aa r X i v : . [ ec on . T H ] A p r Building social networks under consent:A survey ∗ Robert P. Gilles † October 2019Revised: April 2020
During the past two decades there has emerged an extensive literature on game theoretic models ofnetwork formation. Seminally, the fundamentals of such a game theoretic perspective were set outby Aumann and Myerson (1988) in which players are guided by the Myerson value of correspondingcommunication situations. This contribution explored network formation under mutual consentthrough a non-cooperative signalling game: A link between two players is formed if and only ifboth players signal to each other their willingness to form this relationship. The main insight of the
Myerson model (Myerson, 1991) is that the network without any links is always supported througha Nash equilibrium of this signalling game. This theoretical result leads to the conclusion thatnetwork formation under mutual consent has to be considered as difficult, even impossible. Thiswould contradict the well-established understanding of human nature as that of a social networker(Seabright, 2010; Harari, 2014). Nevertheless, paradoxically, the Myerson model is and remains themost natural, straightforward and convincing non-cooperative model of network formation undermutual consent.The relative failure of this natural non-cooperative approach induced Jackson and Wolinsky(1996) to introduce an alternative approach, which is founded on a bilateral cooperative considera-tion. In their approach, Jackson and Wolinsky allow pairs of players to cooperatively deviate from ∗ I am very grateful for the extensive and helpful comments and suggestions of an anonymous referee. I also thankmy co-authors Sudipta Sarangi, Subhadip Chakrabarti and Owen Sims for their contributions and support to explore theparadoxes of social network formation under the hypothesis of mutual consent. The work summarised here could nothave been developed without their contributions, insights and help. † Economics Group, Management School, The Queen’s University of Belfast, Riddel Hall, 185 Stranmillis Road, Belfast,BT9 5EE, UK.
Email: [email protected] The main conclusion is strengthened in the case of costly link formation, in which the empty network is a strong
Nash equilibrium, indicating that starting from an empty network it seems unlikely that rational agents would be able toestablish non-trivial networks. I also refer to Joshi, Mahmut, and Sarangi (2020) for a dynamic model of such non-trivialnetwork formation. An alternative mathematical model emerged with Bala and Goyal (2000) based on one-sided link formation: Oneassumes ex-ante , or implicit, consent among players in the network formation game. The resulting equilibrium net-works are denoted as
Nash networks in the subsequently developed literature. This approach is unsatisfactory due to itsunnatural social foundations with rather limited applicability to explain social and economic phenomena.
1n existing network to modify it. The equilibrium networks under such pairwise modification aredenoted as pairwise stable networks. Pairwise stability provided a fertile foundation for further ex-ploration of network formation under cooperative consent. This resulted in the development andstudy of variations of pairwise stability.Although the Jackson-Wolinsky approach founded on pairwise stability has been very success-ful in explaining the emergence of non-trivial networks, there remained a gap in our understandingconcerning a purely non-cooperative approach to the modelling of mutual consent in network for-mation. This has been more recently explored through the design of bespoke equilibrium conceptapplied in the Myerson model. In particular, van de Rijt and Buskens (2008) and Gilles and Sarangi(2010) introduced models of trusting behaviour in network formation through trust-based belief sys-tems. The equilibrium concepts that are developed from these models have very strong properties,showing that trust in network formation leads to non-trivial equilibrium networks. For example,Gilles and Sarangi (2010)’s notion of monadic stability results in equilibrium networks that form aspecific subclass of pairwise stable networks—denoted as the strictly pairwise stable networks.
Overview of this survey.
This survey explores the various methodologies to properly modelmutual consent in network formation. I compare the different classes of equilibrium networks thatemerge from these different methodologies. After discussing the principles of link formation undermutual consent and Myerson’s seminal model, I turn to the exploration of Jackson-Wolinsky typestability concepts based on pairwise cooperative behaviour. I distinguish different subclasses ofstable network based on hypotheses about how coalitions of certain sizes can modify the currentnetwork. This mainly pertains to pairs of players, but also extends to coalitions of players of ar-bitrary size—resulting in the notion of a strongly stable network (Jackson and van den Nouweland,2005).Subsequently, I turn to the main non-cooperative theory of network formation under mutualconsent, namely extensions of the Myerson model (Myerson, 1991). I survey the results from theliterature that categorise the various classes of equilibrium networks in the Myerson model withtwo- as well as one-sided link formation costs. There emerges a close link to certain classes of stablenetworks in the Jackson-Wolinsky framework.Subsequently, I discuss the idea of equilibrium refinement in the Myerson model to reflect con-siderations of mutual trust in link formation. Indeed, links are representations of socio-economicrelationships that are founded on mutual trust between the interacting parties. This results in theunilateral (van de Rijt and Buskens, 2008) and monadic stability (Gilles and Sarangi, 2010) conceptsin the Myerson model. I explore monadic stability further, which is founded on a conception of mu-tual trust through a belief system in the Myerson model. The properties of these monadically stablenetworks as well as their existence, using Monderer and Shapley (1996)’s theory of game-theoreticpotentials, are also reviewed.I conclude this survey by looking at an alternative method to modelling mutual consent in link-and network formation. This refers to the introduction of correlated strategies in the Myersonmodel as a tool to represent coordinated interaction. The resulting class of “correlated equilibriumnetworks” still needs to be explored in future research.2
Introducing mutual consent: Modelling principles
Throughout this survey, I use a broad class of game theoretic techniques to model how relationships—or “links”—between pairs of socio-economic agents come about. We refer to these socio-economicagents as players in the context of these models. Each player is assumed to be a fully rational indi-vidual decision maker that acts according to a set of behavioural rules described in the developedequilibrium concept.Besides the specific behavioural hypotheses on which these equilibrium concepts are based, itis important to realise that there are some fundamental broad axioms made. These fundamentalaxioms introduce a few fundamental limitations of the approach that is surveyed here:(i) This game-theoretic approach is purely static in nature. This implies that we start from azero state in which no links exist and in which these socio-economic agents decide whetherand which links to build. The end result is a fully formed network in which certain value-generating activities are achieved. It would be more realistic to model the formation ofa network as a dynamic building process. However, in the static conception followedthroughout this survey, one network does not evolve into another.This has major consequences for how we view network formation and which networksactually are identified in these constructions. Indeed, the identified equilibrium networksdo not exhibit the features of large social networks identified in the literature quoted on so-cial networks (Newman, 2010; Barab´asi, 2016). So, these equilibrium networks are usuallyneither scale-free nor small world networks nor satisfying the basic property of assortativemixing. This is a severe limitation of such a static approach. On the other hand, the static approach highlights certain properties of rational decisionmaking in the context of pairwise cooperation, required for building value-generating re-lationships under mutual consent. Rather contradictorily, the main theorem in Myerson’snon-cooperative model shows that rational decision-making does actually not result intoany sensible network formation—the empty network is always supported through a Nashequilibrium in the Myerson model. So, starting from an empty network, fully rational play-ers have no mechanism to create a meaningful interaction structure. Only if we imposethat the decision makers are boundedly rational —and, thus, use animal spirits rather thanoptimisation in decision making—we arrive at the conclusion that non-trivial and sensiblenetworks emerge under mutual consent. This important insight is the main conclusionpresented in this survey.(ii) The game-theoretic approach explored in this survey is founded on a negative stabilitymethodology. Hence, a network is called “stable” if there are no incentives to change the In my discussion in this survey I omit the recent development of incentive-based stochastic models of networkformation. This approach focuses not only on game theoretic incentives in network formation—as the subject matterof this survey—but combines this concept with stochastic processes that describe random meetings. This approach wasseminally developed in Jackson and Rogers (2005, 2007) and further addressed in, e.g., Golub and Livne (2010). This is captured in the notion of a monadically stable network that is founded on trusting behaviour by the play-ers. Such trusting behaviour is fundamentally boundedly rational. Indeed, to trust another player is not founded oncalculation, but on a leap of faith. cannot emerge due to the existing incentives to change the network that theplayers are endowed with. We thus arrive at a class of equilibrium networks that describeconfigurations in which such incentives for deviation are absent.The consequence of the application of this standard game-theoretic methodology is thatreality is only approximated. This approach, for example, does not allow the mixing ofmodes of incentives, which is common in real-life interaction. This, therefore, is anotherreason why the theoretically derived networks do not have the desired features discussedin the literature on large social networks as surveyed by Barab´asi (2016).The next section sets out the basic framework of modelling mutual consent in the formation of arelationship between two players.
We use the basic concepts from the theory of social networks set out in the literature. Followingthe accepted symbolism, the set N = { , . . . , n } represents a set of players . The fundamental issueaddressed here is how these players will build pairwise or binary relationships with other playersand ultimately construct a socio-economic network consisting of such binary relationships.Each player i ∈ N is explicitly endowed with the social ability to build such pairwise relation-ships or links with other players, provided that consent is given by the other party. Again followingthe accepted terminology in the literature (Jackson, 2008), the pairwise subset { i , j } ⊂ N with i , j denotes a pairwise relationship between players i ∈ N and j ∈ N . We follow convention to useshorthand notation and define a link between players i and j as ij = { i , j } ∈ д N , where д N = { { i , j } | i , j ∈ N and i , j } = { ij | i , j ∈ N } (1)denotes the set of all potential links on the player set N . As such the set д N acts as the universalset of all potential links on player set N .A network on N is now an arbitrary subset of links, i.e., any subset д ⊂ д N is a network on N .In particular, д = д = œ is the empty network on N which describes a situation where no links areformed. Furthermore, д = д N is the complete network on N , which is the largest network consistingof all potential links among players in N . We introduce G N = { д | д ⊂ д N } as the collection of allnetworks on N .The neighbourhood of player i ∈ N in network д ∈ G N is given by N i ( д ) = { j ∈ N | ij ∈ д } .The collection of corresponding neighbouring relationships or links is denoted by L i ( д ) = { ij ∈ д | j ∈ N i ( д )} . The complete collection of all potential links that involve player i ∈ N —or that can beformed by player i —is denoted by L i = L i ( д N ) = { ij | j , i } . Adding and deleting links to a network.
In formal models of network formation we considerthe deletion and addition of links to given networks. For this I introduce some well-accepted no-tation (Jackson, 2008). Consider a network д ∈ G N . For every pair of players i , j ∈ N with4 j < д we now denote by д + ij the network that results from д by adding the link ij < д , i.e., д + ij = д ∪ { ij } ∈ G N . Similarly, for some collection of links h ⊂ д N with д ∩ h = œ , we denote д + h = д ∪ h the network that results from adding link collection h to the network д .Next, consider two players i , j ∈ N with ij ∈ д . We denote by д − ij = д \ { ij } ∈ G N the networkthat results from removing the link ij from the network д . Again, for any collection of links h ⊂ д we denote д − h = д \ h the network that results from removing the links in h from the network д .12 34 5Figure 1: Illustration for link addition and deletion. Example 2.1
With these notational conventions we are now equipped to address link formationprocesses. To illustrate this notation, consider the network д = { , , , , } on N = { , , , , } as depicted in Figure 1 above consisting of the red and black links. Considering the green link45 < д , then д ′ = д + = { , , , , , } is depicted in Figure 1 as the network consist-ing of all coloured links. Finally, removing the red link set h = { , } ⊂ д from д results into д ′′ = { , , } , depicted by collection of the black links only in Figure 1. (cid:7) Payoffs.
Throughout the literature on game theoretic approaches to network formation, playersare assumed to be fully incentivised in their drive to build and maintain links as well as delete linksin existing networks. These incentives are introduced as a individualised payoff function. Indeed,for every player i ∈ N we introduce player i ’s network payoff function as φ i : G N → R , which assignsto every network д ∈ G N a value φ i ( д ) that evaluates i ’s situation as a member of the networkedcommunity described by д .We can now capture all payoff information on the population N of players in the network payofffunction given by φ = ( φ , . . . , φ n ) : G N → R N . In particular, I emphasise that the function φ indeed captures all incentives for the decision makers in N in the network formation processes tobe considered next. A network payoff for a player captures all values emanating in the structured community thatis perceived or received by that player. This includes all perceived externalities of third parties. Inthis regard, the network payoff function can capture widespread externalities from relationshipand network formation in that community. The addition of network externalities in the payoffstructure differentiates this inclusive network payoff approach from the more classical cooperativegame theoretic payoff structure employed by Myerson (1977, 1980), Dutta and Mutuswami (1997) We might refer to the multi-dimensional function φ also as representing the network payoff structure . Example 2.2
I illustrate this concept by revisiting the networks depicted in Figure 1. For example,player 1 can be assigned φ ( д ) = φ ( д ′ ) = N ( д ) = N ( д ′ ) = { , } . This, therefore, captures widespreadexternalities from the creation of the link 45 in the network д from the perspective of player 1. (cid:7) The most fundamental and basic model of how networks form under mutual consent was seminallyintroduced as an example in Myerson (1991, page 448). He pointed out that in a very simple networkformation game—known as the
Myerson model —, the resulting networks that are supported by Nashequilibria in this game always include the empty network д . Hence, building no links at all is anequilibrium in the incentive structure generated by player benefits to network formation.Myerson presented this as a negative insight, since it indicates that purely noncooperative gametheory cannot provide a fertile basis for a debate of how non-trivial networks between playersemerge. However, what this really expresses is that networks are not forming if players act purelyselfishly. My contention is throughout that it actually has to be expected that pure selfishness wouldundermine cooperative acts such as forming links between pairs of players.Here I initially explore the seminal Myerson model itself. In subsequent sections I turn to ex-tensions of this basic model with added consideration of link formation costs. For the proper devel-opment of the Myerson model we need to review some basic non-cooperative game theory. Preliminaries: Some game theory.
This section relies heavily on standard noncooperativegame theory. Again we let N = { , . . . , n } be the set of players. A game on N is a pair (A , π ) with A = ( A , . . . , A n ) an ordered collection of strategy sets such that each player i ∈ N is assignedher individual strategy set A i and a game theoretic payoff function π = ( π , . . . , π n ) : A → R N where A = Î i ∈ N A i is the set of all strategy tuples generated in A .Hence, in a non-cooperative game, each player i ∈ N is endowed with her individual strategyset A i and a payoff function π i : A → R . The fundamental idea is that every player selects a strategythat optimises her payoffs, provided that other players also select strategies that affect this payoff.As such, a game is a mathematical representation of a social interaction situation. Game theory isnow a collection of rules and tools that model how players make decisions in the context of suchsocial interaction situations.A strategy tuple is a list a = ( a , . . . , a n ) ∈ A . We use the convention that the list of strategies ofplayers other than i ∈ N are indicated by a − i = ( a , . . . , a i − , a i + , . . . , a n ) ∈ Î j ∈ N : j , i A j . Hence, a = ( a i , a − i ) . Definition 2.3
A strategy tuple a ∗ ∈ A is a Nash equilibrium in the game (A , π ) if for every player i ∈ N and any strategy b i ∈ A i it holds that π I ( a ∗ ) > π i ( b i , a ∗− i ) . In a Nash equilibrium, every playeroptimises her strategy, given the strategic choices of all other players . a i ∈ A i is a best response to strategy tuple a − i ∈ Î j ∈ N : j , i A j if for every strategy a ′ i ∈ A i it holds that π i ( a i , a − i ) > π i ( a ′ i , a − i ) . Hence, a best response is the strategy for player i that optimises herpayoffs given that all other players j , i select the strategy a j ∈ A j .Now a strategy tuple a ∗ ∈ A is a Nash equilibrium if and only if for every player i ∈ N it holdsthat a ∗ i is a best response to a ∗− i . As such a Nash equilibrium is a fixed point of the best responsecorrespondence that is generated by the game. Furthermore, it can be shown that in this respecta Nash equilibrium usually can be interpreted as a saddle point in a well-constructed geometricrepresentation of the game. The Myerson model.
