AA Noncooperative Model of Contest Network Formation ∗ Kenan Huremovi´c
IMT School for Advanced Studies Lucca † May 15, 2020
Abstract
In this paper we study a model of weighted network formation. The bilateral interactionis modeled as a Tullock contest game with the possibility of a draw. We describe stablenetworks under different concepts of stability. We show that a Nash stable network is eitherthe empty network or the complete network. The complete network is not immune tobilateral deviations. When we allow for limited farsightedness, a stable network immune tobilateral deviations must be a complete M -partite network, with partitions of different sizes.We provide several comparative statics results illustrating the importance of the structureof stable networks in mediating the effects of shocks and interventions. In particular, weshow that an increase in the likelihood of a draw has a non-monotonic effect on the level ofwasteful contest spending in the society. To the best of our knowledge, this paper is the firstattempt to model weighted network formation when the actions of individuals are neitherstrategic complements nor strategic substitutes. Key Words: network formation; weighted network; contest; limited farsightedness.
JEL:
D85; D74; C72. ∗ I am grateful to Fernando Vega-Redondo and Piero Gottardi for their advice. I would also like to thankStefano Battiston, Yann Bramoull´e, Arnaud Dragicevic, Matthias Dahm, Joerg Franke, Timo Hiller, AndreaMattozzi, Nicola Pavoni, William H. Sandholm. This paper is based on a chapter of my PhD thesis. I’m gratefulto Cooperazione Italiana allo Sviluppo for the financial support. A previous version of this paper was circulatedunder the name:
Rent Seeking and Power Hierarchies: a Noncooperative Model of Network Formation withAntagonistic Links . Declarations of interest: none. † IMT School for Advanced Studies Lucca, Piazza S. Francesco, 19, 55100 Lucca, Italy. E-mail: [email protected], Website: https://sites.google.com/site/kenanhuremovic a r X i v : . [ ec on . T H ] M a y Introduction
A contest is a strategic interaction in which opposing parties make costly investments in orderto increase their chances of gaining control over scarce resources. Contests have been studiedin different settings, including political rent seeking (Hillman and Riley, 1989), discretionaryspending of top managers (Inderst et al., 2007), competition for funding (Pfeffer and Moore,1980), sport (Szymanski, 2003), litigation (Sytch and Tatarynowicz, 2014), and armed conflict(K¨onig et al., 2017). Agents often compete with several opponents simultaneously. In this case,the set of bilateral contest relations in a population can be described as a network, in which eachagent is a node, and a link indicates the contest between two agents. Contest networks emergein many situations. For instance, (Sytch and Tatarynowicz, 2014) studies the observed networkof patent infringements and antitrust lawsuits among US pharmaceutical firms. (K¨onig et al.,2017) theoretically and empirically demonstrates the importance of the network structure ofconflicts among groups in the Second Congo War. One may also expect that the structure of acontest network has important implications in other settings, including distributional conflictsin a federation as in (W¨arneryd, 1998), lobbying for discretionary spending of top managers asin (Inderst et al., 2007), and appropriation of property rights as in (MacKenzie and Ohndorf,2013).In this paper we propose a model in which players make costly investments (exert costlyeffort) to extract resources from other players in the society. It is a model of weighted networkformation, in which players choose with whom to engage in a bilateral contest and how muchto invest in each of their contests . Our starting point is the model introduced in (Franke andOzturk, 2015). In their model, the set of bilateral contests in the population is given, hencethe structure of contest network is exogenous. The prize of a contest is a fixed transfer fromthe loser to the victor. Our first departure from (Franke and Ozturk, 2015) is in the definitionof the bilateral contest game, where we use a different specification which, being more generalthan one used in (Franke and Ozturk, 2015), allows ties. The main difference between our paperand (Franke and Ozturk, 2015) is that we propose a model in which the structure of the contestnetwork is determined endogenously. We say that a link between two players exists or that theyare engaged in a contest when at least one of them invests a nonzero effort in fighting the other.Our main contribution is that we describe stable network structures under different notions ofstability, and we provide several comparative statics results that highlight the importance of thenetwork structure when assessing how changes in the parameters of the model affect individualand aggregate outcomes.We consider three notions of stability in this paper. Two of them, the
Nash stability and the strong pairwise stability (Bloch and Dutta (2009)), are standard in the literature of weightednetwork formation. The third equilibrium concept, labeled as the limited farsighted pairwisestability (LFPS), is introduced in this paper. The LFPS network is a network which is stableto unilateral and bilateral deviations of limited farsighted players.We show that the Nash stable network is, generically, the complete network in which playersexert the same effort in all contests. The Nash stable network is the complete network, even The empty network is Nash stable, for instance, in the case when the marginal cost of effort, for any levelof effort, is so high that a non-zero investment against an opponent who invests 0 is still not profitable. Weexplicitly state this condition in Proposition 7. strong pairwise stability .Since, in the complete network, any two players prefer to destroy the link between them, itimmediately follows that a non-empty strongly pairwise stable network does not exist.Starting a contest unilaterally is always a profitable action for a player because she doesnot take into account, when making this decision, that the new opponent will fight back. Weconsider an alternative stability concept where we relax this assumption and allow limitedforward looking. We assume that a player, when forming a link, takes into account that thenew opponent will fight back. However, we still assume that players do not take into accountfurther adjustments in other players’ strategies that may be a consequence of the new linkcreation. In that sense, players are limited farsighted . We define a limited farsighted pairwisestable network (LFPS) as a network that is immune to both unilateral and bilateral deviationsof limited farsighted players.The limited farsightedness assumption provides tractability, and we believe it is also sensible.Indeed, calculating the effects of a change in the network structure on the equilibrium investmentprofiles is a very complicated nonlinear problem even when the number of nodes in the network issmall. Assuming that players are able to make these calculations, for any contemplated choiceof opponents and efforts, would be a very strong assumption about their cognitive abilities.Moreover, recent experimental results suggest that, even in a simple bilateral Tullock contestgame, players find it very difficult to anticipate opponents’ best responses to their actions, andeven when the action of an opponent is known, they fail to calculate their own best responsecorrectly (Masiliunas et al., 2014). In (Kirchsteiger et al., 2016) authors find evidence in favor ofthe limited farsightedness in an experimental investigation of much simpler network formationgames.We show that in every LFPS non-empty network, players must be partitioned in M ≥ unequal sizes. Members of the same partition do not have links with each other,but have links with all other players in the network. So, even though players are ex-antehomogeneous, a stable non-empty network is necessarily asymmetric. To understand this result,the concept of a player’s strength is useful. In the model, a player is strong when her opponentsare weak. Thus, the strength of a player can be seen as a recursive measure of her position inthe contest network. In the model, a strong player has an incentive to form a link with a weakplayer, provided that the difference in their strengths is large enough. This is simply becauseit is cheaper to win a contest with a weak player than with a strong player. As the numberof opponents of a weak player increases, she becomes relatively weaker and therefore a moreattractive opponent for other strong players. This mechanism leads to network configurationswith potentially three types of players in a stable network. The strongest players in the society( attackers ) win all of their contests. Hybrid type players are strong enough to win against the Strength is an endogenous concept in our model, and it is a function of the global network structure. victims . They lose all of their contests. We findthat there will always be a single class of attackers and a single class of victims in a stablenon-empty network. The remaining M − This paper contributes to a broad literature of network formation (Jackson and Wolinsky, 1996,Bala and Goyal, 2000, Herings et al., 2009). The issue of network formation has been recognizedand studied in a number of settings, including provision of public goods (Galeotti and Goyal,2010, Kinateder and Merlino, 2017), favor exchange (Masson et al., 2018), collaboration betweenfirms (Goyal et al., 2008), and trade (Mauleon et al., 2010). For a survey of network formationliterature see (Mauleon and Vannetelbosch, 2016). In particular, our paper contributes to theliterature of weighted network formation in which players choose their investment levels specif-ically for each link. Several other papers study network formation with link-specific actions.(Goyal et al., 2008) studies the formation of R&D networks between firms that also compete ina market. (Bloch and Dutta, 2009) and the follow-up work by (Dero¨ıan, 2009) study a model ofnetwork formation in which agents choose how much to invest in each of their communicationlinks. (Baumann, 2017) develops a model of friendship formation in which players choose howmuch time to devote to socializing with each of their friends, and how much time to spend alone.All of these papers consider a bilateral interaction which is directly beneficial to both parties(i.e. collaboration, communication, socializing). Our model deals with a qualitatively differ-ent type of interactions - contests. Moreover, in the model presented in this paper, neighbors’4ctions are neither strategic substitutes nor strategic complements. Since one of the stabilityconcept we use in the paper (LFPS) assumes players are forward-looking, our paper contributesto the branch of the literature on network formation that relaxes the assumption that agentsare myopic when forming connections (Herings et al., 2009, Grandjean et al., 2011, Zhang et al.,2013, Kirchsteiger et al., 2016, Herings et al., 2019)Studying contests has a long tradition in economics, starting from seminal works on rentseeking (Tullock, 1967), and lobbying (Krueger, 1974). A recent comprehensive review of theliterature on contests can be found in (Corch´on and Serena, 2018). This literature is mostlyconcerned with the analysis of single battle n -lateral contest games, or multi-battle contests,with specific (symmetric) contest structures (Kvasov, 2007, Konrad and Kovenock, 2009). Inthis paper we consider a much more complex environment in which a population of playersplays interrelated bilateral contests on a general network structure. We model the bilateralcontest game following (Nti, 1997, Amegashie, 2006) and (Blavatskyy, 2010). Since, in ourmodel, the transfer size does not depend on the number of opponents (same as in (Frankeand Ozturk, 2015)), our model captures the situations in which the prize is relational. Forinstance, this is may be the case in lobbying (Hillman and Riley, 1989), appropriation of propertyrights (MacKenzie and Ohndorf, 2013), and litigation (Sytch and Tatarynowicz, 2014). In ourcomparative statics exercises we show that accounting for the network structure of bilateralcontests when studying the effects of changes in the parameters of the contest model on theequilibrium outcomes (as done for instance in (Nti, 1997) ), may lead to qualitatively differentresults compared to the case when the network structure is ignored.The importance of the structure of a contest network has recently been acknowledged inthe literature, both theoretically and empirically. There are several papers that study contestson a given network structure. (Franke and Ozturk, 2015) develops a model in which playersplay bilateral contests with their neighbors on a given graph. Using the variational inequalityapproach (Xu et al., 2019) generalizes the model of (Franke and Ozturk, 2015) to multilateralcontests on a given hypergraph. In a related paper (Matros and Rietzke, 2018) studies a modelin which there are two types of nodes: players and contests. Players connected to the samecontest play a multilateral contest game. (Dziubi´nski et al., 2016) studies a model in whichconnections between players determine potential conflicts, and agents sequentially choose ifthey wish to start a conflict with their neighbors and the effort level they are going to exert.(K¨onig et al., 2017) studies a model of conflict on a given network with two types of links:enmity links and alliance links. All agents participate in a single n -lateral contest and thenetwork structure is built in the payoff function. They also conduct an econometric analysisusing data on the Second Congo War, and find that there are significant fighting externalitiesacross contests. None of these models consider network formation. The model in this paperendogenizes the network structure in the model of (Franke and Ozturk, 2015), and providesnew comparative static results.There are a few papers that are concerned with formation of contest networks. (Jacksonand Nei, 2015) studies the impact of trade on the formation of interstate alliances and on theonset of war. They show that trade can mitigate conflict. (Grandjean et al., 2017) studies anetwork formation model in which agents form a network of collaboration links and then engage5n a single n -lateral contest. The position of a player in the collaboration network determinesher valuation of the contest prize. The closest paper to ours is (Hiller, 2016), which develops amodel of network formation in which players form positive links (friendship) and negative links(enmity). A negative link indicates that players are involved in a contest. However, contrary toour model, in (Hiller, 2016) players do not choose the fighting effort, and therefore the modelin (Hiller, 2016) is not a model of weighted network formation. (Goyal et al., 2016) provides acomprehensive review of the literature on conflict and networks.The rest of the paper is organized in 5 sections. Section 2 lays out the model. In Section3 we characterize efficient and LFPS networks. In Section 4 we present comparative staticresults. Section 5 provides a characterization of Nash stable networks and strongly pairwisestable networks. We conclude in Section 6. All the proofs are given in Appendix A. In this section we describe our network formation model. In the next paragraph we informallysummarize the model. In Subsection 2.1 we formally introduce the notion of a contest network,and describe the model. In Subsection 2.2 we define stable networks.Informally, we consider a population composed of a finite number of ex-ante identicalplayers . Players can engage in bilateral contests. The outcome of a contest is probabilistic, anddepends on costly investments by both parties. The prize of the contest is a fixed transfer fromthe defeated to the victor. Individuals choose both with whom to engage in a contest and howmuch to invest in each of their contests. We are interested in stable social structures that arisefrom this type of interaction, and how the structure of a stable contest network mediates theeffects of various types of shocks and third party interventions.
Denote with N = { , , ..., n } the set of players. Each player i ∈ N chooses how much to invest inbilateral contests with other players. Strategy of player i is vector s i = ( s i , s i , ..., s i,i − , s i,i +1 , ..., s in ) ∈ R n − ≥ , where s ij denotes the investment of player i in bilateral contest with j .The expected revenue of a bilateral contest between players i and j , π ij ( s ij , s ji ), is definedby: π ij ( s ij , s ji ; r ) = φ ( s ij ) φ ( s ij ) + φ ( s ji ) + r T − φ ( s ji ) φ ( s ij ) + φ ( s ji ) + r T. (1)The expression φ ( s ij ) φ ( s ij )+ φ ( s ji )+ r ∈ [0 ,
1] determines the probability with which i wins the transfer T = 1 from j , and it defines the Contest Success Function (CSF) F : R ≥ → [0 , φ : R ≥ → R ≥ in (1) transforms the investment in the contest (i.e. money, effort) into actualmeans of fighting (i.e. guns, lawyers). The parameter r ≥ r as noise in a transferablecontest, using CSF proposed in (Blavatskyy, 2010) and modeling noise as in (Amegashie, 2006).6n this paper we refer to r simply as the likelihood of a draw . A comprehensive review of contestmodels that allow ties can be found in (Corch´on and Serena, 2018).We make the following assumption about the technology function φ . Assumption 1
Technology function φ : R ≥ → R ≥ is assumed to be: (i) continuous and twice differentiable,(ii) strictly increasing and weakly concave, and (iii) φ (0) = 0 . In Assumption 1 (i) is assumed for analytical convenience, (ii) imposes non-increasing retrunsto scale and (iii) guarantees that zero investment implies zero actual means of fighting.The CSF used in (1) is fairly general, and includes CSFs studied in (Tullock, 1980, Loury,1979, Dixit, 1987) as special cases. In particular, by setting φ to be identity mapping and r = 0we get the CSF used in (Franke and Ozturk, 2015).We say that individuals i and j are linked (connected) or that there is a contest betweenthem when at least one of them exerts positive effort in fighting the other. Since efforts arenon-negative this will be the case if and only if s ij + s ji >
0. Strategy profile s defines (induces) weighted network g ( s ). Weight s ij is assigned to arc i → j . When i and j are linked ( s ij + s ji >
0) we write ij ∈ g ( s ). Clearly ij ∈ g ( s ) if and only if ji ∈ g ( s ). In this paper we use the terms link and contest as synonyms when talking about network g ( s ). We will use N i to denote theneighborhood of node i , so N i = { j ∈ N : ij ∈ g } , and d i = | N i | to denote the degree of node i .An example of contest network is presented in Figure 1. Figure 1: Weighted network g ( s ) with 6 nodes. Weights s ij are indicated next to correspondingarc. To denote that players 2 and 3 are connected we write 23 ∈ g or 32 ∈ g .The expected payoff of agent i from network g ( s ) is defined by: π i ( g ( s )) = (cid:88) j ∈ N i π ij ( s ij , s ji ; r ) − c ( w i ) , (2)where w i = (cid:88) j ∈ N i s ij is the total investment of player i in all of her contests. Function c : R ≥ → R ≥ is the costfunction. We make the following assumption about the cost fuction. For other interpretations of r see (Nti, 1997). To simplify notation, we omit dependence on s whenever there is no danger of ambiguity. Strategy profile s defines the adjacency matrix of network g ( s ). ssumption 2 Function c : R ≥ → R ≥ is continuous, twice continuously differentiable, strictly increasing andstrictly convex, with c (0) = 0 . We conclude this section by specifying what it means to form or destroy a link. Considerstrategy profile s . Suppose that strategies s i and s j are such that s ij = s ji = 0. This means ij / ∈ g ( s ). We say that player i starts a contest with j or that i forms link ij , when i deviatesfrom strategy s i to strategy ˆ s i such that ˆ s ij >
0. If, strategies s i and s j are such that s ij + s ji > i and j to strategies ˆ s i and ˆ s j , we haveˆ s ij + ˆ s ji = 0, we say that players i and j ended contest ij or deleted link ij . In this subsection we introduce two concepts of network stability which we employ in this paper.We first define Nash stable networks, and point out why using this standard equilibrium notionmay be inadequate for the model we study. Then we introduce limited farsighted pairwisestability (LFPS), which circumvents the shortcomings of Nash stability while still allowing fora reasonable tractability in the analysis. In Section 5 we discuss how LFPS relates to otherstability concepts usually employed when studying the formation of weighted networks, and therole of the limited farsightedness assumption.We define Nash stable networks as in (Bloch and Dutta, 2009, Definition 2):
Definition 1 (Nash stable networks)
A network g ( s ) is Nash stable if there is no individual i and strategy s (cid:48) i such that π i (cid:0) g ( s (cid:48) i , s − i ) (cid:1) > π i ( g ( s )) . Network g ( s ) is Nash stable if no player can unilaterally alter her investment pattern andobtain a higher payoff. The Nash equilibrium may not be the most suitable stability conceptfor our model. There are at least two reasons for this. First, we show that starting a contestis profitable for any player, except in extreme cases. Thus, a deviation which leads to theformation of a new link is always profitable. Second, a deviation which results in the destructionof a link is never profitable. The former is a consequence of the lack of forward looking whenstarting a contest. When players are not farsighted, they do not take into account that theopponent will fight back . The latter is a consequence of the fact that Nash stability deals onlywith unilateral deviations. We discuss these points in more detail in Section 5, where we providea characterization of Nash stable networks in Proposition 7.To address the issues pointed out in the previous paragraph, we consider a model in which(i) we assume that when i decides to form a link with j , she takes into account the immediatereaction from j (i.e. anticipates that j will fight back), and (ii) we allow for bilateral deviationsof players. In the following paragraphs we discuss (i) in more detail.Models of network formation usually assume either pure myopia (i.e. Nash stability, pair-wise stability) or complete farsightedness (i.e. pairwise farsighted stability) (Vannetelbosch andMauleon, 2015). In our model, pure myopia implies that starting a contest is always profitable.Given the complexity of the network effects, full farsightedness is too strong of an assumption See Section 5 for more details.
