Prophylaxis of Epidemic Spreading with Transient Dynamics
PProphylaxis of Epidemic Spreading with TransientDynamics
G´eraldine Bouveret a Antoine Mandel b July 16, 2020
Abstract
We investigate the containment of epidemic spreading in networks from a nor-mative point of view. We consider a susceptible/infected model in which agents caninvest in order to reduce the contagiousness of network links. In this setting, westudy the relationships between social efficiency, individual behaviours and networkstructure. First, we exhibit an upper bound on the Price of Anarchy and prove thatthe level of inefficiency can scale up to linearly with the number of agents. Second,we prove that policies of uniform reduction of interactions satisfy some optimal-ity conditions in a vast range of networks. In setting where no central authoritycan enforce such stringent policies, we consider as a type of second-best policy theshift from a local to a global game by allowing agents to subsidise investments incontagiousness reduction in the global rather than in the local network. We thencharacterise the scope for Pareto improvement opened by such policies through anotion of Price of Autarky, measuring the ratio between social welfare at a globaland a local equilibrium. Overall, our results show that individual behaviours canbe extremely inefficient in the face of epidemic propagation but that policy cantake advantage of the network structure to design efficient containment policies.
Keywords: Network, Epidemic Spreading, Public Good, Price of AnarchyJEL codes: D85, D62, I18 a Nanyang Technological University, School of Physical and Mathematical Sciences, 21 NanyangLink, 637371 Singapore, Email: [email protected] b Universit´e Paris 1 Panth´eon-Sorbonne, 17 rue de la Sorbonne, Paris, IL 75005, France, Email: [email protected] a r X i v : . [ ec on . T H ] J u l Introduction
Limiting epidemic spreading is today’s key policy concern in the context of the COVID-19 pandemic. Beyond diseases, a number of socio-economic bads diffuse through socialnetworks via epidemic-like processes, e.g., financial distress or fake news. However, thereis, to our knowledge, no normative analysis of the challenges posed by the containmentof epidemic spreading. This is the issue we address in this paper.The containment of epidemic processes defines a specific class of externality prob-lems: through prophylactic investment, agents can reduce their own contamination riskbut also reduce the risk of contagion of their peers in the network. The external effecthence created has certain features of a public good as the investment of each agentbenefits to all the agents to whom it is connected. However, the magnitude of the effectdepends on the specific connectivity between each pair of agents and thus on the struc-ture of the network. In this setting, we aim to characterise, as a function of the networkstructure, (i) socially optimal containment policies, (ii) inefficiencies induced by indi-vidual strategic behaviours, and (iii) the type of measures that can be implemented toovercome these inefficiencies.We place ourselves in a setting where the network structure is given, each agentcan be initially contaminated with a certain probability, and contagion spreads throughnetwork links proportionally to their contagiousness. Once infected, agents remainso permanently, i.e. we consider a susceptible/infected type of model according to theepidemiological terminology. In this context, agents aim at minimising their probabilityof contagion before a given date. In a narrow interpretation, this date can be seen as theexpected date at which a treatment will be available. In a broader sense, the objectiveof each individual is to reduce the speed of incoming epidemic propagation. We assumethat agents can invest in the network to reduce the speed of contagion. More precisely,they can decrease the contagiousness of links at a fixed linear cost. As the impact ofindividual investments depends on global contagiousness, and hence on the investmentof other players, the situation defines a non-cooperative game. We consider two variantsof the game. The local game in which an agent can only invest in the links throughwhich it is connected. The global game in which an agent can invest in each link of thenetwork. The local game naturally applies to settings where agents are individuals thatcan take individual and costly measures to limit their social interactions. The global2ame corresponds to a more complex setting where agents are usually organisations(regions, countries) that are involved in a scheme that allows one agent to subsidise,directly or indirectly, the investment of other agents in the reduction of contagiousness.Our main results characterise the relationships between social efficiency, individualbehaviours and network structure. First, we derive an upper bound on the Price ofAnarchy (PoA). We show that in worst cases the level of inefficiency can scale up tolinearly with the number of agents. This strongly calls for public policy interventions.In this respect, we study the optimality of a policy of uniform reduction of interactions,akin to social distancing measures put in place during the COVID-19 pandemic, in awide range of networks. This latter result provides normative foundations for the socialdistancing policies implemented during the COVID-19 pandemic. The implementationof such policies nevertheless requires the existence of an authority with sufficient legit-imacy to implement such coercive measures. It can be thus implemented in a domesticcontext but is much harder to implement at the global scale, unless all agents/countrieshave individual incentives to do so. If this is not the case, we regard the shift froma local to a global game as a type of second-best policy. In the latter game, agentscan subsidise investments towards contagiousness diminution in the global rather thanin the local network. The scope for Pareto improvement generated by such policies isthen characterised through a notion of Price of Autarky (PoK), which assesses the ratiobetween social welfare at a global and a local equilibrium. We derive a lower boundon this PoK as a function of the network structure and thus give sufficient conditionsunder which a shift to the global game actually induces a Pareto improvement. Over-all, the results derived not only underline the possible extreme inefficiency of individualbehaviours to limit epidemic propagation, but also the potential for policies to benefitfrom the network structure to design efficient containment policies.The remaining of this paper is organised as follows. Section 2 reviews the relatedliterature. Section 3 introduces epidemic dynamics as well as our behavioural modelof the containment of epidemic spreading. Section 4 provides our main results onthe relationship between individual behaviours, social efficiency and network structure.Section 5 introduces the notion of PoK and applies it to our setting. Section 6 concludes.An appendix with the proofs of the main results is provided.3
Related literature
The paper builds on the very large literature on the optimal design and defense of net-works (see, e.g., Bravard et al. (2017)) and on epidemic spreading in networks. Thelatter literature has been extensively reviewed in Pastor-Satorras et al. (2015) and gen-erally combines an epidemiological model with a diffusion model. The epidemiologicalmodel describes the characteristics of the disease via the set of states each agent canassume, e.g., susceptible/infected (SI), susceptible/infected/susceptible (SIS), suscep-tible/infected/removed (SIR), and the probabilities of transition between these. Thediffusion model considers that the set of agents is embedded in a network structurethrough which the disease spreads in a stochastic manner. Overall, the micro-levelepidemic diffusion model is a continuous-time Markov chain model whose state spacecorresponds to the complete epidemiological status of the population. This state spaceis however too large for the full model to be computationally or analytically tractable.A large strand of the literature has thus focused on the development of good approxima-tions of the dynamics, see, e.g., Chakrabarti et al. (2008); Draief et al. (2006); Ganeshet al. (2005); Mei et al. (2017); Prakash et al. (2012); Ruhi et al. (2016); Van Mieghemet al. (2009); Wang et al. (2003). To the best of our knowledge, the most precise ap-proximation of the dynamics in the SIS/SIR setting is the N ´ intertwined model ofVan Mieghem et al. (2009). This model uses one (mean-field) approximation in theexact SIS model to convert the exact model into a set of N non-linear differential equa-tions. This transformation allows analytic computations that remain impossible withother more precise SIS models and renders the model relevant for any arbitrary graph.The N ´ intertwined model upper bounds the exact model for finite networks of size N and its accuracy improves with N . Van Mieghem and Omic (2008) have extendedthe model to the heterogeneous case where the infection and curing rates depend onthe node. Later, Van Mieghem (2013) has analytically derived the decay rate of SISepidemics on a complete graph, while Van Mieghem (2014) has proposed an exactMarkovian SIS and SIR epidemics on networks together with an upper bound for theepidemic threshold.Most of this literature has focused on SIS/SIR models in which there exists an epi-demic threshold above which the disease spreads exponentially. A key concern has thusbeen the approximation of the epidemic threshold as a function of the characteristics4f the network, and subsequently the determination of immunisation policies that al-low to reach the below-the-threshold regime (see, e.g., Chen et al. (2016, 2015); Holmeet al. (2002); Preciado et al. (2013, 2014); Saha et al. (2015); Schneider et al. (2011);Van Mieghem et al. (2011)).A handful of studies has adopted a normative approach to the issue using a game-theoretic setting. Omic et al. (2009) consider a N ´ intertwined SIS epidemic model,in which agents can invest in their curing rate. They prove the existence of a NashEquilibrium and derive its characteristics as a function of the network structure. Theyprovide a measure of social efficiency through the PoA. They also investigate two typesof policies to reduce contagiousness. The first one plays with the influence of the relativeprices of protection while the second one relies on the enforcement of an upper boundon infection probabilities. Hayel et al. (2014) have also analysed decentralised optimalprotection strategies in a SIS epidemic model. However, in their case, the curing andinfection rates are fixed and each node can either invest in an antivirus to be fullyprotected or invest in a recovery software once infected. They show that the game isa potential one, expressed the pure Nash Equilibrium for a single community/fully-mesh network in a closed form, and establish the existence and uniqueness of a mixedNash Equilibrium. They also provide a characterisation of the PoA. Finally, Goyaland Vigier (2015) examine, in a two-period model, the trade-off faced by individualsbetween reducing interaction