Learning and Selfconfirming Equilibria in Network Games
LLearning and Selfconfirming Equilibria in Network Games ∗ Pierpaolo Battigalli a,c , Fabrizio Panebianco b , and Paolo Pin a,c,da Department of Decision Sciences, Universit`a Bocconi, Milan, Italy b Department of Economics and Finance, Universit`a Cattolica, Milan, Italy c IGIER, Bocconi, Italy d BIDSA, Bocconi, ItalyDecember 2018
Abstract
Consider a set of agents who play a network game repeatedly. Agents may not know thenetwork. They may even be unaware that they are interacting with other agents in a network.Possibly, they just understand that their payoffs depend on an unknown state that in realityis an aggregate of the actions of their neighbors. Each time, every agent chooses an actionthat maximizes her subjective expected payoff and then updates her beliefs according to whatshe observes. In particular, we assume that each agent only observes her realized payoff. Asteady state of such dynamic is a selfconfirming equilibrium given the assumed feedback.We characterize the structure of the set of selfconfirming equilibria in network games and werelate selfconfirming and Nash equilibria. Thus, we provide conditions on the network underwhich the Nash equilibrium concept has a learning foundation, despite the fact that agents mayhave incomplete information. In particular, we show that the choice of being active or inactivein a network is crucial to determine whether agents can make correct inferences about the payoffstate and hence play the best reply to the truth in a selfconfirming equilibrium. We also studylearning dynamics and show how agents can get stuck in non–Nash selfconfirming equilibria. Insuch dynamics, the set of inactive agents can only increase in time, because once an agent findsit optimal to be inactive, she gets no feedback about the payoff state, hence she does not changeher beliefs and remains inactive.
JEL classification codes: C72 , D83 , D85 . ∗ We thank Federico Bobbio, Yann Bramoull´e, Ben Golub, Paola Moscariello, Alessandro Pavan, Yves Zenou, andseminar participants at AMSE, Bocconi, Milano Bicocca, Siena and UTS Sydney. Pierpaolo Battigalli and PaoloPin gratefully acknowledge funding from, respectively, the European Research Council (ERC) grant 324219 and theItalian Ministry of Education Progetti di Rilevante Interesse Nazionale (PRIN) grant 2015592CTH. a r X i v : . [ ec on . T H ] D ec Introduction
Imagine an online social network, like Twitter, with many users. Let us consider a simultaneous-moves game, in which each user i decides her level of activity a i ≥ i receives idiosyncratic externalities, that can be positive and negative, from the other users withwhom she is in contact in the social network. The externality from user i to user j is proportional tothe time that they both spend on the social network, a i and a j . Sticking to a quadratic specification,that allows for linear best replies, let us assume that the payoff of i from this game is u i ( a i , a − i ) = α i a i − a i + (cid:88) j ∈ I \{ i } z ij a i a j . (1)In eq. (1), I is the set of agents in the social network and a i is the level of activity of i ∈ I , while α i represents the individual pleasure of i from being active on the social network in isolation, whichresults in the bliss point of activity in autarchy. Parameter α i can also be negative, and in this case i would not be active in isolation. For each j ∈ I \ { i } , there is some exogenous level of externalityfrom j to i denoted by z ij . We say that j affects i , or that j is a peer of i , if z ij (cid:54) = 0.Later on, in this paper, we will also consider an extra global term in the payoff function u i ( a i , a − i ) = αa i − a i + (cid:88) j ∈ I \{ i } z ij a i a j + β (cid:88) k ∈ j ∈ I \{ i } a k . (2)We can interpret this extra term as an additional pleasure that i gets from being member (even ifnot active) of an online social network that is overall popular .In this paper, the network described by the matrix Z of all the z ij ’s is exogenous . As a firstapproximation, this fits a directed online social network like Twitter or Instagram, where users can-not decide who follows them. Under this interpretation, i receives positive or negative externalitiesfrom those who follow her, that are proportional to her activity. i acquires popularity from beingactive or not in the social network. Payoff represents what i can indirectly observe about her ownpopularity (i.e. likes that she receives, people congratulating with her in real world conversations,and so on. . . ). We imagine that i cannot choose the style of what she writes, since she just followsher exogenous nature. In this interpretation, a i represents the amount of tweets that i writes,and this can make her more or less popular for those who follow her, according to how her stylecombines with the (typically unobserved) tastes of each of her followers.Since we are going to analyze learning dynamics and their steady states, we also have to specifywhat agents observe after their choices, because this affects how they update their beliefs. Twitter This is the class of games originally analyzed by Ballester et al. (2006). Bramoull´e et al. (2014) is one of themore recent papers providing results for such linear-qadratic network games, and they discuss also how to generalizeto games that have the same best–reply functions. Zenou (2016) surveys many applications. i typically observes perfectly her own activity level a i , but she may not observe the signof the externalities and the activity of others. However, she gets indirect measures of her level ofpopularity that come from her conversations and experiences in the real world, where her popularityfrom Twitter affects her social and professional real life. Players of this game may have wrong beliefsabout the details of the game they are playing (e.g. the structure of the network, or the value ofthe parameters) and about the actions of other players. With this, they update their beliefs inresponse to the feedback they receive, which will be their (possibly indirectly measured) payoff.This updating process may lead to a learning dynamic that does not converge to a Nash equilibriumof the game.In this paper we address the following question: Assuming simple updating rules, under whatcircumstances do learning dynamics converge to a Nash equilibrium of the game and when, instead,do they just converge to a selfconfirming equilibrium where agents best reply to confirmed butpossibly wrong beliefs? This question is per se interesting, and with our answers we provide noveltheoretical tools for the analysis of network games. However, the application of the model to onlinesocial networks that we just anticipated can also help in understanding why we may easily observeapparently non–optimal best responses by economic agents in such an environment, such as agentswho get stuck into “inactivity traps.”Section 2 presents our baseline model. For this setting, we characterize the set of selfcon-firming equilibria in Section 3, and we study the learning process in Section 4. In Section 5 weanalyze a more general model that accounts for global externalities. Section 6 concludes. Wedevote appendices to proofs and technical results. Appendix A analyzes properties of feedbackand selfconfirming equilibria in a class of games including as special cases the network games thatwe consider. Appendix B reports existing results in linear algebra, that we use to find sufficientconditions for reaching interior Nash equilibria in network games. Appendix C contains the proofsof our propositions. Consider a set I of agents, with cardinality n = | I | and generic element i , located in a network. Letthe network be characterized by an adjacency matrix Z ∈ R I × I , where entry z ij specifies whetheragent i is linked to agent j (cid:54) = i and the weight of this link, and we let z ii = 0 by convention. Inwhat follows we consider the case of directed networks, so that, given i, j ∈ I , we allow z ij > z ji = 0. Externality weights are an unknown parameter of the model. We assume that thereare commonly known upper and lower bounds ¯ w and w in the weighted externalities, that can bepositive or negative, between players. We let Θ ⊆ [ w, ¯ w ] I × I denote the compact set of possibleweighted networks Z . The network game is parametrized by Z ∈ Θ.Throughout the paper we will play with different properties and specifications of matrix Z . Tosimplify the notation we will often decompose it in a way that distinguishes between the actual3inks, that specify if there is an externality between two players, and the magnitude and the signof this externality. We call Z ∈ { , } I × I the basic underlying representation of the network, theadjacency matrix whose ij element specifies whether the action of j has an externality on i . Wethink of it as a link from i to j because j is one of i ’s peers. Z is a directed network.On top of that we build Z adding weights on the links of Z . This can be done in several ways,depending on how much heterogeneity we want to allow for. We will write Z = γ Z when all linksbear the same level of externality γ ∈ [ w, ¯ w ]. We will write Z = ΓZ , where Γ is a diagonal matrix,when we want to specify that each player i is affected by the same weight γ i ∈ [ w, ¯ w ] from all herpeers, but these γ i ’s are heterogeneous. We will also consider the case in which the existing linksmay have weights of different signs but the same intensity. That is, we write Z = S (cid:12) Z (in whichthe operator (cid:12) is the Hadamard product), for γ ∈ [ w, ¯ w ], and S ∈ {− γ, γ } I × I . Finally, when wewrite simply Z we consider the case of a directed weighted network Z ∈ Θ. Many of our results willhold for this most general case.Each agent i ∈ I chooses an action a i from interval A i = [0 , ¯ a i ], where the upper bound ¯ a i is “sufficiently large”. For each i ∈ I , A − i := × j (cid:54) = i A j denotes the set of feasible action profiles a − i = ( a j ) j ∈ I \{ i } for players different from i . Similarly, defining N i := { j ∈ I : z ij (cid:54) = 0 } as the setof the neighbors of a given agent i , A N i := × j ∈ N i A j denotes the set of feasible action profiles a N i := ( a j ) j ∈ N i of i ’s neighbors.For each i ∈ I , we posit a set (interval) X i = [ x i , ¯ x i ] of payoff states for i , with the inter-pretation that i ’s payoff is determined by her action a i and by her payoff state x i according to acontinuous utility function v i : A i × X i → R . The payoff state x i is in turn determined by theactions of i ’s neighbors and is unknown to i at the time of his choice. For each agent i ∈ I andmatrix Z , we consider a parametrized aggregator of the coplayers’ actions (cid:96) i : A − i × Θ → X i ofthe following form: (cid:96) i is continuous, its range (cid:96) i ( A − i × Θ) is connected , and for each Z ∈ Θ, thesection of (cid:96) i at Z is (cid:96) i, Z : A − i → X i , a − i (cid:55)→ (cid:80) j (cid:54) = i z ij a j .Note, since X i is the codomain of (cid:96) i , we are effectively assuming that, for every Z ∈ Θ, x i ≤ (cid:88) j ∈ N − i z ij ¯ a j , ¯ x i ≥ (cid:88) j ∈ N + i z ij ¯ a j ,where N − i := { j ∈ I : z ij < } denote the set of neighbors of player i that have a negative effect Note that in the network literature it is common to assume A i = R + . However, for the games we consider, wecan always find an upper bound ¯ a on actions such that the problem is unchanged when actions are bounded aboveby ¯ a . In principle we can allow for non–linear aggregators, as in Feri and Pin (2017). However, in this paper, we focuson the linear case.