8o make. Indeed, even for networks with a small number of nodes, solving for the equilibriumrequires finding the roots of a high order polynomial. Thus, calculating all future adjustmentsin other players’ strategies after a deviation is computationally extremely demanding. More-over, experimental results suggest that limited farsightedness may be the most accurate way todescribe players’ behavior in network formation games (Kirchsteiger et al., 2016). In this paperwe adopt a specific form of limited farsightedness, described in the next paragraph.Consider strategy profile s . Let F i = { j ∈ N | ij / ∈ g ( s ) } . Thus, F i is the set of players withwhom player i does not have a contest. Consider a situation in which i contemplates initiatingcontests with players j ∈ L i ⊆ F i . We assume that, when assessing the payoff of starting contest ij with action s ij , player i expects that j will fight back by choosing the best response BR ( s ij ),given j (cid:48) s current contest investments s j . This means that, when i forms links to players from set L i by deviating from s i to s (cid:48) i , her expected payoff is π i ( g ( s (cid:48) i , ˆ s L i , s − i − L i )) where ˆ s L i = (ˆ s j ) j ∈ L i is such that for each j ∈ L i : π j (cid:0) g ( s (cid:48) i , ˆ s j , s L i − j , s − i − L i ) (cid:1) ≥ π j (cid:0) g ( s (cid:48) i , ˆ s (cid:48) j , s L i − j , s − i − L i ) (cid:1) , (3)for each ˆ s (cid:48) j with ˆ s (cid:48) jk = ˆ s jk = s jk for k (cid:54) = i . Here we use − i − L i to denote all players, except i and players from L i . We write L i − j to denote players in L i except player j .We are now ready to state the stability concept we use in this paper. Definition 2 (Limited Farsighted Pairwise Stable Networks)
Weighted network g = g ( s ∗ ) is stable if conditions (U) and (B) hold.(U) For any player i ∈ N , and any, potentially empty, set L i ⊆ F i , and any strategy s i ∈ R n − ≥ , π i ( g ( s ∗ )) ≥ π i (cid:0) g (cid:0) s i , ˆ s L i , s ∗− i − L i (cid:1)(cid:1) . (B) For any pair of players ( i, j ) such that ij ∈ g ( s ∗ ) , any two sets L i ⊆ F i and L j ⊆ F j , andany two strategies s i and s j such that ij / ∈ g ( s i , s j , s ∗− i − j ) , π i ( g ( s i , s j , ˆ s L i , s ∗− i − j − L i ) ≥ π i ( g ( s ∗ )) ⇒ π j ( g ( s j , s i , ˆ s L j , s ∗− j − i − L j ) < π j ( g ( s ∗ )) . Part (U) of Definition 2 states that no player i ∈ N has an incentive to unilaterally deviateand change her pattern of contest investments. The important assumption there is that if thedeviation entails the onset of a contest with player j , player i takes into account that j may fightback, as discussed in the paragraph preceeding equation (3). Part (B) of Definition 2 states notwo players find it profitable to jointly deviate by deleting the link between them, while at thesame time potentially adjusting their strategies in other contests or forming new links. Thus, abilateral deviation may include the deletion of a link, and formation of many other links. This isa feature shared with models of network formation studied in (Goyal and Vega-Redondo, 2007),(Bloch and Dutta, 2009), and (Dev, 2018).It is clear that, in order to start a contest (create a link), the action of one party suffices.This is a natural property, since, for instance, to start a litigation process it is sufficient thatone side files a lawsuit. On the other hand, to end contest ij , both players i and j must choosezero investment. In other words, to make peace, both sides must choose not to fight. Therefore,in our model, the creation of a link is the result of an unilateral action, while the destructionof a link is a result of a bilateral action. 9 Analysis
We start our analysis by outlining important properties of the network formation game inSection 3.1. We then turn our attention to the analysis of stable networks in Section 3.2.
We begin our analysis by outlining the properties of the payoff function and the nature ofstrategic interactions. It is straightforward to verify that the payoff function (2) of player i isincreasing and concave in s i , and decreasing and convex in s − i . The sign of the first and thesign of the second derivative of the payoff function with respect to r depend on s i and s − i .When a player’s probability of winning is greater than the probability of losing in all contests,the payoff function will be decreasing and convex in r . Similarly, if the probability of winningis lower than the probability of losing in all of her contests, the payoff function is increasingand concave in r . The best reply curves of the bilateral contest game are nonlinear and non-monotonic. The bilateral contest game is neither a game of strategic complements nor strategicsubstitutes. To the best of our knowledge, the only papers that consider this type of bilateralstrategic interactions on networks are (Franke and Ozturk, 2015, Matros and Rietzke, 2018, Xuet al., 2019) and (Bourl`es et al., 2017). Neither of these papers studies network formation.
We now turn to describing LFPS network architectures. We start by introducing a few usefulconcepts and observations. Then through a series of intermediate results arrive at our mainresult in this section - a description of stable networks.For our analysis it is useful to introduce unweighted graph ¯ g assigned to network g ( s ). Werefer to ¯ g as the structure of contest network g ( s ) since it fully describes the set of contests inthe population. Definition 3
Graph ¯ g = ( N, E ) with set of vertices N and set edges E ⊆ {{ i, j } : i, j ∈ N ∧ i (cid:54) = j } is said torepresent the structure of network g ( s ) or that it is induced by s if ij ∈ E ⇔ ij ∈ g ( s ) . Graph ¯ g representing the structure of network from Figure 1 is presented in Figure 2. Figure 2: Graph representing the structure of weighted network from Figure 1.To indicate that two nodes are connected in ¯ g we, abusing notation, write ij ∈ ¯ g . Thusgraph ¯ g induced by s can be thought of as an unweighted and undirected projection of network When r = 0 the payoff function is not defined at the point s ij = s ji = 0, however, this does not affect ourresults. ( s ). To avoid confusion, we use the word network to refer to the weighted network g = g ( s ) ,and the word graph to refer to the unweighted network ¯ g . The following proposition states that if g ( s ∗ ) is a LFPS network then there does not existanother LFPS network g ( s (cid:48) ) such that s ∗ (cid:54) = s (cid:48) with the property that ij ∈ g ( s ∗ ) ⇔ ij ∈ g ( s (cid:48) ).Hence, without ambiguity, we can talk about LFPS stability of graph ¯ g . Definition 4 formallyintroduces the notion of stable graph.
Proposition 1
Let g ( s ∗ ) be a LFPS network. If g ( s (cid:48) ) is a LFPS and such that ij ∈ g ( s (cid:48) ) if and only if ij ∈ g ( s ∗ ) then s (cid:48) = s ∗ . If additionally φ (cid:48) (0) = ∞ , or r → then s ∗ ij > and s ∗ ji > whenever ij ∈ g ( s ∗ ) . Definition 4
Graph ¯ g is said to be LFPS stable when there exists a strategy profile s ∗ such that g ( s ∗ ) is LFPS,and s ∗ induces ¯ g . In the remaining part of the analysis of LFPS networks we will, for mathematical conve-nience, maintain the following assumption.
Assumption 3
We assume that φ (cid:48) (0) = ∞ or r → . The importance of Assumption 3 is that it guarantees, according to Proposition 1, that bothplayers involved invest positive amount in each bilateral contest, and hence we do not have todeal with corner solutions. For instance, the contest success function considered in (Franke andOzturk, 2015) satisfies Assumption 3.We now define the strength of a player.
Definition 5
Consider g ( s ) that satisfies (U) for L i = ∅ . Player i ∈ N is said to be stronger than player j ∈ N in g whenever w i < w j . Definition 5 is motivated with the result that for two players i and j , such that ij ∈ g ( s )and g ( s ) satisfies (U) for L i = ∅ , i wins contest ij whenever w i < w j . We state and provethis result formally in Proposition 10 in Appendix A. This seemingly counter-intuitive resultis a direct consequence of the convexity of the cost function – high w j implies that both themarginal cost and per unit cost is high for player j . Therefore j spends less than i in contest ij ,although she spends more overall in all of her contests. Intuitively, high w j may be attributedto strong opponents or a high number of opponents. For instance, if N i ⊂ N j then, additionallyto fighting all rivals of i , j fights with other opponents as well. Hence, it is not surprising thatwe find that in this case j spends more resources overall, and therefore is weaker than i . SinceLFPS network g ( s ∗ ) satisfies (U) for L i = ∅ by definition, w ∗ i determines the strenght of player i in a stable network. It is important to note that w ∗ i is an equilibrium outcome determined bythe underlying network topology. Terms graph and network are practically synonyms. The term graph is more often used to denote a mathe-matical object, while the term network is more often used to denote the graph that represents a real-world object(Jackson, 2008, Estrada and Knight, 2015) This result, in the context of the game on a fixed network, appears in (Franke and Ozturk, 2015, Proposition2) for the case when r = 0, φ ( x ) = x and c ( x ) = x .
11t is useful to partition players in a stable network with respect to their strengths. To dothat. sort ( w ∗ i ) i ∈ N starting from the lowest ( w ∗ < w ∗ < ... < w ∗ M ) , where M ≤ n is the numberof different total equilibrium investment levels. We use W i to denote the class of players thathave the i -th lowest total investment level, and use m i = | W i | do denote the cardinality of class i . Definition 6
Player a ∈ W i is an attacker if all of her contests are with agents from W i = { W j | j > i } . Player a ∈ W i is a hybrid if there exist players b and c such that ab, ac ∈ g and w ∗ b > w ∗ a > w ∗ c . Player a ∈ W i is a victim if she has all of her contests with players from W i = { W j | j < i } . Definition 6 acknowledges the fact that a contest between two players of the same strengthis not profitable to any of the players involved, and hence cannot be part of a stable network.If j is weaker than i in stable network g ( s ∗ ) and ij ∈ g ( s ∗ ), there exists a bilateral deviationwhich is profitable for j in which i and j destroy link ij . This is simply because j loses thecontest ij and thus prefers not to engage in it (see Proposition 10 in Appendix A). Therefore, we say that i controls link ij if i is stronger than j . This in particular implies that in a stablenetwork every attacker must receive a positive payoff. If this were not true for some attacker i and contest ij , then a joint deviation in which i and j choose s ij = s ji = 0 (delete link ij )would be profitable for both i and j .In order to study the network formation, it is important to be able to compare contests inthe network. We now state a result which enables us to do that. Proposition 2
Let g ( s ∗ ) be a LFPS network. Suppose a ∈ W i , b ∈ W j , c ∈ W k such that i < j < k and ab ∈ g, ac ∈ g, bc ∈ g . Then s ∗ ab > s ∗ ac , s ∗ ba > s ∗ ca , s ∗ ca < s ∗ cb and s ∗ ac > s ∗ bc . Furthermore, π ∗ ac > π ∗ ab . Proposition 2 states that a strong player ( a ) engaged in contests with players weaker thanher ( b and c ), spends less, and has a less intensive contest with weaker players among heropponents. On the other hand, a weak player ( c ) spends less in the contest with the strongerof the two opponents (both stronger than her). Intuitively, this result relies on the followingtwo observations. First, the resources are more costly on the margin for weak players. Second,the best reply functions in a bilateral contest are nonmonotonic – s ij is increasing in s ji when s ij > s ji , and is decreasing in s ji when s ij < s ji . Having these two points in mind, compare forinstance contests ab and ac . As the strongest player, a wins in both contests. Since b is strongerthan c , the resources for b are cheaper on the margin, leading to s ∗ ba > s ∗ ca . Since the best replyof a increases with the efforts of b and c , a will spend more in the contest with b than in thecontest with c ( s ∗ ab > s ∗ ac ). Player a spends more in contest with b than in contest in c , but, onthe other hand, b spends more than c in contest with a . Thus, it is a-priory not clear whichcontest, ab or ac , brings higher benefit to a . The last part of the theorem states that player a earns higher expected revenue from contest ac . Since she also spends less in this contest, thecontest with the weaker player of the two is more beneficial for a .We now turn to the identifying necessary conditions for stability of network g ( s ∗ ), whichare stated in Proposition 3. The formal arguments leading to the proof of Proposition 3 are12eveloped through a series of intermediate results. We first argue that a nonempty stablenetwork must be connected, and then – by considering profitable deviations of attackers, hybrids,and victims – we identify network structures that can be stable. In the next few paragraphs weprovide the main intuition behind these results. Formal statements and proofs of the auxiliaryresults are relegated to Appendix A.Our first observation is that a player prefers to be in contest with weaker opponents, sinceresources are more costly (on the margin) to weaker opponents, and therefore they fight backwith less intensity. This implies that g ( s ) cannot be stable if, for some player i and two players j and k such that w i < w j ≤ w k , we have ij ∈ g ( s ) and ik / ∈ g ( s ). If this were the case, then itwould be profitable for i and j to jointly deviate by destroying link ij and for i to form the linkwith player k instead of the link with j . Since this result is important to understand linkingpattern of attackers and hybrids we state it as a separate Lemma. Lemma 1 If ij ∈ g ( s ∗ ) , and g ( s ∗ ) is LFPS, then ik ∈ g ( s ∗ ) ∀ ( k ∈ N ) : w ∗ k ≥ w ∗ j . A direct implication of Lemma 1 is that an attacker is connected with the weakest playersin a stable network. This in turn implies that a non-empty stable network g must be connected.Indeed, if there were two components in a stable network, then there would exist at least oneattacker that is not connected to the weakest player in the network.We now argue that all attackers focus their fighting effort on the same set of weak players.To this end, we rule out the possibility that two attackers i and j have different neighborhoods( N i (cid:54) = N j ) in a stable network. If N i and N j are different and not nested, then, without lossof generality, i has an opponent which is not an opponent of j and is not stronger than allopponents of j . According to Lemma 1, this is incompatible with a stable network. N i ⊂ N j implies that the weaker of the two players ( j ) finds it beneficial to fight, additionally to allplayers from N i , with other players as well. But since these additional contests are profitablefor player j (otherwise j , as an attacker, would have a profitable deviation), initiating them isa profitable deviation for i , because i is stronger than j .All members of the same class of hybrids in a stable network must have the same neigh-borhood as well. To understand this result, it is useful to partition neighborhood of a hybridplayer into the set of rivals that are stronger than her ( strong neighborhood ) and the set of rivalsthat are weaker than her ( weak neighborhood ). From our discussion of attackers, it follows thatmembers of the strongest class of hybrids W have the same strong neighborhood (attackers).Hybrid i ∈ W behaves as an attacker relative to (weaker than her) rivals in the weak neighbor-hood. Hence, we can apply a similar reasoning to the one we used when discussing attackers toargue that hybrids from W have the same weak neighborhood as well. Proceeding analogously,we show that the claim must hold for members of all hybrid classes W k : 2 ≤ k < M , providedthey exist in a stable network.Since there is a finite number of players, there exists the weakest player in a stable network(not necessarily just one player). From Lemma 1 we know that a player who wins at least onecontest must be connected to the weakest players in the network. The set of the weakest playersin the network constitutes the class of victims. See Lemmas 2-4 in Appendix A for formal arguments.