and buying protection, and its impacts on infection rates.They analyse the equilibrium levels of interaction and protection as well as the infectionrate of the population, and show the existence of a unique equilibrium. They highlightthat individuals investing in protection are more willing to interact than those who donot invest, and establish the non-monotonic effects of changes in the contagiousness ofa disease.Yet, most of these contributions focus on situations where (i) some form of vaccine ortreatment is available and (ii) dynamics are of the SIS/SIR type. Our attention is ratheron situations where there is no known cure to the epidemic and where the objectiveis to delay its propagation through investments in the reduction of contagiousness.Therefore, we focus on the transient dynamics of the SI model. In this respect, we buildon the recent contribution of Lee et al. (2019) who provide an analytical framework torepresent the transient dynamics of the SI epidemic dynamics on an arbitrary network.In particular, they derive a tight approximation in closed-form of the solution to the SI5pidemic dynamics over all time t . The latter overcomes the shortfalls of the existinglinearised approximation (see Canright and Engø-Monsen (2006); Mei et al. (2017);Newman (2010)) by means of a thorough mathematical transformation of the systemgoverning the SI dynamics. Lee et al. (2019) have also derived vaccination policiesto mitigate the risks of potential attacks or minimise the consequences of an existingepidemic spread with a limited number of available patches or vaccines over the network.From an economic perspective, our contribution relates to the growing literature onthe private provision of public goods on network. This literature mostly focuses on therelationship between the network structure and the individual provision of public goods.It generally considers a fixed network and that the public good/effort provision of anagent only affects its neighbors. In particular, Allouch (2015) shows the existence of aNash Equilibrium in this setting under very general conditions. Bramoull´e and Kran-ton (2007) prove, in a more specific setting, that Nash Equilibria generically have aspecialised structure in which some individuals contribute and others free ride. A morerecent contribution by Kinateder and Merlino (2017) extends the models of private pro-vision of public goods to a setting with an endogenous network formation process. Yet,the network is formed in view of the benefits provided by the public good/effort offeredby connections. Hence, although related, our focus differs from this strand of literatureas, in our setting, the process of link formation per se is the source of external effects,and effects propagate throughout the network. Another related contribution is Elliottand Golub (2019), which provides a more conceptual view on the relationship betweenthe network structure and public goods. It focuses on the network of external effectsper se and characterises efficient cooperation/bargaining institutions in this framework.Our model could be subsumed into an extended version of their model which considersmulti-dimensional actions. However, their framework abstracts away from the processunderlying the interactions, which is one of our key focus. We consider N the set of natural numbers and N P N . The notation M N (resp. M N p R ` q )denotes the set of N ´ dimensional square matrices with coefficients in R (resp. R ` ).For a given M P M N , we write p M q i,j or m i,j , 1 ď i, j ď N , to refer to its element in the6 th ´ row and j th ´ column. Moreover, for any M P M N , || M || denotes its Frobenius normand for any matrix M and K in M N ˆ M N , we write M ď K if m i,j ď k i,j , @ i, j “ , ..., N . Additionally, the matrix I (resp. O ) stands for the N ´ dimensional squareidentity (resp. null) matrix.Similarly, for a N ´ dimensional column vector u P R N , u i , ď i ď N, refers to itselement in the i th ´ row while u J denotes its transpose and || u || its Euclidean norm.Additionally, for any u and v in R N ˆ R N , we let u ĺ v if u i ď v i , @ i “ , ..., N . Wedefine similarly u ă v . For a function f : R ÞÑ R and a vector u P R N , f p u q denotesthe N ´ dimensional column vector with f p u i q , ď i ď N, as entries. Moreover, isthe N ´ dimensional column vector with one as entries.We also consider diag p u q , the N ´ dimensional square diagonal matrix with u i , ď i ď N, as diagonal entries. Additionally, for the i th ´ vector of the canonical basis of R N , e i , ď i ď N, and any matrix M P M N , we define the product operator ă e i , M ą : “ ˜ N ÿ j “ e ij ˆ m j, , ..., N ÿ j “ e ij ˆ m j,N ¸ , a N ´ dimensional row vector.We define S N (resp. S N p R ` q ) as the subset of elements of M N (resp. M N p R ` q ) thatare symmetric. We observe that S N is a real vector-space of dimension N p N ` q{ p B t i,j u q ď i ď j ď N such that b t i,j u i,j “ b t i,j u j,i “ b t i,j u k,(cid:96) “ t k, (cid:96) u “ t i, j u . Accordingly, given a matrix D P S N , we let d t j,k u : “ d j,k ` d k,j . Moreover, given U Ď S N , a differentiable function φ : U Ñ R , and ¯ D P U, we denote by B φ B d t i,j u p ¯ D q the partial derivative in the direction of B t i,j u , that is B φ B d t i,j u p ¯ D q : “ B φ B d i,j p ¯ D q ` B φ B d j,i p ¯ D q , where B φ B d i,j p ¯ D q and B φ B d j,i p ¯ D q denote the partial derivatives in the directions inducedby the canonical basis of M N . Finally, for a set B , we note Card p B q its cardinal and p B q c the complementary set. We consider a finite set of agents, N “ t , , ..., N u , N ě , connected through aweighted and undirected network. The set of links is given by E Ď tt i, j u | i, j P N u A P S N p R ` q . In particular, for all i P E , a ii “
0. The agents face the risk of shifting from a good/susceptible state toa bad/infected state. This transition occurs in continuous time through an epidemicprocess over the network. At time zero, a subset of agents idiosyncratically shifts to theinfected state. Following this initial shock, infected agents contaminate their neighboursin the network with a probability that is proportional to the weight of the correspondinglink. Infected agents remain so permanently and cannot revert to the susceptible state.As intimated in Section 2, this model is known as the SI model in the epidemiologicalliterature (see, e.g., Pastor-Satorras et al. (2015)). It provides an accurate description ofepidemic dynamics at short time scale and/or when no vaccine or treatment is availableagainst the epidemic.We consider a socio-economic setting in which strategic agents can invest in thenetwork in order to reduce contagion rates. We are concerned with the characterisationof the equilibrium behaviour in this context, its relation to social efficiency, and thepotential impacts of policy on these features. Such setting captures the behaviour ofcountries facing the global propagation of an epidemic as well as that of individualsfacing its local propagation. It can also be applied to other socio-economic contextsuch as the propagation of computer viruses (see, e.g., Pastor-Satorras and Vespignani(2001)) or financial distress (see, e.g., Battiston et al. (2012)).In order to formally define the model, we first provide a detailed description ofthe epidemic dynamics and its approximation (see Section 3.3) and then introduce arepresentation of agents’ prophylactic behaviours (see Section 3.4).
Formally, an exact model of the dynamics of epidemic spreading in the SI frameworkis given by a continuous-time Markov chain p X p t qq t ě with state space X : “ t , u N . A state X p t q P X is defined by the combination of states of all the N agents at time t and is thus described by the set of susceptible agents t i P N | X i p t q “ u and theset of infected agents t i P N | X i p t q “ u . The key specificity of the dynamics of theMarkov chain p X p t qq t ě is the stochastic rate of contagion for a susceptible node i P N ,characterising its transition probability as followslim h Ñ h P r X i p t ` h q “ | X i p t q “ s “ β ÿ j P N a i,j t X j p t q“ u , (3.1)8here β is a unit contagion rate, a i,j , i, j P N , is the contagiousness of the networklink t i, j u , and P denotes the probability on the underlying probability space. Equation(3.1) characterises completely the dynamics, as infected nodes remain so permanently.It highlights the role of the network in the contagion process and the possible heteroge-neous contagiousness of different network links. In this respect, we make the followingassumption about the network structure throughout the paper. Assumption . The adjacency matrix A P S N p R ` q is irreducible and aperiodic.The irreducibility assumption amounts to considering that every agent faces a riskof contagion as soon as at least one agent in the network is infected. Indeed, the net-work is then necessarily connected and the asymptotic behaviour of the Markov chainis trivial: there is an unstable steady state where none of the agent is infected and aunique stable steady state where all agents are contaminated . In the following, weshall actually consider that agents are concerned by the time at which they are likelyto be infected rather than by their asymptotic infection status. Accordingly, we areconcerned with the transient behaviour of the Markov chain. However, as intimatedabove, the infection rate defined by the right-hand side of Equation (3.1) is a randomvariable, making the process doubly stochastic. The random nature of the infectionrate could then be offset by conditioning with respect to all the possible combinationsof the neighbouring states, i.e. X j p t q , j ‰ i. Yet, such conditioning of every nodewould lead to a Markov chain with 2 N states, i.e. with a number of states increasingexponentially with the number of nodes, and thus being neither analytically nor compu-tationally tractable. Therefore, the conventional practice in epidemiological modellingis to consider a mean-field approximation of the infection rate. In particular, the N -intertwined model of Van Mieghem et al. (2009) considers the average behaviour, i.e.the expectation, over states for the infection rate. Therefrom, the latter model derivesan approximate dynamics of the probability for an agent i P N to be in the infectedstate at time t given by x i p t q : “ P r X i p t q “ s . More precisely, using the fact that P r X i p t q “ s ` P r X i p t q “ s “ Stability must be understood in the sense that, for any initial non-null probability distribution, thelimiting distribution of the Markov chain has full support on the full contamination state. Let T i : “ inf t t ě X i p t q “ u valued in R ` . We observe that for t ě x i p t q “ P r T i ď t s . i P N , B x i p t qB t “ p ´ x i p t qq β N ÿ j “ a i,j x j p t q . (3.2)Equation (3.2) thus provides a deterministic approximation of the dynamics of thecontagion probability that takes into account the full network structure. It neverthe-less disregards the positive correlation between the infection status of neighbouringnodes. This implies that Equation (3.2) over-estimates the probability of contagion(see Van Mieghem et al. (2009)). Remark 3.1.