4n the payoff state of i . Similarly, N + i := { j ∈ I : z ij > } denotes the set of neighbors of player i that have a positive effect on the payoff state of i .The overall payoff function that associates each action profile ( a i , a − i ) with a payoff for agent i is thus parametrized by the adjacency matrix Z : u i : A i × A − i × Θ → R ,( a i , a − i , Z ) (cid:55)→ v i ( a i , (cid:96) i ( a − i , Z )). (3)We assume that each agent i knows how her payoff depends on her action and her payoff state,that is, we assume that i knows function v i , but we do not assume that i knows Z . Actually, fromthe perspective of our analysis, agent i might even ignore that the payoff state x i aggregates herneighbors’ activities according to some weighted network structure, because we are not modelinghow i reasons strategically . If v i,x i : A i → R is strictly quasi–concave for each x i , there is a uniquebest reply r i ( x i ) to each payoff state x i . Although the aggregator is linear, if this “proximate” bestreply function r i : X i → A i is non-linear, then also the best reply r i ( (cid:96) i ( a − i , Z )) is non-linear in a − i . Linearity obtains if and only if v i is quadratic in a i and linear in x i . Without substantial lossof generality, among such utility functions we consider the following form, generalizing equation (1)that we discussed earlier: v i : A i × X i → R ,( a i , x i ) (cid:55)→ α i a i − a i + a i x i . (4)Note that v i in eq. (4) is continuous and strictly concave in a i . Thus, G = (cid:10) I, Θ , ( A i , u i ) i ∈ I (cid:11) , with u i defined by eqs. (3)-(4), is a parametrized nice game (see Moulin 1984 for a definition of nicegame, and Appendix A for a generalization, with results for non-linear-quadratic network games).We assume that the game is repeatedly played by agents maximizing their instantaneous payoff.After each play agents get some feedback. Let M be an abstract set of “messages” (e.g., monetaryoutcomes). The information obtained by agent i ∈ I at the end of each period is described by a feedback function f i : A i × X i → M . Assuming that i knows how her feedback is determined bythe payoff state given her action, if she receives message m after action a i she infers that the state x i belongs to the “ex post information set” f − i,a i ( m ) := (cid:8) x (cid:48) i ∈ X i : f i (cid:0) a i , x (cid:48) i (cid:1) = m (cid:9) . This completes the description of the object of our analysis. The structure
N G = (cid:10) I, Θ , ( A i , X i , v i , (cid:96) i , f i ) i ∈ I (cid:11) If the parametrized payoff functions and the parameter space Θ are common knowledge, strategic reasoningaccording to the epistemic assumptions of rationality and common belief in rationality can be captured by a simpleincomplete-information version of the rationalizability concept. See, e.g., Chapter 7 of Battigalli (2018) and thereferences therein. More precisely, not affine.
5s a (parameterized) network game with feedback, or simply network game.
Our analysisdepends on assumptions about the payoff functions and the feedback functions. Here we presentthe strongest assumptions, the Appendix contains a more general analysis.
Definition A network game with feedback
N G is linear-quadratic if the utility function ofeach player has the linear-quadratic form (4). In this case, the proximate best-reply function is r i ( x i ) =
0, if x i ≤ − α i , α i + x i , if − α i < x i < ¯ a i − α i ,¯ a i , if x i ≥ ¯ a i − α i . (5)Even if agent i may play a best reply to the aggregate x i , it is possible to write the derived bestreply to the actions of others as r i ( (cid:96) i ( a − i , Z )) =
0, if (cid:80) j (cid:54) = i z ij a j ≤ − α i , α i + (cid:80) j (cid:54) = i z ij a j , if − α i < (cid:80) j (cid:54) = i z ij a j < ¯ a i − α i ,¯ a i , if (cid:80) j (cid:54) = i z ij a j ≥ ¯ a i − α i . (6) Definition Feedback f i satisfies observability if and only if player i is active (OiffA) ifsection f i,a i is injective for each a i ∈ (0 , ¯ a i ] and constant for a i = 0 ; f i satisfies just observablepayoffs (JOP) relative to v i if there is a function ¯ v i : A i × M → R such that ∀ ( a i , x i ) ∈ A i × X i , v i ( a i , x i ) = ¯ v i ( a i , f i ( a i , x i )) and the section ¯ v i,a i : M → R is injective for each a i ∈ A i . A network game with feedback N G satisfies observability by active players if feedback f i satisfies OiffA, for each player i ∈ I , andit satisfies just observable payoffs if f i satisfies JOP for each player i ∈ I . In a game with just observable payoffs, because of injectivity of the feedback function, agentsinfer their realized payoff from the message they get, but no more than that, that is, inferencesabout the payoff state can be obtained by looking at the preimages of the payoff function. Forexample, the feedback could be a total benefit, or revenue function f i : A i × X i → R ,( a i , x i ) (cid:55)→ α i a i + a i x i ,with the payoff given by the difference between benefit and activity cost C i ( a i ): v i : A i × X i → R ,( a i , x i ) (cid:55)→ f i ( a i , x i ) − C i ( a i ).Under the reasonable assumption that agent i knows her cost function, when she chooses a i andthen gets message m , she infers that her payoff is ¯ v i ( a i , m ) = m − C i ( a i ). Thus, each section ¯ v i,a i ( a i ∈ A i ) is indeed injective. If the feedback/benefit function is f i ( a i , x i ) = α i a i + a i x i , then itsatisifes observability if and only if i is active. 6 emark If N G is linear-quadratic and satisfies just observable payoffs , then it satisfies observ-ability by active players. If N G satisfies observability by active players, then f − i,a i ( f i ( a i , x i )) = (cid:40) X i , if a i = 0 , { x i } , if a i > for every agent i ∈ I and action-state pair ( a i , x i ) ∈ A i × X i . Most of our analysis focuses on linear-quadratic network games with just observable payoffs.This implies that agents who are active get as feedback a message enabling them to perfectlydetermine the state. Conversely, inactive agents get a completely uninformative message.To choose an action, subjectively rational agents must have some deterministic or probabilisticconjecture about the payoff state x i . We refer to conjectures about the state as shallow con-jectures , as opposed to deep conjectures, which concern the specific network topology and theactions of other players ( a − i ). In linear-quadratic network games (more generally, in nice gameswith feedback), it is sufficient to focus on deterministic shallow conjectures. Indeed, for every prob-abilistic conjecture µ i ∈ ∆ ( X i ), there exists a deterministic conjecture ˆ x i ∈ X i that justifies thesame action a ∗ i as the unique best reply (see the discussion in A.1). We analyze a notion of equilibrium which is broader than Nash equilibrium. Recall that ourapproach allows for the possibility of agents who are unaware of the full game around them. Inequilibrium, agents best respond to conjectures consistent with the feedback that they receive,which is not necessarily fully revealing. We believe that this approach fits well to a networkedenvironment where agents’ knowledge and the information they receive are only local. Definition A profile ( a ∗ i , ˆ x i ) i ∈ I ∈ × i ∈ I ( A i × X i ) of actions and (shallow) deterministic con-jectures is a selfconfirming equilibrium (SCE) at Z if, for each i ∈ I ,1. (subjective rationality) a ∗ i = r i (ˆ x i ) ,2. (confirmed conjecture) f i ( a ∗ i , ˆ x i ) = f i (cid:0) a ∗ i , (cid:96) i (cid:0) a ∗− i , Z (cid:1)(cid:1) . The two conditions require that 1) each agent best responds to her own conjectures; 2) theconjectures in equilibrium must belong to the ex-post information set so that the expected feedbackcoincides with the actual feedback at (cid:96) i (cid:0) a ∗− i , Z (cid:1) . We say that a ∗ = ( a ∗ i ) i ∈ I is a selfconfirming In a context of endogenous strategic network formation, McBride (2006) applies the conjectural equilibrium concept, which is essentially the same as selfconfirming equilibrium for games with feedback (see Battigalli et al. (1992) and the discussions in Battigalli et al. et al. (2018) have adopted self–confirming equilibrium notions to describe network games. Their assumptions and theirresults are different and independent from ours. ction profile at Z if there exists a corresponding profile of conjectures (ˆ x i ) i ∈ I such that ( a ∗ i , ˆ x i ) i ∈ I is a selfconfirming equilibrium at Z , and we let A SCE Z denote the set of such profiles. Also, for anyadjacency matrix Z ∈ Θ, we denote by A NE Z the set of (pure) Nash equilibria of the (nice) gamedetermined by Z , that is, A NE Z := (cid:8) a ∗ ∈ × i ∈ I A i : ∀ i ∈ I, a ∗ i = r i (cid:0) (cid:96) i (cid:0) a ∗− i , Z (cid:1)(cid:1)(cid:9) .Nice games satisfy all the standard assumptions for the existence of Nash equilibria. Hence, weobtain the existence of selfconfirming equilibria for each Z ∈ Θ. Indeed a Nash equilibrium is aselfconfirming equilibrium with correct conjectures. To summarize:
Remark For every Z , there is at least one Nash equilibrium, and every Nash equilibrium is aselfconfirming profile of actions: ∀ Z ∈ Θ , ∅ (cid:54) = A NE Z ⊆ A SCE Z . In this section we characterize the set A SCE Z of selfconfirming equilibrium profiles of actions inlinear-quadratic network games with just observable payoffs. All our proofs are derived from theresults in Appendix A and Appendix B, which refer to the case of generic network games withoutthe restriction to linear best replies, and are stated in Appendix C. We start with the simplest casein which every agent necessarily finds it subjectively optimal to be active (that is, being inactive isdominated – see Lemma A in Appendix A). Proposition Consider a network game
N G satisfying observability by active players . Assumethat, for every i ∈ I and for every ˆ x i ∈ X i , r i (ˆ x i ) > . Then, for each Z ∈ Θ , A SCE Z = A NE Z . Assume that α i (from eqs (4) and (5)) is such that α i >
0. Assume further that Z = γ Z ,with γ > Z ∈ { , } I × I . This represents the standard case of local complementaritiesstudied by Ballester et al. (2006). If γ ( n − < I denote the setof players for whom being inactive is justifiable . Note that, by Lemma A in Appendix A, I = { i ∈ I : min r i ( X i ) = 0 } . Since the self-map a (cid:55)→ ( r i ( a − i , Z )) i ∈ I is continuous on the convex and compact set A = × i ∈ I [0 , ¯ a i ], by Brouwer’sTheorem it has a fixed point. Z ∈ Θ and non–empty subset of players J ⊆ I , let A NEJ, Z denote the set of Nashequilibria of the auxiliary game with player set J obtained by imposing a i = 0 for each i ∈ I \ J ,that is, A NEJ, Z = (cid:110) a ∗ J ∈ × j ∈ J A j : ∀ j ∈ J, a ∗ j = r j (cid:16) (cid:96) j (cid:16) a ∗ J \{ j } , I \ J , Z (cid:17)(cid:17)(cid:111) ,where I \ J ∈ R I \ J is the profile that assigns 0 to each i ∈ I \ J . If J = ∅ , let A NEJ, Z = { ∅ } byconvention, where ∅ is the peudo-action profile such that ( ∅ , I ) = I . We relate the set ofselfconfirming equilibria to the sets of Nash equilibria of such auxiliary games.