13o far we have argued that in a non-empty stable network we can partition players into
M < n classes with respect to their strength. There is only one class of attackers and only oneclass of victims. The remaining M − W (cid:96) is in a contest with all players outside W (cid:96) . This meansthat a non-empty stable network must have a complete M-partite structure. Finally we arguethat stronger classes in a stable network are larger (as measured by the number of nodes). Tosee this, compare two classes W i and W i +1 in a stable network. We recall that strong playersspend more per contest relative to weak players when facing the same opponents. On theother hand, by definition, stronger players have a lower total equilibrium spending ( w ∗ ). Thesetwo claims can hold simultaneously in a stable M-partite network only if m i > m j .We are now ready to state the main result about LFPS networks, which follows directlyfrom the intermediate results discussed above. Proposition 3
A non-empty stable network g ( s ∗ ) has a complete M -partite network structure with | W k | > | W k +1 | ∀ k ∈ { , ..., M − } . The empty network is stable. Proposition 3 provides necessary conditions for LFPS. Clearly, not all complete M -partitenetworks with property m k > m k +1 are stable. The difference in strengths, and consequentlyin the class sizes, must be at least large enough to ensure that every bilateral contest in thenetwork is profitable for the stronger opponent.A feature of LFPS networks worth highlighting is that even though players are ex-anteidentical, a non-empty network structure must be asymmetric enough to be stable. Thereason is that the asymmetry in strengths is necessary for a bilateral contest to be profitable.This asymmetry arises through the division of the population in different, mutually exclusive,partitions. Quite remarkably, the division is achieved in a completely non-cooperative fashionand without any direct benefits for players from belonging to a given partition. One way tounderstand how a player ends up in one and not in the other partition is by looking at thedynamics of the network formation process. In Appendix B, we propose a stylized dynamicalprocess of network formation which allows pairs of players to revise the network in sequenceand has a property that settles only in LFPS networks.Providing both the sufficient and the necessary conditions for LFPS stability is a quitecomplicated issue, due to the highly nonlinear and multidimensional nature of interactions weconsider. In Proposition 4 we make a step forward in this direction by focusing on a class ofcomplete bipartite networks M = 2. A stable complete bipartite graph is presented in Figure3. Follows directly from Proposition 2. This is not a unique feature of our model. For instance asymmetric networks in network formation modelsamong ex-ante identical players appear in Jackson and Wolinsky (1996), Bloch and Dutta (2009), Goyal andVega-Redondo (2007). i = π j =- Figure 3: A LFPS complete bipartite graph, with class of attackers A and a class of victims V .For each i ∈ A and j ∈ V s ∗ ij = 0 .
327 and s ∗ ji = 0 .
146 . The equilibrium payoffs (same for eachmember of a given partition) are indicated below the partition.Let K a,v denote a complete bipartite graph with n = a + v nodes and partitions of size a and v . We denote the two partitions by A and V respectively. The following proposition holds. Proposition 4
Suppose that φ ( x ) = x and c ( x ) = x . There exists v ∗ ∈ [1 , n − n √ ] such that K a,v = K n − v,v isLFPS if and only if v ≤ v ∗ and n ≥ . When n < the empty network is only LFPS network. Figure 4 depicts values of v ∗ as a function of the population size ( n ), and correspondingranges of v for which K n − v,v is LFPS. When n < ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶
100 200 300 400 n n - v * , v * ◼ n - v * ✶ v * n Figure 4: K n − v,v is stable when v is in the lower shaded region and corresponding a = n − v isin the upper shaded region.To understand the intuition behind Proposition 4, it is illustrative to think about howthe payoff of an attacker i ∈ A changes after a deviation which involves the destruction ofa link with j ∈ V when the strategy profile played satisfies condition (U) from Definition2. The destruction of a link ij implies that the amount of resources that i can appropriatefrom her opponents decreases. At the same time, i can reallocate the resources from ij to herother contests, and therefore increase her expected revenue in each of the remaining contests.The trade-off between these two effects is illustrated in Figure 5. In the figure we plot thepayoff of player i ∈ A obtained at the strategy profile which satisfies (U) and the network is It is straightforward to check that if g ( s ) satisfies (U) from Definition 2 for L i = ∅ and g ( s ) has a completeM-partite structure, then g ( s ) satisfies (U) for any set L i ⊆ F i (i.e. no player has an incentive to form a link). n − v,v (denoted with π i ) and her maximal payoff after the bilateral deviation which involvesthe destruction of link ij, j ∈ V (denoted with π devi ) when v changes. When v is low, theformer effect dominates, while when v is large enough the latter effect dominates. We find thatthe payoff from the destruction of a link π devi − π i is monotonically increasing with v . v - - Payoff π i , i ∈ A π i dev , i ∈ A Figure 5: The payoff of i ∈ A ( s played is consistent with (U) from Definition 2 and g ( s ) is K − v,v ) and the payoff of i after destroying a link with j ∈ V , as functions of v . Calculationsdone for c ( x ) = x , φ ( x ) = x , and r = 0.In propositon 3 we provide a necessary condition for a network to be LFPS. It is worthwhilenoting that this is done without explicitly solving for the strategy profile s ∗ . Solving for s whichsatisfies condition (U) from Definition 2 is in general infeasible, even for L i = ∅ for all i ∈ N .We devoted special attention to the case when M = 2 in Proposition 4, since this case allowssome tractability. Providing stronger results for cases M ≥ In this section we are primarily interested in the inefficiencies associated with stable networks.We focus on the total wasteful spending w ∗ = (cid:80) i w ∗ i . We analyze the effects of small changesin the parameters of the model on w ∗ and s ∗ while keeping the network structure fixed, and therole of the network structure in mediating the propagation of small shocks hitting a player inthe network. We focus on stable bipartite networks. Unless stated otherwise, in this section wemaintain Assumptions 1-3. We start by analyzing how changes in the likelihood of a draw, themarginal cost, and transfer size affect w ∗ . Not surprisingly, we find that when the effort becomesless expensive at the margin for all players, or when the transfer T increases in all contests, w ∗ increases. Interestingly, when the likelihood of a draw r increases, the total spending inthe equilibrium may both increase and decrease. The direction of the effect crucially dependson how asymmetric the stable network is, and on the value of r . The following propositionsummarizes these comparative static findings: Proposition 5
Consider stable graph K a,v : v < a then: Even solving for s which satisfies (U) with L i = ∅ requires solving a system of 6 nonlinear equations withsix unknowns and 3 additional parameters (sizes of partitions). For a fixed values of the parameters, this systemadmits up to 32 solutions, with only one of them being from R ≥ . It is interesting that our numerical explorationpoints to a conclusion that, in our benchmark case φ ( x ) = x and c ( x ) = x and r = 0, a stable tripartite networkdoes not exits. An example of complete tripartite network is presented in Figure 3a. . If the cost function for each player changes from c to ˜ c such that ˜ c (cid:48) ( x ) < c (cid:48) ( x ) for all x , w ∗ increases.2. If transfer size T changes from T = 1 to ˜ T > , w ∗ increases.3. w ∗ may both increase and decrease with r . In special case when φ ( x ) = λx, λ > , c ( x ) = α x α , α ≥ , and r → , w ∗ will increase in r when a > v . The non-monotonic effect of a change in r on w ∗ is a consequence of the non-monotonicityof the best reply function in r . When r and s ji are small enough, the best reply function of i ∈ A , increases with r , otherwise it deceases with r . Therefore, a priori it is not clear if anincrease in r will result in an increase or a decrease in the equilibrium spending per contest for i ∈ A . To illustrate this point, Figure 6 depicts the best response curves and the equilibriumpoint for a contest ij ∈ K a,v when r takes values 0 and 0 .
05. The left panel is the plot for K , . In this case the change r from 0 to 0 .
05 will lead to the new equilibrium (intersection ofdotted lines) in which both i ∈ A and j ∈ V spend less, and therefore the intensity of contest ij decreases. The situation is different on the right panel, where we consider the effect of thesame change but for K , . In this case, in the new equilibrium i invests more, and the intensityof each contest ij is larger when r = 0 .
05 than when r = 0. s ij s ji a =
4, v = s ij s ji a =
40, v = s ji ( s ij ,r = ) s ij ( s ji ,r = ) s ji ( s ij ,r = ) s ij ( s ji ,r = ) Figure 6: The equilibrium for r = 0 and r = 0 . i ∈ A and j ∈ V . Networks K , and K , are stable with s ∗ ji = s ∗ jk > s ∗ ij = s ∗ kj > k ∈ A for both r = 0 and r = 0 . r increases, the probability of losing for weak players (members of V ), cateris paribus,decreases. Since weak players already have a high marginal cost of spending at their currenttotal investment level, they will have an incentive to decrease their spending. On the otherhand, an increase in r will lead to a decrease in the probability of winning for stronger players(members of A ). When strong players’ total effort is not high, this will lead to an increase intheir per contest effort. An increase in the investment of strong players will further increasethe incentive of weak players to spend less. What will be the final effect on w ∗ depends on therelative magnitudes of the two effects discussed above. In Figure 7 we consider network K , in which an increase in r can lead to an increase in w ∗ .17 .02 0.04 0.06 0.08 0.10 r effort s ij * s ji * w * .Figure 7: Star network ( a = 200, v = 1): Graph depicts the equilibrium efforts of the centernode i and the periphery node j in a single contest, and w ∗ = s ∗ ij + s ∗ ji as functions of r . We scale w ∗ with the inverse of the number of links in K , for a more clear graphical representation. φ ( x ) = x and c ( x ) = x .The effects of changes in the likelihood of a draw on the equilibrium outcomes in contestgames have been already studied in (Nti, 1997) and (Acemoglu and Jensen, 2013). Both of thesepapers find that a decrease in the likelihood of a draw unambiguously leads to an increase in thetotal equilibrium effort. The reason why we find qualitatively different results is that we takeinto account asymmetries implied by the network structure. In (Nti, 1997) the author studiessymmetric n -lateral contests. In (Acemoglu and Jensen, 2013) the authors consider changes in r which are a positive shock to a player. When the network is asymmetric enough, a decrease in r is a negative shock for weak players, and positive shock for strong players. Hence, the resultsfrom (Acemoglu and Jensen, 2013) cannot be applied.In Proposition 5 we have considered changes that simultaneously affect all players in thenetwork. Now we discuss the effects of a change that affects only one player. We contemplatea scenario in which the cost function of player k for an exogenous reason changes to c k ( x ) =(1 + (cid:15) k ) c ( x ). We refer to this change as the cost shock hitting player k . In case of conflict, forinstance, the shock can be a third party intervention which makes it more costly for a party toacquire weapons. We are interested to see how s ∗ and w ∗ change in response to the shock, andhow this depends on the structure of the network. We focus on small shocks, (cid:15) k → φ ( x ) = λx for λ >
0, the totalequilibrium spending is implicitly defined with a system of equations (4), where d i denotes thedegree of node i (see Lemma 8 in Appendix A). w ∗ k = (cid:88) j ∈ N k c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ j ) + (1 + (cid:15) k ) c (cid:48) ( w ∗ k )) − d k r λ ,w ∗ i = (cid:88) j ∈ N i ,j (cid:54) = k c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ j ) + c (cid:48) ( w ∗ i )) + 2(1 + (cid:15) k ) c (cid:48) ( w ∗ k )( c (cid:48) ( w ∗ i ) + (1 + (cid:15) k ) c (cid:48) ( w ∗ k )) ik ∈ g − d i r λ , i (cid:54) = k. (4)System (4) provides the expression for the strength of player i as a function of the strengths ofher neighbors. Taking derivatives of (4) with respect to (cid:15) k and solving for ∂w ∗ i ∂(cid:15) k , i ∈ N we getthe following result: Proposition 6
Suppose φ ( x ) = λx , λ > , and suppose that player k experiences a cost shock in LFPS graph Other types of small shocks can be studied using the same approach. a,v .(i) If k ∈ A then ∂w ∗ k ∂(cid:15) k < , ∂w ∗ i ∂(cid:15) k < i ∈ A, i (cid:54) = k , and ∂w ∗ j ∂(cid:15) k > , j ∈ V . If k ∈ V then ∂w ∗ k ∂(cid:15) k < , ∂w ∗ j ∂(cid:15) k < j ∈ V, j (cid:54) = k , and ∂w ∗ i ∂(cid:15) k < , i ∈ A .(ii) ∂w ∗ ∂(cid:15) k < , k ∈ N. To understand (i) from Proposition 6, notice that, when k ∈ A , the direct effect of the shockhitting k will be that k will decrease her contest investment w ∗ k . Because members of V areweaker than k , their effort in contests with k will increase. At the same time, they will decreasetheir investment in contests with other players from A . When k ∈ V , the direct effect of theshock will again cause a decrease in w ∗ k . Since all opponents of k are stronger than k , they willalso decrease their investment in contests with k , but will increase their investment in contestswith other members of V . This will, in turn, lead to a decrease in the total equilibrium effort ofother members of V . This result is a consequence of the network structure of interactions, andthe property of the best reply function, which increases with the effort of a weaker opponentand decreases with the effort of a stronger opponent. Even though some players may spendmore in contests after the shock, w ∗ still decreases after the shock. In this section we discuss the relation between LFPS and other concepts of stability used inthe analysis of the formation of weighted networks. We point out some issues when theseequilibrium concepts are applied to the formation of contest networks, and argue that LFPSaddresses some of these issues. Two stability concepts employed in the literature on weightednetwork formation are: the Nash stability (Rogers, 2006, Bloch and Dutta, 2009, Baumann,2017), and the strong pairwise stability (Bloch and Dutta, 2009, Baumann, 2017). In thissection we maintain Assumptions 1-2, while Assumption 3 is not needed for the results.We first discuss Nash stable networks in our model (Definition 1). In case when, at zeroinvestment level, the marginal benefit of investing in a contest against player who does not defendherself is greater than the marginal cost, the complete network will be the only Nash stablenetwork structure. Otherwise, the empty network is the only Nash stable network structure.The following proposition holds:
Proposition 7
The Nash stable network is the empty network, when φ (cid:48) (0) r ≤ c (cid:48) (0) . Otherwise the unique Nashstable network g ( s ) is the complete network, with s ij = s ji > , ∀ i, j ∈ N . We note that the condition φ (cid:48) (0) r > c (cid:48) (0) will be satisfied in the special case when φ is theidentity mapping and c is a quadratic function defined with c ( x ) = αx , for any finite r > α > i and j would benefit from ending contest ij . However, the destruction of a link is never19 profitable unilateral deviation. This is a consequence of a coordination problem which oftenarises in non-cooperative models of network formation in which the link formation is a bilateraldecision (Bloch and Dutta, 2009). In our model, the link destruction is essentially a bilateraldecision, which creates similar coordination problem. To address this issue (Bloch and Dutta,2009, Definition 3) introduces the concept of strong pairwise stability, which considers bothunilateral and bilateral deviations. We show that a non-empty strongly pairwise stable contestnetwork does not exist. To see why, recall that the strong pairwise stability is a refinement ofthe Nash stability. According to Proposition 7 the unique non-empty Nash stable network is thecomplete network. In the complete network, each pair of players has an incentive to bilaterallydeviate by destroying the link between them, since they have the same strength. Therefore, thecomplete network is not immune to bilateral deviations. Proposition 8
The strong pairwise stable network is the empty network if φ (cid:48) (0) r ≤ c (cid:48) (0) . Otherwise, it does notexist. One would expect that, when initiating a contest, a player takes into account that therival will fight back. For instance, this is the case in litigation, lobbying, and conflict. Theseconsiderations about the response of a new opponent are absent when one contemplates the Nashequilibrium which, by definition, does not involve the anticipation of future play Therefore, inthe definition of LFPS networks, we allow that a player takes into account the expected effortof an opponent when forming new links. In particular, we assume that, when calculating theexpected payoff of starting contest ij with action s ij , player i assumes that j will fight backby choosing the best response s ji = BR ( s ij ), given j (cid:48) s current total spending w j . Thus, i is limited farsighted, since she does not take into account further adjustments in investmentsthat will take place in the network once ij is formed. Since calculating all the adjustments inequilibrium strategies when forming a link is equivalent to solving a highly nonlinear system ofequations, which is even numerically a very difficult problem, we believe that this is a reasonableassumption. Experimental results suggest that in network formation games players are limitedfarsighted (Kirchsteiger et al., 2016), even in models that are much simpler than the modelconsidered in this paper. Furthermore, experimental evidence indicates that the difficulty informing correct beliefs about the opponent’s best response may be one of the main reasonsbehind the fact that in experiments subjects rarely play Nash strategies in Tullock contestgames (Masiliunas et al., 2014).While the analysis of farsighted stable networks is outside of the scope of this paper and,as argued in the previous paragraph, assuming full farsightedness may be too strong of anassumption about the players’ behavior, in Appendix B we define farsighted stable networks byadapting the notion of farsighted stability (Jackson, 2008, Herings et al., 2009, Vannetelboschand Mauleon, 2015) to our model of contest network formation. The anticipation of the newrival’s action after creating a link is of course present when players are fully farsighted. As Nash equilibrium of course implies logical ”anticipation” that the opponents are rational according to commonknowledge rationality but no anticipation about future actions of opponents. To the best of our knowledge, no concept of farsighted stability has been applied to the formation of weightednetworks so far.