Alternative mean-field approximations used in the literature are generallymuch coarser that the N -intertwined model considered here. Two common approachesare (i) to average over agents and focus on the (approximate) dynamics of the averageprobability of contagion or (ii) to average over agents with equal degree and focus on the(approximate) dynamics of the average probability of contagion for an agent of a givendegree (see Pastor-Satorras et al. (2015) for an extensive review). The non-linear Equation (3.2) does not have an analytical solution. A commonapproach in the literature, used in particular to analyse the outbreak of an epidemic,is to assume x i p t q small enough to discard the factor p ´ x i p t qq and thus focus on thefollowing linear equation B x i p t qB t “ β N ÿ j “ a i,j x j p t q . (3.3)However, this approximation grows exponentially towards `8 , whereas it is assumedto approximate a probability. In a recent contribution, Lee et al. (2019) provide a muchbetter approximation of the solution of Equation (3.2). More precisely, they define forall i P N and t P R ` , y i p t q : “ ´ log p ´ x i p t qq , and observe that ¯ x : “ r ¯ x , ..., ¯ x N s J is a solution of the system defined by Equation (3.2) with initial condition x p q : “r x p q , ..., x N p qs J “ : x , with at least one non-null element to avoid triviality, if andonly if ¯ y : “ r ¯ y , ..., ¯ y N s J is a solution of the system of equations defined for all i P N by B y i p t qB t “ β ÿ j P N a i,j p ´ exp p´ y j p t qqq , (3.4)10ith the corresponding initial condition. They then show that a tight upper bound tothe solution of the system defined by Equation (3.4) when x p q “ x ă y p t q : “ ´ ln p ´ x q ` r exp p βtA diag p ´ x qq ´ I s diag p ´ x q ´ x , (3.5)and accordingly that ˇ x p t q : “ ´ exp p´ ˇ y p t qq is a tight upper bound to the solution ¯ x of the system defined by Equation (3.2) with initial condition x p q “ x in the sensethat one has (see Lee et al. (2019, Theorem 5.1 and Corollary 5.2)) • lim t Ñ`8 || ˇ x p t q ´ x p t q|| “ , • for any t ě
0, ¯ x p t q ĺ ˇ x p t q ĺ ˜ x p t q where ˜ x : “ r ˜ x , ..., ˜ x N s J is the solution of thesystem defined by Equation (3.3) with initial condition x p q “ x . Hence, ˇ x provides an approximation of the probability of contagion that is asymptot-ically exact and more accurate than the standard linear approximation, even at shorttime scale. From now on, we shall consider that agents base their assessment of the dynamics ofcontagion on the approximated contagion probabilities ˇ x associated to a given and fixedinitial condition x p q “ x ă
1, having at least one non-null element. In this sense,they make decisions on the basis of approximate information. This approach provides aconsistent representation of the decision-making situation of actual agents which oughtto base their decisions on similar approximations.In this respect, we recall that in our SI setting, all agents eventually become infected.Thus, agents cannot base their decisions on their asymptotic infection status. Rather,they shall aim at delaying the growth rate of the epidemic. This is notably the strategypursued by most countries during the recent COVID-19 pandemic. More precisely, weconsider that agents consider a target date ¯ t, which can be interpreted as the planninghorizon or the expected date of availability of a treatment, and aim at minimising theprobability of contagion up to that date. We further assume, for sake of analyticaltractability, that they have a logarithmic utility of the form u i p ˇ x i p ¯ t qq : “ δ i log p ´ ˇ x i p ¯ t qq , i P N , x i p ¯ t q is the approximate contagion probability given by Equation (3.5) and δ i ě i . One should note that the utility is non-positive and equal toa benchmark of zero if and only if there is no risk of contagion. In our setting, x , β ,and ¯ t being fixed, Equation (3.5) implies that the contagion probability is completelydetermined by the adjacency matrix A . The utility of agent i P N can thus be expresseddirectly as v i p A q : “ ´ δ i ă e i , exp p β ¯ tA diag p ´ x qq diag p ´ x q ´ x ą , (3.6)where the constant term ln p ´ x q ` diag p ´ x q ´ x has been discarded to simplifythe notations.Equation (3.6) highlights that, for a given admissible initial probability of conta-gion x , the only lever that agents can use to reduce their contagion probability is thedecrease of the contagiousness of the network, i.e. the decrease of the value of the coef-ficients of the adjacency matrix A. This is exactly the strategy put in place during theCOVID-19 pandemic, at the local scale through social distancing measures, and at theglobal scale through travel restrictions and border shutdowns (see Colizza et al. (2006)for an analysis of the role of the global transport network in epidemic propagation).Formally, we consider a strategic game in which each agent can invest in the reductionof contagiousness of network links. We distinguish two alternative settings to accountfor potential constraints on agents’ actions: • In the global game, we assume that each agent can invest in the reduction ofcontagiousness of every network link. Therefore, the set of admissible strategyprofiles is given by S p A q : “ tp D i q i P N P p S N p R ` qq N : A ´ ř i P N D i ě u . • In the local game, we assume that each agent can only invest in the links throughwhich it is connected. Therefore, the set of admissible strategy profiles is givenby K p A q : “ tp D i q i P N P S p A q : @ i P N , @ k, j P N , k, j ‰ i ñ d ik,j “ u . Local games correspond to a setting where agents are individuals that limit their socialinteractions through individual and costly measures. On the other hand, global gamesapply to a more involved setting where agents are organisations (regions, countries)that have the ability to subsidise the investment of other agents in the reduction ofcontagiousness, either directly or indirectly.12 emark 3.2.
Both S p A q and K p A q are non-empty, convex and compact sets. The payoff function is defined in a similar fashion in both settings: • First, a strategy profile p D i q i P N turns the adjacency matrix into A ´ ř i P N D i andthus yields to agent i P N a utility U i p D i , D ´ i q : “ v i p A ´ ÿ i P N D i q “ ´ δ i ă e i , exp ˜ β ¯ t p A ´ ÿ i P N D i q diag p ´ x q ¸ diag p ´ x q ´ x ą , where D ´ i : “ p D j q j P N , j ‰ i is the strategy profile of all agents but i . • Second, we consider that agents face a linear cost for their investment in thereduction of contagion. More precisely, one has for all i P N , C i p D i q : “ ρ J D i “ ρ ÿ j,k P N d ij,k , where ρ ą • Overall, the payoff of agent i P N given a strategy profile p D i q i P N is given by Π i p D i , D ´ i q : “ U i p D i , D ´ i q ´ C i p D i q“ ´ δ i ă e i , exp ˜ β ¯ t p A ´ ÿ i P N D i q diag p ´ x q ¸ diag p ´ x q ´ x ą ´ ρ J D i . A few remarks are in order about the characteristics of the game. First, agents’ strategysets are constrained by the choices of other players. Namely, given a strategy profilefor the other players D ´ i P p S N p R ` qq N ´ , the set of admissible strategies for player i is S i p A, D ´ i q : “ t D i P S N p R ` q | p D i , D ´ i q P S p A qu (resp. K i p A, D ´ i q : “ t D i P S N p R ` q |p D i , D ´ i q P K p A qu ) in the global (resp. local) game. Although, it is not the moststandard, this setting is comprehensively analysed in the literature (see, e.g., Rosen(1965)). Second, linear cost is a natural assumption in our framework. Indeed, themarginal cost paid to decrease the contagiousness of a link should not depend on theidentity of the player investing. Third, the payoff function is always non-positive as itis the combination of both a utility and a cost that are always non-positive. Finally, forlarge ¯ t and homogeneous initial contagion probabilities, the utility can be approximatedthrough the eigenvector centrality of the contagion network, as detailed in the followinglemma. Definition 3.1 ( p α q -Homogeneous Game) . A game is p α q -homogeneous if it involvesan homogeneous initial probability of contagion, i.e. such that for all i P N , x i “ α for some α P p , q . emma 3.1. Consider an p α q -Homogeneous Game, and let D P S p A q be such that A ´ ř i P N D i is irreducible and aperiodic. Let then µ denote the Perron-Frobeniuseigenvalue of p A ´ ř i P N D i q , | µ ´ µ | the spectral gap and v the normalised eigenvectorassociated to µ , corresponding to the eigenvector centrality of the network. One has exp ˜ β ¯ t p ´ α qp A ´ ÿ i P N D i q ¸ “ p ` O p exp p´ β ¯ t p ´ α q| µ ´ µ |qqq exp p β ¯ t p ´ α q µ q vv J . Remark 3.3. As A is irreducible and aperiodic, the condition ř i P N d ij,k ă a j,k for all j, k P N such that a j,k ą , is sufficient for getting the irreducibility and aperiodicity of A ´ ř i P N D i . We now state some properties on the marginal utility and the payoff function (recallthe notations for the partial derivatives of a symmetric matrix in Section 3.1).
Lemma 3.2.
For every i P N , for any strategy profile p D i , D ´ i q P S p A q , and for all k, (cid:96) P N , B U i p¨ , D ´ i qB d i t k,(cid:96) u p D i q “ B U i p¨ , D ´ i qB d ik,(cid:96) p D i q ` B U i p¨ , D ´ i qB d i(cid:96),k p D i q“ δ i β ¯ t ˜ exp p β ¯ t p A ´ ÿ j P N , j “ i D j ´ D i q diag p ´ x qq ¸ i,k x (cid:96) ` δ i β ¯ t ˜ exp p β ¯ t p A ´ ÿ j P N , j “ i D j ´ D i q diag p ´ x qq ¸ i,(cid:96) x k , (3.7) and the marginal utility is non-negative. Moreover, the map Π i p¨ , D ´ i q is concave on S i p A, D ´ i q . From the previous lemma derives the following result relating the marginal utilityof every agent to the utility itself, when the initial probability of infection is constantamong agents.
Lemma 3.3.
Consider an p α q -Homogeneous Game, and let M Ď N . For every i P N , and for any strategy profile p D i , D ´ i q P S p A q , ÿ k,(cid:96) P N k or (cid:96) P M B U i p¨ , D ´ i qB d i t k,(cid:96) u p D i q “ ´ p M q δ i β ¯ tα ÿ k P N ˜ exp p β ¯ t p ´ α qp A ´ ÿ i P N D i qq ¸ i,k “ ´ p M q β ¯ t p ´ α q U i p¨ , D ´ i qp D i q . (3.8)14 .5 Nash Equilibrium and Social Optimum In the following, unless otherwise specified, we consider as implicitly given the utilityweights δ : “ r δ , ..., δ N s J , the time-horizon ¯ t, the unit contagion rate β , the initialcontagion matrix A , the initial contagion probabilities x , and the investment cost ρ .We then define the “local game” L p δ , A, β, ¯ t, x , ρ q as the game with strategy profilesin K p A q and payoff function Π and the “global game” G p δ , A, β, ¯ t, x , ρ q as the one withstrategy profiles in S p A q and payoff function Π . In this setting, a Nash Equilibrium isdefined as follows.
Definition 3.2 (Nash Equilibrium) . • An admissible set of strategies ˇ D : “ p ˇ D i q i P N P S p A q is a Nash Equilibrium forthe global game if @ i P N , @ D i P S i p A, ˇ D ´ i q , Π i p ˇ D i , ˇ D ´ i q ě Π i p D i , ˇ D ´ i q . • An admissible set of strategies ¯ D : “ p ¯ D i q i P N P K p A q is a Nash Equilibrium forthe local game if @ i P N , @ D i P K i p A, ¯ D ´ i q , Π i p ¯ D i , ¯ D ´ i q ě Π i p D i , ¯ D ´ i q . The existence of a Nash Equilibrium follows from standard arguments.