Proposition Suppose that network game with feedback
N G is linear-quadratic and satisfies just observable payoffs . Then, for each Z ∈ Θ , the set of selfconfirming action profiles is A SCE Z = (cid:91) I \ J ⊆ I A NEJ, Z × (cid:8) I \ J (cid:9) ,that is, in each SCE profile a ∗ , a subset I \ J of players for whom being inactive is justifiable choose , and every other player chooses the best reply to the actions of her coplayers. Therefore, in eachSCE profile a ∗ and for each player i ∈ I , a ∗ i = 0 ⇒ x i ≤ − α i , a ∗ i > ⇒ α i + (cid:88) j ∈ I z ij a ∗ j > ∧ a ∗ i = min ¯ a i , α i + (cid:88) j ∈ I z ij a ∗ j . (8)In every SCE we can partition the set of agents in two subsets. Agents in J ⊆ I are active,i.e., they choose a strictly positive action, agents in I \ J instead choose the null action. Startconsidering the latter. Since they play a ∗ i = 0, they get null payoff independently of others’actions. But, since every conjecture ˆ x i ∈ ( −∞ , − α i ] is consistent with this payoff, their conjectureis (trivially) consistent with their feedback. As for agents in J , since they choose a strictly positiveaction a ∗ i >
0, they receive a message that enables them to infer the true payoff state x i ; with this,they necessarily choose the objective best reply to their neighbours actions, whether or not they areaware of them. Note that, if being inactive is justifiable for every agent ( I = I ), then I ∈ A SCE Z for every Z ∈ Θ.This implies that the set of selfconfirming equilibria can be characterized by means of the sets ofNash equilibria of the auxiliary games in which only active agents are considered. If, for example,there is a unique interior Nash equilibrium for the auxiliary game corresponding to every subset ofactive players, then | A SCE Z | = 2 | I | , that is, there are exactly 2 n SCE action profiles. A.3 discussesthe equilibrium characterization for the generalized case of non linear-quadratic network games. As we do in set theory with the empty set, when we consider functions whose domain is a subset of some indexset I , it is convenient to have a symbol for the pseudo-function with empty domain. For example, if I = N , suchfunctions are (finite and countably infinite) sequences, or subsequences, and ∅ is the empty sequence. xample 1. Consider Figure 1, representing a network between 4 nodes. We set α i = 0 . i . Let us first assume that each arrow represents a positive externality of 0 . a = 0 .
1, because she is not affected by any externality. Other players, instead, playdifferently when active, according to who else is active.Figure 1: A network between 4 nodes. Every arrow is for an externality of equal magnitude andsign.
All { , , } { , , } { , , } { , , } { , } { , } { , } . . . ∅ a a a a .
2. The unique Nash Equilibrium is in bold.Consider now the same network, but assume that each arrow represents a negative externality of0 .
6. In this case we have more NEs (there is not a NE where all players are active, but thereare 3 NEs), but less than 16 SCEs (there are 13), because for some subset J of players (such as J = I = { , , , } ) there is no SCE in which all its elements are active. Table 2 reports the actionsof players in each case (we omit redundant pairs and singletons). (cid:78) , , } { , , } { , } { , } { , } . . . ∅ a a a a − .
6. Nash Equilibria are in bold.This simple example shows that moving from a case of full complementarity to a case of fullsubstitability, we may increase the number of Nash Equilibria and decrease the number of SCEs.However, even in the limiting case where substitution effects are extremely strong, the two sets ofequilibria will not coincide, because the strategy profile in which everyone is inactive will be a anSCE but not an NE.
Next, we focus on the network Z . We list below some properties of matrix Z that are not maintainedassumptions. In different parts of the paper we will use some of these assumptions to have sufficientconditions for the existence and stability of selfconfirming equilibria. We refer to Appendix B fora deeper discussion on these assumptions and their implications. Assumption Matrix Z of size n has bounded values , i.e. | z ij | < n for all i and j . Assumption Matrix Z has the same sign property i.e., for every i, j , sign ( z ij ) = sign ( z ji ) ,where the sign function can have values − , or . Assumption Matrix Z is negative , i.e. z ij < for all i and j , We recall here that the spectral radius ρ ( Z ) of Z is the largest absolute value of its eigenvalues. Assumption Matrix Z is limited , i.e. ρ ( Z ) < . In Section 2 we discussed how, in some cases, we can write Z as Z = ΓZ , where Γ is a diagonalmatrix, and Z is the basic underlying representation of the network. When this is possible, matrix Z represents a basic network combined with an additional idiosyncratic effect by which every agent i weights the effects of the others on her. This effect is modeled by the parameter γ i . The next The sign condition is the one used in Bervoets et al. (2016) to prove convergence to Nash equilibria in networkgames, under a particular form of learning. Then the payoff of i ∈ I at a given profile a of the original game is u i ( a ) = αa i − a i + a i γ i (cid:88) j ∈ I z ,ij a j = αa i − a i + a i (cid:88) j ∈ I z ij a j . Z . Assumption Matrix Z is symmetrizable , i.e. it can be written as Z = ΓZ , with Γ diagonaland Z symmetric. Moreover, Γ has all positive entries in the diagonal. Note that if Z is symmetrizable then all its eigenvalues are real. Moreover, since Γ has allpositive entries, Assumption 5 implies the sign condition from Assumption 2.Our final assumption is discussed in Bramoull´e et al. (2014) and combines Assumptions 4 and 5above. Assumption
6. Z = ΓZ is symmetrizable-limited , i.e. Z is symmetrizable and, for every i, j , z ij = z ,ij √ γ i γ j , is limited. Our previous results from Section 3, about the characterization of selfconfirming equilibria,state that we can choose any subset of agents and have them inactive in a SCE. However we cannotensure that the other agents are active, because their best response in the reduced game could benull. The next result goes in the direction of specifying under which sufficient conditions this doesnot happen. Given the matrix Z , and given J ⊆ I , we call Z J the submatrix who has only rowsand columns corresponding to the elements of J . Proposition Consider a set J ⊆ I . Let us assume that Z J satisfies at least one of the threeconditions below:1. it has bounded values (Assumption 1),2. it is negative and limited (Assumptions 3 and 4),3. or it is symmetrizable–limited (Assumption 6).Then, we have the two following results:1. A NEJ, Z = (cid:8) a NEJ (cid:9) , such that a NEJ > ;2. There exists a ∗ ∈ A SCE Z such that a ∗ = (cid:8) a NEJ (cid:9) × (cid:8) I \ J (cid:9) . Proposition 3 provides sufficient conditions to have an arbitrary set of active and inactive playersin a selfconfirming equilibrium. In this case the set of selfconfirming equilibria has cardinality equalto the cardinality of the power set 2 I , that is 2 n .We provide here below two examples, one with all positive externalities, the other with mixedexternalities. Example 2.
Consider n players, and a randomly generated network between them, of the type Z = ΓZ , generated by the following generating process. Z is undirected, generated by an Erdos12nd R´enyi (1960) process for which each link is i.i.d., and such that its expected number of overalllinks (i.e., counted in both directions) is k · n , for some k ∈ R + . This means that the expectednumber of links for each player is k . It is well known that this model predicts, as n goes to infinity,that Z will have no clustering and, when k ≥
2, a connected giant component. Γ is a diagonal matrix, such that each element γ i in the diagonal is positive and is generatedby some i.i.d. random process with mean µ and variance σ .In this case, F¨uredi and Koml´os (1981) prove that the expected highest eigenvalue of Z , as n grows,is E ( λ i ) = kµ + σ µ + O (cid:18) √ n (cid:19) . From Proposition 3, under Assumption 6, as n tends to infinity, Z is symmetrizable–limited if E ( λ i ) <
1, which implies that µ − σ µ > k . Clearly, a necessary condition for previous inequality to hold is that µ > σ .When this happens, as n grows to infinity, we will always have a unique NE of the game where allplayers are active.Note that this limiting result excludes the possibility (because the expected clustering of Z goesto 0) that there is a subset J of players, that have a dense sub–network between them, and a highrealization of γ i ’s, such that there does not exist a ∗ ∈ A SCE Z , for which a ∗ = (cid:8) a NEJ (cid:9) × (cid:8) I \ J (cid:9) .In fact, if this was the case, because of only positive externalities, we would not even have an allactive equilibrium for the whole population of n agents. (cid:78) Example 3.
Proposition 3 provides alternative conditions, that are only sufficient, for interior NEin an auxiliary game in which only agents in J are considered. Figure 2 provides an example ofgame that do not satisfy any of them, but still has a unique interior NE. We set α i = 0 . i . Every blue arrow stands for a positive externality of 0 . . (cid:78) All { , , } { , , } { , , } { , , } { , } { , } { , } { , } . . . ∅ a a a a . − .
2. The unique Nash Equilibrium is in bold.13igure 2: A network between 4 nodes. Blue arrows are for positive externalities, red arrows are fornegative externalities.