Myerson (1991) introduced his approach to modelling the formation ofnetworks as an illustration of the underlying processes that determine the Nash equilibria in anon-cooperative strategic form game. Myerson’s framework is the quintessential model of mu-tual consent in link formation. The Myerson model encompasses a basic signalling game in whichplayers send each other messages about whether they want to form a link or not. Due to its veryfundamental and basic nature, it is a model that acts as the benchmark in any discussion on consentin link formation.In Myerson’s framework, players costlessly signal to each other whether they want to formlinks. Now, a link is established if and only if the two players signal both that they would like toform the link. Formally, the
Myerson model Γ mφ on player set N under network payoff function φ : G N → R N is a non-cooperative game Γ mφ = (A m , π m ) given as follows:• For every player i ∈ N , her strategy set is given by all vectors of signals to other players in N : A mi = (cid:8) ℓ i = ( ℓ i , ℓ i , . . . , ℓ in ) (cid:12)(cid:12) ℓ ij ∈ { , } and ℓ ii = (cid:9) ; (2)Here, ℓ ij is a signal that player i communicates to player j about her intentions to form a linkwith j . If ℓ ij =
1, player i indicates that she is interested in forming the link with player j ; if ℓ ij =
0, player i signals that she wants to remain unattached to player j .• A link ij is now formed if both players i and j signal to each other they want to form the link,i.e., if ℓ ij = ℓ ji =
1. If we denote by ℓ = ( ℓ , . . . , ℓ n ) ∈ A m = A m × · · · × A mn a strategy profile,then the resulting network can be identified as д ( ℓ ) = { ij ∈ д N | ℓ ij = ℓ ji = } . (3)We say that д ( ℓ ) is the network supported by the strategy profile ℓ in the Myerson model.• The Myerson model is completed by the game theoretic payoff function π m : A m → R N defined by π mi ( ℓ ) = φ i ( д ( ℓ )) . (4)7learly, the payoff function π m reflects the property that signalling is costless and that thereare no costs incurred in the formation of a link between any pair of players.In the next discussion, I investigate the networks that are supported through Nash equilibria in theMyerson model. M-networks.
The Nash equilibria in the basic Myerson model form a class of signalling profilesthat support networks on N that are stable against unilateral modification. We denote these Nashequilibrium networks as “M-networks” to distinguish this class of networks from other classes ofnetworks. Definition 2.4
Let φ be a network payoff function on player set N and let Γ mφ = (A m , π m ) be thecorresponding basic Myerson model. A network д ∈ G N is an M-network if there exists a Nashequilibrium strategy tuple ℓ д ∈ A m in Γ mφ such that д ( ℓ д ) = д . Clearly, using the Nash equilibrium conditions and the definition of π m , we get the following M-network requirement: For every player i ∈ N and every signal vector ℓ i ∈ A mi it holds that φ i (cid:0) д ( ℓ i , ℓ д − i ) (cid:1) φ i ( д ( ℓ ) ) . The concept of M-network is at the core of the assessment of network formation itself, since itdescribes the stable outcomes of the basic signalling framework represented in the Myerson model.Crucially, Myerson (1991) already pointed out that the empty network is always supported as anM-network. Formally, this can be expressed as follows.
Proposition 2.5 (Myerson’s Lemma)
In the Myerson model Γ mφ = (A m , π m ) the “no-link” signalprofile ℓ = ( , . . . , ) ∈ A m is a Nash equilibrium. Consequently, the empty network д = д ( ℓ ) is anM-network. Proof.
Let ℓ ij = i , j ∈ N , making up the strategy profile ℓ . Then, for any player i ∈ N ,any signal vector ℓ i ∈ A mi is a best response to ℓ − i , since д ( ℓ − i , ℓ i ) = д irrespective of the selectedsignal vector ℓ i . Therefore, ℓ i itself is a best response to ℓ − i , showing that ℓ is a Nash equilibriumin Γ mφ = (A m , π m ) .This property points out that non-trivial M-networks are very hard to form; rational self-interesteasily results in complete failure and no cooperation might emerge. In this case, Myerson’s Lemmaindicates that, without some supporting mechanism, there simply are no incentives to justify thatany links are formed at all. So, Myerson’s Lemma points to the very fundamental issue of humancooperation: Why would rational human beings be cooperative? In this regard, Myerson’s Lemmais a very succinct expression of this major question in social science and economics. The challenge of modelling non-trivial network formation stated in the discussion of the Myersonmodel as Myerson’s Lemma was taken on by Jackson and Wolinsky (1996). They formulated co- perative equilibrium concepts that are tailored to the specific demands of modelling bilateral linkformation. This resulted in the notion of a “pairwise stable” network.I first discuss a class of cooperative or pairwise concepts of network stability from a link-basedperspective as explored in Gilles, Chakrabarti, and Sarangi (2006, 2012). This concerns four funda-mental link-stability principles, each founding a particular form of cooperative stability, and threefurther derived stability notions—including the seminal pairwise stability concept introduced byJackson and Wolinsky (1996).Central to this approach is that while mutual consent is required for establishing a link, a playeris able to delete her links unilaterally. Here, we focus on link-centred considerations. Hence, howwould the deletion of one or more links affects the players’ payoffs? Similarly, how would theaddition of one or more links affect payoffs? These mutual considerations are brought together intoa link- or network-based notion of stability. Deleting links from networks.
Throughout it is assumed that players have full autonomy orsovereignty over the decision to delete one or more of her links. Indeed, the principle of mutualconsent requires that players control which links they participate in. This implies that every playercan veto her participation in any link or relationship. Based on this consideration, I introduce twofundamental stability concepts concerning the deletion of links.As before, let φ : G N → R N be a network payoff function on the player set N .(i) A network д ∈ G N is link deletion proof (LDP) for φ if for every player i ∈ N and everyneighbour j ∈ N i ( д ) , it holds that φ i ( д − ij ) φ i ( д ) .Link deletion proofness requires that no player has an incentive to sever an existing linkwith one of her neighbours.We denote by D( φ ) ⊂ G N the class of all link deletion proof networks for the given payofffunction φ (Jackson and Wolinsky, 1996).(ii) A network д ∈ G N is strong link deletion proof (SLDP) for φ if for every player i ∈ N and every set of her direct links h ⊂ L i ( д ) , it holds that φ i ( д − h ) φ i ( д ) .Strong link deletion proofness requires that no player has incentives to sever links withone or more of her neighbours simultaneously.We denote by D s ( φ ) ⊂ G N the class of all strong link deletion proof networks for the givenpayoff function φ (Gilles, Chakrabarti, and Sarangi, 2006).From the definition it is clear that any SLDP network is always LDP and, therefore, strong linkdeletion proofness is indeed a stronger notion than (regular) link deletion proofness. As indicated,LDP was seminally introduced in Jackson and Wolinsky (1996), while SLDP was only introducedas a stand-alone concept in early drafts of Gilles, Chakrabarti, and Sarangi (2006).Second, the empty network д = œ on any set of players N is trivially strong link deletion proof.Indeed, this network does not contain any links and, therefore, the deletion of links is vacuouslysatisfied. We can therefore summarise that: 9 roposition 3.1 For any network payoff function φ : G N → R N it holds that д ∈ D s ( φ ) ⊂ D( φ ) ⊂ G N . (5)The first question that I consider is under which conditions link deletion proofness is exactly thesame as strong link deletion proofness. This seems a rather innocuous question, since SLDP is somuch stronger a concept than LDP. Nevertheless, it is enlightening to identify the exact propertyon the network payoff structure φ that allows this equivalence. Theorem 3.2
Strong link deletion proofness and link deletion proofness are equivalent for networkpayoff structure φ in the sense that D( φ ) = D s ( φ ) if and only if the network payoff structure φ is convex on the class of link deletion proof networks D( φ ) ⊂ G N in the sense that for every LDP network д ∈ D( φ ) , every player i ∈ N , every neighbour j ∈ N i ( д ) and every link set h ⊂ L i with h ∩ L i ( д ) = œ it holds that Õ ij ∈ h [ φ i ( д + ij ) − φ i ( д ) ] > implies that φ i ( д + h ) > φ i ( д ) . (6)For a proof of Theorem 3.2 I refer to Appendix A.1 of this survey.The convexity property on the payoff structure φ requires that the sign of the sum of valuesfrom adding one link to a network from a set of links fully determines whether adding all links isbeneficial or not. Hence, looking at links one-by-one gives complete information about whether itis beneficial to add all links to the network or not. Adding links to networks.
Next I consider how players assess the addition of a link to an ex-isting network. Again we take the idea of consent in link formation as central into our reasoninghere. This implies that both parties in the formation of a new link have to agree that adding thislink is beneficial.(iii) A network д ∈ G N is link addition proof (LAP) for φ if for all i , j ∈ N with ij < д , itholds that φ i ( д + ij ) > φ i ( д ) implies φ j ( д + ij ) < φ j ( д ) .Link addition proofness states that there are no incentives for any pair of players to forman additional link. This is based on the requirement of mutual consent in link formation.Indeed, if one player would like to add a link, the other player would have strong objections.In this case this is formulated as that, if one player has benefits from forming the link, theother (consenting) party has losses and, thus, would withhold her consent.We denote by A( φ ) ⊂ G N the class of all link addition proof networks for the given payofffunction φ (Jackson and Wolinsky, 1996).(iv) A network д ∈ G N is strict link addition proof (SLAP) for φ if for all i , j ∈ N , it holdsthat ij < д if and only if φ i ( д + ij ) < φ i ( д ) as well as φ j ( д + ij ) < φ j ( д ) .Strict link addition proofness is a far stronger notion that LAP. Indeed, it requires that bothplayers agree that forming an additional link between them is not beneficial for either ofthem. This agreement is imposed and only a certain very specific type of network payoff10tructures would support such networks to exist. Consequently, it has to be expected that,for an arbitrary regular network payoff function, only a rather small class of networksactually satisfies this property.We denote by A s ( φ ) ⊂ G N the class of all strict link addition proof networks for the givenpayoff structure φ (Gilles and Sarangi, 2010).The introduced notions of link addition proofness require some clarification. These two notionsindeed only seem to partially cover the idea that a network is stable if it satisfies the property that“if i has an incentive to form an additional link with j , then j has no incentive to form a link with i ”.This is subject to the next discussion.To understand link addition proofness in more detail, we can reformulate it. Indeed, a network д is link addition proof if and only if for all players i , j ∈ N with ij < д : φ i ( д + ij ) > φ i ( д ) implies φ j ( д + ij ) φ j ( д ) . (7)This has some interesting consequences regarding the interpretation of the LAP property. First, alink ij < д for some i , j ∈ N is non-discerning if it holds that φ i ( д + ij ) = φ i ( д ) as well as φ j ( д + ij ) = φ j ( д ) . (8)From the derivation above, the definition of link addition proofness is indeed ambiguous whetherany non-discerning link ij should be in the network for it to be LAP or not. Hence, such non-discerning links can arbitrarily be added to or deleted from networks without the LAP propertybeing affected. Thus, the class of non-discerning links makes the determination of LAP networks“fuzzy”.To address this issue of the addition or deletion of non-discerning links, I introduce a third typeof link addition proofness:(v) A network д ∈ G N is ⋆ -link addition proof ( ⋆ -LAP) for φ if for all players i , j ∈ N , itholds that if ij < д , then φ i ( д + ij ) > φ i ( д ) implies φ j ( д + ij ) < φ j ( д ) .We denote by A ⋆ ( φ ) ⊂ G N the class of all ⋆ -link addition proof networks for the givenpayoff structure φ .This minor modification of the definition of link addition proofness simply requires that all non-discerning links should be part of a ⋆ -link addition proof network. This makes the definition unam-biguous. Example 3.3
To delineate the three link addition proofness concepts introduced here, we can ex-plore an example of a network payoff function in which these concepts result in different classesof networks. We consider three players and all possible networks, i.e., N = { , , } and G N = { д | д ⊂ д N } where д N = { , , } . Note that there are exactly eight possible networks on N , i.e., G N = φ on the generated class of networks G N N . All potential network payoffs represented by φ can be represented in an appropriately con-structed table: Network д φ ( д ) φ ( д ) φ ( д ) Stability д = œ д = { } ⋆ -LAP д = { } д = { } д = { , } д = { , } д = { , } д = д N д is link addition proof, but not ⋆ -link addition proof. Indeed, if any link is addedto the empty network, no payoffs are changed for any of the players involved. On the other hand,there are no losses, thus precluding that д is ⋆ -link addition proof.Next, д is ⋆ -link addition proof, but not strong link addition proof. Indeed, any addition of a linkto д results into a loss for player 3. However, adding link 13 results into a strict gain for player 1,implying that д is not strong link addition proof.Third, the complete network д N is strong link addition proof by tautology. Indeed, there are no linksto be added to this network, and therefore vacuously the property of strong link addition proofnessis satisfied.I remark that none of the other networks have any link addition proofness properties. (cid:7) Next I explore the equivalence of these link addition proofness concepts. In order to explore theseequivalences effectively, I introduce two auxiliary properties of the network payoff structure.
Definition 3.4
Consider a network payoff structure φ on G N . Then:• The structure φ is said to be discerning on the class of networks G ⊂ G N if for every network д ∈ G it holds that for any pair i , j ∈ N with ij < д either φ i ( д + ij ) , φ i ( д ) or φ j ( д + ij ) , φ j ( д ) or both.• The structure φ is said to be uniform on the class of networks G ⊂ G N if for every network д ∈ G and for any pair i , j ∈ N with ij < д it holds that φ i ( д + ij ) > φ i ( д ) implies φ j ( д + ij ) > φ j ( д ) . (9)Using these auxiliary concepts we can now show the following equivalences:12 eorem 3.5 Let φ be some network payoff structure on the class of all networks G N on the set ofplayers N . Then the following properties hold: (a) д N ∈ A s ( φ ) ⊂ A ⋆ ( φ ) ⊂ A( φ ) ; (b) It holds that A ⋆ ( φ ) = A( φ ) if and only if φ is discerning on A( φ ) , and; (c) It holds that A s ( φ ) = A ⋆ ( φ ) if and only if φ is uniform on A ⋆ ( φ ) . For a proof of Theorem 3.5 I refer to Appendix A.2 in this survey. Furthermore, from Theorem 3.5it is easily concluded that the following equivalence also holds:
Corollary 3.6
SLAP and LAP are equivalent concepts for payoff structure φ in the sense that A s ( φ ) = A( φ ) if and only if the payoff structure φ is discerning and uniform on A( φ ) . In the previous discussion, I introduced four fundamental stability concepts on adding links to anddeleting links from a network. These four basic notions can be combined to define derived concepts.The first concept—known as pairwise stability (Jackson and Wolinsky, 1996)—combines the weakestlink stability notions and has been the subject of extensive discussion in the literature. This notionimplicitly assumes that players only consider the deletion and addition of one specific link at atime.(vi) Network д is pairwise stable (PS) for φ if д is link deletion proof as well as link additionproof. We denote by P( φ ) = D( φ ) ∩ A( φ ) the family of pairwise stable networks for thepayoff function φ .The original pairwise stability concept—introduced by Jackson and Wolinsky (1996)—only concernsitself with the contemplation of adding a single link to or deleting a single link from a given network.If there are no incentives for players to either add a link to the existing network or delete a linkfrom the network, then the network is “pairwise stable”: There are no incentives present under thehypothesis of mutual consent in link formation that anybody wants to change a single link in thisnetwork.Two further derived stability concepts, which strengthen the notion of pairwise stability, haveparticular relevance in the theory of consent in link formation. Strong pairwise stability (Gilles, Chakrabarti, and Sarangi,2006, 2012) assumes that players can delete an arbitrary collection of links under their control.Hence, they can veto any link in which they participate. On the other hand, the contemplation ofadding links remains confined to adding a single link.Strict pairwise stability (Gilles and Sarangi, 2010) is the strongest notion in this framework. Itnot only considers that players can delete any number of their existing links, but also that they areassumed to be in agreement regarding the addition of a link to an existing network. It is clear thatfor an arbitrary network payoff structure, the collection of such strictly pairwise stable networksmight well be empty. Only for certain network payoff structures such networks might emerge.13vii) Network д is strongly pairwise stable (SPS) for φ if it is strong link deletion proof as wellas link addition proof.We denote by P ⋆ ( φ ) = D s ( φ ) ∩ A( φ ) the family of strongly pairwise stable networks forthe payoff function φ .(viii) Network д is strictly pairwise stable (SPS*) for φ if it is strong link deletion proof as wellas strict link addition proof.We denote by P s ( φ ) = D s ( φ ) ∩ A s ( φ ) the family of strictly pairwise stable networks forthe payoff function φ .These three pairwise stability concepts generate different classes of networks in most cases. I con-sider an example to illustrate this. Example 3.7
Again consider three players and all potentially generated networks, i.e., N = { , , } with д N = { , , } . Now, consider a network payoff function φ on the generated class of net-works G N on N represented in the following table: Network д φ ( д ) φ ( д ) φ ( д ) Stability д = œ д = { } д = { } д = { } д = { , } -1 0 0 д = { , } д = { , } д = д N д is trivially SLDP and in this case as well LAP. Therefore, it isindeed strongly pairwise stable. Second, д is LDP and, therefore, SLDP. Moreover, д is SLAP. Indeed, adding link 13 to д resultsinto strict losses for both players 1 and 3. Similarly, for link 23. Thus, we conclude that д is strictlypairwise stable.Finally, the complete network д N is SLAP due to being the maximal network. Furthermore, д N isLDP. However, д N is not SLDP. player 3 has the strict incentive to delete both her links and revertto network д .We conclude from this discussion that this simple network payoff example induces three distinctclasses of pairwise stable networks. (cid:7) It should be remarked that networks with at most one link are SLDP if they are LDP. Therefore, they are stronglypairwise stable if they are link addition proof and link deletion proof.
