indexed by natural numbers, and an arbitrary node is characterized by the sequence of its parents, i.e., (1, u 1 , u 2 , ⋯, u t ) ∈ N t denotes the node at distance t from the root which is the u t th descendant of the node (1, u 1 , ⋯u t-1 ) (and so on, recursively). We then consider the tree F T of height T, defined recursively as follows:

• The root has n 0 descendants.

• Each of the descendants of the root has n 1 ≥ n 0 descendants, except (1, 1), which has n ′ 1 ≥ n 1 + 3 descendants.

• For each t ∈ N, each node at a distance t from the root has n t ≥ n t-1 descendants, except (1, 1, ⋯, 1), which has n ′ t ≥ n t + 3 descendants.

As above, we denote by K the set of nodes of the network. The adjacency matrix M of the tree is such that m i,j = 1 if i is the descendant of j and m i,j = 0 otherwise. In order to turn F T into a connected network, we add to each node of the network (including the root) one link towards the root, i.e., we set m 1,k = 1 K, for all k ∈ K. The resulting network, denoted by F T , has a primitive adjacency matrix.

It is then straightforward to check that, for any constraint C ∈ C T , the node i * 1 ∶= (1, ⋯, 1) ∈ N T has a strictly greater C-constrained 1-centrality than every other node. Indeed, its C-constrained 1-centrality is at least n T + 1, while any other node has a C-constrained 1-centrality of at most n T . Similarly, the node i * 2 ∶= (1, ⋯, 1) ∈ N T -1 has a C-constrained 2-centrality strictly greater than n T n T -1 , while that of every other node is at most n T n T -1 . More generally, one can show by recursion that, for any constraint C ∈ C T , the node i * t ∶= (1, ⋯, 1) ∈ N T +1-t has a strictly greater C-constrained t-centrality than every other node. According to Proposition 4.15, this implies that, for every t ≤ T , the sequence of nodes (i * t , ⋯, i * 1 ) induces a blind equilibrium in the game Γ t associated with the network F T ,, when strategic players have equal influence.

Any subset of nodes, except 1 and the descendants of (1, 1), can be deleted from the network without affecting this result. We thereby define a family of trees with a "dominant branch" in which there exists a blind equilibrium.

Characterization in the case of small influence

Finally, we focus on the limit case where the influence exerted by the strategic players is vanishingly small. In this setting, the dynamics without external influence provides a useful benchmark: one knows that the vector of opinions converges to a consensus in which the influence of each initial opinion is proportional to the eigenvector centrality of the corresponding node (see e.g. [START_REF] Jackson | Social and Economic Networks[END_REF][START_REF] Golub | Naive learning in social networks and the wisdom of crowds[END_REF]. If strategic players have a small influence, they shall take as given the relative importance of non-strategic agents and thus choose their targets according to their eigenvector centrality. We show below that this is indeed the case and, more broadly, provide an equivalence between dynamic games with small influence and a class of static games.

Fix a game Γ. We first define, for every positive measure y over K, a one-shot game G y where Player 1 chooses a probability distribution a over K and player -1 chooses a probability distribution b over K. The definition is in two steps. First, for every pair of non-strategic agents i, j ∈ K, we define

f 1 y (i, j) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y(i) λ λ+νi if i ≠ j, y(i) λ λ+µ+νi if i = j. and f -1 y (i, j) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y(j) µ µ+νj if i ≠ j, y(j) µ λ+µ+νj if i = j.
Second, given a ∈ A = ∆(K) and b ∈ B = ∆(K), we define the payoff of Player 1 by

g 1 y (a, b) = ∑ i,j∈K a(i)b(j)f 1 y (i, j) ∑ i,j∈K a(i)b(j)(f -1 y (i, j) + f 1 y (i, j))
and the payoff of Player -1 in the opposite way by

g -1 y (a, b) = 1 -g 1 y (a, b)
In line with the preceding section, y can be interpreted as a measure of the influence potential associated with each node. Accordingly, in the game G y , players aim to maximize their expected relative influence given the influence potential y.