We have not considered any dynamics yet. Definition 3 of selfconfirming equilibrium, characterizedalso by the conditions stated in Proposition 2, identifies steady states: If agents happen to haveselfconfirming conjectures and play accordingly, then they have no reason to move away from it.However we may wonder how agents get to play SCE action profiles, and if these profiles are stable.We first notice that SCE has solid learning foundations. The following result is specificallyrelevant for this paper (see Gilli (1999) and Chapter 6 of Battigalli (2018)). Consider a sequencein time of action profiles, given by ( a t ) ∞ t =0 . Then, if ( a t ) ∞ t =0 is consistent with adaptive learning and a t → a ∗ , it follows that a ∗ must be a selfconfirming equilibrium action profile. Of course, the limit of the trajectory may or may not be a Nash equilibrium. Let us now con-sider a best response dynamics. This generates trajectories that—by construction—are consistentwith adaptive learning. With this, we prove convergence (under reasonable assumptions), henceconvergence to an SCE.To ease the analysis we consider best reply dynamics for shallow conjectures. For each period t ∈ N and each agent i ∈ I , a i,t = r (ˆ x i,t ) is the best reply to ˆ x i,t . After actions are chosen, giventhe feedback received, agents update their conjectures. If conjectures are confirmed then an agentkeeps past conjecture, otherwise she updates using as new conjecture the conjecture that would See, for example, Battigalli et al. (1992), Battigalli and Marinacci (2016), Fudenberg and Kreps (1995), and thereferences therein. In a finite game, a trajectory ( a t ) ∞ t =0 is consistent with adaptive learning if for every ˆ t , there exists some T suchthat, for every t > ˆ t + T and i ∈ I , a i,t is a best reply to some deep conjecture µ i that assigns probability 1 to theset of action profiles a − i consistent with the feedback received from ˆ t through t −
1. The definition for compact,continuous games is a bit more complex (cf Milgrom and Roberts (1991)), who assume perfect feedback). x i,t +1 = (cid:40) ˆ x i,t if a i,t = 0 ,(cid:96) i ( a − i,t , Z ) if a ∗ i,t > , (9)and, from (5) (considering that the upper bound ¯ a i is set so that it is never reached) we have simply a i,t +1 = r i (ˆ x i,t +1 ) = (cid:40)
0, if ˆ x i,t ≤ − α i , α i + ˆ x i,t +1 , if ˆ x i,t > − α i .Coherently with the previous analysis, this update rule states that if an agent i at time t is inactive( a i,t = 0), past conjectures are confirmed and thus kept. If instead the agent is active ( a i,t > x i,t = (cid:96) i ( a − i,t , Z ), and so theyupdate conjectures according to (9). This is one possible adaptive learning dynamics. The resultcited above implies that if the dynamics described above converges, then it must converge to aselfconfirming equilibrium, i.e., a rest point where players keep repeating their choices.In this section we analyze the stability of such rest points in the simplest possible case ofrobustness to small perturbations, as in Bramoull´e and Kranton (2007). However, we will notconsider perturbations to the strategy profile, but perturbations on the profile of conjectures. Definition (Learning process) . Each player i ∈ I starts at time with a belief, and beliefs arerepresented by a vector of shallow deterministic conjectures ˆx = (ˆ x i, ) i ∈ I . In each period t playersbest reply to their conjectures: for each i ∈ I , a i,t = max { α i + ˆ x i,t , } .At the beginning of each period t + 1 each player i keeps his t -period shallow conjecture if hewas inactive, and updates his conjecture to period- t revealed payoff state if he was active, that is, ˆ x i,t +1 = u i ( a t ) a i,t − α i + a i,t . Even if we consider the case of linear best replies, from equations (8) and (9), the system is notlinear because ˆ x i,t +1 = (cid:40) ˆ x i,t if ˆ x i,t ≤ − α i , (cid:80) j ∈ I z ij a j,t if ˆ x i,t > − α i , and for every other player j , we have that a j,t = max { α j + ˆ x j,t , } .Clearly an SCE of the game, as defined in the beginning of Section 3, is always a rest point ofthis learning dynamic. We now consider the stability of such rest points a ∗ . Say that a profile ofconjectures ˆx is consistent with a ∗ if a ∗ i = r i (ˆ x i ) for every i ∈ I . Definition A selfconfirming action profile a ∗ ∈ A SCE Z is locally stable if there are a profile ofconjectures ˆx and (cid:15) > consistent with a ∗ such that the learning dynamics starting from any ˆx (cid:48) with (cid:107) ˆx (cid:48) − ˆx (cid:107) < (cid:15) converges back to ˆx . .1 Results Each SCE is characterized by a set of active agents. So, given a strategy profile a = ( a i ) i ∈ I , let I a = { i ∈ I : a i > } denote the set of active players. With this, for each action profile a , Z I a denotes the submatrix with rows and columns corrsponding to players who are active in a . Thisallows us to characterize locally stable selfconfirming equilibria. Proposition Consider a ∗ ∈ A SCE Z . a ∗ is locally stable if • Assumption 4 holds for matrix Z I a ∗ ; • for some ˆx consistent with a ∗ and every i ∈ I \ I a ∗ , α i + ˆ x i < . Intuitively, consider a sufficiently small perturbation of players’ conjectures. The first conditionensure that active players keep being active and their actions converge back to the Nash equilbiriumof the auxiliary game with player set I a ∗ . The second condition ensures that inactive players keepbeing inactive. Next, we provide alternative sufficient conditions that allow to characterize thesubsets of active agents associated to SCEs. Proposition Consider a selfconfirming strategy profile a ∗ ∈ A SCE Z . If Z I a ∗ satisfies at leastone of the three conditions below:1. it has bounded values (Assumption 1),2. it is negative and limited (Assumptions 3 and 4),3. or it is symmetrizable–limited (Assumption 6),then a ∗ is locally stable and, for every J ⊆ I a ∗ , there exists a locally stable selfconfirmingequilibrium a ∗∗ ∈ A SCE Z such that1. A NEJ, Z = (cid:8) a NEJ (cid:9) , with a NEJ > J ;2. { a ∗∗ } = (cid:8) a NEJ (cid:9) × (cid:8) I \ J (cid:9) . The proof is based on results from linear algebra. In fact, if an adjacency matrix satisfies one ofthe conditions from Proposition 5, then also every submatrix of that matrix satisfies that property.We know that there may be SCEs that are not Nash equilibria, because some agents are inactiveeven if this is not a best response to the actions of the others. Proposition 5 tells us two additionalthings. Under the stated conditions, for any given SCE a ∗ with set of active agents I a ∗ , any subset J ⊆ I a ∗ of those agents is associated to a stable SCE where all agents in J are active, and the otheragents are inactive. Second, since the empty subset of agents is trivially associated to the stableSCE where every agent is inactive, for every network game there is always a subset J of agentsassociated to a stable SCE where all and only the agents in J are active.16 .2 Examples and discussion The following example shows that we can reach SCEs that are not NE also if the initial beliefsinduce all positive actions at the beginning of the learning dynamic.
Example 4.
Consider the case of 4 players, with the network matrix Z ∈ {− . , , . } I × I shownin Figure 2, and, for every i , α i = 0 .
1. This is a case of general externalities, that can be positiveor negative. Figure 3 shows the learning dynamics of actions and beliefs that start from differentinitial conditions. In one case (left panels) we converge to the unique Nash equilibrium of this game(the dotted lines), in the other (right panels) the learning dynamics put, after 2 rounds, one playerout from the active agents, and the remaining 3 converge to a selfconfirming equilibrium which isnot Nash. (cid:78)
Figure 3: General strategic externalities. Starting from different beliefs on the same network (fromFigure 2), the learning process may converge to the unique Nash equilibrium (left panels) or toa SCE which is not a Nash equilibrium (right panels). Note that actions are just an upwardtranslation of beliefs, by the quantity α .The next example (which does also not satisfy the local stability conditions of Proposition 5)shows that convergence may not occur even in a simple case of positive externalities. Example 5.
Now consider again the network from Example 1 (Figure 1), with 4 nodes. Even ifthere are only positive externalities, the magnitude of γ may imply convergence or not. If γ < γ ≥ γ = 0 . γ = 1 respectively, starting from the same initial beliefs. Note that, nodes/players 1 and 4reinforce each other, and this gives rise to an oscillating behavior of their beliefs. (cid:78) Figure 4: Only positive externalities. Starting from the same beliefs on the same network structure(from Figure 1), the learning process may converge or not depending on the size of γ : γ = 0 . γ = 1 in the right panel. We report only beliefs because, as in Figure 4, actions arejust an upward translation of beliefs, of amount α ..Our notion of stability with respect to conjectures relates to the standard notion of stabilitywith respect to actions in the following way. First of all, since played actions are justified bysome conjectures, the only reason for these actions to change is a perturbation of the surroundingconjectures, but this is not a sufficient condition. If all agents are active, the two definitions have thesame the consequences in terms of stability, since a perturbation with respect to actions happensif and only if every agent’s conjecture is perturbed. However, if a selfconfirming equilibrium hasinactive agents, then those inactive agents who play a corner solution do not show perturbation inactions when their conjectures are perturbed. This implies that if an action profile is stable withrespect to actions perturbations, then it is also stable under conjectures perturbations, but theconverse does not hold. As anticipated when discussing Eq. (2), we consider now an extension to the case of equation (4),in which we add a global externality term with no strategic effects. For each i ∈ I , we posit aninterval Y i = [ y i , ¯ y i ], a coefficient β ∈ R , and we consider the following aggregator: This aggregator g sums up the actions of all the agents in the network except agent i . We could have consideredagent i as well, but we opted for this specification so as not to change the first order condition with respect to thecase with just local externalities. i,β : A − i → Y i a − i (cid:55)→ β (cid:80) j (cid:54) = i a j . We assume that every agent i ∈ I knows Y i . Then, we let y i = g i ( a − i , β ) and we maintain theassumption that x i = (cid:96) i ( a − i , Z ). The new parametrized utility function is v i : A i × X i × Y i → R ( a i , x i , y i ) (cid:55)→ α i a i − a i + a i x i + y i , (10)where both x i and y i are unknown. The general form of the feedback function is f i : A i × X i × Y i → M .Deterministic shallow conjectures for each i ∈ I are now determined by the pair (ˆ x i , ˆ y i ) ∈ X i × Y i .We provide now the definition of selfconfirming equilibrium for games with global externalities. Definition A profile ( a ∗ i , ˆ x i , ˆ y i ) i ∈ I ∈ × i ∈ I ( A i × X i × Y i ) of actions and (shallow) deterministicconjectures is a selfconfirming equilibrium at Z and β of a linear quadratic network game withfeedback and global externalities if, for each i ∈ I ,1. (subjective rationality) a ∗ i = r i (ˆ x i ) ,2. (confirmed conjecture) f i ( a ∗ i , ˆ x i , ˆ y i ) = f i (cid:0) a ∗ i , (cid:96) i (cid:0) a ∗− i , Z (cid:1) , g i (cid:0) a ∗− i , β (cid:1)(cid:1) . Notice that the rationality condition is unchanged with respect to the case of only local exter-nalities since best-reply conditions are not affected by the global externality term. To compare thisgame with the linear-quadratic network game with only local externalities, we consider the case of just observable payoffs . Then, without loss of generality we can assume that f i = v i for every i ∈ I .With this, we can characterize the SCE set as follows: Proposition Fix Z ∈ Θ and β . Every selfconfirming equilibrium profile (cid:0) a ∗ i , ˆ x i , ˆ y i (cid:1) i ∈ I ∈× i ∈ I ( A i × X i × Y i ) of a linear-quadratic network game with global externalities and just observ-able payoffs is such that, for every i ∈ I ,1. if a ∗ i = 0 , then ˆ x i ∈ ( −∞ , − α i ] , ˆ y i = y i ;2. if a ∗ i > , then a ∗ i = α i + ˆ x i , ˆ y i = y i + a ∗ i ( x i − ˆ x i ) . We discuss how the presence of the global externality term in the utility function changesradically the characterization of selfconfirming equilibria. As before, we assume that players observetheir own realized payoffs. Yet, when global externalities are present, observability by active playersdoes not hold anymore. Inactive players have correct conjectures about the global externality,but may have correct or incorrect conjectures about the local externality. Active players, on the19ther hand, are not able to determine precisely the magnitude of the local effects with respectto the global effects. Given any strictly positive action a ∗ i , the confirmed conjectures conditionyields (ˆ y i − y i ) = a ∗ i ( x i − ˆ x i ). Then, in equilibrium, if agent i overestimates (underestimates) thelocal externality, she must compensate this error by underestimating (overestimating) the globalexternality. Then, compared to the case of only local externalities, we have that: (i) active agentschoose a best response to a (typically) wrong conjecture about x ; thus, (ii) it is not possible tocharacterize SCE by means of Nash equilibria of the auxiliary games restricted to the active players.We present now a simple example showing how wrong conjectures about local and global ex-ternalities may have a big effect on equilibrium actions. Example 6.