Corollary 3.8
Consider a network payoff structure φ on the class of all networks G N on set of players N . Then the following relationships hold: (a) P s ( φ ) ⊂ P ⋆ ( φ ) ⊂ P( φ ) ; (b) Pairwise stability and strong pairwise stability are equivalent concepts for φ in the sense that P( φ ) = P ⋆ ( φ ) if and only if φ is convex on P( φ ) ; (c) Strong pairwise stability and strict pairwise stability are equivalent concepts for φ in the sensethat P ⋆ ( φ ) = P s ( φ ) if and only if φ is discerning and uniform on P ⋆ ( φ ) , and; (d) Pairwise stability and strict pairwise stability are equivalent concepts for φ in the sense that P( φ ) = P s ( φ ) if and only if φ is convex, discerning as well as uniform on P( φ ) . Next I discuss some of the ideas put forward by Jackson and van den Nouweland (2005). They in-vestigated networks that emerge if coalitions of arbitrary size can make changes to the network ina coordinated fashion to the coalition’s overall benefit. As such strong stability is an extension ofthe pairwise stability concept to allow arbitrary coalitions to adjust the network structure undertheir control.As a preliminary we denote a coalition as any subset S of players in N ; hence, a coalition isany S ⊂ N . This includes the empty coalition œ as well as the “grand” coalition N itself. In a non-cooperative game (A , π ) , for any coalition S ⊂ N and strategy profile a ∈ A we denote by a S the S -restriction of a defined by ( a j ) j ∈ S and by a N \ S its complement ( a k ) k < S .Now, in a non-cooperative game (A , π ) a strategy tuple a ∈ A is a strong equilibrium if for every(non-empty) coalition of players S ⊂ N and every coordinated strategic deviation b S = ( b i ) i ∈ S ∈ A S = Î i ∈ S A i it holds that π i (cid:0) a N \ S , b S (cid:1) π i ( a ) for all i ∈ S (10)Next we introduce the strong stability concept put forward by Jackson and van den Nouweland(2005). The next definition essentially transposes strong equilibrium conditions to network forma-tion situations. Definition 3.9
Let φ be a network payoff function on N and consider the corresponding Myersonmodel Γ mφ = (A m , π m ) . (i) A network д ′ ∈ G N can be obtained from network д ∈ G N through the coordinated actionsof coalition S ⊂ N if д ′ = д + h + − h − , where h + ⊂ д S = { ij | i , j ∈ S } and h − ⊂ ∪ i ∈ S L i ( д ) . This approach is akin to the strong equilibrium concept proposed by Aumann (1959) in non-cooperative game theory.Jackson-Nouweland’s concept of strong stability can be viewed as a network theoretical implementation of the ideasbehind Aumann’s strong equilibrium concept.
A network д ∈ G N is strongly stable if for every coalition S ⊂ N and every network д ′ that is obtainable from network д through coordinated actions from coalition S it holds that φ i ( д ′ ) > φ i ( д ) for some player i ∈ S implies that there exists some other player j ∈ S with φ j ( д ′ ) < φ j ( д ) . It should be remarked that Dutta and Mutuswami (1997) introduced a slightly different definitionof “strong stability”. They consider that all members of S need to be made strictly better off for adeviation to be successful. Strong equilibrium is a very demanding concept and these equilibria do not exist in many gametheoretic decision situations. Similarly, the notion of strong stability is equally demanding, resultingthat such networks rather unlikely exist. The next example illustrates these issues and introducesthe notion of costly link formation that will be explored further in the next two subsections.
Example 3.10 (Costly trade networks)
This example of a Walrasian trade network has been introduced seminally in Jackson and Watts(2002) and further developed in Jackson and van den Nouweland (2005) and Gilles, Chakrabarti, and Sarangi(2011). It considers an economy of n players who trade goods through connection paths. Thereare two commodities X and Y and all players are endowed with a Cobb-Douglas utility function u ( x , y ) = √ xy . All players are assumed to have a commodity endowment of either ( , ) or ( , ) with an equal probability of .Players can trade with any other player that they are connected with, directly or indirectly. Hence,there emerge complete markets in each of the components. So, for n = д = { , , } generates two components and two markets, namely 123 separated from 45. Additional links, there-fore, not always contribute to the extent of these markets: д ′ = { , , , } results in exactly thesame markets 123 and 45.The cost c of forming any link ij is uniform and set at c > . The costs of the formation of the tradenetwork are divided equally among the members of a market, being a component of the network.The network payoff function φ is now defined as the expected net benefits from participating in thegenerated market structure. This can be developed as follows.First, consider the case of a market of the size two. There is a probability of that these two playershave opposite endowments and a probability of that they have the same endowment. Hence, theprobability of trade is resulting in a Walrasian allocation of ( , ) resulting in φ = · q − c = − c < k players. The probability of r players havingendowment ( , ) and ( k − r ) players having endowment ( , ) is now k C r (cid:0) (cid:1) k − r · (cid:0) (cid:1) r = k C r (cid:0) (cid:1) k . In the definition used by Jackson and van den Nouweland (2005) a deviation needs to make all members of S to beat least as well off and making one member strictly better off.
16e expected gross payoff from trade is now given by r k · (cid:18) k − rr (cid:19) + k − r k · (cid:16) rk − r (cid:17) = p r ( k − r ) k Hence, taking into account that there are exactly k − k players,the resulting net payoff from this trade network is given by φ = k · k " k − Õ r = k C r p r ( k − r ) − ( k − ) ck . Turning to n = k = φ = √ − c > < c < √ . For n = k = (cid:7) In this section I discussed the stability concept and its variants in the link-based cooperative frame-work as seminally set out by Jackson and Wolinsky (1996). It is clear that these concepts are ratherlimited in their scope, since they are link-based only. Individual and collective incentives are nottruly taken into account. Indeed, considerations are founded on adding and deleting links; theplayers’ incentives are assumed to coincide with the (marginal) benefits generated from these linksrather than the individualised payoffs. Next, I return to Myerson’s original non-cooperative frame-work founded on the direct benefits to players to the formation of links.
In this section I review stability and equilibrium concepts that refine the class of M-networks thatemerges from the Myerson approach to non-cooperative network formation under mutual consent.This literature is founded on the insight that the class of M-networks is very large. This is subjectof the next theorem, which states the equivalence of the class of M-networks with the set of stronglink deletion proof networks.
Theorem 4.1
Let φ be a network payoff function on N and consider the corresponding Myerson model Γ mφ = (A m , π m ) . (a) A network д ∈ G N is an M-network for φ if and only if д is strong link deletion proof for φ . Suppose that the network payoff structure φ is link monotone in the sense that for every player i ∈ N , every network д ∈ G N and every link ij < L i ( д ) it holds that φ i ( д + ij ) > φ i ( д ) . Thenevery network д ∈ G N is supported as an M-network. For a proof of this theorem I refer to Appendix A.3.The fundamental insights presented as Myerson’s Lemma and Theorem 4.1 have motivatedeconomists and social scientists to look into “refinements” of the Nash equilibrium concept in theMyerson model. These refinement equilibrium concepts have been developed particularly for ad-dressing link formation issues from the perspective of consent. These attempts can be divided intotwo classes.First, the standard approach in game theoretic models of network formation is to strictly applymethodological individualistic perspectives. Thus, all motivations emanate from the player decisionmakers and are not considered to be external to the rational decision making process. This hasresulted into a number of equilibrium concepts that simply assume that decision makers have anatural ability to cooperate if the incentives are in favour of such cooperation. Below I presentthe refinements considered by Bloch and Jackson (2006) and Gilles, Chakrabarti, and Sarangi (2011,2012).The second approach is to explicitly assume that decision makers are not fully individualistic,but adhere to some institutional or trusting norms of behaviour. van de Rijt and Buskens (2008) andGilles and Sarangi (2010) explicitly introduce a model of trusting behaviour through the introduc-tion of a individualised belief or conjecture that other decision makers will form links if they benefitfrom that. Thus, the trust in network formation is internalised into the player decision makers; allsuch decision makers adhere to a well-defined norm of decision making that expresses trustingbehaviour. This is fully developed in Section 5.Similarly, certain equilibrium concepts in non-cooperative game theory are founded on institu-tional signalling systems. The main such concept is Aumann’s correlated equilibrium , which can beused to introduce institutional arrangements in the decision making processes of players (Aumann,1974). Here these institutions are explicitly modelled as external to these players. They adhere tothese institutions since they benefit from applying these institutional behavioural rules instead ofacting purely selfish. This is explored fully in Section 6.
Goyal and Joshi (2006) introduced a refinement of the M-network concept that implements the ideaof cooperation between players to modify the network through coordinated actions. Thus, it is as-sumed that decision makers can implement bilateral or pairwise coordinated network modification.So, we consider any pair of players i , j ∈ N who consider how to modify their strategic signals ℓ i and ℓ j to modify the resulting network in their favour.This bilaterally coordinated action can be modelled in two different fashion. First, within theMyerson model as the so-called “pairwise” Nash equilibrium (Goyal and Joshi, 2006) and, second, asa network stability notion, denoted as “bilateral” stability (Gilles, Chakrabarti, and Sarangi, 2011). I remark here that I use a terminology that deviates from the literature. Indeed, the pairwise Nash equilib-
Definition 4.2
Let φ be a network payoff function on N and consider the corresponding Myersonmodel Γ mφ = (A m , π m ) . (i) A signal profile ℓ ∈ A m is a pairwise Nash equilibrium in Γ mφ if ℓ is a Nash equilibriumin Γ mφ and for every pair of players i , j ∈ N it holds that π mi (cid:16) ℓ ′ i , ℓ ′ j , ℓ − i , j (cid:17) > π mi ( ℓ ) implies that π mj (cid:16) ℓ ′ i , ℓ ′ j , ℓ − i , j (cid:17) < π mj ( ℓ ) (11) for all deviations ℓ ′ i ∈ A mi and ℓ ′ j ∈ A mj . (Here, ℓ − i , j refers to the restricted signal profile ( ℓ h ) h , i , j .) (ii) A network д ∈ G N is bilaterally stable for φ if д is strong deletion proof for φ and for everypair of players i , j ∈ N and network д ′ = д + ˆ h − h i − h j with ˆ h ∈ { { ij } , œ} , h i ⊂ L i ( д ) and h j ⊂ L j ( д ) it holds that φ i ( д ′ ) > φ i ( д ) implies that φ j ( д ′ ) < φ j ( д ) . (12)It is not hard to see that in the Myerson model there is a complete equivalence between these twoconcepts. The pairwise Nash equilibrium is simply a strategic formulation of bilateral stability. Igive the following proposition therefore without proof. Proposition 4.3
Let φ be a network payoff function on N and consider the corresponding Myersonmodel Γ mφ = (A m , π m ) . A network д ∈ G N is supported through a pairwise Nash equilibrium ℓ ∈ A m with д ( ℓ ) = д if and only if д is bilaterally stable for φ . Although these concepts are quite natural within the context of network formation, the additionalbenefits are rather limited. Coordinated pairwise activity is well captured by the three pairwisestability concepts that have been introduced in this survey. The notion of unilateral stability (SeeSection 5) also captures coordinated action in the sense that it is assumed that players respondpositively to a player’s proposal to change the network if that is to their benefit. Bilateral stabilitydoes not extend this to pairs of players, but reverts back to the normal best response rationalityprinciple that others keep their actions unchanged.
Stability of higher orders.