Theorem 3.1.
There exists a Nash Equilibrium in both the local and global games.
Remark 3.4.
In our setting, equilibrium is in general not unique as there might be in-determinacy on the identity of the players/neighbours which ought to invest in reducingthe contagion of a link (see the discussions in Section 4.1 below).
The key concern, in the remaining of this paper, is the study of the efficiency ofNash Equilibrium. As commonly done in N -agent games, and in particular in networkgames, we define as Social Optimum, the outcome that maximises the equally-weightedsum of individual utilities. Definition 3.3 (Social Optimum) . An admissible set of strategies ˆ D : “ p ˆ D i q i P N P S p A q is a Social Optimum if ˆ D “ argmax p D i q i P N P S p A q ÿ i P N Π i ` D ´ i , D i ˘ . ř i P N Π i p D i , D ´ i q only depends on the value of ř i P N D i . First, this im-plies that the notion of Social Optimum is the same in the local and global game. Indeed,it is straightforward to check that for every p D i q i P N P S p A q , there exists p ˜ D i q i P N P K p A q such that ř i P N ˜ D i “ ř i P N D i . Second, given a matrix D P S N p R ` q such that D ď A, we shall let ˆΠ p D q : “ ÿ i P N ˆ v i p D q ´ ρ ÿ j,k P N d j,k , where ˆ v i : D ÞÑ v i p A ´ D q , and, with a slight abuse of notation, state that ˆ D is a SocialOptimum if it is such that ˆΠ p D q is maximal over D p A q : “ t D P S N p R ` q : D ď A u .The existence of a Social Optimum directly follows from the continuity of ˆΠ and thecompactness of D p A q . Theorem 3.2.
There exists a Social Optimum in both the local and global games.
We end this section with two lemmas that are the counterparts of Lemma 3.2-3.3 above for the Social Optimum D P D p A q and whose proofs are a straightforwardadaptation of the proofs of the latter lemmas. Lemma 3.4.
For every i P N , for any Social Optimum D P D p A q , and for all k, (cid:96) P N , B ˆ v i p D qB d t k,(cid:96) u “ δ i β ¯ t p exp p β ¯ t p A ´ D q diag p ´ x qqq i,k x (cid:96) ` δ i β ¯ t p exp p β ¯ t p A ´ D q diag p ´ x qqq i,(cid:96) x k ě . Moreover, the map ˆ v i p¨q is concave on D p A q . Lemma 3.5.
Consider an p α q -Homogeneous Game, and let M Ď N . For every i P N , and for any Social Optimum D P D p A q , ÿ k,(cid:96) P N k or (cid:96) P M B ˆ v i p D qB d t k,(cid:96) u “ ´ p M q β ¯ t p ´ α q ˆ v i p D q . (3.9) In this section, we provide a characterisation of equilibrium behaviours as a functionof the network structure. We then measure, through the PoA, potential inefficienciesinduced by individual behaviours, giving insight on how to restore optimality in thelocal game. 16 .1 Characterisation of equilibrium behaviours
Let i P N . Individual prophylactic efforts are characterised by the investment in con-tagion reduction D i P S N p R ` q that the individual performs in the links it has access to( S p A q or K p A q ). As emphasised above, the network of contagion is assumed undirectedand thus investments in the link t k, (cid:96) u P E induce an equal reduction of the contagionweights a k,(cid:96) and a (cid:96),k . More precisely, the marginal impact of the investment made byplayer i is characterised by Equation (3.7) above. This equation highlights the bilateralimpact of agents’ investments on contagion from (cid:96) to k on the one hand and from k to (cid:96) on the other hand. The impact on agent i of reduced contagiousness through d ik,(cid:96) (from (cid:96) to k ) depends on (i) the utility weight δ i , (ii) the initial contagiousness of node (cid:96), x (cid:96) , and (iii) the connectivity between k and i measured, for D : “ p D i q i P N , through C i,k p A, D, β, ¯ t, x q : “ β ¯ t ´ exp p β ¯ t p A ´ ř j P N D j q diag p ´ x qq ¯ i,k , or alternatively C i,k p A, D, β, ¯ t, x q “ n ! ÿ n “ p β ¯ t q n ` ˜ p A ´ ÿ j P N D j q diag p ´ x q ¸ ni,k . Hence, the connectivity from k to i, C i,k p A, D, β, ¯ t, x q , depends on the initial structureof the contagion network A, the strategic investments in the reduction of contagiousness D , the unit contagion rate β , the time-horizon ¯ t, and the initial contagion probabilities x . It corresponds to the discounted sum of paths from k to i where each path is as-signed a weight corresponding to the probability that it propagates a genuine contagionsequence. The weight of each link across the path is thus obtained as the product of(i) the probability that the link propagates the contagion, which is measured throughthe corresponding coefficient p A ´ ř j P N D j q , and (ii) the probability that the end nodeis not initially infected, which is measured through the corresponding coefficient ofdiag p ´ x q . Overall, the marginal impact of agents’ actions on contagiousness dependson the characteristics of the disease, measured through the initial contagion probabil-ity, and on the structure of the contagion network modified by the agents’ investments.Equation (3.7) offers a differential characterisation of Nash Equilibria in both the localand global games, as reported in the following two propositions.
Proposition 4.1.
A strategy profile ¯ D P K p A q is a Nash Equilibrium of the local game L p δ , A, β, ¯ t, x , ρ q if and only if for all t k, (cid:96) u P E , the following two conditions hold:(1) One of the following alternative holds: a) ρ ă max i Pt k,(cid:96) u δ i ` C i,k p A, ¯ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ¯ D, β, ¯ t, x q x k ˘ and ¯ d k t k,(cid:96) u ` ¯ d (cid:96) t k,(cid:96) u “ a t k,(cid:96) u , (b) ρ ą max i Pt k,(cid:96) u δ i ` C i,k p A, ¯ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ¯ D, β, ¯ t, x q x k ˘ and ¯ d k t k,(cid:96) u “ ¯ d (cid:96) t k,(cid:96) u “ , (c) ρ “ max i Pt k,(cid:96) u δ i ` C i,k p A, ¯ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ¯ D, β, ¯ t, x q x k ˘ and ¯ d k t k,(cid:96) u ` ¯ d (cid:96) t k,(cid:96) u P r , a t k,(cid:96) u s . (2) For any i P N , one has ¯ d i t k,(cid:96) u ą only if δ i ` C i,k p A, ¯ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ¯ D, β, ¯ t, x q x k ˘ ě ρ . Proposition 4.2.
A strategy profile ˇ D P S p A q is a Nash Equilibrium of the global game G p δ , A, β, ¯ t, x , ρ q if and only if for all t k, (cid:96) u P E , the following two conditions hold:(1) One of the following alternative holds: (a) ρ ă max i P N δ i ` C i,k p A, ˇ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ˇ D, β, ¯ t, x q x k ˘ and ř i P N ˇ d i t k,(cid:96) u “ a t k,(cid:96) u , (b) ρ ą max i P N δ i ` C i,k p A, ˇ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ˇ D, β, ¯ t, x q x k ˘ and ř i P N ˇ d i t k,(cid:96) u “ , (c) ρ “ max i P N δ i ` C i,k p A, ˇ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ˇ D, β, ¯ t, x q x k ˘ and ř i P N ˇ d i t k,(cid:96) u P r , a t k,(cid:96) u s . (2) For any i P N , one has ˇ d i t k,(cid:96) u ą only if δ i ` C i,k p A, ˇ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, D, β, ¯ t, x q x k ˘ ě ρ . The difference between Proposition 4.1 and 4.2 stems from the fact that differentsets of agents can invest in a given link: the agents at the nodes corresponding to thatlink in the local case and all agents in the global case. Otherwise, their interpretationis similar. If the investment cost is large with respect to the marginal utility of playersat equilibrium, there is no investment in the link. If the investment cost is smaller thanthe marginal utility of at least one player at equilibrium, there is full investment in thelink, i.e. it is completely suppressed. Finally, there is the “interior” case in which onlythe agents with the largest marginal utility invest in the link. They do so up to thepoint where their individual marginal utility equals the investment cost. The followingdefinition summarises the different characterisations of equilibria.
Definition 4.1 (Equilibrium characterisation) . • A local (resp. global) Full Investment Equilibrium is an equilibrium that satisfies,for all t k, (cid:96) u P E , case (a) of Proposition 4.1 (resp. 4.2). • A local (resp. global) No Investment Equilibrium is an equilibrium that satisfies,for all t k, (cid:96) u P E , case (b) of Proposition 4.1 (resp. 4.2). • A local (resp. global) Interior Equilibrium is an equilibrium that satisfies, for all t k, (cid:96) u P E , case (c) of Proposition 4.1 (resp. 4.2). A local (resp. global) Homogeneous Interior Equilibrium is a special case of local(resp. global) Interior Equilibrium where, for each t k, (cid:96) u P E , the marginal utilitiesof agents k, (cid:96) (resp. all agents) are equal. We observe that in the case of a Full Investment Equilibrium or an Interior Equilib-rium, there can be an indeterminacy on the identities of the agents that invest. Namely,let E G t k,(cid:96) u p D q : “ t i P N | δ i p C i,k p A, D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, D, β, ¯ t, x q x k q ě ρ u , be theset of players susceptible to invest in the link t k, (cid:96) u P E at an equilibrium D of the globalgame and E L t k,(cid:96) u p D q : “ t i P t k, (cid:96) u | δ i p C i,k p A, D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, D, β, ¯ t, x q x k q ě ρ u , be the set of players susceptible to invest in the link t k, (cid:96) u P E at an equilibrium D of the local game. Proposition 4.3 below, resulting from Proposition 4.1-4.2, highlightsa form of substitutability of investments that arises at equilibrium. Proposition 4.3.