20 consequence, starting a contest will not always be a profitable deviation as was the casewith myopic players. Therefore, we expect that farsighted stable networks look differently thanNash stable networks or strong pairwise stable networks. We demonstrate this by means of anexample in Appendix B, which shows that the set of farsightedly stable networks and the setof Nash stable networks are different and non-nested. The relation between farsightedly stablecontest networks and LFPS networks is not clear, and while it may be an interesting issue tostudy, the complexity of the model may prove to be too big of a hurdle to overcome.
To the best of our knowledge this is the first model of weighted network formation in whichthe interaction between neighbors is an antagonistic one. Moreover, in the model, actions ofneighbors are neither strategic substitutes nor strategic complements. This type of strategicinteraction has not been considered in the literature on weighted network formation so far.In the paper, we describe stable networks using different notions of stability. We also deriveseveral comparative statics results illustrating the fact that taking into account the structureof the contest network may lead to very different results compared to cases when the networkstructure is ignored. We believe that the qualitative insights of the model are applicable to manysituations, including competitions between divisions in companies, lobbying, and allocation ofproperty rights.There are several promising directions for further research. First, our model considersonly enmity links. It would be interesting to extend the model by allowing the formation ofweighted friendship links that imply positive spillovers (i.e. reduction of cost of fighting), andsee if this leads to different stable network configurations. Introducing heterogeneity is a stepwhich is necessary to make the model’s predictions empirically testable. Heterogeneity in theeffectiveness of the contest technology (function φ ), cost of fighting, and transfers can be directlyincluded in the model. Furthermore, one could consider a position in the network as a sourceof heterogeneity. For instance, we can imagine that the amount of resources each enemy of acountry expects to extract decreases with the number of opponents of that country. Finally, wefocus on bilateral contests. It would be interesting to study contest network formation allowingalso for multilateral contests. A starting point for this may be the model presented in this paperand (Matros and Rietzke, 2018). 21 eferences Acemoglu, D. and Jensen, M. K. (2013). Aggregate comparative statics.
Games and EconomicBehavior , 81:27–49.Amegashie, J. A. (2006). A contest success function with a tractable noise parameter.
PublicChoice , 126(1-2):135–144.Bala, V. and Goyal, S. (2000). A noncooperative model of network formation.
Econometrica ,68(5):1181–1229.Baumann, L. (2017). A model of weighted network formation.Blavatskyy, P. R. (2010). Contest success function with the possibility of a draw: axiomatization.
Journal of Mathematical Economics , 46(2):267–276.Bloch, F. and Dutta, B. (2009). Communication networks with endogenous link strength.
Gamesand Economic Behavior , 66(1):39–56.Bourl`es, R., Bramoull´e, Y., and Perez-Richet, E. (2017). Altruism in networks.
Econometrica ,85(2):675–689.Corch´on, L. C. and Serena, M. (2018). Contest theory. In
Handbook of Game Theory andIndustrial Organization, Volume II: Applications , page 125.De la Fuente, A. (2000).
Mathematical methods and models for economists . Cambridge Univer-sity Press.Dero¨ıan, F. (2009). Endogenous link strength in directed communication networks.
Mathemat-ical Social Sciences , 57(1):110–116.Dev, P. (2018). Group identity in a network formation game with cost sharing.
Journal ofPublic Economic Theory , 20(3):390–415.Dixit, A. (1987). Strategic behavior in contests.
The American Economic Review , pages 891–898.Dziubi´nski, M., Goyal, S., and Minarsch, D. E. (2016). Dynamic conflict on a network. In
Proceedings of the 2016 ACM Conference on Economics and Computation , pages 655–656.ACM.Estrada, E. and Knight, P. A. (2015).
A first course in network theory . Oxford University Press,USA.Franke, J. and Ozturk, T. (2015). Conflict networks.
Journal of Public Economics , 126:104 –113.Galeotti, A. and Goyal, S. (2010). The law of the few.
The American Economic Review , pages1468–1492. 22oodman, J. C. (1980). Note on existence and uniqueness of equilibrium points for concaven-person games.
Econometrica , 48(1).Goyal, S., Moraga-Gonz´alez, J. L., and Konovalov, A. (2008). Hybrid r&d.
Journal of theEuropean Economic Association , 6(6):1309–1338.Goyal, S. and Vega-Redondo, F. (2007). Structural holes in social networks.
Journal of EconomicTheory , 137(1):460–492.Goyal, S., Vigier, A., and Dziubinski, M. (2016). Conflict and networks. In
The Oxford Handbookof the Economics of Networks .Grandjean, G., Mauleon, A., and Vannetelbosch, V. (2011). Connections among farsightedagents.
Journal of Public Economic Theory , 13(6):935–955.Grandjean, G., Tellone, D., and Vergote, W. (2017). Endogenous network formation in a tullockcontest.
Mathematical Social Sciences , 85:1–10.Herings, P. J.-J., Mauleon, A., and Vannetelbosch, V. (2009). Farsightedly stable networks.
Games and Economic Behavior , 67(2):526–541.Herings, P. J.-J., Mauleon, A., and Vannetelbosch, V. (2019). Stability of networks underhorizon-k farsightedness.
Economic Theory , 68(1):177–201.Hiller, T. (2016). Friends and enemies: a model of signed network formation.
TheoreticalEconomics .Hillman, A. L. and Riley, J. G. (1989). Politically contestable rents and transfers*.
Economics& Politics , 1(1):17–39.Inderst, R., M¨uller, H. M., and W¨arneryd, K. (2007). Distributional conflict in organizations.
European Economic Review , 51(2):385–402.Jackson, M. O. (2008).
Social and economic networks . Princeton university press Princeton.Jackson, M. O. and Nei, S. (2015). Networks of military alliances, wars, and international trade.
Proceedings of the National Academy of Sciences , 112(50):15277–15284.Jackson, M. O. and Wolinsky, A. (1996). A strategic model of social and economic networks.
Journal of economic theory , 71(1):44–74.Kinateder, M. and Merlino, L. P. (2017). Public goods in endogenous networks.
AmericanEconomic Journal: Microeconomics , 9(3):187–212.Kirchsteiger, G., Mantovani, M., Mauleon, A., and Vannetelbosch, V. (2016). Limited farsight-edness in network formation.
Journal of Economic Behavior & Organization , 128:97–120.K¨onig, M. D., Rohner, D., Thoenig, M., and Zilibotti, F. (2017). Networks in conflict: Theoryand evidence from the great war of africa.
Econometrica , 85(4):1093–1132.23onrad, K. A. and Kovenock, D. (2009). Multi-battle contests.
Games and Economic Behavior ,66(1):256–274.Krueger, A. O. (1974). The political economy of the rent-seeking society.
The Americaneconomic review , 64(3):291–303.Kvasov, D. (2007). Contests with limited resources.
Journal of Economic Theory , 136(1):738–748.Loury, G. C. (1979). Market structure and innovation.
The quarterly journal of economics ,pages 395–410.MacKenzie, I. A. and Ohndorf, M. (2013). Restricted coasean bargaining.
Journal of PublicEconomics , 97:296–307.Masiliunas, A., Mengel, F., and Reiss, J. P. (2014). Behavioral variation in tullock contests.Technical report, Working Paper Series in Economics, Karlsruher Institut f¨ur Technologie(KIT).Masson, V., Choi, S., Moore, A., and Oak, M. (2018). A model of informal favor exchange onnetworks.
Journal of Public Economic Theory , 20(5):639–656.Matros, A. and Rietzke, D. (2018). Contests on networks.Mauleon, A., Song, H., and Vannetelbosch, V. (2010). Networks of free trade agreements amongheterogeneous countries.
Journal of Public Economic Theory , 12(3):471–500.Mauleon, A. and Vannetelbosch, V. (2016). Network formation games.Nti, K. O. (1997). Comparative statics of contests and rent-seeking games.
InternationalEconomic Review , pages 43–59.Pfeffer, J. and Moore, W. L. (1980). Power in university budgeting: A replication and extension.
Administrative Science Quarterly , pages 637–653.Rogers, B. (2006). A strategic theory of network status.
Preprint, under revision .Rosen, J. B. (1965). Existence and uniqueness of equilibrium points for concave n-person games.
Econometrica: Journal of the Econometric Society , pages 520–534.Sytch, M. and Tatarynowicz, A. (2014). Friends and foes: The dynamics of dual social struc-tures.
Academy of Management Journal , 57(2):585–613.Szymanski, S. (2003). The economic design of sporting contests.
Journal of economic literature ,41(4):1137–1187.Tullock, G. (1967). The welfare costs of tariffs, monopolies, and theft.
Economic Inquiry ,5(3):224–232.Tullock, G. (1980).
Efficient Rent-Seeking, in J.M. Buchanan, R.D. Tollison and G. Tullock(eds.) Towards a Theory of a Rent-Seeking Society . Texas A & M Univ Pr.24annetelbosch, V. and Mauleon, A. (2015). Network formation games. In
The Oxford Handbookof the Economics of Networks .W¨arneryd, K. (1998). Distributional conflict and jurisdictional organization.
Journal of PublicEconomics , 69(3):435–450.Xu, J., Zenou, Y., and Zhou, J. (2019). Networks in conflict: A variational inequality approach.
Available at SSRN 3364087 .Zhang, J., Xue, L., and Zu, L. (2013). Farsighted free trade networks.
International Journal ofGame Theory , 42(2):375–398. 25 ppendix A: Proofs
Contest Game on a Given Network
To understand the proofs in Appendix A, it is useful to revisit the case when the set contests is exoge-neously given and fixed. This is the case studied in Franke and Ozturk (2015). So, let the set of possiblecontest in the society be defined with graph ¯ g . The contest game on ¯ g is defined by: C (¯ g ) = { N, { S i (¯ g ) } ni =1 , { π i } ni =1 } . (5)In (5), N is the set of players, payoff functions π i are defined in (2), and the strategy space of player i is given by: S i (¯ g ) ≡ { s i ∈ R n − ≥ : s ij = 0 whenever ij / ∈ ¯ g } . The following proposition, which is a version of the existence and uniqueness result for the contestgame on a given network (Franke and Ozturk, 2015, Proposition 1 and Lemma 1), holds as well whenthe payoff function are given with (2).
Proposition 9
There exists a unique pure strategy Nash equilibrium of game C (¯ g ) , ¯ s . The equilibrium ¯ s is interior( ¯ s ij > ∀ ij ∈ ¯ g ) if φ (cid:48) (0) = ∞ . When φ ( x ) = x and c ( x ) = x the equlibrium will be interior for r smallenough. Proof of Proposition Existence and Uniqueness.
It is enough to follow the same steps as in the proof of (Franke andOzturk, 2015, Proposition 1 and Lemma 1) when the payoff function are given with (2). The main partof the proof is showing that game C (¯ g ) is a concave game, as defined in Rosen (1965), and then directlyapplying Rosen’s result. Interiority.
Assume that ij ∈ ¯ g , in the Nash equilibrium, ¯ s , of C (¯ g ), and that ¯ s ij = 0 ∨ ¯ s ji = 0.We show that when this logical disjunction is true, there is a profitable deviation for either player i orplayer j . Hence, ¯ s ij = 0 ∨ ¯ s ji = 0 cannot be a part of the Nash equilibrium of game C (¯ g ) when ij ∈ ¯ g .We consider the case when φ (cid:48) (0) = ∞ , and the case φ ( x ) = x, c ( x ) = x separately. Case 1: φ (cid:48) (0) = ∞ .Suppose, without loss of generality, that ¯ s ij = 0. There is a profitable deviation in which i invests (cid:15) > j . The marginal cost of this deviation c (cid:48) ( ¯ w i + (cid:15) ). The marginal benefit of thedeviation is r +2 φ (¯ s ji )( φ (0)+ φ (¯ s ji )+ r ) φ (cid:48) (0) ( which becomes r ( r + φ ( (cid:15) )) φ (cid:48) ( (cid:15) ) in case when also ¯ s ji = 0). It is clearthat the marginal benefit at (cid:15) = 0 is infinite, while the marginal cost remains bounded. Case 2: φ ( x ) = x and c ( x ) = x .(i) Suppose first that ¯ s ij = ¯ s ji = 0. We show that this cannot happen for any finite r > w i = 0 consider a deviation in which player i invests (cid:15) > ij . The cost of thisdeviation is (cid:15) . The benefit is (cid:15)(cid:15) + r . It is easy to see that the benefit is larger than the cost,for (cid:15) small enough, since (cid:15)(cid:15) + r − (cid:15) = (cid:15) (cid:18) − r(cid:15) − (cid:15) r + (cid:15) (cid:19) . (b) If ¯ w i > ik such that ¯ s ik >
0. Consider a deviation in which player i reallocates (cid:15) > ik to ij (keeping ¯ w i fixed). The marginal benefit of this deviation or player i , calculated at (cid:15) = 0 is: ∂∂(cid:15) (cid:15)(cid:15) + r (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = 1 r . The marginal cost of the deviation is: ∂∂(cid:15) (¯ s ik − (cid:15) ) − ¯ s ki ¯ s ik − (cid:15) + ¯ s ki + r (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = − r + 2¯ s ki ( r + ¯ s ik + ¯ s ki ) . It is easy to check that the marginal benefit outweights the marginal cost. Indeed1 r − r + 2¯ s ki ( r + ¯ s ik + ¯ s ki ) = 2 r ¯ s ik + (¯ s ik + ¯ s ki ) r ( r + ¯ s ik + ¯ s ki ) > . (ii) We now show that when ij ∈ ¯ g it cannot be that ¯ s ij = 0 and ¯ s ji > r becomes infinitesimal.Suppose otherwise, so suppose that this is the case for some two players i and j .(a) If ¯ w i = 0 then a profitable deviation for player i is to exert (cid:15) > ij . The marginalcost of this deviation, 2 (cid:15) , approaches to 0 when (cid:15) →
0. The marginal benefit of the proposeddeviation, r +2¯ s ji r +¯ s ji + (cid:15) , is positive and bounded away from 0. Hence for (cid:15) small enough, theproposed deviation is profitable.(b) Finally, we consider the case when ¯ w i >
0. First we show that when r → s ji → i of investing in contest ij calculatedat 0 becomes unbounded when r approaches 0.Since, by assumption, ¯ s ji is greater than zero, it must satisfy the first order optimality(sufficient and necessary) conditions. Thus, the following holds: r (¯ s ji + r ) = 2 (cid:88) (cid:96) ¯ s j(cid:96) ⇒ r = (cid:88) (cid:96) (cid:54) = i ¯ s j(cid:96) + 2¯ s ji ( r + ¯ s ji ) . From the last equation above, it is clear that when r → s ji →
0. In this case ( r → i of investing in contest against player j calculated at 0 ( equal to r +2¯ s ji ( r +¯ s ji ) ) becomes unbounded. Indeed, it can be verified that:lim r → r + 2¯ s ji ( r )( r + ¯ s ji ( r )) = + ∞ , (6)since lim r → ¯ s ji ( r ) = 0. The marginal cost of this deviation is obviously bounded from above.Therefore, it cannot be that ¯ s ij = 0 and ¯ s ji > r is small enough. Proofs of Claims from Section 3
Proof of Proposition 1.