Consider three agents in a line network. Let agent 2 be at the center of the line.Then, for every ( a ∗ , Z , β ), (cid:96) (cid:0) a ∗− , Z (cid:1) is proportional to g (cid:0) a ∗− , β (cid:1) , always with the same ratio,while this is not true for agents 1 and 3. We assume that each agent thinks to be playing in acomplete network, so every i ∈ I thinks that (cid:96) i (cid:0) a ∗− i , Z (cid:1) is always proportional to g i (cid:0) a ∗− i , β (cid:1) , withthe same ratio. In this case agents 1 and 3 think to be more central than what they actually are.Table 4 provides the Nash equilibria for the actual network and for the complete network, and theselfconfirming equilibrium actions for the case described above.Line NE Complete NE SCE a a a α = 0 . γ = 0 .
2, and β = 1. Columns refer to 1) NashEquilibrium of the line network; 2) Nash equilibrium of complete network; 3) SCE in the linenetwork in which each i ∈ I believes that (cid:96) i (cid:0) a ∗− i , Z (cid:1) = γβ g i (cid:0) a ∗− i , β (cid:1) .Simulations show that if agents overestimate the impact of local externalities this generates a multiplier effect that makes equilibrium actions increase at a level even larger that what would bepredicted in a complete network by Nash equilibrium. This is the result of how agents misinterprettheir feedbacks. In details, thinking to be in a complete network makes agents 1 and 2 overestimatelocal externalities. Take for instance agent 1. Given any a − , she chooses a best reply higherthan the Nash equilibrium one since she overestimates the local externality. This high action hasthe effect of increasing the global externality term for agent 3. Agent 3, by overestimating localexternality, partly attributes this higher global externality to the local externality term, and choosesan action larger than predicted by Nash equilibrium. The choice of agent 3 increases in turns theglobal externality perceived by agent 1, and so on. At the same time agent 2, as neighbors choosehigher actions, increases her own action level. This effect goes on and a multiplier effect seems tobe at place. In the limit, selfconfirming equilibrium actions are almost ten times larger than the20omplete network Nash equilibrium. (cid:78) We now consider the learning process that originates from an adaptive updating of conjectures, aswe did for the case of only local externalities. For an easy reference, we rewrite here Eq. (2) as apayoff function that depends on players’ actions, with the time index and specifying x i,t and y i,t asfunctions of co-players’ actions: u i,t ( a i,t , a − i,t ) = αa i,t − a i,t + a i,t (cid:88) j ∈ I \{ i } z ij a j,t (cid:124) (cid:123)(cid:122) (cid:125) x i,t + β (cid:88) k ∈ I \{ i } a k,t (cid:124) (cid:123)(cid:122) (cid:125) y i,t .To ease the analysis, we assume the same parameter α for each player and we focus on the caseof strictly positive justifiable actions. We obtain this by assuming that α > and that all theelements of Z are nonnegative . This case, however, is a bit more complex since, at each time, thereare infinitely many collections of feasible pairs (ˆ x i,t , ˆ y i,t ) i ∈ I compatible with adaptive learning. Forevery i ∈ I , and each time t , let m i,t = f i ( a i,t , x i,t , y i,t ) = u i ( a i,t , a − i,t ) be the message agentsreceive. Then, given ˆ x i,t , ˆ y i,t is uniquely determined. In details, at each time period, agent i ’sconjecture is a pair (ˆ x i,t , ˆ y i,t ) consistent with the message received at the previous period. Weobtain ˆ y i,t +1 = m i,t − αa i,t + 12 ( a i,t ) − a i,t ˆ x i,t +1 . Given message m i,t − , and considering that agents perfectly recall their past actions, ˆ y i,t is uniquelydetermined as a function of ˆ x i,t . We can just focus on the dynamics of ˆ x i,t . The dynamics of ˆ x i,t is given by the following equationˆ x i,t +1 = m i,t − ˆ y i,t +1 a i,t − α + 12 a i,t (11)To avoid bifurcations at each time period, we need to use simplifying assumptions. We define c i,t := ˆ x i,t ˆ y i,t . (12)Then, Assumption For each i ∈ I and for each t ∈ N , c i,t = c i,t +1 = c i . We call c i the perceived centrality of player i . For each player, this parameter describeswhat she thinks to be the share of the activity in her neighborhood with respect to the sum of all In doing so, we implicitly assume that players think that not all the other players play the null action a k,t = 0.This is actually a reasonable assumption, because under positive externalities any best response a k,t should be atleast α . a ∗ of the game, where all actions are positive, we have a ∗ i = α + x i = α + (cid:88) j ∈ I \{ i } z ij a ∗ j . The profile of
Bonacich centrality measures b is the unique solution of the linear system b i = α + (cid:88) j ∈ I \{ i } z ij b j . So, when beliefs are correct, as in the Nash equilibrium, we have b i = a i and c i = b i − αy i . Now, in theNash equilibrium we have also y i − y j = β a j − a i y i y j . If the number of players is large, we have y i (cid:29) a i and y j (cid:29) a j , which implies y i (cid:39) y j , and so every c i is roughly the same linear rescaling of b i .From equation (11), and expressing the message as the observed payoff, we get that the followinglearning dynamic ˆ x i,t +1 = x i,t + y i,t a i,t − ˆ y i,t +1 a i,t . (13)Plugging in c i,t = ˆ x i,t ˆ y i,t we get ˆ x i,t +1 = c i,t c i,t a i,t ( a i,t x i,t + y i,t ) . (14)We define the true centrality of player i at time t as c (cid:48) i,t = x i,t y i,t . Note that c (cid:48) i,t ∈ [0 , (cid:80) j (cid:54) = i z ij β ]. For this reason, we also assume that the perceived centrality ofeach player i is such that c i ∈ (cid:16) , (cid:80) j (cid:54) = i z ij β (cid:105) , and this specifies the set of all admissible perceivedcentralities. The dynamic, then, can be written asˆ x i,t +1 = c i y i,t a ∗ i,t c (cid:48) i,t + 1 a ∗ i,t c i + 1 , which implies that the conjecture is correct only when c i = c (cid:48) i,t .We look at best responses a i,t +1 = α + ˆ x i,t +1 , and study existence and characterization of thesteady state of this learning process. Recall that y i,t = β (cid:80) j (cid:54) = i a j,t . To find a fixed point we lookat the system of n equations H i ( a ∗ , c , β, Z ) := α + c i β (cid:88) j (cid:54) = i a ∗ j a ∗ i c (cid:48) i + 1 a ∗ i c i + 1 − a ∗ i = 0 . (15) In general, independently of any game defined on the network, Bonacich centrality is a network centrality measurethat depends on a paramater α >
0. It is defined exactly as the solution of that same linear system. For a detaileddiscussion on this see Dequiet and Zenou 2017 F i ( a ∗ , β, Z ) := α + (cid:88) j ∈ I z ij a ∗ j − a ∗ i = 0 . (16)Let A ⊂ [ α, ∞ ) I denote the set of the solutions of the system (15). We have the following result. Proposition If the system defined by (16) admits a solution, then for each vector c of perceivedcentralities also the system defined by (15) admits a solution. Moreover, the system implies ahomeomorphism Φ between all profiles c and A . Homeomorphism Φ is monotone with respect tothe lattice order of the two sets. The previous result provides information only on the steady states of our dynamical system.Note however that the homeomorphism is implied by the particular learning dynamic that we areassuming, which is based on constant belief centralities. Here below we show a result that providessufficient conditions for convergence of the learning dynamic. We impose as a sufficient conditionthat local and global externalities are not too large.
Proposition If, for each player i ∈ I , < c i β ( n − < (cid:80) j (cid:54) = i z ij < , then the dynamic definedby the learning process (15) always converges to its unique solution, which is stable. It should be noted that we are not requiring that | (cid:80) j (cid:54) = i z ij | <
1, which would imply thatAssumption 4 hold.
Example 7.
Under the conditions of Proposition 8, we use equation (14) to run dynamical systemsconverging to the SCE implicitely defined by (15). This allows us to provide a graphical illustrationof Proposition 7, for the case of three nodes. As in Example 6, we do this for the case of a linenetwork (where each of the two links is bidirectional), and for the case of a complete network.Figure 5 shows the results. We can start from any pattern of perceived centralities for the threenodes. The left panel shows the profile of perceived centralities when at least one node has maximalperceived centrality (the three faces of the cube have different colors, according to which node hasthe maximal centrality). The central panel shows the corresponding SCE conjectures ˆx when thenetwork is a line (the node that has perceived centrality 1 in the red dots is the central node). Theright panel shows the corresponding SCE beliefs ˆx when the network is a complete triangle. Thefigure suggests that homeomorphism Φ (from Proposition 7) is highly non linear, because of theself reinforcement process in beliefs that we discussed in Example 6. The figure also shows that, asstated by Proposition 7, homeomorphism Φ respects the lattice order on the two sets. (cid:78) Proposition 7 tells us that a non-negative shift in each perceived centrality will always resultin a non-negative shift in each agent’s action in the resulting SCE. However, Proposition 8 givesan implicit warning. Too high perceived centralities may imply that the sufficient conditions for23igure 5: Simulations showing the homeomorphism of Proposition 8 for the case of 3 nodes. Theleft panel shows vectors of preceived centralities. The central panel shows the corresponding SCEbeliefs ˆ x when the network is a line (the node that has perceived centrality 1 in the red dots isthe central node). The right panel shows the corresponding SCE beliefs ˆ x when the network is acomplete triangle.stability are lost, and convergence to the corresponding SCE may be lost. Note also that, summingup equation (2) for all the players, the aggregate welfare is maximized if the vector of actionssatisfies the linear system a ∗ i = α + ( n − β + (cid:88) j ∈ I \{ i } ( z ij + z ij ) a ∗ j . Social platforms like Facebook and Twitter often provide information to users about the activityof their peers. A rationale for this marketing strategy can be that these companies want to changethe beliefs of players, making them feel more important (i.e. central) in the social network. Even abenevolent social planner may want to set the perceived centralities to the level for which the socialoptimum is achieved. However, according to our model, if perceived centralities are too high, the24ystem may become unstable.