Remark 4.19 Following [START_REF] Acemoglu | Opinion fluctuations and disagreement in social networks[END_REF], the influence potential y n and the overall intertemporal impact ← γ n defined in Section 4.2 can be interpreted in the context of an n-stage ball-catching game in which players try to catch balls that move backwards in time in the network. More precisely, we consider that a new ball is added to the network uniformly and at random in every period, and that, given a pair of actions (i, j), Player 1 captures a ball present in node i with probability A 1 (i, j) and player -1 captures a ball present in node j with probability A -1 (i, j). Conditionally on the pair of actions, the capture of different balls is supposed to be independent. Then, considering that time flows backwards from period n to period 1, given a sequence of actions h = ((i 1 , j 1 ), ⋯, (i n , j n )) ∈ H a n , one has: • The initial distribution of balls in the network is given by y n = 0.

• For every t < n, the expected number of balls in period t is given by y t = y t+1 B(i t+1 , j t+1 ) + 1 K e ′ . • The expected numbers of balls caught by Player 1 and Player -1 in period t are given by y t A 1 (i t , j t ) and y t A -1 (i t , j t ), respectively.

In this setting, ← γ n (h), corresponds to the difference between the numbers of balls caught by Player 1 and player -1 across time.

The game G y can then be interpreted with this terminology. Assume that a large number of balls are distributed in the network along the measure y. Then, given that Player 1 targets i and Player -1 targets j ∈ K, f 1 y (i, j) and f -1 y (i, j) are respectively the expected number of balls caught by Player 1 and by Player -1. Then g 1 y (a, b) is the relative proportion of balls caught in one stage by Player 1 and 1g 1 y (a, b) is the relative proportion of balls caught by Player -1. Notice that this is not the linear extension of the proportion of balls obtained through pure actions.

Using Sion's minimax theorem [START_REF] Sion | On general minimax theorems[END_REF], one can show that the game G y has a value. We denote by a * y an optimal strategy of Player 1 and by b * y an optimal strategy of Player -1 in this one-shot game.

In line with the dynamic scenario without external influence, we shall focus on games in which the influence potential is proportional to the eigenvector centrality. We denote the eigenvector centrality of M by χ ∈ ∆(K), defined as the unique invariant probability distribution associated with M, i.e., one has χM = χ and ∑ k∈K χ(k) = 1.

One can then show that the equilibrium of the auxiliary game G χ induces an equilibrium in the infinite influence game Γ. In particular, if a player plays an optimal strategy in G χ at every stage, then this strategy is ε-optimal in the infinite game. More formally, we obtain the following result.

Theorem 4.21 Assume that for every k ∈ K, ν k > 0. Let σ * (resp. τ * ) be the strategy of Player 1 (resp. Player -1) that plays a * χ (resp. b * χ ) at every stage in the opinion game Γ. There exists λ 0 ∈ R + such that, for every λ, µ ≤ λ 0 , (σ * , τ * ) is ε-optimal in the infinite game Γ(x 1 ), for every x 1 ∈ X and every information structure. Moreover, one has:

∀x 1 ∈ X, lim n→+∞ v n (x 1 ) = 2w * χ -1.
Note that, although Theorem 4.21 requires the influence to be arbitrarily small, strategic players nevertheless have a real impact on the outcome because the game is arbitrarily long. It is also worth noting that Theorem 4.21 implies that Γ admits an equilibrium in pure strategies whenever the auxiliary game G χ admits an equilibrium in pure strategy. As emphasized above, this is notably the case when players have equal influence and there is a unique node with maximum eigenvector centrality. One thus has the following corollary:

Corollary 4.22 Assume that λ = µ and that there exists ν > 0 such that, for every k ∈ K, ν k = ν. Further, assume that there exists a unique non-strategic agent k * ∈ K such that

χ(k * ) = max k∈K χ(k).
Then there exists λ 0 ∈ R + such that, for every λ ≤ λ 0 , it is an ε-optimal strategy in the infinite game Γ(x 1 ) to target the node k * for every x 1 ∈ X and every information structure.

Note that such an "eigenvector targeting strategy" is another instance of a purely forward-looking strategy: it is optimal independently of the actions of the other player. Strategic agents focus on the diffusion of their own opinion. This also provides evidence for the importance of eigenvector centrality as a measure of influence.