Uniqueness.
Since both g ( s ∗ ) and g ( s (cid:48) ) are stable, condition (U) from Definition 2 must hold. Inparticular, it must hold for any player i and L i = ∅ . Then, Proposition 9 implies that, if g ( s ∗ ) and g ( s (cid:48) )are two stable networks with the same network structure, ¯ g , then s (cid:48) = s ∗ = ¯ s . Interiority.
Follows directly from the proof of the interiority part of Proposition 9. he following proposition is an extension of Proposition (Franke and Ozturk, 2015, Proposition 2)and provides a foundation for definition of strength (Definition 5). Proposition 10
Suppose that conditions for the interiority from Proposition 1 are satisfied, and let g ( s ) satisfy condition(U) for L i = ∅ . Then w i ≥ w j ⇒ s ij ≤ s ji , with equality when w i = w j .Proof of Proposition 10. The following first-order conditions for contest ij ∈ g ( s ) must hold: (cid:18) ( r + 2 φ ( s ji )) φ (cid:48) ( s ij )( r + φ ( s ij ) + φ ( s ji )) − c (cid:48) ( w i ) = 0 (cid:19) ∧ (cid:18) ( r + 2 φ ( s ij )) φ (cid:48) ( s ji )( r + φ ( s ij ) + φ ( s ji )) − c (cid:48) ( w j ) = 0 (cid:19) . (7)From (7) we get: ( r + 2 φ ( s ji )) φ (cid:48) ( s ij )( r + 2 φ ( s ij )) φ (cid:48) ( s ji ) = c (cid:48) ( w i ) c (cid:48) ( w j ) . Since φ (cid:48) ( x ) > φ (cid:48)(cid:48) ( x ) ≤ c (cid:48)(cid:48) ( x ) > w i ≥ w j ⇒ c (cid:48) ( w i ) c (cid:48) ( w j ) ≥ ⇒ ( r + 2 φ ( s ji )) φ (cid:48) ( s ij )( r + 2 φ ( s ij )) φ (cid:48) ( s ji ) ≥ ⇒ s ji ≥ s ij , (8)where the last implication in (8) follows from the facts that φ is an increasing function and φ (cid:48) is adecreasing function. The equality holds when w i = w j . Proof of Proposition 2 . To prove the claim, we compare the solutions of the FOC system associatedto links ab and ac . To do this, it is helpful to first consider the following parameterized system ofequations on R ≥ with unknowns x and y , and positive parameters β and β :( r + 2 φ ( y )) φ (cid:48) ( x )( r + φ ( x ) + φ ( y )) − c (cid:48) ( β ) = 0 , ( r + 2 φ ( x )) φ (cid:48) ( y )( r + φ ( x ) + φ ( y )) − c (cid:48) ( β ) = 0 . (9)It is easy to verify that (9) satisfies the conditions of the implicit function theorem. Note that when β = w ∗ a and β = w ∗ b , then x = s ∗ ab and y = s ∗ ba is the unique solution of system (9). Taking thederivative of x and y defined by (9) with respect to β we get: ∂x∂β = c (cid:48)(cid:48) ( β ) ( r + 2 φ ( x )) ( r + φ ( x ) + φ ( y )) (cid:2) φ (cid:48)(cid:48) ( y )( r + φ ( x ) + φ ( y )) − φ (cid:48) ( y ) (cid:3) Den ,∂y∂β = 2 c (cid:48)(cid:48) ( β )( φ ( x ) − φ ( y )) φ (cid:48) ( x ) φ (cid:48) ( y )( r + φ ( x ) + φ ( y )) Den , where
Den =2 φ (cid:48) ( x ) (cid:0) φ (cid:48) ( y ) ( r + φ ( x ) + φ ( y )) − ( r + 2 φ ( x ))( r + 2 φ ( y )) φ (cid:48)(cid:48) ( y ) (cid:1) + ( r + 2 φ ( x ))( r + 2 φ ( y )) φ (cid:48)(cid:48) ( x ) (cid:0) φ (cid:48)(cid:48) ( y )( r + φ ( x ) + φ ( y )) − φ (cid:48) ( y ) (cid:1) . For positive x and y , expression Den will be positive, given the properties of functions φ and c statedin Assumptions 1-2. Furthermore, the numerator of ∂x∂β is always negative, while the numerator of ∂y∂β is negative when φ ( x ) < φ ( y ) (and therefore when x < y ), and otherwise positive. Therefore, for the nique solution ( x, y ) of system 9 the following holds comparative statics result holds: ∂x∂β < ,∂y∂β ≤ x ≤ y,∂y∂β > x > y. (10)We now prove that s ∗ ab > s ∗ ac . The other inequalities stated in the claim of the Proposition areproven analogously. Consider (7) associated to ab and (7) associated to ac , which must hold in aninterior equilibrium s ∗ .( r + 2 φ ( s ∗ ba )) φ (cid:48) ( s ∗ ab )( r + φ ( s ∗ ab ) + φ ( s ∗ ba )) − c (cid:48) ( w ∗ a ) = 0 , ( r + 2 φ ( s ∗ ab )) φ (cid:48) ( s ∗ ba )( r + φ ( s ∗ ab ) + φ ( s ∗ ba )) − c (cid:48) ( w ∗ b ) = 0 . (7 ab)( r + 2 φ ( s ∗ ca )) φ (cid:48) ( s ∗ ac )( r + φ ( s ∗ ac ) + φ ( s ∗ ca )) − c (cid:48) ( w ∗ a ) = 0 , ( r + 2 φ ( s ∗ ac )) φ (cid:48) ( s ∗ ca )( r + φ ( s ∗ ac ) + φ ( s ∗ ca )) − c (cid:48) ( w ∗ c ) = 0 . (7 ac)We can think of (7 ab) as a system of equations (9) with unknowns s ∗ ab , s ∗ ba , where w ∗ a and w ∗ b areplaying a role of β and β , and analogously for (7 ac). By assumption w ∗ a < w ∗ b and w ∗ a < w ∗ c . Then,Proposition 10 implies that s ∗ ab > s ∗ ba , and s ∗ ac > s ∗ ca respectively. Taking this into account and comparingsystems (7 ab) and (7 ac), the second inequality in (10) implies that s ∗ ab > s ∗ ac .To prove that π ∗ ac > π ∗ ab we show that the equilibrium expected revenue of player i in contest ij increases with total spending of her opponent j . To do this, we first express φ ( s ∗ ab ) from system (7) forcontest ij , and get: φ ( s ∗ ij ) = 2 (cid:2) φ (cid:48) ( s ∗ ij ) (cid:3) c (cid:48) ( w ∗ i ) φ (cid:48) ( s ∗ ji ) (cid:0) φ (cid:48) ( s ∗ ij ) c (cid:48) ( w ∗ j ) + φ (cid:48) ( s ∗ ji ) c (cid:48) ( w ∗ i ) (cid:1) − r . (11)Plugging in (11) we get that the expected revenue in equilibrium from contest ij for player i becomes(after some algebra): φ ( s ∗ ij ) − φ ( s ∗ ji ) φ ( s ∗ ij ) + φ ( s ∗ ji ) + r = 1 − φ (cid:48) ( s ∗ ji ) c (cid:48) ( ¯ w i ) φ (cid:48) ( s ∗ ij ) c (cid:48) ( w ∗ j ) + φ (cid:48) ( s ∗ ji ) c (cid:48) ( w ∗ i ) . (12)Since c is convex, it is straightforward to check now that φ ( s ∗ ij ) − φ ( s ∗ ji ) φ ( s ∗ ij )+ φ ( s ∗ ji )+ r increases with w ∗ j .We now state and prove an important corollary of Proposition 2 which states that the total equilib-rium investment w ∗ is increasing with the neighborhood of a player, with respect to the relation of setinclusion. Corollary 1 (of Proposition 2)
Let N i (cid:40) N j in stable network g , then w ∗ i < w ∗ j .Proof of Corollary 1. Suppose the claim does not hold. So, suppose that N i (cid:40) N j and w ∗ i ≥ w ∗ j . Then,from Proposition 2 it follows that for every k ∈ N i ∩ N j s ∗ ik ≤ s ∗ jk . But then w ∗ i = (cid:80) k ∈ N i s ∗ ik ≤ (cid:80) k ∈ N i s ∗ jk < (cid:80) k ∈ N j s ∗ jk = w ∗ j , which is in contradiction with w ∗ i ≥ w ∗ j .We now state and prove Lemmas 1 to 4 which are concerned with attackers in a stable network. Ourmain goal is to show that there can be only one class of attackers in LFPS network. For clarity, we do thisin several steps, each step being a separate lemma. We first show that attackers always have links with eakest players in the network (Lemma 1). We use Lemma 1 extensively in proofs of subsequent claimsin the paper. An useful corollary of this lemma is that a stable network must be connected. We continueby showing that members of the same class of attackers must have the same neighborhood (Lemma 2),and that two different class of attackers cannot have nested neighborhoods (Lemma 3). Finally, usingLemmas 1 - 3 we show that there can be only one class of attackers (Lemma 4). Proof of Lemma 1 . Assume that g ( s ∗ ) is stable, and such that for some player i and two other players j, k with w ∗ j < w ∗ k we have ij ∈ g ( s ∗ ) and ik / ∈ g ( s ∗ ). We show that in this case there exists a profitabledeviation for players i and j , hence g ( s ∗ ) cannot be stable.First note that if contest ij is not profitable for i , then it cannot be part of the stable network ((B)does not hold).When ij is profitable for i , it must be w ∗ i < w ∗ j . We show that there is a profitable bilateral deviationfor i and j . Consider a deviation in which j deviates from s ∗ j to s (cid:48) j such that s (cid:48) ji = 0 and s (cid:48) j(cid:96) = s ∗ j(cid:96) forall (cid:96) (cid:54) = i . At the same time, i deviates to s (cid:48) i such that s (cid:48) ik = s ∗ ij , s (cid:48) ij = 0 and s (cid:48) i(cid:96) = s ∗ i(cid:96) for all (cid:96) / ∈ { j, k } .It is clear that this deviation is profitable for j . We prove that it is also profitable for i . It is enough toprove that the expected reaction of k to the proposed deviation, denoted by ˆ s ki , is such that ˆ s ki < s ∗ ji .To do this, we note that s ∗ ji must satisfy the following optimality condition:( r + 2 φ ( s ∗ ij )) φ (cid:48) ( s ∗ ji )( r + φ ( s ∗ ij ) + φ ( s ∗ ji )) = c (cid:48) ( w ∗ j ) . (13)The expected reaction of player k to the proposed deviation is determined with the following condi-tion: ( r + 2 φ ( s ∗ ij )) φ (cid:48) (ˆ s ki )( r + φ ( s ∗ ij ) + φ (ˆ s ki )) = c (cid:48) ( w ∗ k + ˆ s ki ) . (14)Since w ∗ k + ˆ s ki > w ∗ k ≥ w ∗ j it must be that c (cid:48) ( w ∗ k + ˆ s ki ) > c (cid:48) ( w ∗ j ). This is due to strict convexity of c . Thus the right hand side of (14) is strictly larger than the right hand side of (13). The same relationmust hold for the left hand sides of (13) and (14). Since φ is an increasing function and φ (cid:48) is a decreasingfunction, this holds only when ˆ s ki < s ∗ ji . Corollary 2 (of Lemma 1)
A non-empty stable network g ( s ∗ ) is connected. Proof of Corollary 2 : We use a proof by contradiction. Assume that the claim does not hold, so thereare at least two components in stable network g . Choose two components ( C and C ) from g such thatthe weakest player in the network ( v ) belongs to C . All opponents of v must find the contest with v profitable, otherwise the network would not be stable ((B) would not hold). Then, the strongest playerin C (denote her with a ) by Lemma 1 has an incentive to form a link with v instead of a link withone of her current opponents, who by definition is not weaker than v . If | C | = 1, a does not haveany opponents. Then, she has an incentive to form a link with v with action s ∗ a ,v , since a v ∈ g is aprofitable contest for a . Lemma 2
Two players that belong to the same class of attackers W a have the same neighborhood in stable network g . Proof of Lemma 2 : Let g be a stable network. Consider any two attackers i, j ∈ W a , and suppose,contrary to what is asserted, that N i (cid:54) = N j . It cannot be that N i ⊂ N j because then the total spending of i and j would not be equal (by Corollary 1). Since N i (cid:54) = N j , there exist nodes h ∈ N i \ N j and k ∈ N j \ N i . uppose that, without loss of generality, w ∗ k ≥ w ∗ h . Then it is profitable for player i to replace ih withlink ik according to Lemma 1. This is in contradiction with the assumption that g is stable. Lemma 3
Let i and j be two attackers in stable network g ( s ∗ ) . It cannot be that N i ⊂ N j . Proof of Lemma 3 . If i and j belong to the same class, then Lemma 2 implies N i = N j . Considernow the case when i and j belong to different classes of attackers. We assume that N i ⊂ N j and showthat there will always exist a profitable deviation. We will use N i to denote the neighborhood of i innetwork g ( s ∗ ).Since N i ⊂ N j , by Corollary 1 it must be w ∗ i < w ∗ j .Suppose first that π j ( g ( s ∗ )) ≥ π i ( g ( s ∗ )). We show that in this case i can form links to all playersin L i = N j \ N i , and obtain a payoff greater than π j ( g ( s ∗ )). To show this, consider the deviationin which player i deviates to ˜ s i = s ∗ j . Let us denote the payoff of player i after this deviation with π i ( g (˜ s i , ˆ s L i , s ∗− i − L i )) where ˆ s L i is defined in (3). We proceed by showing that π i ( g (˜ s i , ˆ s L i , s ∗− i − L i )) >π j ( g ( s ∗ )).Because w ∗ i < w ∗ j , Proposition 2 implies that s ∗ ki < s ∗ kj k ∈ N i ∩ N j . The convexity of the costfunction implies that ˆ s ki < s ∗ kj for all k ∈ L i under the contemplated deviation. This means that after thedeviation the expected cost of i will be equal to the cost of j , i and j will have the same set of opponents,and φ (˜ s ik ) − φ (ˆ s ki ) φ (˜ s ik )+ φ (ˆ s ki )+ r > φ ( s ∗ jk ) − φ ( s ∗ kj ) φ ( s ∗ jk )+ φ ( s ∗ kj )+ r ∀ k ∈ N j . Therefore π i ( g (˜ s i , ˆ s L i , s ∗− i − L i )) > π j ( g ( s ∗ )) ≥ π j ( g ( s ∗ )).Suppose now that π i ( g ( s ∗ )) > π j (( s ∗ )), and suppose that j does not have an incentive to update herstrategy (otherwise the network would not be stable). From π i ( g ( s ∗ )) > π j (( s ∗ )) it follows that: (cid:88) k ∈ N i π ik ( s ∗ ik , s ∗ ki ; r ) > − c ( w ∗ j ) + c ( w ∗ i ) + (cid:88) k ∈ N j π jk ( s ∗ jk , s ∗ kj ; r ) . (15)Consider now the same deviation of player i , as contemplated in the first part of the proof. We get (using N i to denote neighborhood of i in network g ( s ∗ )): π i ( g (˜ s i , ˆ s L i , s ∗− i − L i )) − π i ( g ( s ∗ )) = (cid:88) k ∈ N i π ik ( s ∗ jk , s ∗ ki ; r ) + (cid:88) k ∈ L i π ik ( s ∗ jk , ˆ s ki ; r ) − (cid:88) k ∈ N i π ik ( s ∗ ik , s ∗ ki ; r ) − c ( w ∗ j ) + c ( w ∗ i ) > (cid:88) k ∈ N i π ik ( s ∗ jk , s ∗ ki ; r ) + (cid:88) k ∈ L i π ik ( s ∗ jk , ˆ s ki ; r ) − − c ( w ∗ j ) + c ( w ∗ i ) + (cid:88) k ∈ N j π jk ( s ∗ jk , s ∗ kj ; r ) − c ( w ∗ j ) + c ( w ∗ i ) = (cid:88) k ∈ N i π ik ( s ∗ jk , s ∗ ki ; r ) + (cid:88) k ∈ L i π ik ( s ∗ jk , ˆ s ki ; r ) − (cid:88) k ∈ N j π jk ( s ∗ jk , s ∗ kj ; r ) > , where the first inequality comes directly from (15) and the last inequality comes from the fact thatˆ s ki < s ∗ kj for all k ∈ L i . This completes the proof. Lemma 4