Let D be an equilibrium of the global game G p δ , A, β, ¯ t, x , ρ q (resp.local game L p δ , A, β, ¯ t, x , ρ q ). Assume that ˜ D P S p A q (resp. ˜ D P K p A q ) is such thatfor all t k, (cid:96) u P E , one has:(1) ř i P N ˜ d i t k,(cid:96) u “ ř i P N d i t k,(cid:96) u ,(2) For any i P N , ˜ d i t k,(cid:96) u ą only if i P E G t k,(cid:96) u (resp. i P E L t k,(cid:96) u ).Then ˜ D is an equilibrium of the global (resp. local) game. Hence, each player that has a large enough marginal utility is willing to invest ina link up to the equilibrium level independently of the actions of other players. Thisleads to indeterminacy on the allocation of investments (and thus of the related costs)among players that have a large enough marginal utility.
Remark 4.1.
Consider the game G p δ , A, β, ¯ t, x , ρ q (resp. L p δ , A, β, ¯ t, x , ρ qq and itsequilibrium D . We observe that, whenever for some i, j, k, (cid:96) P N , δ i p C i,k p¨q x (cid:96) ` C i,(cid:96) p¨q x k qp A, D, β, ¯ t, x q ě δ j p C j,k p¨q x (cid:96) ` C j,(cid:96) p¨q x k qp A, D, β, ¯ t, x q , then j P E G t k,(cid:96) u (resp. j P E L t k,(cid:96) u ) implies i P E G t k,(cid:96) u (resp. i P E L t k,(cid:96) u provided that i “ k or (cid:96) ). Moreover, Proposition 4.1-4.3 imply that Full Investment Equilibria and InteriorEquilibria have a notable property: they induce equilibria in each network that is morestrongly connected than the equilibrium network (i.e. with a weight on each link higherthan the one of the corresponding link in the equilibrium network). This property isformally stated in the following proposition.19 roposition 4.4.
Let D be a Full Investment Equilibrium or an Interior Equilibriumof the global game G p δ , A, β, ¯ t, x , ρ q (resp. local game L p δ , A, β, ¯ t, x , ρ q ). Then forall ˜ A ě A ´ ř i P N D i , any strategy profile ˜ D P S p ˜ A q (resp. ˜ D P K p ˜ A q ) such that ř i P N ˜ D i : “ ř i P N D i ` ˜ A ´ A is a Full Investment Equilibrium or an Interior Equilibriumof G p δ , ˜ A, β, ¯ t, x , ρ q (resp. L p δ , ˜ A, β, ¯ t, x , ρ q ). Finally, both equilibria induce the sameequilibrium network. We end this section discussing a salient example of existence of global equilibria inlocal strategies, that of a symmetric game and an almost fully-connected network.
Definition 4.2 ( p α, δ q -Symmetric Game & Complete/ p a q -Complete Network) . • A game is p α, δ q -symmetric if for all i P N , δ i “ δ for some δ ą and x i “ α for some α P p , q . • A network is complete if for all k, (cid:96) P N , k ‰ (cid:96), a k,(cid:96) ą . In particular, thenetwork is p a q -complete if for all k, (cid:96) P N , k ‰ (cid:96), a k,(cid:96) “ a for some a ą . Example 4.1.
Consider an p α, δ q -Symmetric Game and an p a q -Complete Network.The game is then symmetric and, using the non-emptiness, convexity and compactnessproperties of the strategy space as well as the continuity and concavity properties ofthe payoff, there exists a symmetric equilibrium in both the local and global games (seeCheng et al. (2004, Theorem 3)). We shall show that both equilibria coincide and that,for ρ in an appropriate range, they are interior. Indeed, let ˇ D P S p A q , be a symmetricequilibrium of the global game. According to Equation (3.7), one has for all i, k, (cid:96) P N , B U i p¨ , ˇ D ´ i qB d i t k,(cid:96) u p ˇ D i q “ δβ ¯ tα p exp p H q i,k ` exp p H q i,(cid:96) q , (4.1)where H is of the form H : “ ¨˚˚˚˚˚˝ h . . . hh . . . . . . ...... . . . . . . hh . . . h ˛‹‹‹‹‹‚ with h : “ β ¯ t p ´ α qp a ´ ÿ i P N ˇ d ik,(cid:96) q , k, (cid:96) P N , k ‰ (cid:96) . (4.2)20sing a Taylor expansion, one can prove that exp p H q is of the formexp p H q : “ ¨˚˚˚˚˚˝ χ p h q χ p h q ´ exp p´ h q . . . χ p h q ´ exp p´ h q χ p h q ´ exp p´ h q . . . . . . ...... . . . . . . χ p h q ´ exp p´ h q χ p h q ´ exp p´ h q . . . χ p h q ´ exp p´ h q χ p h q ˛‹‹‹‹‹‚ , where χ p h q : “ ` ř k ě u k { k !, with p u k q k P N zt u satisfying the following recursive system $&% u “ v “ hu k “ h p N ´ q v k ´ and v k “ h rp n ´ q v k ´ ` u k ´ s for k ě . As χ p h q ą χ p h q ´ exp p´ h q , it follows from Equation (4.1) that for all distinct elements i, k, (cid:96) P N , one has B U i p¨ , ˇ D ´ i qB d i t i,(cid:96) u p ˇ D i q ą B U k p¨ , ˇ D ´ k qB d k t i,(cid:96) u p ˇ D k q . Using Proposition 4.2, this yields the following characterisation:(1) The symmetric equilibrium is such that h “ , or equivalently ř i P N ˇ D i “ A, leading to χ p q “
1, if only if ρ ď δβ ¯ tα. (2) The symmetric equilibrium is such that h “ β ¯ t p ´ α q a, or equivalently ř i P N ˇ D i “ O if and only if ρ ě δβ ¯ tα ˜ χ p β ¯ t p ´ α q a q where ˜ χ p β ¯ t p ´ α q a q : “ χ p β ¯ t p ´ α q a q ´ exp p´ β ¯ t p ´ α q a q . (3) The symmetric equilibrium is interior if and only if ρ {p δβ ¯ tα q P p , ˜ χ p β ¯ t p ´ α q a qq .In the third case, the equilibrium is a local Homogeneous Interior Equilibrium, while,in the first two cases, there exists an equilibrium in local strategies that is equivalentto ˇ D in the sense of Proposition 4.3.As stated below, in view of Proposition 4.4, Example 4.1 induces a Full InvestmentEquilibrium or an Interior Equilibrium in each network that is sufficiently connected. Proposition 4.5.
Consider an p α, δ q -Symmetric Game where there exists ε ą suchthat for all k, (cid:96) P N , k ‰ (cid:96), a k,(cid:96) ě ε. Then, one has in both the local and global games: This system has a closed-form solution that can be determined by elementary methods. However,its expression is too inconvenient to report it in full length here.
1) If ρ {p δβ ¯ tα q ă ˜ χ p β ¯ t p ´ α q ε q , there exists a Full Investment Equilibrium or anInterior Equilibrium.(2) If, moreover ρ {p δβ ¯ tα q ą , there exists an Interior Equilibrium. Using Lemma 3.4, one can provide a differential characterisation of Social Optima, asreported in the following proposition.
Proposition 4.6.
A strategy profile ˆ D P D p A q is a Social Optimum if and only if forall t k, (cid:96) u P E , one of the following alternative holds: (a) ρ ă ř i P N δ i ´ C i,k p A, ˆ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ˆ D, β, ¯ t, x q x k ¯ and ˆ d t k,(cid:96) u “ a t k,l u , (b) ρ ą ř i P N δ i ´ C i,k p A, ˆ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ˆ D, β, ¯ t, x q x k ¯ and ˆ d t k,(cid:96) u “ , (c) ρ “ ř i P N δ i ´ C i,k p A, ˆ D, β, ¯ t, x q x (cid:96) ` C i,(cid:96) p A, ˆ D, β, ¯ t, x q x k ¯ and ˆ d t k,(cid:96) u P r , a t k,l u s . Hence at a Social Optimum, there is investment in a link only if the sum of marginalutilities induced by the investment is larger than or equal to the investment cost. If thecost is smaller than the sum of marginal utilities, then the link is completely suppressed(case (a)). On the other hand, if the solution is interior, then the level of investment issuch that the sum of marginal utilities is exactly equal to the investment cost (case (c)).By analogy with the case of Nash Equilibria, we introduce the following definition.
Definition 4.3 (Social Optimum characterisation) . • A Full Investment Optimum is an optimum that satisfies, for all t k, (cid:96) u P E , case(a) of Proposition 4.6. • A No Investment Optimum is an optimum that satisfies, for all t k, (cid:96) u P E , case(b) of Proposition 4.6. • An Interior Optimum is an optimum that satisfies, for all t k, (cid:96) u P E , case (c) ofProposition 4.6. The comparison between Proposition 4 . Loc : “ | Worst social welfare at a local Nash Equilibrium || Social welfare at a Social Optimum | , (4.3)PoA Glo : “ | Worst social welfare at a global Nash Equilibrium || Social welfare at a Social Optimum | . (4.4)By construction, the PoA is greater or equal to 1 and equal to 1 only when all NashEquilibria of the game are socially optimal. An increasing PoA corresponds to anincreasing social inefficiency of individual behaviours at a Nash Equilibrium.Theorem 4.1 and 4.2 below provide a partial characterisation of the PoA in the caseof an p α q -Homogeneous Game and Complete Network. Theorem 4.1.
Consider an p α q -Homogeneous Game and a Complete Network. Assumethat the global game G p δ , A, β, ¯ t, x , ρ q is such that: the worst Nash Equilibrium ˇ D isnot a Full Investment Equilibrium and one of the corresponding Social Optima ˆ D is nota Null Investment Optimum. Then PoA
Glo ď N β ¯ t p ´ α q ` J ř i P N ˇ D i N β ¯ t p ´ α q ` J ˆ D ď N ` β ¯ t p ´ α q N J A . The upper bound for PoA
Loc has a stronger dependence on the structure of thenetwork.
Theorem 4.2.