In Corollary 4.22, we focus on arbitrarily long games. One can, however, build on Proposition 4.10 to characterize equilibrium in pure strategies in "short" games, i.e., of a given fixed length N ∈ N, with small influence. Indeed, if influence is sufficiently small and the time-horizon is bounded, strategic actions do not modify the ordering of nodes in terms of t-centrality. In this context, let us assume that there exists a unique sequence of nodes that maximize t-centrality in the following sense.

Assumption 4.23 For all 1 ≤ t ≤ n, there exists a unique i n-t ∈ K such that

i n-t = argmax k∈K ζ n-t-1 k (M ).
For a fixed n, this assumption holds generically, i.e., up to an arbitrarily small modification of the network weights. It suffices to guarantee the existence of a unique equilibrium in pure strategies in the game of length n with equal and small influence. Proposition 4.24 Suppose that Assumption 4.23 is satisfied, x 1 = 0, λ = µ, and there exists ν > 0 such that, for every k ∈ K, ν k = ν. Then, there exists λ 0 ∈ R + such that, for every λ < λ 0 , the sequence (i 1 , ⋯, i n ) ∈ K n of nodes with maximal t-centrality is the only blind equilibrium of the game Γ n (up to an arbitrary choice of i n ).

Comparison between static and dynamic strategies

The influence game considered in this paper is a dynamic extension of [START_REF] Grabisch | Strategic influence in social networks[END_REF]. We consider the same model for the diffusion of opinions, but we allow strategic agents to update their targets dynamically in the network, whereas [START_REF] Grabisch | Strategic influence in social networks[END_REF] treat the case where, once chosen, the target agent is fixed. The model of [START_REF] Grabisch | Strategic influence in social networks[END_REF] can be emulated in our framework by considering static strategies in the following sense.

Definition 5.1 We say that a pure strategy τ of Player -1 is static if there exists j * ∈ K such that, for every history h ∈ ∪ +∞ n=1 H -1 n , τ (h) = j * . A strategy is static if it is a mixture of pure static strategies. Informally, Player -1 chooses a target once and for all at the beginning of the game.

This allows one to compare the efficiency of dynamic and static strategies. In particular, let us note that, in the class of regular networks considered in Section 4.1, there is no equilibrium in pure strategies. In fact, one has: Proposition 5.2 Suppose Assumption 4.1 holds with K ≥ 3, α > 0, φ is such that the orbit of every element is at least 3 (i.e., for all k ∈ K, φ(k) = k and φ 2 (k) = k), and n ≥ 3. If the players are constrained to use a static strategy, there does not exist an equilibrium in pure strategies in the game Γ n .

The non-existence of pure equilibria with static strategies was already noted by [START_REF] Grabisch | Strategic influence in social networks[END_REF] in the case of the circular network. The comparison between this result and Theorem 4.2 shows that there might exist equilibria in pure strategies for networks for which there is no equilibrium in pure strategies in a static setting. Updating targeting strategies through time is, in a sense, akin to mixing strategies in the static game. It provides the players with sufficient flexibility to counteract the actions of their opponents.

Another corollary of this result is that, in regular networks in the sense of Assumption 4.1, a player adopting a dynamic strategy necessarily gains an edge against an opponent using a static strategy, i.e., if Player 1 is "dynamic" while Player -1 is "static", Player 1 can guarantee a strictly positive value (assuming both players have equal influence). Indeed, from periods 2 onward, Player 1 knows the action played by Player -1 for the rest of the game and can thus gain an advantage by best-replying as in the proof of Proposition 5.2. In order to investigate the relative performance of dynamic and static targeting strategies in a broader range of networks, we ran a series of numerical simulations. In particular, we have drawn a sample of Erdos-Renyi random graphs with linkage probability varying in {0.1, 0.25, 0.5} and number of nodes varying in {10, 25, 50}. We investigated whether dynamic strategies perform better than static strategies in games of length 10, 25, and 50 using the following algorithm. We applied a genetic algorithm3 to see whether there was a sequence of actions of Player 1 that would guarantee a payoff strictly greater than 0 against every fixed action of player -1. If such a sequence of actions exists, it implies that Player 1 is strictly better off using the corresponding dynamic strategy than any static strategy, i.e., there is indeed an advantage in using dynamic strategies. Table 5 gives the results of these simulations. It shows first that, in a large proportion of networks, dynamic strategies do improve upon static ones. Second, it provides insights into the influence of network structure on the relative performance of dynamic and static targeting strategies. In this respect, the share of runs in which the genetic algorithm identified an improvement upon static strategies can be interpreted as a lower bound on the number of networks for which such an improvement exists, or a measure of the computational complexity of finding such an improvement 4 . From this perspective, our results suggest that the relative performance of dynamic strategies decreases with the size and density of the network, as well as with the length of the game. This might be due to the increasing complexity of finding such strategies or to the emergence of more central nodes, which ought to be the sole target of players, as the size and density of the network grows.