In this paper we lay the basis for a novel approach to network games. Many of the applications ofthose games mimic large societies with million of nodes and non regular distribution of connections.It is natural to assume that players are not aware of the complete structure of the network; thus,they do not perform sophisticated strategic reasoning possibly leading to a Nash equilibrium, butjust best–respond to some to subjective beliefs affected by information feedback they receive. Weanalyze simple adaptive dynamics and show that in some cases they converge to stable Nashequilibria. However, we characterize also those situations in which feasible stable outcomes are notNash equilibria, but rather selfconfirming equilibria in which some (if not all ) players have wrongbeliefs and yet the feedback they receive is consistent with such beliefs. We also show that simplebiases in the perception of own centrality in the network may lead players to play action profilesthat are very far from the unique Nash equilibrium of the game.The natural application of this approach is to online social platforms like Facebook and Twitter.Using a linear quadratic structure for the payoff function we have also laid the ground for a tractablewelfare analysis of the model. However, policy implications are not straightforward if we want toconsider the long–run benefits of connections and not only about the instantaneous payoffs of theusers of those platforms.Our analysis does not account for the strategic reasoning that agents can perform given somecommonly know features of the network. For example, known results about rationalizability implythat, if the (nice) network game has strategic complementarities and is common knowledge, thensophisticated strategic reasoning leads to Nash equilibrium. If only some aspects of the networkgame are commonly known, then both strategic reasoning and learning affect the long-run outcome,which is a kind of rationalizable self-confirming equilibrium. This is a topic we are working on. On nice games with strategic complementaries see, e.g., Chapter 5 of Battigalli (2018) and the references therein. ppendix A Selfconfirming equilibria in parametrized nice gameswith aggregators In this section we develop a more general analysis of selfconfirming equilibria in a class of gamesthat contains the linear-quadratic network games with feedback. To ease reading, we make thissection self-contained repeating some definitions from the main text.A parametrized nice game with aggregators and feedback is a structure G = (cid:10) I, Θ , ( A i , (cid:96) i , v i , f i ) i ∈ I (cid:11) where • I is the finite players set , with cardinality n = | I | and generic element i . • Θ ⊆ R m is a compact parameter space . • A i = [0 , ¯ a i ] ⊆ R + , a closed interval , is the action space of player i with generic element a i ∈ A i . • X i = [ x i , ¯ x i ] ⊆ R , a closed interval , is the a space of payoff states for i . • (cid:96) i : A − i × Θ → X i (where A − i = × j ∈ I \{ i } A j ) is a continuous parametrized aggregator ofthe actions of i ’s coplayers such that its range (cid:96) i ( A − i × Θ) is connected . • v i : A i × X i → R is the payoff (utility) function of player i , which is strictly quasi-concave in a i and continuous, and from which we derive the parameterized payoff function u i : A i × A − i × Θ → R ,( a i , a − i , θ ) (cid:55)→ v i ( a i , (cid:96) i ( a − i , θ )).Thus, x i = (cid:96) i ( a − i , θ ) is the payoff relevant state that i has to guess in order to choosea subjectively optimal action. With this, for each θ ∈ Θ, (cid:10) I, ( A i , u i,θ ) i ∈ I (cid:11) is a nice game(Moulin, 1979), and (cid:10) I, Θ , ( A i , u i ) i ∈ I (cid:11) is a parametrized nice game. We let r i : X i → A i x i (cid:55)→ arg max a i ∈ A i v i ( a i , x i )denote the best reply function of player i . Since the range of each section (cid:96) i,θ must be a closed interval, we require that the union of the closed intervals (cid:96) i,θ ( A − i ) ( θ ∈ Θ) is also an interval, which must be closed because Θ is compact and (cid:96) i continuous. That is, v i is jointly continuous in ( a i , x i ) and, for each x i ∈ [ x i , ¯ x i ], the section v i,x i : [0 , ¯ a i ] → R has a uniquemaximizer a ∗ i (that typically depends on x i ), it is strictly increasing on [0 , a ∗ i ], and it is strictly decreasing on [ a ∗ i , ¯ a i ].Of course, the monotonicity requirement holds vacuously when the relevant subinterval is a singleton. Let M ⊆ R be a set of “messages,” f i : A i × X i → M is a feedback function that describeswhat i observes (a “message,” e.g., a monetary outcome) after taking any action a i given anypayoff state x i . On top of the formal assumptions stated above, we maintain the following informal assumption about players’ knowledge of the game: • Each player i knows v i and f i .Unless we explicitly say otherwise, we instead do not assume that i knows θ , or function (cid:96) i , oreven that i understands that his payoff is affected by the actions of other players. However, since i knows the feedback function f i : A i × X i → M and the action he takes, what i infers about thepayoff state x i after he has taken action a i and observed message m is that x i ∈ f − i,a i ( m ) := (cid:8) x (cid:48) i : f i (cid:0) a i , x (cid:48) i (cid:1) = m (cid:9) . A.1 Conjectures
Definition A. A shallow conjecture for i is a probability measure µ i ∈ ∆ ( X i ) . A (deep) conjecture for i is a probability measure ¯ µ i ∈ ∆ ( A − i × Θ) . An action a ∗ i is justifiable if thereexists a shallow conjecture µ i such that a ∗ i ∈ arg max a i ∈ A i (cid:90) X i v i ( a i , x i ) µ i (d x i ) ;in this case we say that µ i justifies a ∗ i . Similarly, we say that (deep) conjecture ¯ µ i ∈ ∆ ( A − i × Θ) justifies a ∗ i if the shallow conjecture induced by ¯ µ i ( µ i = ¯ µ i ◦ (cid:96) − i ∈ ∆ ( X i ) ) justifies a ∗ i . Remark If a i (cid:55)→ v i ( a i , x i ) is strictly concave for each x i , then also a i (cid:55)→ (cid:82) X i v i ( a i , x i ) µ i (d x i ) is strictly concave and the map µ i (cid:55)→ arg max a i ∈ A i (cid:90) X i v i ( a i , x i ) µ i (d x i ) is a continuous function. The following lemma summarizes well known results about nice games (see, e.g., Battigalli 2018)and some straightforward consequences for the more structured class of nice games with aggregatorsconsidered here:
Lemma A. The best reply function r i : X i → A i is continuous, hence its range r i ( X i ) is a closedinterval, just like X i . Furthermore, for each given a ∗ i ∈ A i , the following are equivalent: Here the assumption that M is a set of real numbers is without loss of generality, because the same holds for theset of payoff states X i . When ∆ ( X i ) is endowed with the topology of weak convergence of measures. a ∗ i is justifiable, • a ∗ i ∈ r i ( X i ) (that is, a ∗ i is justified by a deterministic shallow conjecture), • there is no a i such that v i ( a ∗ i , x i ) < v i ( a i , x i ) for all x i ∈ X i (that is, a ∗ i is not dominated byany other pure action). Corollary A. Suppose that the aggregator (cid:96) i is onto . Then, an action of player i is justifiable ifan only if it is justified by a deep conjecture. Proof.
The “if” part is trivial. For the “only if” part, fix a justifiable action a ∗ i arbitrarily. ByLemma A, there is some x i ∈ X i such that a ∗ i = r i ( x i ). Since the aggregator (cid:96) i is onto, there issome ( a − i , θ ) ∈ (cid:96) − i ( x i ) such that a ∗ i ∈ arg max a i ∈ A i u i ( a i , a − i , θ ) .Hence a ∗ i is justified the deep conjecture δ ( a − i ,θ ) , that is, the Dirac measure supported by ( a − i , θ ). (cid:4) With this, from now on we restrict our attention to (shallow, or deep) deterministic conjectures . A.2 Feedback properties
Definition B. Feedback f i satisfies observable payoffs (OP) relative to v i if there is a function ¯ v i : A i × M → R such that v i ( a i , x i ) = ¯ v i ( a i , f i ( a i , x i )) for all ( a i , x i ) ∈ A i × X i ; if the section ¯ v i,a i is injective for each a i ∈ A i , then we say that f i satisfies just observable payoffs (JOP) relative to v i . Game G satisfies (just) observable payoffs if, foreach player i ∈ I , feedback f i satisfies (J)OP relative to v i . If f i satisfies JOP, we may assume without loss of generality that f i = v i , because, for eachaction a i , the partitions of X i induced by the preimages of v i,a i and f i,a i coincide: Remark Feedback f i satisfies JOP relative to v i if and only if ∀ a i ∈ A i , (cid:110) v − i,a i ( u ) (cid:111) u ∈ v i,ai ( X i ) = (cid:110) f − i,a i ( m ) (cid:111) m ∈ f i,ai ( X i ) . (a) Proof. (Only if) Fix a i ∈ A i . Since f i satisfies JOP relative to v i , v i,a i ( X i ) = (¯ v i,a i ◦ f i,a i ) ( X i )(by OP), for each u ∈ v i,a i ( X i ) there is a unique message m a i ,u = ¯ v − i,a i ( u ) (by injectivity of ¯ v i,a i ),and v − i,a i ( u ) = { x i ∈ X i : v i ( a i , x i ) = u } = { x i ∈ X i : ¯ v i ( a i , f i ( a i , x i )) = u } = { x i ∈ X i : f i ( a i , x i ) = m a i ,u } = f − i,a i ( m a i ,u ) ,28hich implies eq. (a).(If) Suppose that eq. (a) holds. For every a i ∈ A i and m ∈ f i,a i ( X i ) select some ξ i ( a i , m ) ∈ f − i,a i ( m ). Let D := (cid:91) a i ∈ A i { a i } × f i,a i ( X i )With this, ξ i : D → X i is a well defined function. Domain D is the set of action-message pairs for which the definition of¯ v i matters. Define ¯ v i as follows:¯ v i ( a i , m ) = (cid:40) v i ( a i , ξ i ( a i , m )) if ( a i , m ) ∈ D ,0 otherwise.By construction, eq. (a) implies that ∀ ( a i , x i ) ∈ A i × X i , ¯ v i ( a i , f i ( a i , x i )) = v i ( a i , x i ) .Hence, OP holds. Furthermore, for all a i ∈ A i , m (cid:48) , m (cid:48)(cid:48) ∈ f a i ( X i ), m (cid:48) (cid:54) = m (cid:48)(cid:48) ⇒ ξ i (cid:0) a i , m (cid:48) (cid:1) (cid:54) = ξ i (cid:0) a i , m (cid:48) (cid:1) ⇒ v i (cid:0) a i , ξ i (cid:0) a i , m (cid:48) (cid:1)(cid:1) (cid:54) = v i (cid:0) a i , ξ i (cid:0) a i , m (cid:48)(cid:48) (cid:1)(cid:1) ⇒ ¯ v i (cid:0) a i , m (cid:48) (cid:1) (cid:54) = ¯ v i (cid:0) a i , m (cid:48) (cid:1) where the first and the second implications follow from eq. (a) ( ξ i ( a i , m (cid:48) ) and ξ i ( a i , m (cid:48) ) belongto different cells of the coincident partitions, hence yield different utilities), and the third holds byconstruction. Therefore, ¯ v i,a i is injective for every a i , which means the JOP holds. (cid:4) Definition C. Feedback f i satisfies observability if and only if i is active (OiffA) if section f i,a i is injective for each a i > and constant for a i = 0 . Game G satisfies observability by activeplayers if OiffA holds for each i . Remark If N G is linear-quadratic and satisfies just observable payoffs , then it satisfies observ-ability by active players. Proof.