Consider an p α q -Homogeneous Game and a Complete Network with N ě . Assume that the local game L p δ , A, β, ¯ t, x , ρ q is such that: the worst NashEquilibrium ¯ D is not a Full Investment Equilibrium and one of the corresponding SocialOptima ˆ D is not a Null Investment Optimum. Then PoA
Loc ď β ¯ t p ´ α q ” N ρ ` p N ´ q α p N ´ q ř i P N K i p δ i , A, ¯ D, ¯ D, β, ¯ t, x q ı ` ρ J ř i P N ¯ D i ρ ” N β ¯ t p ´ α q ` J ˆ D ı ď N ` p N ´ q αN p N ´ q ρ ÿ i P N K i p δ i , A, O, A, β, ¯ t, x q ` β ¯ t p ´ α q N J A , here for any adjacency matrix B and strategy profiles D, D , K i p δ i , B, D, D , β, ¯ t, x q : “ ř k P N k ‰ i δ i p C i,k p B, D, β, ¯ t, x q ´ C i,i p B, D , β, ¯ t, x qq .In particular, for all i, k P N , k ‰ i , ď ρ ` αδ i r C i,k p A, ¯ D, β, ¯ t, x q ´ C i,i p A, ¯ D, β, ¯ t, x qs ď r ρ ´ αδ i C i,i p A, ¯ D, β, ¯ t, x qs . Theorem 4.1 and 4.2 imply that the PoA grows at most linearly with the number ofagents. As highlighted in the proposition below (that recalls the notations of Example4.1), the case of a Symmetric Game shows that one cannot improve this linear bound.
Proposition 4.7.
Consider an p α, δ q -Symmetric Game and an p a q -Complete Networkwith N ě . Assume that ρ {p δβ ¯ tα q P p , ˜ χ p β ¯ t p ´ α q a qq , and let ˇ D (resp. ˆ D ) be theworst Nash Equilibrium (resp. one of the corresponding Social Optima). Then PoA
Glo “ PoA
Loc “ N ρ β ¯ t p ´ α q ´ N p N ´ q δα exp p´ h q p ´ α q ` ρ J ř i P N ˇ D i Nρ β ¯ t p ´ α q ` ρ J ˆ D ď N ´ p N ´ q δβ ¯ tα exp p´ h q ρ ` β ¯ t p ´ α q N J A , where h : “ β ¯ t p ´ α qp a ´ ř i P N ˇ d ik,(cid:96) q ą , for any k, (cid:96) P N , k ‰ (cid:96) . In particular, ρ ´ δβ ¯ tα exp p´ h q ě . Our previous results highlight the fact that individual strategic behaviours can leadto major inefficiencies in the containment of epidemic spreading. In particular, therecan be complete free-riding of other players on the investment of the agent that is themost affected by the epidemic (Proposition 4.1 and 4.2) and the inefficiency can scaleup to linearly with the number of agents (Theorem 4.1 and 4.2 and Proposition 4.7).In other words, individual strategic behaviours can be highly inefficient in terms ofsocial welfare as soon as there is a large number of agents involved. This is the case inreal-world applications whether one considers epidemic spreading between individualsat the domestic scale or between countries at the global scale.Against this backdrop, it is natural to search for a public policy response for theprevention of epidemic spreading. During the recent COVID-19 outbreak, a widespreadpolicy response has been the implementation of social distancing measures that havereduced, in a uniform way, the scale of social interactions. Formally, we can de-fine the social distancing policy at level κ P R ` as restricting social interactions to24 p A, κ q : “ p c i,j q i,j P N such that for all i, j P N , c i,j : “ κa i,j { ř k P N a i,k (or equiva-lently c i,j : “ κa i,j { ř k P N a k,i as A is symmetric). The level κ P R ` must be such that A ´ C p A, κ q ě O . Hence, the social distancing policy amounts to bounding the level ofsocial interactions of each agent to a fixed level. In practice, this has been implementedby massive restrictions on socio-economic activities such as interdiction of public gath-erings, closing of schools and businesses, and travel restrictions. A formal analysis ofthis policy in our framework shows it can be socially efficient, at least if the initial con-tagion probability and the disutility are assumed to be uniform. Namely, it is optimalin the following sense. Proposition 4.8.
Consider a p α, δ q -Symmetric Game and assume that ř k P N a i,k ą for all i P N .1. If δβ ¯ tα ě ρ, then κ “ is optimal and the optimal social distancing measureinvolves the suppression of every link.2. If δβ ¯ tα ă ρ, then for every ε ą , there exists ¯ T ą such that for ¯ t ě ¯ T , onecan find δβ ¯ tα ă ρ ď δβ ¯ tα exp p β ¯ t p ´ α q ˆ min i P N ř k P N a i,k q for which thereexists an admissible κ ą satisfying ˆΠ p A ´ C p A, κ qq ě max D P D p A q ˆΠ p D q ´ ε . In this case, for every ε ą and for ¯ t large enough, there is an ε ´ optimal socialdistancing measure which does not involve any suppression of link. Hence, uniform reduction of social interactions appears as being an extremely ef-ficient policy in our framework. This appears as a natural counterpart to existingresults in the literature that emphasise the role of highly connected nodes, e.g.,“superspreaders”, in epidemic propagation (see, e.g., Pastor-Satorras and Vespignani (2001);Pastor-Satorras et al. (2015)). Our result would benefit from further research on theextent to which one could relax the assumption that the “implicit” costs of connectiv-ity reduction are homogeneous. This assumption indeed neglects the fact that certainactors might value more social interactions because of their economic, psychological,or social characteristics. Moreover, results on the determination of the optimal levelof social distancing would also be welcome. In practice, the level of social distanc-ing has been determined according to policy decisions about the socially/economicallyacceptable rate of contagion, rather than inferred from individual preferences.25
The PoK
Social distancing measures can be implemented at the domestic scale in order to re-duce the propagation of epidemics between individuals. However, at the internationalscale, there is no authority entitled to implement such coercive measures. Furthermore,individual countries can take measures to reduce their interactions with other coun-tries, e.g., border closures, but cannot directly reduce interactions between two othercountries. They are thus, by default, in the framework of a local game. One couldnevertheless consider schemes in which countries with a higher disutility from infectionsubsidise investments in other parts of the network to reduce global contagiousness.This would turn the problem into a global game. In order to compare outcomes inthese two situations, we introduce the notion of PoK, which corresponds to the ratiobetween the social welfare at the worst equilibrium of the local game and at the bestequilibrium of the global game.PoK : “ |
Worst social welfare at a Nash Equilibrium of the local game || Best social welfare at a Nash Equilibrium of the global game | . In particular, if PoK ą
1, then global equilibria are better than local ones. On thecontrary if global and local equilibria coincide (as in Example 4.1), PoK “
1. Thelatter implies in particular that an increase in the set of admissible strategies does notnecessarily induce an increase in individual welfare. Sometimes it could even lead tomore free riding. In general, the value of the PoK is determined by the network structureand the individual disutilities associated to contagion, measured by the coefficients δ i , i P N . In particular, following the lines of the proofs of Theorem 4.1-4.2, one canprovide an explicit lower bound on the PoK, as detailed in the theorem below. Theorem 5.1.
Consider an p α q -Homogeneous Game and a Complete Network with N ě . Assume that the local game L p δ , A, β, ¯ t, x , ρ q and global game G p δ , A, β, ¯ t, x , ρ q are such that: the worst local Nash Equilibrium ¯ D is an Homogeneous Interior Equi-librium and the best global Nash Equilibrium ˇ D is not a Full Investment Equilibrium. hen PoK ě β ¯ t p ´ α q “ N p N ´ q ρ ´ p N ´ q α ř i P N δ i C i,i p A, ¯ D, β, ¯ t, x q ‰ ` ρ J ř i P N ¯ D i ρ ” N β ¯ t p ´ α q ` J ř i P N ˇ D i ı (5.1) ě Nβ ¯ t p ´ α q N β ¯ t p ´ α q ` J A . In particular, for all i, k P N , k ‰ i , ď ρ ` αδ i r C i,k p A, ¯ D, β, ¯ t, x q ´ C i,i p A, ¯ D, β, ¯ t, x qs “ r ρ ´ αδ i C i,i p A, ¯ D, β, ¯ t, x qs . (5.2)As hinted above, in the case of a Complete Network, under the assumptions ofProposition 4.7, the conditions of Theorem 5.1 are satisfied and PoK “
1. Moreover,Equation (5.1)-(5.2) above highlight the fact that the PoK increases when there existsagents i P N with a large disutility of contagion δ i that are highly connected to othernodes in the network, as measured by C i,k p¨q , i, k P N , k ‰ i. A salient example of suchcase is that of an p a q -Complete Network where one of the agents has a much higherdisutility of contagion than its peers. We thus intend to study this example and morespecifically to consider the limit case where δ “ δ e i for some i P N , i.e. where thedisutility of all other agents is negligible with respect to that of agent i. In this settingthere is no indeterminacy on the agent that is playing and we can give a necessarycondition for global strategies to dominate local ones. This is the aim of the followingproposition.
Proposition 5.1.
Fix i P N . Consider the local game L p δ e i , A, β, ¯ t, x , ρ q for an p α q -Homogeneous Game and an p a q -Complete Network with N ě . If δ i β ¯ tα ă ρ ă δ i β ¯ tα ˆ sinh p? N ´ β ¯ t p ´ α q a q? N ´ ` cosh p? N ´ β ¯ t p ´ α q a q ˙ , then one has:(1) Any local equilibrium ¯ D is an Homogeneous Interior Equilibrium and is such that ¯ D j “ O , for all j P N , j ‰ i, and ¯ d ii,k “ ¯ d ik,i “ h for all t i, k u P E for some h P r , a s .