Conclusion

We consider the problem of maximizing influence through the choice of a sequence of targets when two players compete for predominance over a social network. We show that the problem can be modelled by a stochastic game with compact state space. This raises the question of the existence of a uniform value, which we answer affirmatively under the assumption that the adjacency matrix of the network is primitive. Hence, strategic players can design optimal targeting strategies independently of the length of the game.

Further analysis of the structure of equilibrium strategies shows that they depend strongly on the level of information available to each player about the actions of his opponents, and hence about the state of opinions in the network. In the absence of information, there exist pure equilibria only if players can devise strategies that are efficient independently of the current state of the network, i.e., they must adopt a purely forward-looking perspective on the diffusion of their influence. When players are well informed, they can implement more flexible strategies that take into account the state of opinions in the network and implement a trade-off between a forward-looking perspective, according to which they try to maximize the future spread of their opinion in the network, and a backward-looking perspective, according to which they try to counteract their opponent's previous actions. This flexibility sustains the existence of equilibria in pure strategies in networks where there is no such equilibrium with low levels of information or in the static game à la [START_REF] Grabisch | Strategic influence in social networks[END_REF] with fixed targets.

These results emphasize the relevance of adopting a dynamic approach, in particular in view of empirical applications: sequential strategies allow one to solve the targeting problem in a wide range of networks and independently of the length of the game. It is likely impossible to obtain a complete analytical characterization of these strategies, but it seems to us that an algorithmic approach could be fruitfully applied to the problem in future work. The model could also be extended, analytically or numerically, along with a number of dimensions, such as number of strategic players, number of targets, or cost of targeting.

  is the sum of the weights of those paths of length less than n ∈ N leading to k and satisfying the constraintC, i.e., ζ n,C k (M ) = e ′ ∑ n t=2 ∑ p∈P t,C k (M)m p (where the constraints are truncated according to path length). In vector form, one has ζ n,C (M ) = e ′ ∑ n-1 t=1 ∏ t s=1 M -Cs , where, given C s ∈ C, M -cs denotes the matrix obtained from M by substituting 0 for those rows whose indices are in C s .

  Proposition 4.20 For every positive measure y ∈ R K + over K, the game G y admits a value w * y ∶

  Number of runs (out of 20 Monte-Carlo simulations for each parameter combination) in which the genetic algorithm identifies a dynamic strategy that dominates any static strategy in games of length 10 (left), 25 (center), and 50 (right).

In the following, we shall also use the notation Γn(x 1 ) to emphasize the dependence on the initial vector of opinions x 1 .

This fact was already noticed in a weaker form by[START_REF] Amir | Stochastic games in economics and related fields: an overview[END_REF] in deterministic stochastic games with complete information.

The default genetic algorithm implemented in matlab.

The fact that the genetic algorithm did not always find an improvement upon static strategies does not mean that such an improvement did not exist, but rather that the algorithm hadn't been able to identify one. Only a comprehensive exploration of the set of all dynamic strategies could allow one to make a categorical statement about non-existence, but this would rapidly lead to combinatorial explosion.

Acknowledgement: Mandel acknowledges the support of the EU H2020 FET project DOLFINS (grant nb. 640772). Venel acknowledges the support of the Agence National de la Recherche [ANR CIGNE ANR-15-CE38-0007-01].