The notion of bilateral stability can easily be extended to stabilityof higher orders. Indeed, under bilateral stability it is assumed that coalitions of two players canmodify the network as proposed above. This can be extended to coalitions of at most r members,where r ∈ N is the assumed maximum size of the coalition under consideration. This is referred to as“stability of order r ” in Gilles, Chakrabarti, and Sarangi (2011). In particular, if r = n , we arrive at the rium concept in the Myerson model was seminally introduced in Goyal and Joshi (2006) and explored further byBloch and Jackson (2006) and Joshi, Mahmut, and Sarangi (2020). It refers to M-networks that are additionally link ad-dition proof. Therefore, I use the notion of pairwise Nash equilibrium here in a slightly different way as introduced inGilles, Chakrabarti, and Sarangi (2011). Example 3.10 introduced the idea that there are normally link formation costs. In this particular casethe costs of network formation are borne equally among all players that participate in the network.This signifies a collective approach to the allocation of network formation costs. It is more naturalto assume that players only bear the costs of the links that they participate in. Next, I develop theidea of link formation costs further and refine the notion of M-networks to capture this.In particular, I consider a modification of the Myerson model where the “intent to form links”is costly in the sense that approaching another player to form a link involves explicit investmentof time, effort and energy. Hence, the act of sending a signal is costly. However, if the other playerdoes not reciprocate and the link does not materialise, the player choosing to “reach out” still incursthis cost. This means that if player i ∈ N contemplates building a link ij with player j ∈ N andsends a message ℓ ij =
1, she incurs a cost of c ij >
0. On the other hand, ℓ ij = no costs on player i .Formally, a link formation cost structure can therefore be represented by a function c : N × N → R + where c ( i , j ) = c ij > i ∈ N incurs for sending a message to player j ∈ N , using the convention that c ( i , i ) = i ∈ N . Hence, player i incurs a cost c ij > j that she wants to form a link. In particular, this cost refers to the effortto respond to messages sent by others. Obviously, if c ij =
0, then there is no cost to communicatingand sending messages from i to j .This construction introduces the consent model with two-sided link formation costs as a modifi-cation of the (basic) Myerson model Γ mφ given as a non-cooperative game Γ aφ ( c ) = (A a , π a ) , whereplayer i ’s strategy set is given by A ai = A mi and player i ’s payoff for any strategy tuple ℓ ∈ A a isgiven by π ai ( ℓ ) = φ i ( д ( ℓ )) − Õ j , i ℓ ij · c ij = π mi ( ℓ ) − Õ j , i ℓ ij · c ij , (13)where φ : G N → R N is the network payoff function representing the gross benefits from networkformation without taking into account the costs of link formation.Our first result develops a complete characterisation of the Nash equilibria in the consent modelwith two-sided link formation costs. Part of this equivalence theorem was already stated withoutproof in Gilles and Sarangi (2010) and as stated here is taken from Gilles, Chakrabarti, and Sarangi(2012). There are some preliminaries that need to be developed before stating the main assertion. Definition 4.4
Let φ be a network payoff function on player set N and let c : N × N → R + a linkformation cost structure on N . Furthermore, let Γ aφ ( c ) = (A a , π a ) be the associated consent model with For results concerning these intermediate stability concepts, I refer to the quoted papers. This model of two-sided link formation costs was introduced in Gilles, Chakrabarti, and Sarangi (2006) and developedfurther by Gilles and Sarangi (2010) and Gilles, Chakrabarti, and Sarangi (2012). wo-sided link formation costs.A strategy tuple ℓ ∈ A a = A m is non-superfluous in the consent model with two-sided link formationcosts Γ aφ = (A a , π a ) if for all pairs of players i , j ∈ N , ℓ ij = if and only if ℓ ji = .We call a non-superfluous strategy tuple ℓ ∈ A a that is a Nash equilibrium a non-superfluous Nashequilibrium . The main theorem states that in Γ aφ ( c ) the networks that are supported by Nash equilibria are exactlythe strong link deletion proof networks for a network payoff function that takes account of the linkformation costs. For a proof of the next theorem I refer to Appendix A.4. Theorem 4.5
Let φ be a network payoff function on player set N and let c : N × N → R + be a linkformation cost structure on N . Furthermore, let Γ aφ ( c ) = (A a , π a ) be the associated consent model withtwo-sided link formation costs.Then for every network д ∈ G N the following three statements are equivalent: (a) Network д is supported by a Nash equilibrium of the consent model with two-sided link forma-tion costs Γ aφ ( c ) . (b) Network д is supported by a non-superfluous Nash equilibrium of the consent model with two-sided link formation costs Γ aφ ( c ) . (c) Network д is strong link deletion proof with regard to the network payoff function φ a : G N → R N given by φ ai ( д ) = φ i ( д ) − Õ j ∈ N i ( д ) c ij (14)Theorem 4.5 provides a complete and detailed characterisation of the set of all Nash equilibria of theconsent model with two-sided link formation costs. Furthermore, Theorem 4.5 clearly generalisesthe insight that the class of M-networks in the basic Myerson model is exactly the class of strongdeletion proof networks under network payoff function φ .In particular, each Nash equilibrium network is actually supported by a unique non-superfluousstrategy profile if the cost structure is non-trivial in the sense that all link formation costs arepositive. Gilles, Chakrabarti, and Sarangi (2012) also discuss that there actually exist superfluousNash equilibria if costs of link formation are zero for one of the players. Example 4.6 (Gilles, Chakrabarti, and Sarangi, 2012)
Consider the binary network formation situation with N = { , } and the network payoff functiongiven by φ ( д ) = φ ( д ) = φ ( д N ) = φ ( д N ) =
1. Link formation costs are given by c = c =
1. Hence, we can derive that under two-sided link formation costs that φ ai ( д ) = φ ai ( д N ) =
0, for i = , д is both (strong) link deletion proof for the net payoff function φ a andsupported by the superfluous Nash equilibrium characterised by ℓ = ℓ =
0. Of course, д is also supported as a Nash equilibrium through its non-superfluous strategy profile ℓ = ℓ = ( A a , π a ) . (cid:7) .3 One-sided link formation costs It is a natural extension to consider a network formation process under a one-sided cost structure.In this approach, one of the two linking players acts as the initiator and sends an initiation mes-sage to the other. If the other player, called the responder , chooses to reciprocate positively, thelink materialises; otherwise, not. This link formation process has a similar nature as the processconsidered in Bala and Goyal (2000), except that here the responder has to consent to the formationof the link, while in Bala-Goyal’s model this is not required. There the initiator can create a linkwith the respondent in the absence of consent.The decision making process is more complex than that under two-sided link formation costs.Consequently, the action set has to be constructed differently. Following Gilles, Chakrabarti, and Sarangi(2012), for each player i , we introduce a strategy set given by A bi = (cid:8) ( l ij , r ij ) j , i (cid:12)(cid:12) l ij , r ij ∈ { , } (cid:9) . (15)This means that player i chooses to act as an initiator in forming a link with j if she initiates amessage to j indicated as l ij =
1. In this case, player j acts as the respondent and responds positivelyto this initiative if r ji =
1. On the other hand, player j rejects the initiated link with i if r ji = l ij = r ji =
1. This isformalised as follows.Let A b = Î i ∈ N A bi be the set of such communication profiles. Given the link formation processset out above, for any profile ( l , r ) ∈ A b , the resulting network is now given by д b ( l , r ) = { ij ∈ д N | l ij = r ji = } . (16)To delineate the one-sided model from the two-sided model, it is preferred to use a different notationfor the incurred link formation costs. Instead, I introduce the function γ : N × N → R + as the one-sided link formation cost structure. Here, when i initiates a link with j —represented by l ij = i incurs a cost of γ ij >
0, regardless of whether the initialised link is accepted by j or not. On theother hand, responding to a link initialisation message is costless, i.e., j incurs no cost in respondingto any message ℓ ij sent by i in the link formation process.For a given network payoff function φ on N this now results in the following net payoff functionfor player i : π bi ( l , r ) = φ i (cid:16) д b ( l , r ) (cid:17) − Õ j , i l ij · γ ij . (17)Formally, let φ be a network payoff function on N and let γ : N × N → R + be a given one-sidedlink formation cost structure. Then we refer to the non-cooperative game in strategic form Γ bφ ( γ ) = (A b , π b ) as the consent model of network formation with one-sided link formation costs . Nash equilibria of the consent model with one-sided link formation costs.
As before, wecan now introduce a non-superfluous strategy tuples in the consent model with one-sided link22ormation costs:
Definition 4.7
Let φ be a network payoff function on N and let γ : N × N → R + be a given one-sidedlink formation cost structure. Consider the corresponding consent model with one-sided link formationcosts Γ bφ ( γ ) = (A b , π b ) .Then a strategy profile ( l , r ) ∈ A b is non-superfluous if for all pairs i , j ∈ N it holds that l ij = implies that r ji = as well as l ji = r ij = , and (18) r ij = implies that l ji = as well as l ij = r ji = . (19)Unlike for the consent model with two-sided link formation costs, each network is no longer sup-ported by a unique non-superfluous strategy profile. Indeed, it depends on who of the two playersinvolved initiates and who responds in the link formation process.On the other hand, under a non-superfluous strategy profile, only one player bears the estab-lishment cost of each existing link, and every initialisation is responded to positively. As a first stepin the analysis of this one-sided approach, I explore the relationship between the Nash equilibriaof the two-sided and the one-sided model. Secondly, I present a full characterisation of the Nashequilibria of the one-sided model in terms of network stability properties. These results are takenfrom Gilles, Chakrabarti, and Sarangi (2011).The main question to be considered here is whether there is a network payoff function whichwould provide equivalence between Nash equilibria of the one-sided model and strong link deletionproofness with regard to a payoff function in a similar fashion as Theorem 4.5 for two-sided linkformation costs. In particular, I follow efficiency logic and consider a payoff function which onlyassigns link formation costs to the player with the lower cost of link formation. If link formationcosts are equal, a tie-breaking rule is applied.Let M i ( д ) = { j ∈ N i ( д ) | γ ij < γ ji or γ ij = γ ji , i < j } ⊂ N i ( д ) be the potential links that player i should finance based on incurring the lowest link formation costs. The corresponding payofffunction φ b is defined for i ∈ N by φ bi ( д ) = φ i ( д ) − Õ j ∈ M i ( д ) γ ij given the network payoff function φ representing benefits without taking into account costs of linkformation. We can show the following implication, which proof is relegated to Appendix A.5. Theorem 4.8
Let φ be a network payoff function on N and let γ : N × N → R + be a given one-sidedlink formation cost structure. If network д ∈ G N is strong link deletion proof for the net payoff function φ b , then д can be supported by a non-superfluous Nash equilibrium in the consent model with one-sidedlink formation costs Γ bφ ( γ ) = ( A b , π b ) . The converse of Theorem 4.8 does not hold as shown by the following counter-example.
Example 4.9
Consider the minimal binary network formation situation with N = { , } and net-work payoffs given by φ ( д ) = φ ( д ) = φ ( д N ) = φ ( д N ) =
10. Link formation costs are23iven by γ = γ = i = , φ bi ( д ) = φ b ( д N ) = − φ b ( д N ) =
3. Clearly, the complete network д N is not link deletion proof for the network payoff function φ b , since player 1 would benefit fromsevering the unique link 12.However, there is a Nash equilibrium of the one-sided consent model Γ bφ ( γ ) = ( A b , π b ) that supportsthe complete network д N : l = r = l = r = (cid:7) One might expect that a network payoff function that assigns a link initiator role to the player withthe higher marginal net benefits as a result of formation of the link in question might resolve theissue of characterising the supported equilibrium networks in Γ bφ ( γ ) = ( A b , π b ) . Below it is shownthat this is actually not the case. Example 4.10
Consider a situation with three players, N = { , , } . The following table givesthe benefits for each of the three players in the case of the formation of one of only three relevantnetworks: Network д φ ( д ) φ ( д ) φ ( д ){ }
10 10 0 { }
10 0 10 { , }
15 20 20All other networks generate no benefits to any of the three players, i.e., φ i ( д ) = д not listed in the table.Consider the following one-sided link formation cost structure: γ = γ = γ = γ = γ = γ =
10. Within this context, player 1 has the highest marginal net benefit from forminglinks 12 as well as 13, namely φ ({ }) − γ = φ ({ }) − γ =
1, while the other players have nopositive marginal benefits from forming links 12 and 13.Now, the network { , } is not link deletion proof for the network payoff function that is basedon the property that the player with the highest net marginal benefit is assumed to finance theformation of a link. Indeed, player 1—who has the highest net marginal benefits from both links—has a negative net return from forming network { , } and would prefer to sever one of the twolinks to increase her net benefit to 1.On the other hand, { , } is supported by a non-superfluous Nash equilibrium strategy profileunder one-sided link formation costs with l = r = l = r = (cid:7) These examples show that the problem of finding a reasonable payoff function that completelycharacterises all Nash equilibria of the one-sided consent model in terms of network stability re-mains open. The issues are such that it can be argued that there is actually no reasonable networkpayoff function that characterises all supported equilibrium networks in the consent model underone-sided link formation costs. Note that in the case of two-sided link formation costs, the cost of link formation is a total of γ + γ = + = д N not being supported by a Nash equilibrium in Γ aφ ( γ ) . This indicates theunderlying reason why two-sided link formation costs shrink the set of supported networks in comparison with the caseof one-sided link formation costs. ulti-stage network formation under one-sided link formation costs. One can ask whethercertain other approaches can resolve the coordination and free riding issues that are indicated inthe discussion of the converse of Theorem 4.8 above. Here, I consider a two-stage network formation process to restore equivalence between equi-libria of that model under one-sided costs and strong link deletion proofness with respect to somewell-constructed network payoff function. This is motivated by the fact that often sequential de-cision making solves coordination problems. With this in mind, consider the following naturaltwo-stage process:(i) In the first stage, every players i ∈ N initiates links by selecting initiation messages ( l ij ) j , i .(ii) In the second stage, all players respond to links initiated in the first stage and select ( r ij : l ji = ) j , i .The question is whether the subgame perfect Nash equilibria of this game are strong link deletionproof with regard to φ b . We show that this is not necessarily the case. Example 4.11
Reconsider the simple binary linking situation in Example 4.9. We showed earlierthat the complete network д N = { } is not (strong) link deletion proof for the net payoff function φ b but that there is a Nash equilibrium communication profile of the one-sided model that supportsit, namely, l = r = l = r = l = l = r r , − , −
71 2 , , r = r =
0. This is exactlythe second part of the indicated communication profile. Thus, the given communication profile isindeed a subgame perfect equilibrium in the two-stage link formation process. (cid:7)
The reason why sequential decision making cannot resolve the coordination problem is that herethe problem stems from costs not being transferable. Complete transferability of costs and benefitswould take us into the framework of Jackson and Wolinsky (1996) and, in particular, Bloch and Jackson(2006, 2007). This discussion requires knowledge of multi-stage, sequential games and the notion of subgame perfection. Thisdiscussion can be skipped without any difficulty. For more elaborate discussion of multi-stage and sequential games Irefer to Osborne (2004), Harrington (2008) and Maschler, Solan, and Zamir (2013). formal comparison of one-sided and two-sided link formation costs. Since the two mod-els that we considered in this section have different philosophical bases, we must make some sim-plifying assumptions to enable a more formal comparison. In particular, we have to address howthe two different link formation cost formulations are related. This simply requires us to formulatethe one-sided cost structure γ in terms of the two-sided cost structure c . Hence, we consider γ tobe a particular functional form of c .I look at two simplified cases that facilitate this comparison. Case A:
The initiator bears all.
Suppose that the initiator in the model with one-sided costsbears both his cost and the cost of the responder in the context of the two-sided consent model. So,initiation is tantamount to bearing the total cost of link formation, i.e., γ ij = c ij + c ji for all i , j .Benefits described by φ remain individualised and are not transferable.In this case, it is quite obvious that the Nash equilibria of the two models are not comparable,which is shown in the next simple example. Example 4.12
Consider again a binary link formation situation with N = { , } and φ i ( д N ) = φ i ( д ) = i = ,
2. Moreover, let c = c =
50. Hence, γ = γ = д N = { } issupported by a Nash equilibria of the two-sided model, namely through ℓ = ℓ =
1. But there isno Nash equilibrium in the one-sided model that would support it because no one would be willingto pay a cost of 100 in order to sustain this link.Next, modify the situation to let φ ( д N ) = φ ( д N ) = φ i ( д ) = i = , c = c = γ = γ =
10. Then, д N = { } is now supported by a Nash equilibrium of the one-sidedmodel, namely through l = r = l = r =
0. The strategy supporting this network is not aNash equilibrium in the two-sided model. (cid:7)
Case B:
A sunk cost formulation.
Next, we consider the case in which the link formationcosts are not transferable and that the initiator has to bear only his own cost. This corresponds toa scenario where the costs of the responding party are sunk and, thus, not relevant to the decisionmaking process.Hence, we assume that γ ij = c ij for all i , j . In this case, it can be shown that networks sup-ported by Nash equilibria of the two two-sided model are also supported by some Nash equilibriumof the one-sided model, while the converse does not hold. For a proof of the next theorem I refer toAppendix A.6. Theorem 4.13
Let φ be a network payoff function on player set N and let c : N × N → R + a two-sidedlink formation cost structure on N .If a network д ∈ G N is supported by a Nash equilibrium of the consent model with two-sided linkformation costs Γ aφ ( c ) , then there exists a non-superfluous Nash equilibrium supporting network д inthe consent model with one-sided link formation costs Γ bφ ( c ) , i.e., for one-sided link formation coststructure γ given by γ ij = c ij for all i , j ∈ N . We show that the converse of Theorem 4.13 does not hold.26 xample 4.14
Consider again the binary link formation situation with N = { , } . Furthermore,assume now that φ ( д ) = φ ( д ) = φ ( д N ) = φ ( д N ) =
4. Let two-sided costs of linkformation be uniform, given by c ij = i , j ∈ N .The complete network д N = { } initiated by player 1 is supported by a Nash equilibrium in theone-sided model for γ ij = c ij . But the strategy tuple ℓ = ℓ = (cid:7) This discussion shows that one-sided link formation processes require a very careful analysis anddo not necessarily result in very delineated conclusions.