There is only one class of attackers ( W ) in stable network g ( s ∗ ) . Members of W are connected to allplayers outside W . Proof of Lemma 4 : Suppose, contrary to what is asserted, that there are two different classes ofattackers W and W in LFPS network g ( s ∗ ). Since Lemma 2 implies that all members of the same classof attackers have the same neighborhood, we restrict our attention to representative nodes i ∈ W and j ∈ W .Since w ∗ j > w ∗ i there are 2 possible situations that we need to consider: Recall that since j is an attacker, any of her opponents would be better off by destroying a link with j . i) N i ⊂ N j is ruled out by Lemma 3.(ii) N i (cid:54)⊂ N j = ⇒ ( ∃ k ∈ N i \ N j ∧ ∃ h ∈ N j \ N i ) . If w ∗ k ≥ w ∗ h Lemma 1 implies that j has a profitabledeviation. If w ∗ k < w ∗ h the same lemma implies that i has a profitable deviation.We now prove a lemma which is concerned with hybrids. In the proof we rely on arguments whichare analogous to those used in the proof of Lemma 4. Lemma 5
In a stable network g ( s ∗ ) all members of a hybrid class are connected to all other nodes in the networkthat do not belong to their class. Proof of Lemma 5 : If there are only two classes of nodes in a stable network ( W and W ) then thereare no hybrids. Suppose there are more than two classes of nodes in a stable network. First, let usconsider the strongest hybrid class ( W ). A node h ∈ W must be connected to all nodes from W . Thisis because hybrid h must be connected to at least one player that is stronger than her, who must be anattacker since h ∈ W . Then, Lemma 4 implies that h must be connected to all players from W , sinceall nodes in W have the same neighborhood. This holds for any h ∈ W .Let ¯ N i = { j ∈ N i : w ∗ j < w ∗ i } and ¯ N i = { j ∈ N i : w ∗ j ≥ w ∗ i } denote the strong and the weakneighborhood of player i respectively.We now prove that all members of the class W have the same neighborhood. Suppose this is nottrue. Let h and h be two players from W such that N h (cid:54) = N h . The following implication holds:( W ⊂ N h ∧ W ⊂ N h ) ⇒ (( N h /N h ) ∪ ( N h /N h )) ∩ W = ∅ . Thus, ¯ N h = ¯ N h and ¯ N h (cid:54) = ¯ N h . Itcannot be ¯ N h ⊂ ¯ N h ∨ ¯ N h ⊂ ¯ N h because then it would be w ∗ h (cid:54) = w ∗ h by Corollary 1. Consider twonodes, k ∈ ¯ N h \ ¯ N h and (cid:96) ∈ ¯ N h \ ¯ N h . If w ∗ k ≥ w ∗ (cid:96) then h and (cid:96) have a profitable deviation (link h (cid:96) is destroyed, link h k is formed). If w ∗ k < w ∗ (cid:96) , then h and k have an analogous profitable deviation.Let W be the third strongest class in the network. If M = 3 then, by definition, all players in W must be connected to some players from W , because otherwise they would not be hybrid types. Notethat if player i ∈ W is connected to some player from class W then she is connected to all players fromclass W - because we have shown that all members of class W have the same neighborhood. If thereexists player j ∈ W who is not connected to all players from W , then j is only connected to all playersfrom W . But then i and j cannot belong to the same class. So, for K = 3 the claim of the lemma holds.Suppose M >
3. Lemma 1 implies that all members of W must be connected to all members of W since they are connected to all members of W . We now show that all players from W are connectedto all players from W . Again we proceed by using a proof by contradiction. Suppose that there existplayers i ∈ W and j ∈ W such that ij / ∈ g ( s ∗ ). We show that in this case there is a profitable deviation.Player i loses only in contests with players from W . Hence, i has control over all of her links exceptlinks with players from W . Furthermore, w ∗ i < w ∗ j ⇒ N i (cid:54) = N j . There are two possibilities for relationbetween N i and N j that we need to consider:(i) N i ⊂ N j case can be ruled out by applying the same argument as in Lemma 3 to ¯ N i and ¯ N j .(ii) N i (cid:54)⊂ N j ⇒ ( ∃ k ∈ N i \ N j ∧ ∃ h ∈ N j \ N i ) . But then, if w ∗ k ≥ w ∗ h Lemma 1 implies that j has aprofitable deviation, and if w ∗ k < w ∗ h , the same Lemma implies that i has a profitable deviation.We have shown that in a stable network it cannot happen that there are no links between membersof W and W . If two players from W and W are connected, than all players from W and W areconnected, because all players from W have the same neighborhood, and because of Lemma 1.Using the same reasoning as above, we can show that all players from W k must be connected to allplayers from W k +1 . Since the number of nodes in the network is finite, the number of classes is finiteand this procedure reaches W M in a finite number of steps. orollary 3 There is only one class of victims in a stable network g and all victims have the same neighborhood Proof of Corollary 3:
Follows from Lemma 4 and Lemma 5.We show now that classes must be of different sizes, and that stronger players belong to morenumerous classes.
Lemma 6
Let | W k | denote the number of nodes that belong to class W k in stable network g ( s ∗ ) . Then | W k | > | W k +1 |∀ k ∈ { , , ..., M − } . Proof of Lemma 6 : Suppose that the claim does not hold, so | W k | ≤ | W k +1 | for some k = 1 , ..., M − a , s ∗ ac = s ∗ ad whenever d and c belong to the same class.Therefore, for any two players a, b such that a ∈ W k and b ∈ W k +1 , we have that w ∗ a = (cid:80) i (cid:54) = k,c ∈ W i | W i | s ∗ ac and w ∗ b = (cid:80) i (cid:54) = k +1 ,c ∈ W i | W i | s ∗ bc . Since w ∗ a < w ∗ b , Proposition 2 implies s ∗ ac > s ∗ bc , c ∈ { W , W .., W K } \{ W k , W k +1 } . Furthermore, since w ∗ a < w ∗ b we have that s ∗ ab > s ∗ ba according to Proposition 10. But then | W k | < | W k +1 | ⇒ (cid:80) i (cid:54) = k,c ∈ W i | W i | s ∗ ac > (cid:80) i (cid:54) = k +1 ,c ∈ W i | W i | s ∗ bc ⇒ w ∗ a > w ∗ b . This is in contradiction with a ∈ W k and b ∈ W k +1 . Proof of Proposition 3:
From Lemma 4, Lemma 5 and Corollary 3 it directly follows that a nonemptystable network g must be a complete M -partite network. Lemma 6 directly implies the asymmetry insizes. Proof of Proposition 4 : When g ( s ∗ ) satisfies (U) for L i = ∅ ∀ i ∈ N and g ( s ∗ ) has a complete bipartitestructure then g ( s ∗ ) satisfies (U) for any L i ⊆ F i . To see this, consider game C ( K n − v,v ). Proposition9 states that there is a unique pure strategy Nash equilibrium ¯ s of game C ( K n − v,v ). The equilibriumis interior under Assumption 1-3. Since ¯ s is the NE of C ( K n − v,v ), g (¯ s ) satisfies (U) for L i = ∅ . Theonly new links that can be formed in g (¯ s ) are with members of own partition. It is easy to see that noplayer will have an incentive to form a link with a member of own partition in g (¯ s ), since all membersof the same partition have the same total spending, as they play the same strategy. Hence, ¯ s satisfiescondition (U) from Definition 2. In the remaining part of the proof we show that part (B) of Definition2 will be satisfied when v < v ∗ .First note that a deviation in which players i ∈ A and j ∈ V destroy link ij is profitable for player j ∈ V , simply because she is a victim. We will now show that there exists v ∗ > i ∈ A prefers to destroy link with j ∈ V in K n − v,v whenever v ≥ v ∗ . To this end, let us define functions h : R ≥ → R ≥ and f : R ≥ → R ≥ by: h ( v, s, r ) = max x (cid:26) x − sx + s + r v − α ( vx ) (cid:27) , (16) f ( n, v, r ) = h ( v − , ¯ s v,n − v , r ) − h ( v, ¯ s v,n − v , r ) , (17)where ¯ s v,n − v denotes the Nash equilibrium per-contest investments of a member of V in C ( K n − v,v ). Dueto symmetry, all members of the same partition will play the same strategy in ¯ s . Note that f ( n, v, r ) isthe expected benefit of destroying a link of an attacker in network g (¯ s ), where ¯ s is the Nash equilibriumof C ( K n − v,v ).We now show that function f is monotonically increasing in v ∈ [1 , a ] and that it takes a positivevalue when v is big. We will treat v as a continuous variable in the remaining part of the proof. e show now that for v ∈ [1 , a ] = [1 , n − v ], f ( n, v − , r ) < f ( n, v, r ) . In order to do this, we first show that h decreases with s , and that it decreases faster with s forhigher values of v ( ∂h∂s decreases with v ). Indeed, taking the derivative of h with respect to s we get: ∂h∂s = ∂h∂x ∂x∂s + ∂h∂s = − x + r ( x + s + r ) v, (18)where we used the fact that ∂h∂x = 0, since x is the maximizer of h . Differentiating with respect to v weget: ∂ h∂v∂s = 2 (cid:20) − x + r ( x + s + r ) + v x − s ( x + s + r ) ∂x∂v (cid:21) . (19)The above derivative will be negative for all positive values of s and x such that x ≥ s and ∂x∂v <
0. Thiswill hold in particular when v ∈ [1 , a ] - since in the Nash equilibrium of C ( K n − v,v ), the attackers exerta higher effort than the victims ( x ≥ s ) and the investment of members of A decreases with v ( ∂x∂v < v ∈ [1 , a ) ∂ [ h ( v − , s, r ) − h ( v, s, r )] ∂s > . (20)Since ¯ s v − ,n − v +1 < ¯ s v − ,n − v < ¯ s v,n − v from (20) directly follows that: h ( v − , ¯ s v,n − v , r ) − h ( v, ¯ s v,n − v , r ) > h ( v − , ¯ s v − ,n − v +1 , r ) − h ( v, ¯ s v − ,n − v +1 , r ) ⇒ f ( n, v, r ) > h ( v − , ¯ s v − ,n − v +1 , r ) − h ( v, ¯ s v − ,n − v +1 , r ) . Finally, using the fact that h is concave in v (directly follows from the concavity of payoff function x − sx + s + r v − ( vx ) in v , see (De la Fuente, 2000, Theorems 2.12. and 2.13 in Section 7) for the formalargument), the following holds: h ( v − , ¯ s v − ,n − v +1 , r ) − h ( v, ¯ s v − ,n − v +1 , r ) > h ( v − , ¯ s v − ,n − v +1 , r ) − h ( v − , ¯ s v − ,n − v +1 , r ) , and therefore: f ( n, v, r ) > f ( n, v − , r ) , which is what we wanted to prove.If for v = 1, f takes a positive value, than no K n − v,v is stable. If for v = 1 f takes a negative value,this means that star network is stable.Suppose now that r = 0. By solving for the equilibrium of game C ( K n − v,v ) (see Franke and Ozturk(2015) for details) we verify that when n ≥ h (0 , s, r ) = 0, we have that f (3 , , < K , is stable. In the same way we verify thatwhen f ( n, n − n √ , > n ≥ f is strictly monotone, and that it changes sign implies that there exists v ∗ ∈ (cid:104) , n − n √ (cid:105) such that f ( n, v, r ) ≥ v ≥ v ∗ and f ( n, v, r ) < v < v ∗ , which completes theproof. roofs of Claims from Section 4 We first show that the contest game on a complete bipartite network is a nice aggregative game (Acemogluand Jensen, 2013), so we can use results from that paper for some of our comparative statics exercises.For the cases when results from (Acemoglu and Jensen, 2013) cannot be directly applied, we rely onthe implicit function derivation of the equilibrium conditions. We use the fact that when stable network g ( s ∗ ) has a complete bipartite structure K a,v , then s ∗ is the Nash equilibrium of game C ( K a,v ). Thisis a direct consequence of the fact that stable g ( s ) satisfies (U) for L i = ∅ . Lemma 7
The contest game on a complete bipartite network C ( K a,v ) can be represented as a nice aggregative gameas defined in (Acemoglu and Jensen, 2013). Proof of Lemma 7:
The pure strategy Nash equilibrium of game C ( K a,v ) is such that all players fromthe same class play the same strategy and invest the same amount of effort in each of their contest. Theconditions which determine the equilibrium investments in C ( K a,v ) are equivalent to the system of FOCsthat pins down the pure strategy Nash equilibrium of two players contest game in which the strategyspace of each player is the set of nonnegative real numbers and the payoffs are defined by: π i ( s ij , s ji ; r ) = φ ( s ij ) − φ ( s ji ) φ ( s ij ) + φ ( s ji ) + r − v c ( vs ij ) ,π j ( s ji , s ij ; r ) = φ ( s ji ) − φ ( s ij ) φ ( s ij ) + φ ( s ji ) + r − a c ( as ij ) . Since φ ( s ij ) − φ ( s ji ) φ ( s ij )+ φ ( s ji )+ r = − φ ( s ij )+ rφ ( s ij )+ φ ( s ji )+ r and φ ( · ) is strictly increasing it is straightforward toverify that this game is a nice aggregative game studied in (Acemoglu and Jensen, 2013) (see Definition1 and Definition 6 in Acemoglu and Jensen (2013)). Proof of Proposition 5 . According to Lemma 7, the contest game on a complete bipartite network can be represented as anice aggregative game. For completeness we define the notion of positive shock relevant for our model asintroduced in (Acemoglu and Jensen, 2013). Definition 7 (Positive shock)
Consider the payoff functions π i = π i ( s i , s − i , t ) with s i ∈ S i ⊆ R and t ∈ R . Then an increase in t iscalled a positive shock if each π i exhibits increasing differences in s i and t .