By Remark 4 JOP implies that, for each a i ∈ A i , (cid:110) v − i,a i ( u ) (cid:111) u ∈ v i,ai ( X i ) = (cid:110) f − i,a i ( m ) (cid:111) m ∈ f i,ai ( X i ) .The linear-quadratic form of v i implies that, for every x i ∈ X i , v − i, ( v i, ( x i )) = X i a i > , v − i,a i ( v i,a i ( x i )) = { x i } .These equalities imply that f i, is constant and f i,a i is injective for a i >
0, that is,
N G satisfiesobservability by active players. (cid:4)
Definition D. Feedback f i satisfies own-action independence (OAI) of feedback about the stateif, for all justifiable actions a ∗ i , a oi and all payoff states ˆ x i , x i ∈ X i , f i ( a ∗ i , ˆ x i ) = f i ( a ∗ i , x i ) ⇒ f i ( a oi , ˆ x i ) = f i ( a oi , x i ) .Game G satisfies own-action independence of feedback about the state if, for each player i ∈ I ,feedback f i satisfies OAI. In other words, OAI says that if player i cannot distinguish between two payoff states ˆ x i and x i when he chooses some given justifiable action a ∗ i , then he cannot distinguish between these twostates when he chooses any other justifiable action a oi . This is equivalent to requiring that thepartitions of X i of the form (cid:110) f − i,a i ( m ) (cid:111) m ∈ f i,ai ( X i ) coincide across justifiable actions, i.e., acrossactions a i ∈ r i ( X i ) (see Lemma A).The following lemma—which holds for any game, not just nice games—states that, under payoffobservability and own-action independence, an action is justified by a confirmed conjecture if andonly if it is a best reply to the actual payoff state: Lemma B. If f i satisfies payoff observability relative to v i and own-action independence of feedbackabout the state, then for all ( a ∗ i , x i ) ∈ A i × X i the following are equivalent:1. there is some ˆ x i ∈ X i such that a ∗ i ∈ arg max a i ∈ A i v i ( a i , ˆ x i ) and f i ( a ∗ i , ˆ x i ) = f i ( a ∗ i , x i ) ,2. a ∗ i ∈ arg max a i ∈ A i v i ( a i , x i ) . Proof. (Cf. Battigalli et al. f i . To prove that (1) implies (2), suppose that f i satisfies OP-OAI and let ˆ x i be such that (1) holds. Let a oi be a best reply to the actual state x i . We must show that also a ∗ i isa best reply to x i . Note that both a ∗ i and a oi are justifiable; hence, by OAI, f i ( a ∗ i , ˆ x i ) = f i ( a ∗ i , x i )implies f i ( a oi , ˆ x i ) = f i ( a oi , x i ). Using OP, condition (1), and OAI as shown in the following chainof equalities and inequalities, we obtain v i ( a ∗ i , x i ) (OP) = ¯ v i ( a ∗ i , f i ( a ∗ i , x i )) (1) = ¯ v i ( a ∗ i , f i ( a ∗ i , ˆ x i )) (OP) = v i ( a ∗ i , ˆ x i ) (1) ≥ v i ( a oi , ˆ x i ) (OP) = ¯ v i ( a oi , f i ( a oi , ˆ x i )) (1 , OAI) = ¯ v i ( a oi , f i ( a oi , x i )) (OP) = v i ( a oi , x i ) .Since a o is a best reply to x i and v i ( a ∗ i , x i ) ≥ v i ( a oi , x i ), it must be the case that also a ∗ i is a bestreply to x i . (cid:4) orollary B. Suppose that G satisfies payoff observability and own-action independence offeedback about the state, then the sets of selfconfirming action profiles and Nash equilibrium actionprofiles coincide for each θ : ∀ θ ∈ Θ , A SCEθ = A NEθ . Proof
By Remark 2, we only have to show that A SCEθ ⊆ A NEθ . Fix any a ∗ = ( a ∗ i ) i ∈ I ∈ A SCEθ andany player i . By definition of SCE, there is some ˆ x i ∈ X i such that a ∗ i ∈ r i (ˆ x ∗ i ) and f i ( a ∗ i , ˆ x i ) = f i (cid:0) a ∗ i , (cid:96) i (cid:0) a ∗− i , θ (cid:1)(cid:1) . By Lemma B a ∗ i ∈ r i (cid:0) (cid:96) i (cid:0) a ∗− i , θ (cid:1)(cid:1) . This holds for each i , hence a ∗ ∈ A NEθ . (cid:4) Corollary B provides sufficient conditions for the equivalence between SCE and NE. Next, wegive sufficient conditions that allow a characterization of A SCEθ by means of Nash equilibria ofauxiliary games.
A.3 Equilibrium Characterization If a i ∈ [0 , ¯ a i ] is interpreted as an activity level (e.g., effort) by player i , then it makes sense to saythat i is active if a i > inactive otherwise. Let I denote the set of players for whombeing inactive is justifiable . Note that, by Lemma A, I = { i ∈ I : min r i ( X i ) = 0 } .Also, for each θ ∈ Θ and nonempty subset of players J ⊆ I , let A NEJ,θ denote the set of Nashequilibria of the auxiliary game with players set J obtained by letting a i = 0 for each i ∈ I \ J , thatis, A NEJ,θ = (cid:110) a ∗ J ∈ × j ∈ J A j : ∀ j ∈ J, a ∗ j = r j (cid:16) (cid:96) j (cid:16) a ∗ J \{ j } , I \ J , θ (cid:17)(cid:17)(cid:111) ,where I \ J ∈ R I \ J is the profile that assigns 0 to each i ∈ I \ J . If J = ∅ , let A NEJ,θ = ∅ by convention. Lemma C. Suppose that the parametrized nice game with aggregators and feedback G satisfies observability by active players . Then, for each θ , the set of selfconfirming action profiles is A SCEθ = (cid:91) I \ J ⊆ I A NEJ,θ × (cid:8) I \ J (cid:9) . Proof
Let J be the set of players i such that a ∗ i >
0. Fix θ ∈ Θ arbitrarily. Let a ∗ ∈ A SCEθ andfix any i ∈ I . If a ∗ i = 0, then 0 is justifiable for i , that is i ∈ I . If a ∗ i >
0, OiifA implies that f i,a ∗ i is injective, that is, action a ∗ i reveals the payoff state, hence the (shallow) conjecture justifying a ∗ i is correct: a ∗ i = r i (cid:0) (cid:96) i (cid:0) a ∗− i , θ (cid:1)(cid:1) . Thus, a ∗ = (cid:16) a ∗ J , a ∗ I \ J (cid:17) so that a ∗ i = 0 for each i ∈ I \ J ⊆ I , and a ∗ j = r j (cid:16) (cid:96) j (cid:16) a ∗ J \{ j } , I \ J , θ (cid:17)(cid:17) > j ∈ J . Hence, a ∗ = (cid:16) a ∗ J , a ∗ I \ J (cid:17) ∈ A NEJ,θ × (cid:8) I \ J (cid:9) with I \ J ⊆ I .31et I \ J ⊆ I and (cid:16) a ∗ J , a ∗ I \ J (cid:17) ∈ A NEθ × (cid:8) I \ J (cid:9) . Since G satisfies OiffA, for each i ∈ I \ J , anyconjecture justifying a ∗ i = 0 (any ˆ x i ∈ r − i (0)) is trivially confirmed. For each j ∈ J , a ∗ j > x ∗ j = (cid:96) i (cid:16) a ∗ J \{ j } , I \ J , θ (cid:17) .Hence, (cid:16) a ∗ J , a ∗ I \ J (cid:17) = (cid:0) a ∗ J , I \ J (cid:1) ∈ A SCEθ . (cid:4) Appendix B Interior Nash equilibria
Propositions 1 and 2 in Section 3 show that in our framework there exists an equivalence betweenany selfconfirming equilibrium and the Nash equilibrium of a reduced game in which only activeagents are considered and there is also
Oif f A . Moreover, we can set any subset of agents to beinactive. We now provide some results about existence of these selfconfirming equilibria, that willbe useful in proving Proposition 3 in Section 3. We first present sufficient conditions that arepresent in the literature for the existence of interior Nash equilibria, then we provide some originalresults.In this appendix we formulate the problem as a linear algebra problem. We consider a squarematrix Z ∈ R n × n such that z ii = 0 for all i ∈ { , . . . , n } . We call I the identity matrix, λ max ( Z )the maximal eigenvalue of Z , ρ ( Z ) the spectral radius of Z (i.e. the largest absolute value of itseigenvalues), is the vector of all 1’s, is the vector of all 0’s, and (cid:29) is the strict partial orderingbetween vectors (meaning that all the elements in the first vector are pairwise strictly greater thanthe elements in the second vector). Proposition C. If for all i , z ii = 0 , for all j (cid:54) = i , z ij ≤ , and if ρ ( Z ) < , then ( I − Z ) − (cid:29) . There are also results when the sign of the externalities are mixed. We recall that the matrix Z issymmetrizable if there exists a diagonal matrix Γ and a symmetric matrix Z such that Z = ΓZ .Note that if Z is symmetrizable then all its eigenvalues are real. If for all i , z ii = 0, and Z issymmetrizable, we define the symmetric matrix ˜ Z to be such that ˜ z ij = z ij √ γ i γ j . Proposition D. If for all i , z ii = 0 , Z is symmetrizable, and if | λ max ( ˜ Z ) | < , then ( I − Z ) − (cid:29) . We provide here below an alternative condition, which does also guarantee all positive solutions.