In this section I review some concepts that try to capture the fundamental idea that “trust buildsnetworks”. These concepts go beyond the approaches that I have reviewed thus far, being Myer-son’s model and its variations as well as the Jackson-Wolinsky approach to incorporate cooperativeconceptions into a network formation setting.I discuss two different implementations of trusting behaviour into network formation. First,van de Rijt and Buskens (2008) consider the notion of unilateral stability that is founded on theprinciple that players attempt the formation of links even if their correspondents did not signalthat they would necessarily agree to the formation of these links. Thus, players follow the rule thatone should certainly try to form links if one expects the correspondent to benefit from its formation.This leads to a refinement of the class of M-networks.A very similar conception has been developed by Gilles and Sarangi (2010). Within the consentmodel under two-sided link formation costs Gilles and Sarangi (2010) developed a belief-based sta-bility concept denoted as monadic stability for understanding a purely non-cooperative process ofnetwork formation based on trusting behaviour. Again players are assumed to pursue the forma-tion of links if they perceive the correspondents to benefit from their creation. However, monadicstability is defined as a self-confirming equilibrium (Fudenberg and Levine, 1993) based on thesebelief systems, deviating considerably from van de Rijt and Buskens (2008)’s conception of trustingbehaviour.
The mathematical sociologists van de Rijt and Buskens (2008) proposed a refinement of the Nashequilibrium concept that considers expanding a player’s ability to affect the network that is formedin a broader way than allowed through best response rationality underlying the Nash equilibriumconcept. They recognised that the multitude of Nash equilibria in the Myerson model is due toa simple (mis-)coordination problem: Players are indifferent between proposing or not proposinga link if the other player actually does not propose the link herself already. This resulted in arefinement of the Nash equilibrium concept that takes account of the idea that players trust thatmutually beneficial link formation will indeed be pursued by other players.27 efinition 5.1
Let φ be a network payoff function on N and consider the corresponding Myersonmodel Γ mφ = (A m , π m ) . A network д ∈ G N is unilaterally stable if there exists a strategy profile ℓ ∈ A m in the Myerson model with д ( ℓ ) = д such that (i) for all i ∈ N and ℓ ′ i ∈ A mi : π mi ( ℓ ) > π mi ( ℓ ′ i , ℓ − i ) (Nash equilibrium condition) , and (ii) for every i ∈ N and every alternative strategy ℓ ′ i ∈ A mi , it holds that π mi ( ℓ ⋆ ) > π mi ( ℓ ) implies that there is some j ∈ N with ℓ ′ ij = and ℓ ij = for whom π mj ( ℓ ⋆ ) < π mj ( ℓ ) , where ℓ ⋆ ∈ A m is given by ℓ ⋆ i = ℓ ′ i , ℓ ⋆ jk = ℓ jk for j , i , k and ℓ ⋆ ji = ℓ ′ ij = for j , i . A network is unilaterally stable if it is supported through a Nash equilibrium in the Myerson modelunder the additional provision that every player can modify her direct neighbourhood provided thatthis modification can be constructed with the consent of her chosen neighbours. So, if i ’s proposalwould make herself better off, then all newly selected neighbours would have no objections andwould not receive lower payoffs as a consequence of this modification of the network.Unilateral stability introduces a form of trusting behaviour into the Myerson approach to net-work formation under mutual consent. The consent of any player’s neighbours is reasoned by thatplayer is conducted in such a way that it reflects trusting behaviour by that particular player. Insome sense it introduces a bounded form of rationality of any player in her consideration of howother players respond to changes in her behaviour. As such the notion of unilateral stability can becategorised as a model of trusting behaviour in network formation under mutual consent.An alternative definition of unilateral stability is also possible as captured in the propositionbelow. It reflects the idea to add trusting behaviour to the M-network concept. Proposition 5.2 (An alternative definition of unilaterally stable networks)
A network д ∈ G N is unilaterally stability if and only if д is an M-network such that for every player i ∈ N and all link sets h − i ⊂ L i ( д ) and h + i ⊂ L i ( д N \ д ) it holds that either φ i ( д − h − i + h + i ) φ i ( д ) or φ i ( д − h − i + h + i ) > φ i ( д ) implies there is some j ∈ N such that ij ∈ h + i and φ j ( д − h − i + h + i ) < φ j ( д ) . Unilateral stability is the strongest individualistic or “monadic” network formation concept thathas been proposed in the literature. Indeed, going beyond the unilateral formation of links underconsent as formulated here would actually involve active participation of multiple players.Next, we turn to discussing some simple properties of unilateral stability.
Proposition 5.3
Let φ be a network payoff function on N and consider the corresponding Myersonmodel Γ mφ = (A m , π m ) . Then the following properties hold: (a) Every unilaterally stable network is strongly pairwise stable.
There exist strictly pairwise stable networks that are not unilaterally stable. (c)
If the network payoff structure φ is link monotone, then д N ∈ G N is the unique unilaterallystable network for φ . I prove all three assertions in Proposition 5.3 in an informal fashion, rather than a rigorous mathe-matical way.First, from Proposition 4.1 it follows that every M-network д is strong link deletion proof. Fur-thermore, applying the unilateral stability condition to a single link ij ∈ д reduces to the LAPproperty. This immediately shows Proposition 5.3(a).Next, if the network payoff structure is link monotone, then there are no objections of anyplayer to add more links to an existing network. Hence, the complete network д N is the only M-network that satisfies the unilateral stability condition, implying the assertion stated as Proposition5.3(c).Finally, to show Proposition 5.3(b), I device an example for the case of three players. This ex-ample also has an important role to assess the relationship between unilateral stability and otherstability concepts, introduced further down in these lecture notes. Example 5.4
Here, consider three players N = { , , } and a network payoff structure φ given inthe next table. Network д φ ( д ) φ ( д ) φ ( д ) Stability д = œ д = { } д = { } д = { } д = { , } -1 0 0 д = { , } д = { , } д = д N д is strongly pairwise stable, but is not unilaterally stable. Indeed, player 3 can add bothlinks 13 and 23 to make д without objection of the other players.Furthermore, д is strictly pairwise stable and again not unilaterally stable. As before, player 3 canadd links 13 and 23 to move to д N without any objections of the other two players. This showsassertion 5.3(b).Also, it is clear from the table that the complete network д N is unilaterally stable, since it is stronglink deletion proof. Note that in this case д N is strictly pairwise stable as well.29inally, I refer to Example 5.10 for a detailed discussion of an example in which assertion of Propo-sition 5.3(b) is strengthened in the sense that the class of strictly pairwise stable networks is com-pletely disjoint from the class of unilaterally stable networks. (cid:7) To assess unilateral stability, it is clear that van de Rijt and Buskens (2008) introduce it as an ex-pression of firmly methodological individualistic behavioural principles: Players act selfishly only,but conjecture that other players will consent to the creation of links that directly benefit them.It builds on the hypothesis that players offer no objections to the formation of links that directlybenefit them.However, an alternative interpretation can easily be applied here as well. Indeed, the unilateralstability concept can be interpreted to be an application of a principle of trusting behaviour: playerstrust others to consent to forming links if it does not hurt them. This is closely akin to the modelof trusting behaviour. An alternative model of trusting behaviour founded on belief systems inMyerson’s framework is discussed next.
Gilles and Sarangi (2010) introduced a belief-based conception of trusting behaviour in the settingof the consent model with two-sided link formation costs. Their approach imposes minimal infor-mational requirements. Unlike other models of strategic network formation, players need not beaware of the payoffs associated with every network. For any given network д ∈ G N to emergein such a setting, a player is required to know the payoffs associated with any change (creation ordeletion) only involving their own direct links ij ∈ L i ( д ) .This results in an amendment of Myerson’s consent game such that, based on their information,players form simple, myopic beliefs about the direct benefits other players will receive from estab-lishing links with them. According to these myopic beliefs, each player i ∈ N assumes that anotherplayer j ∈ N is willing to form a new link with i if j stands to benefit from it in the prevailingnetwork. Similarly i also assumes that j will break an existing link ij in the prevailing network if j does not benefit from having this link. Thus, in this process player i assumes that all other links inthe prevailing network remain unchanged.Therefore, these monadic beliefs are indeed “myopic” in the sense that they only pertain todirect effects of the addition or removal of a link in the network. Hence, these beliefs disregardhigher order effects on the payoffs of all players in the network due to the addition or removal ofsuch a link. As such these behavioural standards reflect a bounded form of rationality in decisionmaking, implying that the boundedly rational foundation of monadic stability is fundamentallydifferent from the rational standard imposed by unilateral stability.Such myopic beliefs essentially capture the idea that network formation primarily occurs be-tween acquaintances with sufficiently large an amount of information about each other to assessfirst order effects of network changes. This concept is a normal form implementation of the self- That social relations are mainly formed between acquaintances is confirmed empirically byWellman, Carrington, and Hall (1988) using data from the East York area. This principle also forms the founda-tion of the model in Brueckner (2006), who models friendship as building links between players chosen from a given setof acquaintances. We now formalise these myopic belief systems for the consent model under two-sided linkformation costs.
Defining monadic stability.
Throughout we assume there is a given network payoff function φ : G N → R N and we impose a two-sided link formation cost structure c = ( c ij ) i , j ∈ N . Based onthis data, consider the corresponding consent model under two-sided link formation costs Γ aφ ( c ) = (A a , π a ) . We can introduce specific belief systems in this setting that represent the trusting be-havioural principle as discussed above. Definition 5.5
Let ℓ ∈ A a be an arbitrary communication profile resulting in network д = д ( ℓ ) . Forevery player i ∈ N we define i ’s monadic belief system concerning ℓ as a communication profile ℓ i ⋆ ∈ A a given by (i) for every j , i with ij ∈ д let• ℓ i ⋆ ji = if φ j ( д − ij ) + c ji > φ j ( д ) and• ℓ i ⋆ ji = if φ j ( д − ij ) + c ji φ j ( д ) ; (ii) for every j , i with ij < д let• ℓ i ⋆ ji = if φ j ( д + ij ) − c ji < φ j ( д ) and• ℓ i ⋆ ji = if φ j ( д + ij ) − c ji > φ j ( д ) ; (iii) and for all j , k ∈ N with j , i , k let ℓ i ⋆ jk = ℓ jk . A monadic belief system reflects that a player believes that other players are myopically selfish andwill act in their myopic self-interest. Hence, links are consented to if that directly benefits the otherplayer and are refused if deleting that link benefits the other player.Now monadic stability simply requires that each player acts rationally in view of these beliefs.
Definition 5.6
Let φ and c be given with the corresponding consent model under two-sided link for-mation costs Γ aφ ( c ) = (A a , π a ) . It is clear that this approach is akin to the notion of unilateral stability introduced before. A comparison of monadicstability with unilateral stability is, therefore, called for. This is further developed here as well.
A network д ∈ G N is weakly monadically stable for ( φ , c ) if there exists some communicationprofile ℓ ∈ A a with д = д ( ℓ ) such that for every i ∈ N : ℓ i ∈ A ai is a best response to her monadicbeliefs ℓ i ⋆ − i ∈ A a − i for payoff function π a ; thus, π ai (cid:0) д ( ℓ ′ i , ℓ i ⋆ − i ) (cid:1) π ai (cid:0) д ( ℓ i , ℓ i ⋆ − i ) (cid:1) (20) for all ℓ ′ i ∈ A ai . (b) A network д ∈ G N is monadically stable for ( φ , c ) if there exists some communication profile ℓ ∈ A a with д = д ( ℓ ) such that for every i ∈ N : ℓ i ∈ A ai is a best response to her monadicbeliefs ℓ i ⋆ − i ∈ A a − i for payoff function π a and player i ’s monadic belief system ℓ i ⋆ is confirmedin the sense that for every j , i it holds that ℓ i ⋆ ji = ℓ ji . Weak monadic stability of a network is founded on the principle that every player i ∈ N anticipates—as captured by her (monadic) expectations about direct links—that other players will respond my-opically selfishly to her attempts to form a link with them. Note that ℓ − i is fully replaced by theplayer’s belief system ℓ i ⋆ − i in the standard best-response formulation of Nash equilibrium for player i and is therefore irrelevant for the decision making process of i .Monadic stability strengthens the above concept by requiring that the beliefs of each playerare confirmed in the resulting equilibrium. Hence, monadic stability imposes a self-confirmingcondition on the weakly monadic equilibrium. This describes the situation that all players are fullysatisfied with their beliefs; the observations that they make about the resulting network confirmtheir beliefs about the other players’ payoffs. This amounts to updating one’s initial beliefs. As such,monadic stability is an implementation of a self-confirming equilibrium based on the monadic beliefsystem in the context of consent model with two-sided link formation costs (Fudenberg and Levine,1993).To delineate the two monadic stability concepts for networks, we discuss a three player example.This example shows that the class of monadically stable networks is usually strictly larger than theclass of the weakly monadically stable networks. Example 5.7
Consider N = { , , } and assume uniform link formation costs with c ij = i , j ∈ N . Let the network payoff function φ be given in the table below:32 etwork д φ ( д ) φ ( д ) φ ( д ) Stability д = œ M w д = { } д = { } д = { } д = { , } д = { , } д = { , } M w д = д N M w and M This table identifies whether the network in question is weak monadically stable—indicated by M w —or whether it is monadically stable—indicated by M .Within this example we now consider some of the networks given and analyse their stability prop-erties. Network д : We show that this network is weakly monadically stable for a supporting communi-cation profile that is superfluous. Indeed, select ℓ = ( ( , ) , ( , ) , ( , ) ) ∈ A a with д ( ℓ ) = д = œ . Observe here that player 1 incurs link formation costs with π a ( ℓ ) = −
2, while π a ( ℓ ) = π a ( ℓ ) =
0. Then we can determine the monadic belief systems for all players as ℓ ⋆ = (− , ( , ) , ( , ) ) ℓ ⋆ = (( , ) , − , ( , ) ) ℓ ⋆ = (( , ) , ( , ) , − ) It should be emphasised that in this case player 1 believes that both other players are willingto make links with her, because there are direct benefits from forming such links. However,the other players believe that player 1 will not attempt to make a link with them, because shehas no direct (net) benefits from doing so. This refers to a classical coordination problem.Now we determine that the best responses for all players are given by • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 1. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 2. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 3.This confirms that д is indeed weakly monadically stable for ℓ . However, д is not monad-ically stable, since in the communication profile ℓ , player 1’s beliefs are not confirmed. Sheexpects the other two players to be willing to form links with her, although they do not doso. 33 etwork д : This network is neither weakly monadically stable, nor monadically stable. The non-superfluous communication profile ℓ = ( ( , ) , ( , ) , ( , ) ) is an obvious candidate to sup-port this network. For this profile we compute that ℓ ⋆ = (− , ( , ) , ( , ) ) ℓ ⋆ = (( , ) , − , ( , ) ) ℓ ⋆ = (( , ) , ( , ) , − ) This results into the following best response configuration: • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 1. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 2. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 3.From this it is clear that д cannot be supported by ℓ . This illustrates that weak monadicstability requires selecting a best response to a specific set of beliefs for each player i ∈ N .Without such a restriction on the beliefs it would be possible to support any strategy asweakly monadic stable. Moreover, observe that players only form beliefs about the behaviourof their acquaintances with regard to direct links, making it myopic but realistic. In fact,because of this, it is possible that monadically stable equilibria do not exist.Finally, we can complete the argument by checking that other communication profiles canbe ruled out in similar fashion. Network д : We argue that this network is weakly monadically stable as well. We can show that д is supported by the action tuple ℓ = ( ( , ) , ( , , ) , ( , ) ) . Again we compute ℓ ⋆ = (− , ( , ) , ( , ) ) ℓ ⋆ = (( , ) , − , ( , ) ) ℓ ⋆ = (( , ) , ( , ) , − ) Note here that player 1 is indifferent between д and д in terms of her net payoff π a . Thus,in the computation of ℓ ⋆ we use the bias of player 1 towards having more links rather thanfewer in player 2’s belief system.This results into the following best response configuration: • β (cid:0) ℓ ⋆ (cid:1) = { ( , ) , ( , ) } is the set of best responses to ℓ ⋆ for player 1, i.e., ( , ) and ( , ) are both best responses for this player. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 2. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 3.This shows that ℓ is indeed supported as a weak monadically stable communication profile.On the other hand, д is not monadically stable, since the beliefs of player 2 are not confirmed.34 etwork д : First, we claim that this network is strictly pairwise stable. Strong link deletionproofness follows trivially from the payoffs listed. Indeed, the net payoffs in other networks( д , . . . , д ) are at most the net payoff in д for all players. Second, strict link addition proof-ness is trivially satisfied since there are no links that are not part of д = д N .Furthermore, the complete network д = д N is weakly monadically stable. We claim that д issupported by the only communication profile supporting this network, ℓ = ( ( , ) , ( , , ) , ( , ) ) .We can determine that the monadic belief systems are given by ℓ ⋆ = (− , ( , ) , ( , ) ) ℓ ⋆ = (( , ) , − , ( , ) ) ℓ ⋆ = (( , ) , ( , ) , − ) From this we conclude that • β (cid:0) ℓ ⋆ (cid:1) = { ( , ) , ( , ) } is the set of best responses to ℓ ⋆ for player 1. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 2. • β (cid:0) ℓ ⋆ (cid:1) = ( , ) is the unique best response to ℓ ⋆ for player 3.So, ℓ is indeed a best response profile with regard to the generated monadic belief systems.Hence, д is indeed weakly monadically stable.Finally, all players’ monadic belief systems are confirmed here. So, in fact, д is monadicallystable.In this example, it is made clear that the introduced monadic belief systems require only that playersuse minimal information about each other’s payoffs to formulate appropriate expectations abouteach other’s linking behaviour. Indeed, monadic stability only considers players to use first-ordereffects of forming new links and deleting existing links to formulate their monadic beliefs. (cid:7) This example clarifies the relationship between the notion of weak monadic stability and the monadicstability concept. Next, I provide a more general characterisation.