1. To prove the claim it is enough to show that the change in cost function from c ( · ) to ˜ c ( · ) is a positiveshock for both players. Denote with ˜ π i the payoff function of player i ∈ A (and symmetrically for j ∈ V ) after c becomes ˜ c . It is straightforward to see that ∂ ˜ π i ∂s ij ≤ ∂π i ∂s ij when ˜ c (cid:48) ( vs ij ) ≤ c (cid:48) ( vs ij ),which according to Definition 7 implies that the contemplated change is a positive shock.2. To prove the claim it is enough to show that an increase in transfer T is a positive shock for bothplayers. Differentiating we get that for player i (symmetrically for j ) ∂ π i ∂s ij ∂T = ( r + 2 φ ( s ji ) φ (cid:48) ( s ij )( r + φ ( s ij ) + φ ( s ji )) > , which implies that an increase in T is a positive shock. See (Acemoglu and Jensen, 2013) for a more general statement. . To conduct a comparative statics exercise with respect to r we cannot apply the result for ag-gregative games, as an increase in r can be a positive shock for one player, and, at the same time,a negative shock for some other player. Indeed, ∂ π i ∂s ij ∂r = φ ( s ij ) − φ ( s ji ) − r ( φ ( s ij ) + φ ( s ji ) + r ) φ (cid:48) ( s ij ) , does not have the same sign for all non-negative arguments. Therefore, we rely on the implicitfunction theorem. The strategy profile s ∗ satisfies the first order optimality conditions: r + 2 φ ( s ∗ ji )( φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) φ (cid:48) ( s ∗ ij ) = c (cid:48) (cid:32)(cid:88) k ∈ V s ∗ ik (cid:33) , i ∈ A,r + 2 φ ( s ∗ ij )( φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) φ (cid:48) ( s ∗ ji ) = c (cid:48) (cid:32)(cid:88) k ∈ A s ∗ jk (cid:33) , j ∈ V. (21)Taking the derivative of (21) with respect to r . To make expressions short, in writing we omit thedependence of s ∗ ij and s ∗ ji on r . We get the following system of equations: s ∗ ij (cid:48) (2 φ ( s ∗ ji ) + r ) φ (cid:48)(cid:48) ( s ∗ ij )( φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) + φ (cid:48) ( s ∗ ij ) (cid:0) s ∗ ji (cid:48) φ (cid:48) ( s ∗ ji ) + 1 (cid:1) ( φ ( s ∗ ji ) + φ ( s ∗ ji ) + r ) − φ ( s ∗ ji ) + r ) φ (cid:48) ( s ∗ ij ) (cid:0) s ∗ ij (cid:48) φ (cid:48) ( s ∗ ij ) + s ∗ ji (cid:48) φ (cid:48) ( s ∗ ji ) + 1 (cid:1) ( φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) = c (cid:48)(cid:48) (cid:32)(cid:88) k ∈ V s ∗ ik (cid:33) (cid:88) k ∈ V s ∗ ik (cid:48) , i ∈ A,s ∗ ji (cid:48) (2 φ ( s ∗ ij ) + r ) φ (cid:48)(cid:48) ( s ∗ ji )( φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) + φ (cid:48) ( s ∗ ji ) (cid:0) s ∗ ij (cid:48) φ (cid:48) ( s ∗ ij ) + 1 (cid:1) ( φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) − φ ( s ∗ ij ) + r ) φ (cid:48) ( s ∗ ji ) (cid:0) s ∗ ij (cid:48) φ (cid:48) ( s ∗ ij ) + s ∗ ji (cid:48) φ (cid:48) ( s ∗ ji ) + 1 (cid:1) ( φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) = c (cid:48)(cid:48) (cid:32)(cid:88) k ∈ A s ∗ jk (cid:33) (cid:88) k ∈ A s ∗ jk (cid:48) , j ∈ V Using symmetry ( s ∗ ik = s ∗ i(cid:96) , i ∈ A, k, (cid:96) ∈ V and s ∗ jk = s ∗ j(cid:96) , j ∈ V, k, (cid:96) ∈ A ) , and solving for s ∗ ij (cid:48) ( r )and s ∗ ji (cid:48) ( r ) we get: s ∗ ij (cid:48) ( r ) = − φ (cid:48) ( s ∗ ij ) 2 kφ (cid:48) ( s ∗ ji ) + (cid:0) − φ ( s ∗ ij ) + 3 φ ( s ∗ ji ) + r (cid:1) (cid:2) ak c (cid:48)(cid:48) − ( r + 2 φ ( s ∗ ij ) φ (cid:48)(cid:48) ( s ∗ ji ) (cid:3) Ω ,s ∗ ji (cid:48) ( r ) = − φ (cid:48) ( s ∗ ji ) 2 kφ (cid:48) ( s ∗ ij ) + (cid:0) φ ( s ∗ ij ) − φ ( s ∗ ji ) + r (cid:1) (cid:2) vk c (cid:48)(cid:48) − ( r + 2 φ ( s ∗ ji ) φ (cid:48)(cid:48) ( s ∗ ij ) (cid:3) Ω , (22)where Ω = (cid:0) c (cid:48)(cid:48) k v − ( r + 2 φ ( s ∗ ji )) φ (cid:48)(cid:48) ( s ∗ ij ) (cid:1) (cid:2) ac (cid:48)(cid:48) k + ( r + 2 φ ( s ∗ ij )) (cid:0) φ (cid:48) ( s ∗ ji ) − kφ (cid:48)(cid:48) ( s ∗ ji ) (cid:1)(cid:3) +2 φ (cid:48) ( s ∗ ij ) (cid:2) ac (cid:48)(cid:48) k r + 2 ac (cid:48)(cid:48) k φ ( s ∗ ji ) + 2 φ (cid:48) ( s ∗ ji ) k − ( r + 2 φ ( s ∗ ij ))( r + 2 φ ( s ∗ ji )) φ (cid:48)(cid:48) ( s ∗ ij ) (cid:3) ,c (cid:48)(cid:48) = c (cid:48)(cid:48) (cid:0)(cid:80) k ∈ V s ∗ ik (cid:1) , c (cid:48)(cid:48) = c (cid:48)(cid:48) (cid:16)(cid:80) k ∈ A s ∗ jk (cid:17) , and k = ( r + φ ( s ∗ ij ) + φ ( s ∗ ji )).The expression Ω is positive, since c is convex function, φ is concave function, and φ ( x ) ≥ , ∀ x ≥ a > v and s ∗ ij > s ∗ ji together with (22) imply that s ∗ ji (cid:48) ( r ) is always negative. On the ther hand, the sign of s ∗ ij (cid:48) ( r ) is ambiguous, and s ∗ ij (cid:48) ( r ) is positive whenever: φ ( s ∗ ij ) > φ ( s ∗ ij ) + φ ( s ∗ ji ) + r ) φ (cid:48) ( s ∗ ij ) a ( r + φ ( s ∗ ij ) + φ ( s ∗ ji )) c (cid:48)(cid:48) − ( r + 2 φ ( s ∗ ij )) φ (cid:48)(cid:48) ( s ∗ ji ) + r + 3 φ ( s ∗ ji ) , which will, since ( r + 2 φ ( s ∗ ij )) φ (cid:48)(cid:48) ( s ∗ ji ) <
0, hold whenever: φ ( s ∗ ij ) > φ (cid:48) ( s ∗ ij ) a ( r + φ ( s ∗ ij ) + φ ( s ∗ ji )) c (cid:48)(cid:48) + r + 3 φ ( s ∗ ji ) . We now discuss the sign of ∂w ∗ ∂r . From (22) we get: ∂w ∗ ( r ) ∂r > ⇔− φ (cid:48) ( s ∗ ij ) (cid:0)(cid:0) − φ ( s ∗ ij ) + 3 φ ( s ∗ ji ) + r (cid:1) (cid:2) ak c (cid:48)(cid:48) − ( r + 2 φ ( s ∗ ij ) φ (cid:48)(cid:48) ( s ∗ ji ) (cid:1)(cid:3) − φ (cid:48) ( s ∗ ji ) (cid:0)(cid:0) φ ( s ∗ ij ) − φ ( s ∗ ji ) + r (cid:1) (cid:2) vk c (cid:48)(cid:48) − ( r + 2 φ ( s ∗ ji ) φ (cid:48)(cid:48) ( s ∗ ij ) (cid:1)(cid:3) > kφ (cid:48) ( s ∗ ij ) φ (cid:48) ( s ∗ ji ) (cid:2) φ (cid:48) ( s ∗ ij ) + φ (cid:48) ( s ∗ ji ) (cid:3) . (23)When c ( x ) = α x α , and φ ( x ) = λx , with α ≥ λ >
0, equation (23) simplifies to: − ( α − (cid:104) (3 s ∗ ji − s ∗ ij + rλ ) a α − s ∗ jiα − + (3 s ∗ ij − s ∗ ji + rλ ) v α − s ∗ ijα − (cid:105) > s ∗ ij + s ∗ ji + rλ . In a specific case when r → − ( α − (cid:104) (3 s ∗ ji − s ∗ ij ) a α − s ∗ jiα − + (3 s ∗ ij − s ∗ ji ) v α − s ∗ ijα − (cid:105) > s ∗ ij + s ∗ ji . In this case (see Proposition 12 in Appendix B) s ∗ ji = ¯ s ji = (cid:2) va (cid:3) α − α s ∗ ij and s ∗ ij + s ∗ ji = ( av ) − ( α − α ( a α − α + v α − α ) α − α , so the above inequality can be written as (cid:20)(cid:0) a α − − v α − (cid:1) + (cid:104) va (cid:105) α − α (cid:0) v α − − a α − (cid:1)(cid:21) s ∗ ijα − > α − av ) ( α − α ( a α − α + v α − α ) − αα . (24)Since s ∗ ijα − = a ( α − α ( a α − α + v α − α ) α − α ( av ) − ( α − α (see Proposition 12 in Appendix B), (24) after somealgebra becomes: (cid:20)(cid:0) a α − − v α − (cid:1) + (cid:104) va (cid:105) α − α (cid:0) v α − − a α − (cid:1)(cid:21) > α − (cid:16) a α − α + v α − α (cid:17) v ( α − α . (25)When α = 2, this inequality holds whenever a ≥ v . We show in Lemma 9 in Appendix B thatif this inequality holds for α = 2 it holds for any α ≥ Lemma 8
The total spending of each node in the equilibrium is defined as a solution of system (4) . Proof of Lemma 8 . Expressing s ∗ ij from (7), when φ ( x ) = λx we get that in the equilibrium: ∗ ij = 2 c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) − r λ . (26)Summing over all contests of player i , and accounting for the fact that the cost funciton of player k is c k ( x ) = (1 + (cid:15) k ) c ( x ) we get (4). Proof of Proposition 6 . Suppose first that k ∈ A . Due to the symmetry, (4) is reduced to thefollowing system of equations: w ∗ k = v c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ j ) + (1 + (cid:15) k ) c (cid:48) ( w ∗ k )) − v r λ ,w ∗ i = v c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ j ) + c (cid:48) ( w ∗ i )) − v r λ , i ∈ A and i (cid:54) = k,w ∗ j = ( a −
1) 2 c (cid:48) ( w ∗ i )( c (cid:48) ( w ∗ j ) + c (cid:48) ( w ∗ i )) + 2(1 + (cid:15) k ) c (cid:48) ( w ∗ k )( c (cid:48) ( w ∗ j ) + (1 + (cid:15) k ) c (cid:48) ( w ∗ k )) − a r λ , j ∈ V. (27)Differentiating with respect to (cid:15) k , letting (cid:15) k →
0, and using the fact that when (cid:15) k → w ∗ k = w ∗ i we get the following linear system in first derivatives: (cid:32) v c (cid:48) ( w ∗ j ) c (cid:48)(cid:48) ( w ∗ i )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) (cid:33) w ∗ k (cid:48) =2 v c (cid:48) ( w ∗ i ) − c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) c (cid:48)(cid:48) ( w ∗ j ) w ∗ j (cid:48) − v c (cid:48) ( w ∗ i ) c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) , (cid:32) v c (cid:48) ( w ∗ j ) c (cid:48)(cid:48) ( w ∗ i )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) (cid:33) w ∗ i (cid:48) =2 v c (cid:48) ( w ∗ i ) − c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) c (cid:48)(cid:48) ( w ∗ j ) w ∗ j (cid:48) , (cid:32) a c (cid:48) ( w ∗ i ) c (cid:48)(cid:48) ( w ∗ j )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) (cid:33) w ∗ j (cid:48) =2( a − c (cid:48) ( w ∗ j ) − c (cid:48) ( w ∗ i )( c (cid:48) ( w ∗ j ) + c (cid:48) ( w ∗ i )) c (cid:48)(cid:48) ( w ∗ i ) w ∗ i (cid:48) + 2 c (cid:48) ( w ∗ j ) − c (cid:48) ( w ∗ k )( c (cid:48) ( w ∗ j ) + c (cid:48) ( w ∗ i )) c (cid:48)(cid:48) ( w ∗ i ) w ∗ k (cid:48) − c (cid:48) ( w ∗ i ) − c (cid:48) ( w ∗ j ) c (cid:48) ( w ∗ i )( c (cid:48) ( w ∗ j ) + c (cid:48) ( w ∗ i )) . (28)When r → w ∗ i and w ∗ j simplify to w ∗ i = 2 v c (cid:48) ( w ∗ j )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) , w ∗ j = 2 a c (cid:48) ( w ∗ i )( c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j )) . We plug these expressions into (28) and after some algebra (28) becomes. (cid:0) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) + 2 w ∗ i c (cid:48)(cid:48) ( w ∗ i ) (cid:1) w ∗ k (cid:48) = vw ∗ j − aw ∗ i a c (cid:48)(cid:48) ( w ∗ j ) w ∗ j (cid:48) − w ∗ i c (cid:48) ( w ∗ i ) , (cid:0) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) + 2 w ∗ i c (cid:48)(cid:48) ( w ∗ i ) (cid:1) w ∗ i (cid:48) = vw ∗ j − aw ∗ i a c (cid:48)(cid:48) ( w ∗ j ) w ∗ j (cid:48) , (cid:0) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) + 2 w ∗ j c (cid:48)(cid:48) ( w ∗ j ) (cid:1) w ∗ j (cid:48) = aw ∗ i − vw ∗ j va c (cid:48)(cid:48) ( w ∗ i ) w ∗ k (cid:48) + ( a − aw ∗ i − vw ∗ j va c (cid:48)(cid:48) ( w ∗ i ) w ∗ i (cid:48) + c (cid:48) ( w ∗ i ) aw ∗ i − vw ∗ j va . (29) olving for w ∗ i (cid:48) , w ∗ j (cid:48) and w ∗ k (cid:48) we get : w ∗ j (cid:48) = c (cid:48) ( w ∗ i ) (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) (cid:3) ( aw ∗ i − vw ∗ j ) den > ,w ∗ i (cid:48) = − c (cid:48) ( w ∗ i ) (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) (cid:3) c (cid:48)(cid:48) ( w ∗ j )( aw ∗ i − vw ∗ j ) den < ,w ∗ k = a w ∗ i − (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) (cid:3) v − w ∗ i (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) (cid:3) (cid:2) c (cid:48)(cid:48) ( w ∗ j ) + 4 vc (cid:48)(cid:48) ( w ∗ i ) (cid:3) − aw ∗ i c (cid:48)(cid:48) ( w ∗ i ) c (cid:48)(cid:48) ( w ∗ j ) den − a vw ∗ i w ∗ j c (cid:48)(cid:48) ( w ∗ i ) c (cid:48)(cid:48) ( w ∗ j ) + v c (cid:48)(cid:48) ( w ∗ j ) (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) + 2 aw ∗ i c (cid:48)(cid:48) ( w ∗ i ) (cid:3) w ∗ j den − a (2 a − vw ∗ i w ∗ j (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) (cid:3) c (cid:48)(cid:48) ( w ∗ j ) den < , (30)where den c (cid:48)(cid:48) ( w ∗ i ) c (cid:48)(cid:48) ( w ∗ j )( a w ∗ i + v w ∗ j ) + av (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) + 2 c (cid:48)(cid:48) ( w ∗ i ) w ∗ i (cid:3) (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) + 2 c (cid:48)(cid:48) ( w ∗ j ) w ∗ j (cid:3) > .den a (cid:2) c (cid:48) ( w ∗ i ) + c (cid:48) ( w ∗ j ) + 2 w ∗ i c (cid:48)(cid:48) ( w ∗ i ) (cid:3) den > . The respective signs follow directly from the fact that c is convex, and aw ∗ i > vw ∗ j . The case when k ∈ V is analogous (just switch v and a ). Finally, from (30) we get: ∂w ∗ ∂(cid:15) k | (cid:15) k =0 = − av c (cid:48) ( w ∗ i ) (cid:0) aw ∗ i + vw ∗ j (cid:1) (cid:2) aw ∗ i c (cid:48)(cid:48) ( w ∗ j )+ vw ∗ j c (cid:48)(cid:48) ( w ∗ j )+ v (cid:0) c (cid:48) ( w ∗ i )+ c (cid:48) ( w ∗ j ) (cid:1)(cid:3) den < , where the inequality follows directly from Assumptions 1-2. This completes the proof. Proofs of Claims from Section 5
Proof of Proposition 7 . Consider contest network g ( s ) such that ij / ∈ g for some players i and j . Weshow that g ( s ) is not Nash stable when φ (cid:48) (0) r > c (cid:48) (0).(i) Consider first the case when player i is not involved in any contest, thus w i = 0. The marginalbenefit of investing (cid:15) > ij calculated at (cid:15) = 0 is φ (cid:48) (0) r . The marginal cost of this actionis c (cid:48) (0). As long as φ (cid:48) (0) r > c (cid:48) (0) player i will wish to start a contest with player j .(ii) When w i >
0, there must exist some some k such that s ik >
0. We discuss two possible cases:(a) There exists a contest ik ∈ g ( s ) such that s ik ≥ s ki . Consider a deviation in which i reallocates (cid:15) > ik to start contest with j . The marginal benefit of this actionfor i is φ (cid:48) (0) r . The marginal cost of a proposed deviation is ( r +2 φ ( s ki )) φ (cid:48) ( s ik )( r + φ ( s ik )+ φ ( s ki )) . The followingchain of inequalities holds:( r + 2 φ ( s ki )) φ (cid:48) ( s ik )( r + φ ( s ik ) + φ ( s ki )) ≤ r + 2 φ ( s ki )( r + 2 φ ( s ki )) φ (cid:48) ( s ik ) ≤ r + 2 φ ( s ki ) φ (cid:48) (0) < r φ (cid:48) (0) , (31)where we have used the fact that φ is increasing and concave function. So, in this case, themarginal benefit of the proposed deviation is greater than its marginal cost.(b) There is no ik ∈ g ( s ) such that s ik ≥ s ki . In this case consider a deviation in which i reallocates s ik from contest ik to ij . The change in payoff due to this deviation is equal to Details od derivation are available upon request. φ ( s ik ) φ ( s ik )+ r − φ ( s ki ) φ ( s ki )+ r (cid:17) − φ ( s ik ) − φ ( s ki ) φ ( s ik )+ φ ( s ki )+ r . Simplifying we get: φ ( s ik ) φ ( s ik ) + r − φ ( s ki ) φ ( s ki ) + r = φ ( s ik ) − φ ( s ki ) φ ( s ik ) φ ( s ki ) r + φ ( s ik ) + φ ( s ki ) + r > φ ( s ik ) − φ ( s ki ) φ ( s ik ) + φ ( s ki ) + r , where for the last inequality we used the fact that s ik < s ki , and φ is increasing.Hence, provided that φ (cid:48) (0) r > c (cid:48) (0), Nash stable network g ( s ) must be such that s ij + s ji >