2) Global strategies dominate local ones if and only if the parameters β, ¯ t, α, and a are such that sinh p? N ´ β ¯ t p ´ α qp a ´ h qq ą ? N ´ p? N ´ β ¯ t p ´ α qp a ´ h qq . In this paper, we have investigated the prophylaxis of epidemic spreading from a nor-mative point of view in a game-theoretic setting. Agents have the common objective toreduce the speed of propagation of an epidemic of the SI type through investments inthe reduction of the contagiousness of network links. Despite this common objective,strategic behaviours and free-riding can lead to major inefficiencies. We have shownthat the PoA can scale up to linearly in our setting. This strongly calls for public inter-vention to reduce the speed of diffusion. In this respect, we have shown that a policy ofuniform reduction of social interactions, akin to the social distancing measures enforcedduring the COVID-19 pandemic, can be ε -optimal in a wide range of networks. Suchpolicies thus have strong normative foundations. Our results however assume that thecost of reducing interactions is uniform among agents. The validity of this assumptionstrongly depends on the scope of the analysis: it is a much more benign approxima-tion when the focus is on public health than in the case where economic and financialconsiderations ought to be taken into account.We have partly accounted for heterogeneity as far as the benefits of prophylaxis areconcerned. In this setting, we have shown that allowing agents to subsidise investmentsin the reduction of contagiousness in distant parts of the network can be Pareto im-proving. This result calls for further research on the design of mechanisms to improvethe efficiency of cooperation against epidemic spreading. Appendix
Proof. [ of Lemma 3.1 ] The proof follows from the spectral decomposition of A ´ ř i P N D i and a direct application of the Perron-Frobenius theorem. See Lee et al. (2019,Appendix C) for details. Proof. [ of Lemma 3.2 ] Let i P N and p D i , D ´ i q be a strategy profile in S p A q . For28 , (cid:96) P N , one has B U i p¨ , D ´ i qB d i t k,(cid:96) u p D i q “ δ i ă e i , β ¯ t exp p β ¯ t p A ´ ÿ j P N , j “ i D j ´ D i q diag p ´ x qqp ˆ I k,(cid:96) ` ˆ I (cid:96),k q x ą , where for any k, j P N , ˆ I k,j is the N ´ dimensional square matrix with null entriesexcept on the k th ´ row and j th ´ column for which the entry is equal to one. This leadsto Equation (3.7). Moreover, for all k, (cid:96), p, q P N , we compute B U i p¨ , D ´ i qB d ik,(cid:96) B d ip,q p D i q“ ´ ă e i , p β ¯ t q exp p β ¯ t p A ´ ÿ j P N , j “ i D j ´ D i q diag p ´ x qq ˆ I p,q diag p ´ x q ˆ I k,(cid:96) x ą“ $&% ´p β ¯ t q ´ exp p β ¯ t p A ´ ř j P N , j “ i D j ´ D i q diag p ´ x qq ¯ i,p p ´ x q q x (cid:96) if k “ q . Therefore the Hessian matrix of U i p¨ , D ´ i q on S i p A, D ´ i q is the matrix of a quadraticform that is negative semi-definite. The concavity property thus follows. Proof. [ of Theorem 3.1 ] We know from Remark 3.2 that the sets S p A q and K p A q of admissible strategies are compact and convex, and from Lemma 3.2 that the ob-jective function Π is concave on S i p A, D ´ i q and K i p A, D ´ i q with i P N and D ´ i Pp S N p R ` qq N ´ . Moreover, since Π is continuous in its arguments, we therefore concludefrom Rosen (1965, Theorem 1) that a Nash Equilibrium exists. Proof. [ of proposition 4.1 ] For 1 ď i ď N , the optimisation programme for charac-terising the Nash Equilibria writesmax D i P M N Π i p D i , ¯ D ´ i q subject to: d ik,(cid:96) “ d i(cid:96),k , @ k, (cid:96) P N , (cid:96) ă k , (A-1) d ik,(cid:96) “ , @ k, (cid:96) P N , (cid:96) ą k, k and (cid:96) ‰ i ,d ik,(cid:96) ě , @ k, (cid:96) P N , (cid:96) ą k, k or (cid:96) “ i , p A ´ ¯ D ´ i ´ D i q k,(cid:96) ě , @ k, (cid:96) P N , (cid:96) ě k . (A-2)Applying the Karush-Kuhn-Tucker conditions, we obtain that ¯ D i P K i p A, ¯ D ´ i q is asolution if and only if, for any t k, (cid:96) u P E , one of the following cases holds, assumingwithout loss of generality that B U k p¨ , ¯ D ´ k q{B d k t k,(cid:96) u p ¯ D k q ď B U (cid:96) p¨ , ¯ D ´ (cid:96) q{B d (cid:96) t k,(cid:96) u p ¯ D (cid:96) q :29a) (i) ρ ă B U k p¨ , ¯ D ´ k q{B d k t k,(cid:96) u p ¯ D k q and ¯ d k t k,(cid:96) u ` ¯ d (cid:96) t k,(cid:96) u “ a t k,(cid:96) u , (ii) B U k p¨ , ¯ D ´ k q{B d k t k,(cid:96) u p ¯ D k q ă ρ ă B U (cid:96) p¨ , ¯ D ´ (cid:96) q{B d (cid:96) t k,(cid:96) u p ¯ D (cid:96) q and ¯ d k t k,(cid:96) u “
0, ¯ d (cid:96) t k,(cid:96) u “ a t k,(cid:96) u , (iii) B U k p¨ , ¯ D ´ k q{B d k t k,(cid:96) u p ¯ D k q “ ρ ă B U (cid:96) p¨ , ¯ D ´ (cid:96) q{B d (cid:96) t k,(cid:96) u p ¯ D (cid:96) q and ¯ d k t k,(cid:96) u P r , a t k,(cid:96) u s , ¯ d (cid:96) t k,(cid:96) u “ a t k,(cid:96) u ´ ¯ d k t k,(cid:96) u , (b) ρ ą B U (cid:96) p¨ , ¯ D ´ (cid:96) q{B d (cid:96) t k,(cid:96) u p ¯ D (cid:96) q and ¯ d k t k,(cid:96) u “ ¯ d (cid:96) t k,(cid:96) u “ , (c) (i) B U k p¨ , ¯ D ´ k q{B d k t k,(cid:96) u p ¯ D k q ă ρ “ B U (cid:96) p¨ , ¯ D ´ (cid:96) q{B d (cid:96) t k,(cid:96) u p ¯ D (cid:96) q and ¯ d k t k,(cid:96) u “
0, ¯ d (cid:96) t k,(cid:96) u P r , a t k,(cid:96) u s , (ii) B U k p¨ , ¯ D ´ k q{B d k t k,(cid:96) u p ¯ D k q “ ρ “ B U (cid:96) p¨ , ¯ D ´ (cid:96) q{B d (cid:96) t k,(cid:96) u p ¯ D (cid:96) q and 0 ď ¯ d k t k,(cid:96) u ` ¯ d (cid:96) t k,(cid:96) u ď a t k,(cid:96) u . Proof. [ of Proposition 4.2 ] For 1 ď i ď N , the optimisation programme for charac-terising the Nash Equilibria writesmax D i P M N Π i p D i , ˇ D ´ i q subject to:Condition (A-1)-(A-2) ,d ik,(cid:96) ě , @ k, (cid:96) P N , (cid:96) ą k . Applying the Karush-Kuhn-Tucker conditions, we obtain that ˇ D i P S i p A, ˇ D ´ i q is asolution if and only if, for any t k, (cid:96) u P E , one of the following cases holds:(a) (i) ρ ă min i P N ´ B U i p¨ , ˇ D ´ i q{B d i t k,(cid:96) u p ˇ D i q ¯ and ř q P N ˇ d q t k,(cid:96) u “ a t k,(cid:96) u , (ii) there exists i, j P N , such that B U i p¨ , ˇ D ´ i q{B d i t k,(cid:96) u p ˇ D i q ă ρ ă B U j p¨ , ˇ D ´ j q{B d j t k,(cid:96) u p ˇ D j q andˇ d q t k,(cid:96) u “ q P A i p ˇ D q , ř q P ¯ A j p ˇ D q ˇ d q t k,(cid:96) u “ a t k,(cid:96) u , where for a given h ě A h p ˇ D q : “ ď r ď N : B U r p¨ , ˇ D ´ r qB d r t k,(cid:96) u p ˇ D r q ď B U h p¨ , ˇ D ´ h qB d h t k,(cid:96) u p ˇ D h q + , and¯ A h p ˇ D q : “ ď r ď N : B U r p¨ , ˇ D ´ r qB d r t k,(cid:96) u p ˇ D r q ě B U h p¨ , ˇ D ´ h qB d h t k,(cid:96) u p ˇ D h q + , (iii) there exists i, j P N , such that B U i p¨ , ˇ D ´ i q{B d i t k,(cid:96) u p ˇ D i q “ ρ ă B U j p¨ , ˇ D ´ j q{B d j t k,(cid:96) u p ˇ D j q and 0 ď ř q P ˆ A i p ˇ D q ˇ d q t k,(cid:96) u ď a t k,(cid:96) u , ˇ d q t k,(cid:96) u “ q P ` ¯ A i p ˇ D q ˘ c , ř q P ¯ A j p ˇ D q ˇ d q t k,(cid:96) u “ a t k,(cid:96) u ´ ř q P ˆ A i p ˇ D q ˇ d q t k,(cid:96) u , where for a given h ě A h p ˇ D q : “ ď r ď N : B U r p¨ , ˇ D ´ r qB d r t k,(cid:96) u p ˇ D r q “ B U h p¨ , ˇ D ´ h qB d h t k,(cid:96) u p ˇ D h q + , (b) ρ ą max i P N ´ B U i p¨ , ˇ D ´ i q{B d i t k,(cid:96) u p ˇ D i q ¯ and ř q P N ˇ d q t k,(cid:96) u “ , c) there exists j P N , such that ρ “ B U j p¨ , ˇ D ´ j q{B d j t k,(cid:96) u p ˇ D j q and ˇ d q t k,(cid:96) u “ q P ` ¯ A j p ˇ D q ˘ c ,0 ď ř q P ˆ A j p ˇ D q ˇ d q t k,(cid:96) u ď a t k,(cid:96) u . Proof. [ of Proposition 4.6 ] For 1 ď i ď N , the optimisation programme for charac-terising the Social Optima writesmax D P M N ˆΠ p D q subject to: d k,(cid:96) “ d (cid:96),k , @ k, (cid:96) P N , (cid:96) ă k ,d k,(cid:96) ě , @ k, (cid:96) P N , (cid:96) ą k , p A ´ D q k,(cid:96) ě , @ k, (cid:96) P N , (cid:96) ě k . The proof is then a straightforward adaptation of the proof of Proposition 4.1 and 4.2.