Proposition 5.8
Let the network payoff function φ and the link formation cost structure c be given.Every monadically stable network д ∈ G N for ( φ , c ) satisfies the following two properties: (i) д is weakly monadically stable, and (ii) д is supported by a monadic belief system ℓ д that is non-superfluous in the sense that ℓ дij = ℓ дji for all pairs i , j ∈ N . Proof.
Let д ∈ G N be monadically stable and let action tuple ℓ д ∈ A a support д as such. Supposethat ij < д with ℓ дij = ℓ дji =
0. Then from the property that ℓ дi ∈ A ai is a best response to thebelief system ℓ д i ⋆ − i it can be concluded that ℓ дij = ℓ д i ⋆ ji =
1. But this would then implythat ℓ дji , ℓ д i ⋆ ji , violating the monadic stability self-confirmation condition.35e reverse of the assertion of Proposition 5.8 is not true. Simple examples can be constructed inwhich weakly monadically stable networks exist that satisfy the stated property, but which are notmonadically stable.A few comments regarding the relationship between weak monadic stability and network-basedstability concepts are in order here. First, weakly monadically stable networks are not necessarilystrong link deletion proof or link addition proof. Second, a network that is strong link deletion proofas well as link addition proof is not necessarily weakly monadically stable. We refer to network д inExample 5.7, which is weakly monadically stable, but not link addition proof. The other comparisonscan also be shown by properly constructed counterexamples. An equivalence result.
The main insight from this approach is that trust indeed builds verystrong networks. This is exemplified by the equivalence of the class of monadically stable andstrictly pairwise stable networks. For a proof I refer to Appendix A.7.
Theorem 5.9
Let the network payoff function φ and the link formation cost structure c = ( c ij ) i , j ∈ N be given such that c ij > for all i , j ∈ N with i , j . Then a network д ∈ G N is monadically stable for ( φ , c ) if and only if д is strictly pairwise stable for the network payoff function φ a given by φ ai ( д ) = φ i ( д ) − Õ ij ∈ L i ( д ) c ij (21)Through the monadic stability concept we have considered the notion of confidence—as a form ofmutual trust—into an advanced equilibrium concept, specifically designed for network formation.Confidence is introduced as an internalised feature into the behaviour of the players in network for-mation. Thus, trusting behaviour is as such a individualised feature rather than a social normativephenomenon.The strength as well as the weakness of the monadic stability approach is the myopic nature ofthe belief systems. Players do not apply very sophisticated reasoning; they only look at the firstorder effects of link formation. Natural future extensions of this line of theoretical research shouldexplore the possibility of introducing forward looking behaviour to understand how farsightedlystable networks arise. As mentioned in the introduction to this section, unilateral and monadic stability seem to be foundedon the same principles of trusting behaviour: Players attempt to form links with other players ifthey perceive these players to benefit from these links.Recall that a network is unilaterally stable if there is no player who can induce changes to thenetwork based on the belief that other players will consent to these changes if they are not harmfulto them. Note here that unilateral stability assumes a fully rational form of farsightedness in the This can be compared with existing models of farsighted network formation developed in Dero¨ıan (2003),Dutta, Ghosal, and Ray (2005), Page, Wooders, and Kamat (2005), Herings, Mauleon, and Vannetelbosch (2009), Navarro(2014), Kirchsteiger, Mantovani, Mauleon, and Vannetelbosch (2016), F¨orster, Mauleon, and Vannetelbosch (2016) andSong and van der Schaar (2020).
A formal comparison.
Next I consider a more technical comparison of the two concepts. Fromthe discussion above it cannot be expected that the application of monadic stability and unilateralstability results in exactly the same class of stable network. The next example shows that these twoconceptions can lead to completely different sets of stable networks.
Example 5.10
Again consider the by-now familiar case of three players N = { , , } . Let thenetwork payoff function φ be given in the table below and assume that link formation is costless,i.e., c ij = i , j ∈ N . 37 etwork д φ ( д ) φ ( д ) φ ( д ) Stability д = œ д = { } д = { } д = { } д = { , } д = { , } д = { , } д = д N д , д and д = д N . I discuss these in detail below: Network д : We investigate the stability properties of this network. First, note that д is not uni-laterally stable. Indeed, player 3 prefers to propose the formation of links 13 and 23 to createnetwork д N , which represents a strict Pareto improvement for all players in N .Second, network д is supported by a non-superfluous communication profile that is repre-sented as ℓ = ( ( , ) , ( , ) , ( , ) ) . This results into a monadic belief system given by ℓ ⋆ = (− , ( , ) , ( , ) ) ℓ ⋆ = (( , ) , − , ( , ) ) ℓ ⋆ = (( , ) , ( , ) , − ) Clearly ℓ constitutes a best response profile to the given monadic belief system and themonadic belief system is confirmed through ℓ , showing that д is supported as a monadicallystable network. Network д : First, note that д is strongly pairwise stable as well as unilaterally stable. Indeed,only player 1 has an incentive to add link 12 to form the complete network д = д N , whichis rejected by player 2 due to a loss in payoff. There are no players who have incentives tosever any of the two existing links.Next, д is not monadically stable. Indeed, take the non-superfluous communication profilethat supports it, given by ℓ = ( ( , ) , ( , ) , ( , ) ) . Then the corresponding monadic belief Similarly, note that д is actually a strictly pairwise stable network. The equivalence theorem shows that, therefore, д has to be monadically stable. ℓ ⋆ = (− , ( , ) , ( , ) ) ℓ ⋆ = (( , ) , − , ( , ) ) ℓ ⋆ = (( , ) , ( , ) , − ) Obviously, the communication profile ℓ is a best response to the monadic belief system above.This implies that д is weakly monadically stable. However, it is not monadically stable. In-deed, player 2 believes that player 1 would pursue the creation of a link with her—as repre-sented by ℓ ⋆ =
1. This is not as described by ℓ ; player 1 does not propose a link to player 2and, as such, the belief system of player 2 is not confirmed in the equilibrium communicationprofile. Network д = д N : To conclude the discussion of the situation described in this example, we con-sider the complete network д = д N , which is uniquely supported by the communicationprofile ℓ = ( ( , ) , ( , ) , ( , ) ) . The resulting monadic belief systems can now be repre-sented by ℓ ⋆ = (− , ( , ) , ( , ) ) ℓ ⋆ = (( , ) , − , ( , ) ) ℓ ⋆ = (( , ) , ( , ) , − ) Obviously, the communication strategy ℓ = ( , ) is not a best response to ℓ ⋆ , since player1 expects player 2 not to form a link with her. Therefore, ℓ is not supported as a monadicallystable communication profile. Thus, д is not weakly monadically stable.Furthermore, this network is neither unilaterally stable; in particular, it is not link deletionproof. Indeed, player 2 has an incentive to break the link with player 1 to move to network д .This example clearly shows that the class of unilaterally stable networks can be completely disjointfrom the class of monadically stable networks. In this example, however, the unilaterally stablenetwork is weakly monadically stable. This implies that in a unilaterally stable network monadicbeliefs can destabilise the network, leading to unending improvement attempts by the players inthe network. Thus, boundedly rational belief formation can undermine a farsightedly rational foun-dation for the network; as such, it represents an example of a direct conflict between farsighted orfull and boundedly rational behaviour. (cid:7) The question of existence of monadically stable networks is an important one. The previous discus-sion already identified the class of monadically stable networks to be exactly equal to the class ofstrictly pairwise stable networks. Obviously, this class is empty for a large collection of network39ayoff structures. Here I investigate certain conditions under which the class of monadic networksis non-empty.These conditions are related to the notion of a network potential as seminally developed byChakrabarti and Gilles (2007). There it is explored what the consequences are of founding networkpayoffs on an underlying link-based payoff function—denoted as a network potential . Network pay-off functions that admit a potential impose a payoff structure in which players assess the value oflinks in a similar fashion. It can be shown that for network payoff structures that are founded onsuch potentials, there exist strictly pairwise stable networks.In the subsequent discussion, I summarise the main insights from Chakrabarti and Gilles (2007).For details of the proofs of the main theorems I also refer to that paper and its appendices. Beforestating the main definitions and the resulting properties, I recall the definition of two potential con-cepts in the context of a non-cooperative game (A , π ) on the player set N as seminally introducedby Monderer and Shapley (1996). Definition 5.11
Let (A , π ) be a non-cooperative game on player set N . Then: (a) The game (A , π ) admits an exact potential in the sense of Monderer and Shapley (1996) ifthere exists a function P : A → R such that π i ( a ) − π i ( b i , a − i ) = P ( a ) − P ( b i , a − i ) (22) for every player i ∈ N , every strategy tuple a ∈ A and every strategy b i ∈ A i . (b) The game (A , π ) admits an ordinal potential in the sense of Monderer and Shapley (1996)if there exists a function P : A → R such that π i ( a ) > π i ( b i , a − i ) if and only if P ( a ) > P ( b i , a − i ) (23) for every player i ∈ N , every strategy tuple a ∈ A and every strategy b i ∈ A i . Based on these two notions of game-theoretic potentials, we can now consider how network payoffstructures might be founded on similar constructs.
Network potentials.
There are two main conceptions of the notion of a potential as a foundingdevice in the determination of network payoffs. Again we refer to these notions as an “exact po-tential” and an “ordinal potential”, following the accepted terminology in the literature. The nextdefinition introduces these two notions.
Definition 5.12
Let φ : G N → R N be a network payoff function. (a) The network payoff function φ admits an exact potential if there exists a function Λ : G N → R such that φ i ( д ) − φ i ( д − ij ) = Λ ( д ) − Λ ( д − ij ) (24) for every network д ∈ G N , every player i ∈ N and every link ij ∈ L i ( д ) . The network payoff function φ admits an ordinal potential if there exists a function Λ : G N → R such that the following conditions hold: φ i ( д ) > φ i ( д − ij ) if and only if Λ ( д ) > Λ ( д − ij ) (25) φ i ( д ) < φ i ( д − ij ) if and only if Λ ( д ) < Λ ( д − ij ) (26) φ i ( д ) = φ i ( д − ij ) if and only if Λ ( д ) = Λ ( д − ij ) (27) for every network д ∈ G N , every player i ∈ N and every link ij ∈ L i ( д ) . An exact potential imposes that the network payoff structure exhibits a cardinally uniform way ofhow players assess the addition or deletion of a link to a network. It is clear that the admittance ofan exact potential is a very strong condition on the network payoff structure. This is confirmed bythe following insight from Chakrabarti and Gilles (2007, Theorem 3.3):
Lemma 5.13
A network payoff function φ admits an exact potential if and only if the correspondingMyerson model Γ mφ admits an exact potential in the sense of Monderer and Shapley (1996). The admittance of an ordinal potential in a network payoff structure imposes a uniform assessmentof deleting and adding links to networks by all players in purely ordinal terms. Although this prop-erty is significantly weaker than the admittance of an exact potential, it remain a rather demandingcondition on the network payoff structure. The next lemma makes clear that there is again a rela-tionship with the notion of an ordinal potential in the sense of Monderer and Shapley (1996). Thenext lemma is stated as Theorem 4.3 in Chakrabarti and Gilles (2007). For a proof I refer to thatsource.
Lemma 5.14
Let φ be some network payoff structure. If the corresponding Myerson model Γ mφ admitsan ordinal potential in the sense of Monderer and Shapley (1996), then φ admits an ordinal potential. The reverse of the assertion stated in Lemma 5.14 is not true, as shown in Chakrabarti and Gilles(2007, Example 4.4).
Properties of network payoff structures that admit potentials.
Using the introduced no-tions of game-theoretic and network potentials, we can now distinguish three essential classes ofnetwork payoff structures. First, those network payoff structures that admit an exact potential;second, those network payoff structures for which the corresponding Myerson game admits an or-dinal potential; and, finally, those network payoff structures that admit an ordinal potential. Eachof these classes is larger than the previous.The next propositions collect some properties of the third class, namely those network payoffstructures that admit an ordinal potential. For proofs of these assertions I again refer to Chakrabarti and Gilles(2007).
Proposition 5.15
Let φ be some network payoff structure that admits an ordinal potential Λ . Thenthe following properties hold: (i) There exists at least one pairwise stable network.
The sets of strongly pairwise stable and strictly pairwise stable networks coincide.