Proposition E. Consider a square matrix Z ∈ R n × n such that: • z ii = 0 for all i ∈ { , . . . , n } ; This is Theorem 1 in Ballester et al. (2006). The same result is in Appendix A in Sta´nczak et al. (2006). See Section VI of Bramoull´e et al. (2014), generalizing Proposition 2 therein. Note that in their payoff specificationexternalities have a minus sign, while in (4) we have a plus sign: this is why we have a condition on the maximaleigenvalue and not on the minimal eigenvalue. | z ij | < n for all i, j ∈ { , . . . , n } .Then ( I − Z ) − (cid:29) . Proof:
Call B = ( I − Z ). First of all, by Gershgorin circle theorem , B has all eigenvaluesstrictly between 0 and 2, so det ( B ) (cid:54) = 0.Consider the n vectors b , . . . , b n given by the n rows of B , and take the hyperplane in R n passingby those n points: H = { h ∈ R n : ∃ α ∈ R n with α (cid:48) · = 1 and h = B (cid:48) α } . Now, consider the following vector v = B − . Component v i of v is exactly the sum of the elements in i th row of B − . However, v is also a vectorperpendicular to H . That is because for any h ∈ H we have h · v = (cid:0) B (cid:48) α (cid:1) (cid:48) · B − = α (cid:48) = n (cid:88) i =1 α i = 1 , which is a constant.Now, we want to show that H does not pass thorugh the convex region of vectors with allnon-postitive elements: H ∩ ( −∞ , n = ∅ .In fact, it is impossible to find α ∈ R n , such that α (cid:48) · = 1 and B (cid:48) α (cid:28) .If it was the case, by absurdum, we could take k = arg max i ∈{ ,...,n } { α i } ( α k > (cid:80) ni =1 α i =1), and write α b k = α k + (cid:88) j (cid:54) = k α j b jk > α k − (cid:88) j (cid:54) = k | α j || z jk | > α k − (cid:88) j (cid:54) = k | z jk | > , which would be a contradiction.Finally, we show that if an hyperplane H satisfies H ∩ ( −∞ , n = ∅ , then its perpendicularvector from the origin has all positive elements, and this would close the proof .We do so by induction on n .1. n = 2: This is easy to show graphically;2. Induction hypothesis : Suppose it is true for n = m − https://en.wikipedia.org/wiki/Gershgorin_circle_theorem Induction step : In R m , a vector v from the origin which is perpendicular to an hyperplane H not passing through the origin can be obtained in the following way. For each dimension i ∈ { , . . . , m } take V ¬ i = { v ∈ R m : v i = 0 } . Call H ¬ i the intersection of H with V ¬ i ,and take a vector v ¬ i ∈ V ¬ i from the origin that is perpendicular to H ¬ i . By the inductionhypothesis v ¬ i has all positive elements. We can obtain the vector v from the origin that isperpendicular to H by rescaling each v ¬ i , such that v ¬ i is the projection of v on H ¬ i . Byconstruction, v will have all positive elements.Notice that, if Z satisfies the conditions of Proposition E, then it must also hold that | λ max ( Z ) | <
1, because of
Gershgorin circle theorem . However, the condition that | λ max ( Z ) | < I − Z ) − (cid:29) . (cid:4) Appendix C Proofs
Proof of Proposition 1
Proof.
Since every agent is active, state observability by active players implies own actionindependence of the feedback about the state . Then, the result derives from Corollary B in AppendixA. (cid:4)
Proof of Proposition 2
Proof.
By Remark 5,
N G satisfies observability by active players. Hence, Lemma C in AppendixA and the best reply equation (6) yield the result. (cid:4)
Proof of Proposition 3
Proof.
Condition ( i ), ( ii ) and ( iii ) correspond, respectively, to the conditions in Propositions E,C and D from Appendix B. (cid:4) Proof of Proposition 4
Proof.
If for every i ∈ I \ I a ∗ we have that α + ˆ x i <
0, then changing their ˆ x i such that theinequality is still strict, will not make them become active.So, let us focus on the subset I a ∗ of active agents. If we perturb locally the beliefs, we will perturblocally also their actions. Assumption 4 garantees that the discrete dynamical system defined foractions by (8) and (9) is stable. So, the variation to beliefs can always be small enough such that:all their actions remain strictly positive;we are in a neighborhood of a ∗ in the actions’ space, such that the discrete dynamical systemdefined for actions by (8) and (9) converges back to a ∗ . (cid:4) roof of Proposition 5 Proof.
When we remove elements from J a and set them to 0, it is as if we delete correspondingrows and columns in the Z J a matrix. By the Cauchy interlace theorem applied to symmetrizablematrices (see Kouachi 2016) we know that the eigenvalues of the new matrix are between theminimal and the maximal eigenvalues of the old matrix. (cid:4) Proof of Proposition 6
A selfconfirming equilibrium is such that, for all i ∈ I , rationality implies a ∗ i = max { , α i + ˆ x i } . Each agent then thinks that m ∗ = α i a ∗ i −
12 ( a ∗ i ) + a ∗ i ˆ x i + ˆ y i , so that ˆ y i = m ∗ − α i a ∗ i + 12 ( a ∗ i ) − a ∗ i ˆ x i . Substituting the expression of the true payoff function m ∗ = α i a ∗ i −
12 ( a ∗ i ) + a ∗ i x i + y i into it, we get the dependence between ˆ y i and ˆ x i :ˆ y i = y i + a ∗ i ( x i − ˆ x i ) . The first and second items in the proposition are derived, respectively, if a ∗ i = 0 or a ∗ i > (cid:4) Proof of Proposition 7
Proof. First, we derive some properties.
Each equation in the system given by (15) can bewritten also as a parabola b a i + b a i + b = 0, in the following way H i ( a , c , Z ) = c i (cid:124)(cid:123)(cid:122)(cid:125) ≡ b a i + − αc i − c i (cid:88) j ∈ I z ij a j,t (cid:124) (cid:123)(cid:122) (cid:125) ≡ b a i − c i β (cid:88) j (cid:54) = i a j,t (cid:124) (cid:123)(cid:122) (cid:125) ≡ b = 0 . (b)35o, the solution a ∗ i to (cid:96) i ( a , c , Z ) = 0 lies in the right–arm of an upward parabola, where d(cid:96) i da i (cid:12)(cid:12)(cid:12) a i = a ∗ i >
0. With respect to c i , each (cid:96) i ( a , c , Z ) is a linear equation.Note also that each a i is bounded in the interval α < a i < α + (cid:88) j ∈ N i z ij a j + β (cid:80) k (cid:54) = i a k a i . Considering that a ∗ i is increasing in b and decreasing in b , it is easy to see that each a ∗ i increasesin each a j , with j (cid:54) = i . Second, we show that there is a homeomorphism.
There is a continuous function definedfrom each c ∈ [0 , n to an element a ∈ A , that is because • either c i = 0 and then a ∗ i = α ; • or c i > a ∗ i is continuously increasing in each x j with j (cid:54) = i .lim c i → a ∗ i = α .a ∗ i is bounded above by α + (cid:88) j ∈ N i z ij a j + β (cid:80) j (cid:54) = i a j a ∗ i . Since the system defined by (16) admits a solution, also this system has a finite solution.This function is one–to–one and invertible, because for each a ∈ A , we obtain a unique vector c ∈ [0 , n , and since we obtain it from a linear system of equations, also the inverse function from A to [0 , n is continuous.To analyze the relation between a ∗ and c , we can apply the implicit function theorem to F i ( a , c , Z ).We can compute dF i dc i = β (cid:80) j (cid:54) = i a j,t ( a i c i + 1) Now, since (cid:96) i ( a , c , Z ) = − ( a i c i + 1) F i ( x , c ) , we have that (cid:96) i ( a , c , Z ), with respect to a i , has the same zeros as F i ( a , c , Z ), and that, for each a i , (cid:96) i ( a , c , Z ) is negative if and only if F i ( a , c , Z ) is positive. As they are both continuous functions,this means that since d(cid:96) i da i (cid:12)(cid:12)(cid:12) a i = a ∗ i >
0, we have dF i da i (cid:12)(cid:12)(cid:12) a i = a ∗ i <
0. So, we obtain that da i dv i (cid:12)(cid:12)(cid:12)(cid:12) a i = a ∗ i = − ∂F i /∂c i ∂F i /∂a i (cid:12)(cid:12)(cid:12)(cid:12) a i = a ∗ i > . (c)This shows that a ∗ i is increasing with v i , and the other way round. (cid:4) roof of Proposition 8 Proof.
We consider the system (15) F i ( a , v , Z ) = α + c i β (cid:88) j (cid:54) = i a j,t a i c (cid:48) i + 1 a i c i + 1 − a i = 0 , with c (cid:48) i,t = (cid:80) j ∈ I z ij a j,t β (cid:80) j (cid:54) = i a j,t . We can compute its Jacobian, with respect to a , and check that each rowof the Jacobian sum to less than 1, so that the process is always a contraction. The Jacobian J issuch that: (cid:40) J ij = v i a i c i +1 ( β + a i z ij ) J ii = c i (cid:16) β (cid:80) j (cid:54) = i a j (cid:17) (cid:16) c (cid:48) i a i c i +1 − c i a i c (cid:48) i +1( a i c i +1) (cid:17) − (cid:88) j J ij = c i a i c i + 1 β (cid:88) j (cid:54) = i a j (cid:18) c (cid:48) i − c i a i c (cid:48) i + 1 a i c i + 1 (cid:19) + a i (cid:88) j (cid:54) = i z i,j + β ( n − − a i , for any a i ≥ a i → ∞ , we have that expression (d) is equal to (cid:88) j (cid:54) = i z i,j − , (e)whose absolute value is less than one by assumption.If a i →
0, espression (d) becomes c i β (cid:88) j (cid:54) = i a j (cid:0) c (cid:48) i − c i (cid:1) + ( n − − . (f)An interior maximum or minimum of the numerical expression (d), with respect to a i , must satisfyfirst order condition − (cid:18) c i a i c i + 1 (cid:19) β (cid:88) j (cid:54) = i a j (cid:18) c (cid:48) i − c i a i c (cid:48) i + 1 a i c i + 1 (cid:19) + a i (cid:88) j (cid:54) = i z i,j + β ( n − + c i a i c i + 1 β (cid:88) j (cid:54) = i a j (cid:18) c i a i c i + 1 (cid:19) (cid:18) c (cid:48) i − c i a i c (cid:48) i + 1 a i c i + 1 (cid:19) + (cid:88) j (cid:54) = i z i,j = 0Last expression can be simplified and results in v i β ( n −
1) = (cid:88) j (cid:54) = i z i,j , a i . So, the only candidates for being minima or maxima for espression (d)are its value in the extrema, namely (e) and (f).Also, the sign of the first derivative of (d) with respect to a i is equal to the sign of (cid:80) j (cid:54) = i z i,j − c i β ( n − c i β ( n − < (cid:80) j (cid:54) = i z i,j we have that (d) is strictly increasing in a i , and then (e)is strictly greater than (f).The value of (e) is between − < (cid:80) j (cid:54) = i z i,j < v i →
0; and c (cid:48) i →
0. In this case (f) goes to − c i > −
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