Proof. [ of Theorem 4.1 ] It follows from the assumption on ˇ D that for all i P N , ÿ k,(cid:96) P N k ‰ (cid:96) B U i p¨ , ˇ D ´ i qB d i t k,(cid:96) u p ˇ D i q ď N p N ´ q ρ . (A-3)Therefore appealing to Equation (3.8) and Equation (A-3), we obtain ÿ i P N Π i p ˇ D i , ˇ D ´ i q “ ÿ i P N U i p ˇ D i , ˇ D ´ i q ´ ρ J ÿ i P N ˇ D i “ ´ p N ´ q β ¯ t p ´ α q ÿ i P N ÿ k,(cid:96) P N k ‰ (cid:96) B U i p¨ , ˇ D ´ i qB d i t k,(cid:96) u p ˇ D i q ´ ρ J ÿ i P N ˇ D i ě ´ ρ « N β ¯ t p ´ α q ` J ÿ i P N ˇ D i ff . (A-4)Similarly, it follows from the assumption on ˆ D that it is such that for all k, (cid:96) P N , k ‰ (cid:96) , ÿ i P N B ˆ v i p ˆ D qB d t k,(cid:96) u ě ρ . (A-5)31e deduce from Equation (3.9) and Equation (A-5),ˆΠ p ˆ D q “ ÿ i P N ˆ v i p ˆ D q ´ ρ J ˆ D “ ´ p N ´ q β ¯ t p ´ α q ÿ i P N ÿ k,(cid:96) P N k ‰ (cid:96) B ˆ v i p ˆ D qB d t k,(cid:96) u ´ ρ J ˆ D ď ´ ρ „ N β ¯ t p ´ α q ` J ˆ D . (A-6)Combining Equation (A-4) and (A-6) and using Equation (4.4), we obtain the result. Proof. [ of Theorem 4.2 ] We know from Equation (3.7) that for all i, k, (cid:96) P N , B U i p¨ , ¯ D ´ i qB d i t k,(cid:96) u p ¯ D i q “ αδ i r C i,k p A, ¯ D, β, ¯ t, x q ` C i,(cid:96) p A, ¯ D, β, ¯ t, x qs . (A-7)Therefore, it follows from the assumption on ¯ D that for all i, (cid:96) P N , i ‰ (cid:96) , B U i p¨ , ¯ D ´ i qB d i t i,(cid:96) u p ¯ D i q “ αδ i r C i,i p A, ¯ D, β, ¯ t, x q ` C i,(cid:96) p A, ¯ D, β, ¯ t, x qs ď ρ . (A-8)Hence Equation (A-7)-(A-8) give that for all distinct i, k, (cid:96) P N , B U i p¨ , ¯ D ´ i qB d i t k,(cid:96) u p ¯ D i q ď ρ ` αδ i r C i,k p A, ¯ D, β, ¯ t, x q ´ C i,i p A, ¯ D, β, ¯ t, x qs (A-9) ď r ρ ´ αδ i C i,i p A, ¯ D, β, ¯ t, x qs . (A-10)We deduce from Equation (A-9)-(A-10) that for all i P N , ÿ k,(cid:96) P N k ‰ (cid:96) B U i p¨ , ¯ D ´ i qB d i t k,(cid:96) u p ¯ D i q “ ÿ (cid:96) P N (cid:96) ‰ i B U i p¨ , ¯ D ´ i qB d i t i,(cid:96) u p ¯ D i q ` ÿ k P N k ‰ i B U i p¨ , ¯ D ´ i qB d i t k,i u p ¯ D i q ` ÿ k,(cid:96) P N k ‰ (cid:96)k,(cid:96) ‰ i B U i p¨ , ¯ D ´ i qB d i t k,(cid:96) u p ¯ D i qď N p N ´ q ρ ` p N ´ q αδ i ÿ k P N , k ‰ i p C i,k p¨q ´ C i,i p¨qqp A, ¯ D, β, ¯ t, x q (A-11) ď p N ´ qrp N ´ q ρ ´ p N ´ q αδ i C i,i p A, ¯ D, β, ¯ t, x qs . In particular, we observe from the non-decreasing property of marginal utilities (recallLemma 3.2), that for all i P N , ρ ` αδ i r C i,k p¨q ´ C i,i p¨qsp A, ¯ D, β, ¯ t, x q ě @ k P N , k ‰ i, and ρ ´ αδ i C i,i p A, ¯ D, β, ¯ t, x q ě . ÿ i P N Π i p ¯ D i , ¯ D ´ i q “ ÿ i P N U i p ¯ D i , ¯ D ´ i q ´ ρ J ÿ i P N ¯ D i “ ´ p N ´ q β ¯ t p ´ α q ÿ i P N ÿ k,(cid:96) P N k ‰ (cid:96) B U i p¨ , ¯ D ´ i qB d i t k,(cid:96) u p ¯ D i q ´ ρ J ÿ i P N ¯ D i ě ´ β ¯ t p ´ α q « N ρ ` p N ´ q α p N ´ q ÿ i P N K i p δ i , A, ¯ D, ¯ D, β, ¯ t, x q ff ´ ρ J ÿ i P N ¯ D i . Appealing to Equation (A-6) and Equation (4.3), the result follows.
Proof. [ of Proposition 4.7 ] According to Example 4.1, under the holding assump-tions, ˇ D is a local Homogeneous Interior Equilibrium which yields an equilibrium net-work of the form given by Identity (4.2). More precisely, for all distinct i, k, (cid:96) P N , B U i p¨ , ˇ D ´ i qB d i t i,(cid:96) u p ˇ D i q “ δβ ¯ tα p χ p h q ´ exp p´ h qq “ ρ and B U i p¨ , ˇ D ´ i qB d i t k,(cid:96) u p ˇ D i q “ ρ ´ δβ ¯ tα exp p´ h q . In particular, it follows from the non-decreasing property of marginal utilities (recallLemma 3.2) that ρ ´ δβ ¯ tα exp p´ h q ě
0. It is then straightforward to check that ÿ i P N Π i p ˇ D i , ˇ D ´ i q “ ´ N ρ β ¯ t p ´ α q ` N p N ´ q δα exp p´ h q p ´ α q ´ ρ J ÿ i P N ˇ D i . On the other hand, one can assume without loss of generality that ˆ D is of the formˆ D : “ ¨˚˚˚˚˚˝ d . . . ˆ d ˆ d . . . . . . ...... . . . . . . ˆ d ˆ d . . . ˆ d ˛‹‹‹‹‹‚ , for some ˆ d ě . Indeed, by concavity of ˆΠ , we know the set of Social Optima is convex. Moreover, giventhe symmetry of the game, the set of Social Optima shall be invariant by permutation.Thus, the average of all socially optimal profiles is socially optimal and must be sym-metric, i.e. of the form ˆ D. The proof is thus concluded proceeding as in the proof ofTheorem 4.1 to prove Equation (A-6), and recalling Equation (4.4).
Proof. [ of Proposition 4.8 ] Let us first remark that in the case where 2 δβ ¯ tα ě ρ, one can check that ˆ D “ A is a Social Optimum. This amounts to saying that C p A, q
33s optimal and thus allows to conclude. We now consider the case where 2 δβ ¯ tα ă ρ .One can easily check that for every ¯ t ą δβ ¯ tα ă ρ ď δβ ¯ tα exp p β ¯ t p ´ α q ˆ min i P N ř k P N a i,k q such that there exists 0 ă ¯ κ ď min i P N ř k P N a i,k satisfying2 δβ ¯ tα exp p β ¯ t p ´ α q ¯ κ q “ ρ . (A-12)Let us then recall that for any κ ě
0, and k, (cid:96) P N , B ˆΠ B d t k,(cid:96) u p A ´ C p A, κ qq “ ÿ i P N δβ ¯ tα exp p β ¯ t p ´ α q C p A, κ qq i,k ` ÿ i P N δβ ¯ tα exp p β ¯ t p ´ α q C p A, κ qq i,(cid:96) ´ ρ . Now, it is straightforward to check that, for κ ą
0, the largest eigenvalue of β ¯ t p ´ α q C p A, κ q is µ : “ β ¯ t p ´ α q κ. According to the Perron-Frobenius theorem, this largesteigenvalue is simple. Furthermore, the associated normalised eigenvector is v “ p {? N q . Thus, applying Lemma 3.1, one getsexp p β ¯ t p ´ α q C p A, κ qq “ exp p β ¯ t p ´ α q κ q V p ` O p exp p´| µ ´ µ |qqq , where V : “ v v J and µ denotes the second largest eigenvalue in module.One shall then notice that for all i, j P N , p V q i,j “ { N, so that B ˆΠ B d t k,(cid:96) u p A ´ C p A, κ qq “ δβ ¯ tα exp p β ¯ t p ´ α q κ q p ` O p exp p´| µ ´ µ |qqq ´ ρ . Noting that, all the other parameters being fixed, the spectral gap | µ ´ µ | is increasingwith respect to ¯ t and κ, one can assume that for every ε ą , there exists ¯ T ą t ě ¯ T ,2 δβ ¯ tα exp p β ¯ t p ´ α q κ q p ` O p exp p´| µ ´ µ |qqq ď δβ ¯ tα exp p β ¯ t p ´ α q κ q ` ε { } A } , for all κ ą . Combining the latter with Equation (A-12), one concludes that C p A, ¯ κ q is an approximate critical point in the sense that for all t k, (cid:96) u P E , ˇˇˇˇˇ B ˆΠ B d t k,(cid:96) u p A ´ C p A, ¯ κ qq ˇˇˇˇˇ ď ε { } A } . (A-13)Furthermore, if ˆ D denotes the Social Optimum, one has by construction } A ´ C p A, ¯ κ q ´ ˆ D } ď } A } . (A-14)34ow, ˆΠ being continuous and differentiable, one gets through the mean value theorem | ˆΠ p A ´ C p A, ¯ κ qq ´ ˆΠ p ˆ D q| ď ˇˇˇˇˇ B ˆΠ B d t k,(cid:96) u p A ´ C p A, ¯ κ qq ˇˇˇˇˇ ˆ } A ´ C p A, ¯ κ q ´ ˆ D } , leading, using Equations (A-13) and (A-14), to the required result | ˆΠ p A ´ C p A, ¯ κ qq ´ ˆΠ p ˆ D q| ď ε . Proof. [ of Proposition 5.1 ] We assume without loss of generality that i “
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