The class of network payoff structures for which the corresponding Myerson game admits an ordinalpotential is particularly interesting. Indeed, Chakrabarti and Gilles (2007, Theorem 5.7) show thatfor this class of network payoff structures there exist strictly pairwise stable networks. I state forcompleteness the complete assertion:
Proposition 5.16
Let φ be a network payoff function for which the corresponding Myerson model Γ mφ admits an ordinal potential in the sense of Monderer and Shapley (1996). Then there exists at least onestrictly pairwise stable network for φ . This property gives rise to the main conclusion regarding the existence of a monadically stablenetwork. Indeed, the admittance of an ordinal potential in the Myerson model gives rise to theexistence of a strictly pairwise stable network, which in turn is monadically stable due to the fun-damental equivalence theorem. As a consequence, we can formulate the following main existencetheorem:
Theorem 5.17
Let φ : G N → R N be a network payoff structure and let c : N × N → R + be a linkformation cost structure. If the corresponding consent model with two-sided link formation costs Γ aφ ( c ) admits an ordinal potential in the sense of Monderer and Shapley (1996), then there exists at least onemonadically stable network for ( φ , c ) . The previous section focussed mainly on the internalisation of trust in the behaviour of players toresult into so-called “trusting behaviour” in link formation. We chose to internalise trusting be-haviour in the form of belief systems (monadic stability) or through stability concepts themselves(unilateral stability). However, there is rather different an approach possible in which trusting be-haviour is explicitly modelled through an externally determined institutional arrangement. Theseinstitutional arrangements are implemented collectively and are endowed with a form of collec-tively accepted self-enforcement.In my discussion I mainly considered behavioural rules that can be viewed as being part of atrusted governance system. All players are assumed to be embedded in such a governance system,expressing this in the formulated monadic stability concept as embedded monadic belief systems.Hence, we use game theoretic concepts to give this embeddedness an explicit, institutional form asa generally accepted behavioural rule, to behave according to the stated monadic belief system.
Correlation devices.
Next, I turn to a much more explicit conception of behavioural sociality.One can model guiding behavioural norms also as being external to the players, rather than fullyinternalised—as is the case for the notion of monadic belief systems. This refers to the possibilityto let external “devices” guide and coordinate decision-making in a game theoretic setting. In par-ticular, one can consider the question: “Can external guidance let decision makers achieve a higherpayoff than that is achieved through the set of supported Nash equilibria?”42 seminal study by Aumann (1974) introduced an innovative way to exactly introduce a formalway to establish mutually beneficial coordination among players. These external arrangements aredenoted as correlation devices . The basic idea is that the decisions made by players are influencedby things that are external to the decision problem itself, but are situated in their immediate sur-rounding. The classical example is that of a traffic light. The game theoretic representation is a form of the
Game of Chicken as explored extensively inthe literature. Two drivers approach a road crossing. At the crossing, each driver can either “stop”(action S ) or “continue” (action C ). If both continue there will result a crash; if both stop, both lookfoolish and need to coordinate their passing through prolonged negotiation (with hand gestures);and if one stops and the other continues, there is regret of the stopper and maximal payoff to theone who continues. The resulting payoffs can be captured by the following game-theoretic payoffmatrix: S CS C ( , ) and ( , ) depending on who actually stops—and the case inwhich both players stop or continue with equal probability—resulting into the expected payoffvector (cid:0) , (cid:1) . The latter includes a probability of of a crash, due to both players continuing.Now consider that there is an outside regulator—represented as a correlation device—added tothis situation in the form of a traffic light. The most important assumption of this arrangement isthat both drivers are fully informed about what fraction of time the traffic light is in what colour.Hence, both drivers know the probability distribution that is implemented through the traffic light.We investigate two traffic light arrangements: • First, consider that with equal probability the traffic light gives a red light to one player anda green light to the other. Adopting the normal rule to stop for red and to continue for green,we actually coordinate between the two Nash equilibria ( S , C ) and ( C , S ) , resulting into anexpected payoff computed as E π = ( , ) + ( , ) = (cid:0) , (cid:1) . Here there no positive probability of a crash and both drivers are reasonably content withtheir expected payoff.Would this traffic light be self-enforceable within the given social decision situation? We needto check whether this traffic light arrangement is indeed beneficial to both player driversif it is implemented as suggested by these two drivers. Obviously, if any of these drivers The following discussion is mainly based on the excellent account of correlated equilibrium in Chapter 9 ofMaschler, Solan, and Zamir (2013). I recommend the interested reader to look at their presentation. • In comparison with our regular traffic light, we can even increase the expected payoff byintroducing a more complicated coordination device. Indeed, consider a traffic light that canstop both drivers simultaneously with a given probability. In that case, the drivers negotiatethemselves and proceed with caution. So, the traffic light can give both drivers simultane-ously the signal “red”, at which both drivers are suggested to stop and proceed with caution.This allows the mixing of three outcomes in this decision situation. Suppose now that the traf-fic light gives both drivers simultaneously “red” with probability and one driver “red” andthe other driver “green” with equal probabilities . We can depict the resulting probabilitydistribution over all outcomes in a probability matrix:
S CS
12 14 C E π = ( , ) + ( , ) + ( , ) = (cid:0) , (cid:1) ≫ (cid:0) , (cid:1) = E π . Again we can ask whether this traffic light is self-enforcing. If one driver receives “red”, heknows that the other driver receives “red” with probability and “green” with probability .So, if he continues there is a crash with probability and he receives an expected payoff of · + · = < , the latter being the expected payoff if he follows the recommendation ofthe traffic light. Again, we conclude that the traffic light arrangement is indeed self-enforcing;no player has an incentive to deviate from the provided arrangement and recommendations.One can ask whether this reasoning can be extended to even higher payoffs. Indeed, Aumannshowed that this is the case up to payoff level 5. The arrangement that both drivers always face ared light—that is, “red” with probability 1—is, of course, not self-enforcing. Using correlation devices in network formation.
Correlation devices can also be introducedin the processes of network formation. I return to the network formation process under consentthat we discussed thus far and consider how external correlation devices in the form of externalrecommender systems can guide players to form “good” networks. We first take a look at a by-nowfamiliar network formation situation with three players.
Example 6.1
As before, let N = { , , } be the set of three players. Also, we choose φ to be a minormodification of the network payoff function studied in Example 5.10, given in the table below, and This means that both drivers get private recommendations from the traffic light; they do not know what the colourto the other driver is. This is the usual arrangement in modern traffic law. c ij = i , j ∈ N .As reported in the table below, there are actually five M-networks, namely all strong link deletionproof networks given by M = { д , д , д , д , д } . These five M-networks correspond only to threepayoff vectors, namely ( , , ) , ( , , ) and ( , , ) . Network д φ ( д ) φ ( д ) φ ( д ) M-network д = œ д = { } д = { } д = { } д = { , } д = { , } д = { , } д = д N д ? Indeed, д is the most obvious M-network that the players can aim for. Therefore, the payoffvector ( , , ) acts as a benchmark in relationship to any correlation device.Consider an external recommender system based on the three networks д , д , and д = д N . In par-ticular, assume that this correlation device recommends (i) all three players to execute signallingstrategy ℓ a = ( ( , ) , ( , ) , ( , ) ) resulting in network д with probability α = ; (ii) the signallingstrategy ℓ b = ( ( , ) , ( , ) , ( , ) ) resulting in network д with probability β = ; and (iii) the sig-nalling strategy ℓ c = ( ( , ) , ( , ) , ( , ) ) resulting in network д = д N with probability γ = . Theexpected payoffs under this system are now given by E π ( ℓ ) = α · φ ( д ) + β · φ ( д ) + γ · φ ( д ) = · φ ( д ) + · φ ( д ) + · φ ( д ) = (cid:0) , , (cid:1) ≫ ( , , ) = φ ( д ) Hence, coordinating the link building actions through this recommender system results into a strict
Pareto improvement over the best M-network. It remains to show that all three players have noincentives to deviate from the recommended correlated strategy: • Player 1: The only plausible alternative signalling strategy is to play ℓ = ( , ) to achieve thehigh paying network д . This results actually in no changes to the recommended networks,due to the recommended strategies executed by the two other players under the selected cor-relation device. Hence, player 1 has no gain from deviating from the recommended strategy. • Player 2: The only plausible alternative signalling strategy for this player is to execute ℓ ′ = , ) to establish network д . But this results in a lower expected payoff for player 2 if theother players follow the recommended strategies in ℓ : E φ (cid:0) ℓ ′ (cid:1) = · φ ( д ) + · φ ( д ) + · φ ( д ) = · + · = < = E π ( ℓ ) . This recommender system uses two non-M-networks, д and д = д N . Therefore, this correlationdevice is founded on considerations outside the realm of the stability concepts that we have con-sidered thus far. It shows that inefficient networks and non-stable networks play a role in networkformation processes. (cid:7) The example above shows just a single application of the correlated equilibrium concept to networkformation analysis. The application of this concept opens the way to further exploration, eventhough the multitude of correlated equilibria is discouraging. Indeed, Aumann showed that thecollection of expected payoff vectors supported by correlated equilibria includes the convex hull ofall Nash equilibrium payoff vectors. This is rather daunting and discouraging from the perspectivethat correlation will not lead to a smaller class of supported networks.However, the main research question that is still open is whether there exists a specific class ofcorrelation devices that could guide players to highly productive networks. Throughout our history,humans have in fact found ways to implement very effective correlation devices to build effectiveand high-paying networks. This includes recommender systems such as job recommendation re-ferrals and socio-economic recommendations through friendship networks. Further exploration ofthese systems from a Aumannian perspective is required to develop a theory that interprets thesepractical systems as correlation devices.
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Proofs of the main theorems
A.1 Proof of Theorem 3.2.
If:
Let φ be convex on D( φ ) . Obviously from the definitions and the discussions it follows that D s ( φ ) ⊂ D( φ ) . Thus, we only have to show that D( φ ) ⊂ D s ( φ ) .Now let д ∈ D( φ ) . Then for every player i ∈ N and link ij ∈ L i ( д ) it has to hold that φ i ( д ) > φ i ( д − ij ) due to link deletion proofness of д . In particular, for any link set h ⊂ L i ( д ) : Í ij ∈ h [ φ i ( д ) − φ i ( д − ij ) ] >
0. Since φ is convex on D( φ ) and д ∈ D( φ ) , it follows that φ i ( д ) > φ i ( д − h ) for every linkset h ⊂ L i ( д ) . In other words, д is strong link deletion proof, i.e., д ∈ D s ( φ ) . Only if:
Assume that D( φ ) = D s ( φ ) . Suppose further to the contrary that the payoff structure φ isnot convex on D( φ ) . Then there exists some network д ∈ D( φ ) and some player i ∈ N such that forsome link set h ⊂ L i ( д ) we have that Í ij ∈ h [ φ i ( д ) − φ i ( д − ij ) ] > φ i ( д ) < φ i ( д − h ) . Butthen this implies straightforwardly that player i would prefer to sever all links in h , i.e., д < D s ( φ ) .Thus, д cannot be strong link deletion proof giving us the necessary contradiction.This completes the proof of the assertion of Theorem 3.2. A.2 Proof of Theorem 3.5.
Assertion (a) is trivial and a proof is therefore omitted.
Proof of (b).
If:
Let φ be discerning on A( φ ) . Suppose that д is LAP. Furthermore, assume that i , j ∈ N with ij < д are such that φ i ( д + ij ) > φ i ( д ) . Now, if φ j ( д + ij ) = φ j ( д ) , then by definition of φ beingdiscerning, φ i ( д + ij ) > φ i ( д ) . This contradicts the hypothesis that д is LAP. Thus, φ j ( д + ij ) < φ j ( д ) ,confirming that д is indeed ⋆ -LAP. Only if:
Suppose that φ is not discerning on A( φ ) . Then there exists some network д that is LAPand for some i , j ∈ N with ij < д it holds that φ i ( д + ij ) = φ i ( д ) as well as φ j ( д + ij ) = φ j ( д ) . Butthis immediately implies that д can in fact not be ⋆ -LAP, since the link ij should be in д for it to be ⋆ -LAP. This is a contradiction. Proof of (c).