0, for anypair of players i and j , that is ij ∈ g, ∀ i, j ∈ N .We have proved that a Nash stable network must be the complete network. We now argue that thereis a unique strategy profile s such that the complete network g ( s ) is Nash stable. Moreover, s is suchthat s ij = s ji = s >
0, for any two players i and j .To do that, we recall that there exists a unique pure strategy Nash equilibrium of the game C (¯ g )when ¯ g is the complete network. In this equilibrium each player must play the symmetric strategy, asotherwise the uniqueness result would not hold. Condition φ (cid:48) (0) r > c (cid:48) (0) ensures that ¯ s (cid:54) = , by sameargument as used in (i) of this proof. It directly follows from the definition of a Nash stable networkthat it must be s ij = ¯ s ij , where ¯ s is the Nash equilibrium of the contest game on the complete network.Finally, when φ (cid:48) (0) r ≤ c (cid:48) (0) exerting positive amount of resources in contest against opponent whoinvests 0 is never profitable. Furthermore, if for any pair of players we have s ij > s ji > s ij ≥ s ji , then the marginal loss of i in decreasing s ij is always smaller thenthe marginal gain measured by the cost decrease, as long as φ (cid:48) (0) r ≤ c (cid:48) (0). Indeed, the following chain ofinequalities hold: ( r + 2 φ ( s ji )) φ (cid:48) ( s ij )( r + φ ( s ij ) + φ ( s ji )) < r φ (cid:48) (0) ≤ c (cid:48) (0) < c (cid:48) ( w i ) , where the first inequality comes from (31). This completes the proof.For completnesss, we provide definition of strongly pairwise stable network. Definition 8 (strongly pairwise stable network (Bloch and Dutta, 2009))
Network g ( s ) is strongly pairwise stable if it is Nash stable and there is no pair of individuals ( i, j ) andjoint deviation ( s (cid:48) i , s (cid:48) j ) such that: π k ( s (cid:48) i , s (cid:48) j , s − i − j ) > π k ( s ) for k = i, j. Proof of Proposition 8 . When φ (cid:48) (0) r ≥ s (cid:29)
0. Consider any two players i and j , and jointdeviation in which i chooses s (cid:48) ij = 0, and s (cid:48) ik = s ik = s for all k (cid:54) = j , and j deviates by setting s (cid:48) ji = 0and s (cid:48) j(cid:96) = s j(cid:96) = s for all (cid:96) (cid:54) = i . This deviation is profitable for i and j , resulting in benefit φ (0) − φ (0) φ (0) + π (0) + r − φ ( s ) − φ ( s ) φ ( s ) + π ( s ) + r − c (( n − s ) + c ( ns ) > . Thus, when φ (cid:48) (0) r ≥ φ (cid:48) (0) r < If ¯ s were asymmetric, by relabeling players we could find more than one pure strategy NE of the contestgame on the complete network, which would contradict Proposition 9. nline Appendix B LPFS as a resting point of a dynamic process of network formation
One can think of a stable network as defined in Definition 2 as a stable state of a coupled dynamicprocess we present in this section. Over time, players make decisions about their links and about actionsassigned to these links. We assume that a link between players i and j is formed if one player decidesto form it (unilateral), while link ij is destroyed if both agents agree to destroy it (bilateral). Time isindexed with t ∈ N ∪ { } . In t = 0 an arbitrary contest network g ( s t ) is given.For each period t :(i) At the beginning of period t strategy profile s t − is a pure strategy Nash equilibrium of game C (¯ g t − ), where ¯ g t − describes the set of contests in the population at the end of period t − i and j are chosen randomly from the population. They jointly choose their linking patternswhich leads to graph ¯ g t . Players evaluate the expected benefit from forming a link as described inSubsection 2.2.(iii) The second dynamic process ( action adjustment process ) starts, and all agents update their actionsgiven ¯ g t according to the action adjustment process formally described below. This process settlesat the pure strategy NE of game C (¯ g t ). We assume that this process takes place in continuoustime and therefore on a faster time-scale than the network formation process. In other words,players infinitely more often revise their investment in ongoing contests compared to contemplatingstarting/ending a contest.We now formally describe the action adjustment process mentioned in (iii) above. Let ∇ i π i denotethe gradient of the payoff function with respect to s i ∈ S i (¯ g ). Define function J : (cid:81) i R n ≥ → (cid:81) i R n ≥ with: J ( s ) = ∇ π ( s ) ∇ π ( s ) ... ∇ n π n ( s ) . The action adjustment process is defined with: ˙s = J ( s ) , (32)According to (32) each player changes his strategy at a rate proportional to the gradient of her payofffunction with respect to her strategy. It is clear that the Nash equilibrium of game C (¯ g ) (¯ s ) is the stablestate of this process. We also prove that ¯ s is a globally asymptotically stable state of (32). Hence if everyplayer adjusts her actions according to the adjustment process in (32), the action adjustment processconverges, irrespective of the initial conditions. Proposition 11
The action adjustment process given by equation (32) is globally asymptotically stable.Proof.
To prove the claim, we show that the rate of change of || J || = JJ (cid:48) is always negative (and equalto 0 at ¯ s ). Denote with G the Jacobian of J . The following holds:˙ JJ (cid:48) = ( G˙s ) (cid:48) J + J (cid:48) G˙s = ( J (cid:48) G (cid:48) J + J (cid:48) G J ) = J (cid:48) ( G (cid:48) + G ) J < , where G (cid:48) denotes the transpose of G . The last inequality follows from the fact that ( G (cid:48) + G ) is anegative definite matrix. To show that ( G (cid:48) + G ) is negative definite we use Lemma 1 from Goodman(1980) which states that ( G (cid:48) + G ) is negative definite if, for each player i : a) π i ( s ) is strictly concave in s i , (b) π i ( s ) is convex in s − i , (c) σ ( s , z ) = (cid:80) ni =1 z i π i ( s ) is concave in s for some z ∈ R ≥ . To show (a) we note that ∂ π i ∂s ij = ( r + 2 φ ( s ji )) (cid:0) φ (cid:48)(cid:48) ( s ij )( r + φ ( s ij ) + φ ( s ji )) − φ (cid:48) ( s ij ) (cid:1) ( r + φ ( s ij ) + φ ( s ji )) − c (cid:48)(cid:48) ( s i ) < . (33)The inequality in (33) holds as the first term in the difference is negative (due to properties of function φ stated in Assumption 1) and the second term is positive (due to the strict convexity of function c ).Furthermore ∂ π i ∂s ij ∂s ik = − c (cid:48)(cid:48) ( s i ) < ∀ j, k ∈ N i . Thus, the Hessian matrix H i of function π i with respect to s i is the sum of diagonal matrix H i withelements on the diagonal equal to:( r + 2 φ ( s ji )) (cid:0) φ (cid:48)(cid:48) ( s ij )( r + φ ( s ij ) + φ ( s ji )) − φ (cid:48) ( s ij ) (cid:1) ( r + φ ( s ij ) + φ ( s ji )) < , and matrix H i which has all the elements equal to − c (cid:48)(cid:48) ( s i ) < . H i is a negative definite matrix and H i is a negative semidefinite matrix, therefore H i = H i + H i is a negative definite matrix.To see that (b) holds, note that (when ij ∈ ¯ g ) : ∂ π i ∂s ji = ( r + 2 φ ( s ij )) (cid:0) φ (cid:48) ( s ji ) − φ (cid:48)(cid:48) ( s ji )( r + φ ( s ij ) + φ ( s ji )) (cid:1) ( r + φ ( s ij ) + φ ( s ji )) > ∀ jk ∈ ¯ g : k (cid:54) = i ) , ∂ π i ∂s jk = 0 and ∂ π i ∂s jk ∂s (cid:96)m = 0 for any other combination of players j, k, (cid:96) and m. Thus, the Hessian of π i with respect to s − i is a diagonal matrix with all entries positive or zeroand therefore positive semi-definite.Finally, to prove (c) choose z = . Then: σ ( s , ) = n (cid:88) i =1 (cid:88) j ∈ N i (cid:18) φ ( s ij ) φ ( s ij ) + φ ( s ji ) + r − φ ( s ji ) φ ( s ij ) + φ ( s ji ) + r − c ( s i ) (cid:19) = − n (cid:88) i =1 c ( w i )The last equality above holds since in the first sum above φ ( s ij ) φ ( s ij )+ φ ( s ji )+ r appears exactly once with apositive sign (as a part of payoff function π i ) and exactly once with a negative sign (as a part of function π j ). Function − (cid:80) i c ( w i ) is strictly concave due to the strict convexity of the cost function c. Hence, (c) also holds, which completes the proof.We do not study the properties of the dynamical process of network formation. However, it is clearfrom the definition that if this process settles on a single network configuration, then this network mustbe LPFS. It is interesting to note that Proposition 11 has a very practical application. It provides anefficient way to numerically calculate the Nash equilibrium of game C (¯ g ). Farsightedly stable network
Let ¯ g + { i j , i j , ..., i m j m } denote a graph obtained from ¯ g by adding links { i j , i j , ..., i m j m } , and¯ g − { i j , i j , ..., i m j m } the graph obtained from ¯ g by deleting links { i j , i j , ..., i m j m } . We will use ( i(cid:96) ) (cid:96) ∈ L } to denote set of links i(cid:96) with (cid:96) ∈ L . Finally, we define the payoff of player i from graph ¯ g asthe payoff from strategy profile s that induces ¯ g (Definition 3) and is such that each player i choosesstrategy s i consistently with (U) for L i = ∅ . In other words, it is the payoff of player i obtained at thepure strategy Nash equilibrium of game C (¯ g ) . We define farsightedly improving path in our model following (Jackson, 2008, Vannetelbosch andMauleon, 2015). As in Definition 2 we allow players to unilaterally form many links at the same time,and bilateral deviations.
Definition 9 (Farsightedly improving path)
A farsightedly improving path from graph ¯ g to graph ¯ g (cid:48) (denoted with ¯ g → ¯ g (cid:48) ) is a finite sequence ofgraphs { ¯ g , ¯ g , ..., ¯ g K } with ¯ g = ¯ g and ¯ g K = ¯ g (cid:48) such that for any k ∈ { , , ..., K − } either:(i) ¯ g k +1 = ¯ g k + { ( i(cid:96) ) (cid:96) ∈ L i } for some i ∈ N such that π i (¯ g K ) > π i (¯ g k ) , or(ii) ¯ g k +1 = ¯ g k − { ij } + { ( i(cid:96) ) (cid:96) ∈ L i } + { ( j(cid:96) ) (cid:96) ∈ L j } for some two players i, j ∈ N such that π i (¯ g K ) > π i (¯ g k ) and π j (¯ g K ) ≥ π i (¯ g k ) . We are now ready to define farsightedly stable contest network.
Definition 10
Let F (¯ g ) = { ¯ g (cid:48) : ¯ g → ¯ g (cid:48) } . Graph ¯ g is farsightedly stable if F (¯ g ) = ∅ . Network g ( s ) is farsightedly stableif graph ¯ g induced by s farsightedly stable. The analysis of farsightedly stable networks is beyond the scope of this paper. Here, by means of anexample, we demonstrate that there exists a farsightedly graph that is not Nash stable, and that thereexists Nash stable network which is not farsightedly stable.
Example 1
Consider population of 3 individuals, and suppose that φ ( x ) = x , c ( x ) = x and r = 0 . Up to isomor-phism, there are 4 different network structures in this case presented in Figure 8. By Proposition 7, ¯ g is Nash stable. The complete graph, however, is not farsightedly stable, since ¯ g ∈ F (¯ g ) . Indeed,players 2 and 3 find it optimal to deviate and destroy link . One can easily check that ¯ g is the uniquefarsightedly stable graph in this case. Graph ¯ g is also LFPS. - - - g - - - g - - g
10 203 0 g Figure 8: Different network structures with 3 players. The payoff obtained by player i at graph¯ g j ( π i (¯ g j ) ) is written next to corresponding node. Additional auxiliary results
Proposition 12
Let φ ( x ) = λx and c ( x ) = α x α , and r = 0 . The equilibrium strategy profile ¯ s of game C ( K a,v ) is given ith: ¯ s ij = a α − α ( a α − α + v α − α ) α ( av ) − ( α − α i ∈ A, j ∈ V, ¯ s ji = a α − α ( v α − α + v α − α ) α ( av ) − ( α − α j ∈ V, i ∈ A. Proof.
The equilibrium is defined with the system of equations:2¯ s ji (¯ s ji + ¯ s ij ) = 2( v ¯ s ij ) α − , s ij (¯ s ji + ¯ s ij ) = 2( a ¯ s ji ) α − , (34)where i ∈ A, j ∈ V. From (34) it follows that: ¯ s ji = (cid:2) va (cid:3) α − α ¯ s ij . Plugging this back into the second equation in (34) weget ¯ s αij = a α − α ( a α − α + v α − α ) a − α (cid:104) av (cid:105) ( α − α . Since a α − α a − αα = a α − α − α α = a α − αα a α − α − α α = a α − α a − ( α − α , we write ¯ s ij = a α − α ( a α − α + v α − α ) α (cid:104) av (cid:105) ( α − α a − ( α − α = a α − α ( a α − α + v α − α ) α ( av ) − ( α − α . Symmetrically: ¯ s ji = v α − α ( a α − α + v α − α ) α ( av ) − ( α − α . Lemma 9
Let a > v ≥ . If (cid:104)(cid:0) a α − − v α − (cid:1) + (cid:2) va (cid:3) α − α (cid:0) v α − − a α − (cid:1)(cid:105) > α − (cid:16) a α − α + v α − α (cid:17) v ( α − α for α ≥ then it holds for any α + δ with δ ≥ .Proof. Suppose that inequality holds for some α ≥
2. We show that it holds for α + δ for some δ ≥ αα − < α + δ − α + δ < v < a we have that (cid:2) va (cid:3) α − α > (cid:2) va (cid:3) α + δ − α + δ , and thus: (cid:20)(cid:0) a α + δ − − v α + δ − (cid:1) + (cid:104) va (cid:105) α + δ − α + δ (cid:0) v α + δ − − a α + δ − (cid:1)(cid:21) > (cid:20)(cid:0) a α + δ − − v α + δ − (cid:1) + (cid:104) va (cid:105) α − α (cid:0) v α + δ − − a α + δ − (cid:1)(cid:21) > (cid:20)(cid:0) a α − a δ − v α − a δ (cid:1) + (cid:104) va (cid:105) α − α (cid:0) v α − a δ − a α − a δ (cid:1)(cid:21) = a δ (cid:20)(cid:0) a α − − v α − (cid:1) + (cid:104) va (cid:105) α − α (cid:0) v α − − a α − (cid:1)(cid:21) , (35)where the last inequality is due to the fact that v δ < a δ and − v α − + (cid:2) va (cid:3) α − α v α − < α + δ − (cid:16) a α + δ − α + δ + v α + δ − α + δ (cid:17) v ( α + δ − α + δ =2 α + δ − (cid:16) a α − α a δ ( δ + α ) δ + v α − α v δ ( δ + α ) δ (cid:17) v ( α − α v δ − δα ( α + δ ) < α − (cid:16) a α − α + v α − α (cid:17) v ( α − α a δ ( δ + α ) δ a δ − δα ( α + δ ) =2 α − (cid:16) a α − α + v α − α (cid:17) v ( α − α a δ . (36)Since by assumption (cid:20)(cid:0) a α − − v α − (cid:1) + (cid:104) va (cid:105) α − α (cid:0) v α − − a α − (cid:1)(cid:21) > α − (cid:16) a α − α + v α − α (cid:17) v ( α − α and a δ >0 the claim directly follows from (35) and (36).