If:
Suppose that φ is uniform on A ⋆ ( φ ) and take some д ∈ A ⋆ ( φ ) . Assume that i , j ∈ N with ij < д .Then first suppose that φ i ( д ) φ i ( д + ij ) . (28)Then by д being ⋆ -LAP it has to hold that φ j ( д ) > φ j ( д + ij ) . (29)But also by uniformity of φ it has to hold that φ j ( д ) φ j ( д + ij ) . (30)But (29) is in direct contradiction to (30). Thus, we conclude that (28) cannot hold. Therefore, forany ij < д it has to hold that φ i ( д ) > φ i ( д + ij ) as well as φ j ( д ) > φ j ( д + ij ) . Hence, we conclude that д is actually SLAP, i.e., д ∈ A s ( φ ) . Only if:
Assume that A s ( φ ) = A ⋆ ( φ ) . Now take д ∈ A ⋆ ( φ ) to be ⋆ -LAP. Then from д being SLAP,it follows that φ i ( д ) > φ i ( д + ij ) as well as φ j ( д ) > φ j ( д + ij ) . This implies that φ indeed has to beuniform for д . 49is proves the assertion of Theorem 3.5. A.3 Proof of Theorem 4.1
First, we show assertion (a).Suppose that there is an M-network д ∈ G N supported by a Nash equilibrium strategy profile ℓ ∈ A m that is not strong link deletion proof. Then there is some i ∈ N and h i ⊂ L i ( д ) with φ i ( д − h i ) > φ i ( д ) . But then player i can modify his linking strategy as ℓ ′ ij = ij ∈ h i and ℓ ′ ij = ℓ ij . Then д ( ℓ ′ i , ℓ − i ) = д − h i implying that π mi ( ℓ ′ i , ℓ − i ) > π m ( ℓ ) . Therefore, ℓ cannot be a Nashequilibrium in (A m , π m ) . This is a contradiction, showing that M-networks are strong link deletionproof.Next, let д ∈ D s ( φ ) be a strong link deletion proof network for the network payoff function φ on N . Suppose that д is not an M-network. Then the corresponding signalling tuple ℓ д —where ℓ дij = ij ∈ д and ℓ дij = Γ mφ . Hence, there is a player i ∈ N and an alternative strategy ℓ i ∈ A i with ℓ i , ℓ дi suchthat π mi ( ℓ д ) < π mi ( ℓ i , ℓ д − i ) . If we denote by h + i = { ij | ℓ дij = ℓ ij = } , then it is clearthat д ( ℓ i , ℓ д − i ) = д − h i ⊂ L i ( д ) . Using the definition of the Myerson payoff function π m , we haveestablished that φ i ( д ) < φ i ( д − h i ) , which contradicts the hypothesis that д is strong link deletionproof.To show assertion (b), suppose that φ is link monotone. Take any network д ∈ G N and construct astrategy profile ℓ д ∈ A m by ℓ дij = ij ∈ д , for all i , j ∈ N . It is easy to see that ℓ д isindeed a Nash equilibrium in (A m , π m ) due to φ being link monotone: For any i ∈ N , any deviation ℓ i from ℓ дi induces the link set L i (cid:0) д ( ℓ i , ℓ д − i ) (cid:1) ⊆ L i ( д ) for i . This implies by link monotonicity that π mi ( ℓ i , ℓ д − i ) = φ i (cid:0) д ( ℓ i , ℓ д − i ) (cid:1) φ i ( д ) = π m ( ℓ д ) . A.4 Proof of Theorem 4.5. (a) implies (c):
Let ℓ ⋆ be an arbitrary Nash equilibrium in (A a , π a ) . Then denote д ⋆ = д m ( ℓ ⋆ ) = { ij ∈ д N | ℓ ⋆ ij = ℓ ⋆ ji = } . We show that д ⋆ is strong link deletion proof for the derived networkpayoff function φ a .Suppose player i deletes a certain link set h i ⊂ L i ( д ⋆ ) . Define ℓ i ∈ A ai as ℓ ij = ij ∈ д ⋆ − h i and ℓ ij = ij < д ⋆ − h i . Then by ℓ ⋆ being a Nash equilibrium in ( A a , π a ) it follows that д m ( ℓ i , ℓ ⋆ − i ) = д ⋆ − h i and π ai ( ℓ ⋆ ) > π ai ( ℓ i , ℓ ⋆ − i ) . Hence, φ ai ( д ⋆ ) = φ i ( д ⋆ ) − Õ j ∈ N i ( д ⋆ ) c ij = π ai ( ℓ ⋆ ) + Õ k : ℓ ⋆ ik = ,ℓ ⋆ ki = c ik > π ai ( ℓ ⋆ ) > π ai ( ℓ i , ℓ ⋆ − i ) = φ i ( д m ( ℓ i , ℓ ⋆ − i )) − Õ k , i ℓ ik · c ik = φ i ( д ⋆ − h i ) − Õ k ∈ N i ( д ⋆ − h i ) c ik = φ ai ( д ⋆ − h i ) . This proves that д ⋆ is strong link deletion proof for φ a . (c) implies (b): Suppose that д ⋆ ⊂ д N is a strong link deletion proof network for φ a . We show thatit is supported by a non-superfluous Nash equilibrium strategy in (A a , π a ) . Consider the uniquenon-superfluous strategy profile ℓ ⋆ ∈ A a such that д m (cid:0) ℓ ⋆ (cid:1) = д ⋆ . We proceed to show that ℓ ⋆ is aNash equilibrium in (A a , π a ) and ℓ ⋆ ij = ij ∈ д ⋆ . Indeed, π ai ( ℓ ⋆ ) = φ i ( д m ( ℓ ⋆ )) − Õ k , i ℓ ⋆ ik · c ik = φ i ( д ⋆ ) − Õ k ∈ N i ( д ⋆ ) c ik = φ ai ( д ⋆ ) .50ext, for some player i consider some deviation ℓ i , ℓ ⋆ i . Define h i = { ik ∈ д ⋆ | ℓ ik = } . Then, д m ( ℓ i , ℓ ⋆ − i ) = д ⋆ − h i . Since д ⋆ is strong link deletion proof with respect to φ a , it follows that φ ai ( д ⋆ − h i ) φ ai ( д ⋆ ) . Thus, π ai ( ℓ i , ℓ ⋆ − i ) = φ i ( д m ( ℓ i , ℓ ⋆ − i )) − Õ k , i ℓ ik · c ik = φ i ( д ⋆ − h i ) − Õ k ∈ N i ( д ⋆ − h i ) c ik − Õ k : ℓ ik = ,ℓ ⋆ ki = c ik φ i ( д ⋆ − h i ) − Õ k ∈ N i ( д ⋆ − h i ) c ik = φ ai ( д ⋆ − h i ) φ ai ( д ⋆ ) = π ai ( l ⋆ ) .This proves that the non-superfluous signal profile ℓ ⋆ is indeed a Nash equilibrium.Trivially (b) implies (a), which proves the assertion and completes the proof of Theorem 4.5. A.5 Proof of Theorem 4.8
Let д ⋆ be strong link deletion proof under the net payoff function φ b . For д ⋆ , define a non-superfluouscommunication profile λ ⋆ = ( l ⋆ , r ⋆ ) ∈ A b as follows:(i) l ⋆ ij = r ⋆ ji = ij ∈ д ⋆ and γ ij < γ ji , or(ii) l ⋆ ij = r ⋆ ji = ij ∈ д ⋆ , γ ij = γ ji and i < j , or(iii) l ⋆ ij = r ⋆ ji = ij < д ⋆ .Obviously, д b (cid:0) l ⋆ , r ⋆ (cid:1) = д ⋆ and π bi ( λ ⋆ ) = φ i (cid:16) д b ( λ ⋆ ) (cid:17) − Õ j , i l ⋆ ij · γ ij = φ i ( д ⋆ ) − Õ j ∈ M i ( д ⋆ ) γ ij = φ bi ( д ⋆ ) . Now, for player i ∈ N consider an arbitrary deviation b λ i = (cid:16)b l i , b r i (cid:17) , (cid:0) l ⋆ i , r ⋆ i (cid:1) = λ ⋆ i . In any suchdeviation, no new links will be formed because if ij < д ⋆ , it follows that l ⋆ ji = r ⋆ ji =
0. However,links in i ’s neighbourhood link set L i ( д ⋆ ) can be deleted. Hence, let д b (cid:16) b λ i , λ ⋆ − i (cid:17) = д ⋆ − h i where h i ⊂ L i ( д ⋆ ) .We prove that j ∈ N i ( д ⋆ − h i ) and (cid:2) γ ij < γ ji or γ ij = γ ji , i < j (cid:3) implies that b l ij =
1. In other words, j ∈ M i ( д ⋆ − h i ) ⊂ N i ( д ⋆ − h i ) implies that b l ij = j ∈ M i ( д ⋆ − h i ) : b l ij =
0. Now, j ∈ N i ( д ⋆ − h i ) ⇔ b l ij = r ⋆ ji = b r ij = l ⋆ ji = . (31)But l ⋆ ji = γ ij > γ ji or γ ij = γ ji , i > j . Furthermore, r ⋆ ji = γ ij < γ ji or γ ij = γ ji , i < j . Since b l ij = , by (31), it follows that b r ij = l ⋆ ji = γ ij > γ ji or γ ij = γ ji with i > j . This contradicts j ∈ M i ( д ⋆ − h i ) completing the proofof the claim stated above.Now, the proven claim implies that Õ j ∈ M i ( д ⋆ − h i ) γ ij Õ j ∈ N i ( д ⋆ − h i ) b l ij · γ ij Õ j , i b l ij · γ ij . (32)51ence, π bi (cid:16) b λ i , λ ⋆ − i (cid:17) = φ i (cid:16) д b ( b λ i , λ ⋆ − i ) (cid:17) − Õ j , i b l ij · γ ij = φ i (cid:0) д ⋆ − h i (cid:1) − Õ j , i b l ij · γ ij φ i (cid:0) д ⋆ − h i (cid:1) − Õ j ∈ M i ( д ⋆ − h i ) γ ij = φ bi ( д ⋆ − h i ) φ bi ( д ⋆ ) = π bi (cid:0) l ⋆ , r ⋆ (cid:1) . The first inequality follows from (32) and the second follows from the fact that д ⋆ is strong linkdeletion proof with respect to φ b . This completes the proof of Theorem 4.8. A.6 Proof of Theorem 4.13
Let д ⋆ be supported by a Nash equilibrium signalling profile ℓ ⋆ ∈ A a in the consent model with two-sided link formation costs (A a , π a ) . We now construct a non-superfluous strategy tuple (cid:16)b l , b r (cid:17) ∈ A b in the consent model with one-sided link formation costs such that д b (cid:16)b l , b r (cid:17) = д ⋆ and (cid:16)b l , b r (cid:17) is aNash equilibrium in (A b , π b ) .From Theorem 4.5, we can assume without loss of generality that ℓ ⋆ ∈ A a is non-superfluous. Given ℓ ⋆ , we define b λ = (cid:16)b l , b r (cid:17) ∈ A b by(i) b l ij = b r ji = b l ji = b r ij = ℓ ⋆ ij = ℓ ⋆ ji =
1, and either c ij < c ji , or c ij = c ji with i < j . (ii) b l ij = b l ji = b r ij = b r ji = ℓ ⋆ ij = ℓ ⋆ ji = b λ = (cid:16)b l , b r (cid:17) is a non-superfluous communication profile in A b supporting д b (cid:16)b l , b r (cid:17) = д ⋆ .It remains to be shown that b λ is a Nash equilibrium of the consent model with one sided linkformation costs. We sketch the proof of this assertion.Now, if b λ is not a Nash equilibrium, then it has to be because some player prefers to delete one ormore of her links. Also, any link delivers the same benefit to the player as under two-sided linkformation costs, while it would cost no more to establish the link. Thus, preferring to keep a linkunder two-sided link formation costs, implies that the player would prefer to keep the link underone-sided link formation costs. Mathematical details of this argument are left to the reader.This completes the proof of Theorem 4.13. A.7 Proof of Theorem 5.9
We first develop some simple auxiliary insights for weakly monadically stable networks. Supposethat д ∈ G N is weakly monadically stable relative to the data φ and c = ( c ij ) i , j ∈ N . Then there existssome action tuple ˆ ℓ ∈ A a such that д = д ( ˆ ℓ ) and for every player i ∈ N : ˆ ℓ i ∈ A ai is a best responseto the monadic belief system ˆ ℓ i ⋆ − i ∈ A a − i for the payoff function π a .For this setting we state two auxiliary results. Lemma A.1 If ˆ ℓ i ⋆ ji = and c ij > , then ℓ ij = is the unique best response to ˆ ℓ i ⋆ − i . Proof.
Clearly, if player i selects ℓ ij = i only incurs strictly positive costs c ij > i makes a loss from trying to establish link ij . Hence, ℓ ij = ℓ i ⋆ − i . Lemma A.2 If ij ∈ д ( ˆ ℓ ) with c ij > as well as c ji > , then ˆ ℓ i ⋆ ji = ˆ ℓ j ⋆ ij = . roof. We remark that ij ∈ д = д ( ˆ ℓ ) if and only if ˆ ℓ ij = ˆ ℓ ji =
1. The negation of the assertion statedin Lemma A.1 applied to ˆ ℓ i j = ℓ ji = ℓ i ⋆ ji = ˆ ℓ j ⋆ ij = Lemma A.3
Let the cost structure c ≫ be strictly positive. Then every weakly monadically stablenetwork д ∈ G N in the consent model with two-sided link formation costs ( A a , π a ) is link deletionproof for the network payoff function φ a . Proof.
Suppose that д ∈ G N is weakly monadic in the consent model with two-sided link formationcosts ( A a , π a ) . Then there exists some communication profile ˆ ℓ ∈ A a such that д = д ( ˆ ℓ ) and forevery player i ∈ N : ˆ ℓ i ∈ A ai is a best response to ˆ ℓ i ⋆ − i for the game theoretic payoff function π a .Suppose now that д is not link deletion proof for φ a . Then there exists some i ∈ N with ij ∈ д forsome j , i and φ a ( д − ij ) > φ ai ( д ) , implying that φ i ( д − ij ) + c ij > φ i ( д ) . By definition, ˆ ℓ j ⋆ ij = ℓ ji = ℓ j ⋆ for player j . Since ij ∈ д byassumption it has to hold that ˆ ℓ ji =
1. This contradicts the hypothesis that ˆ ℓ j is a best response toˆ ℓ j ⋆ − j .This contradiction indeed shows that д has to be link deletion proof relative to φ a .The proof of Theorem 5.9 now proceeds as follows.First we show that strict pairwise stability for φ a implies monadic stability in ( A a , π a ) under thehypothesis that c ≫ д ∈ G N be a network that is strictly pairwise stable with regard to the network payoff function φ a as given in the assertion. Then д is strong link deletion proof and satisfies the property that ij < д implies that φ ai ( д + ij ) < φ ai ( д ) as well as φ aj ( д + ij ) < φ aj ( д ) . Hence, this can be rewritten as ij < д implies φ i ( д + ij ) − c ij < φ i ( д ) as well as φ j ( д + ij ) − c ji < φ j ( д ) . (33)With д we define for all i ∈ N :ˆ ℓ ij = ij ∈ д ˆ ℓ ij = ij < д Hence, д ( ˆ ℓ ) = д and ˆ ℓ is non-superfluous. We now investigate whether the given communicationprofile ˆ ℓ is indeed a best response to the monadic belief system ˆ ℓ i ⋆ for all i ∈ N as required by thedefinition of weak monadic stability. Case A: ij < д .From (33) it follows immediately that ˆ ℓ i ⋆ ji = ˆ ℓ j ⋆ ij =
0. From the hypothesis that c ij > c ji > ℓ ij = ℓ i ⋆ − i and that ˆ ℓ ji = ℓ j ⋆ − j .Hence, for Case A the communication strategy ˆ ℓ satisfies the condition of weak monadicstability. Case B: ij ∈ д .In this case ˆ ℓ ij = ˆ ℓ ji =
1. Link deletion proofness of д now implies that ˆ ℓ i ⋆ ji = ij ∈ д if and only if ˆ ℓ i ⋆ ji = ˆ ℓ j ⋆ ij = . (34)53pplying strong link deletion proofness and the insight for Case A leads us to the conclusion thatˆ ℓ i is indeed the unique best response to ˆ ℓ i ⋆ − i . This in turn implies that ˆ ℓ supports д as a weaklymonadically stable network.Finally, it is immediately clear from (34) and the definition of ˆ ℓ that for all i , j ∈ N : ˆ ℓ i ⋆ ji = ˆ ℓ ij ,implying that the monadic beliefs are indeed confirmed.Thus, we conclude that ˆ ℓ supports д as a monadically stable network. This completes the proof ofthe first part of the assertion.Second, we show that the monadic stability of a network for ( A a , π a ) implies strict pairwise stabilityfor φ a under the hypothesis that c ≫ д ∈ G N be monadically stable. Then there exists some action tuple ˆ ℓ ∈ A a such that д = д ( ˆ ℓ ) andfor every player i ∈ N : ˆ ℓ i ∈ A ai is a best response to ˆ ℓ i ⋆ − i for the payoff function π a . Furthermore,ˆ ℓ i ⋆ − i = ˆ ℓ − i .From Lemma A.3 we already know that д has to be link deletion proof for φ a since д is weaklymonadically stable. Hence, for every ij ∈ д we have that φ i ( д − ij ) + c ij φ i ( д ) . Now through thedefinition of the monadic belief systems and the self-confirming condition of monadic stability weconclude that for every ij ∈ д :ˆ ℓ ij = ˆ ℓ j ⋆ ij = ˆ ℓ ji = ˆ ℓ i ⋆ ji = . (35)Let i ∈ N and h ⊂ L i ( д ) . Now we define ℓ h ∈ A ai by ℓ hij = ˆ ℓ ij if ij < h ij ∈ h . Then д (cid:16) ℓ h , ˆ ℓ − i (cid:17) = д − h . Since ˆ ℓ i is a best response to ˆ ℓ i ⋆ − i = ˆ ℓ − i it has to hold that π ai (cid:16) ℓ h , ˆ ℓ − i (cid:17) π ai ( ˆ ℓ ) . Hence, φ i ( д − h ) + Õ ij ∈ h c ij φ i ( д ) . (36)This in turn implies that φ ai ( д − h ) φ ai ( д ) .Since, i ∈ N and h are chosen arbitrarily, the network д has to be strong link deletion proof.Next, let ij < д . Then ˆ ℓ ij = ℓ ji =
0. Suppose that ˆ ℓ ji =
0. Then by the confirmationcondition of monadic stability it follows that ˆ ℓ i ⋆ ji = ˆ ℓ ji =
0. Hence by Lemma A.1, ˆ ℓ ij =
0. Thus weconclude that for every ij < д :ˆ ℓ ij = ˆ ℓ j ⋆ ij = ˆ ℓ ji = ˆ ℓ i ⋆ ji = . (37)This in turn implies through the definition of the monadic belief system that φ i ( д + ij ) − c ij < φ i ( д ) as well as φ j ( д + ij ) − c ji < φ j ( д ) . Or φ ai ( д + ij ) < φ ai ( д ) as well as φ aj ( д + ij ) < φ aj ( д ) . This showsthe assertion that д is indeed strictly pairwise stable.This completes the proof of Theorem 5.9. Here we again apply the confirmation condition for monadic stability that is satisfied by ˆ ℓ ..