Mutual Insurance Networks and Unequal Resource Sharing in Communities, First Draft, February 2015

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1 Mutual Insurance Networks and Unequal Resource Sharng n Communtes, Frst Draft, February 2015 Pascal Blland a, Chrstophe Bravard b, Sudpta Sarang c a Unversté de Lyon, Lyon, F-69003, France ; Unversté Jean Monnet, Sant-Etenne, F-42000, France ; CNRS, GATE Lyon St Etenne, Sant-Etenne, F-42000, France. emal: pascal.blland@unv-st-etenne.fr. b Unversté Grenoble 2 ; UMR 1215 GAEL, F Grenoble, France ; CNRS, GATE Lyon St Etenne, Sant-Etenne, F-42000, France. emal: chrstophe.bravard@unv-st-etenne.fr. c DIW Berln and Lousana State Unversty. emal: sarang@lsu.edu Abstract We study formaton of mutual nsurance networks n a model where agents who obtan more resources share a fxed amount of resources wth all drectly lnked agents that obtan fewer resources. We dentfy the parwse stable networks and effcent networks n a basc model where agents are dentcal. Then, we ntroduce n the model two types of heterogenety: an exogenous one, where agents dffers n ther ncome or n ther preferences over the transfer scheme, and an endogenous heterogenety where the costs of lnkng to an agent depends on the number of lnks the latter has already formed n the network. We examne the mpact of these heterogenetes on stablty and effcency. Keywords: Mutual nsurance networks, Parwse stable networks, Effcent networks. JEL classfcaton: C72, D85. Ths paper has beneftted from the comments of Matt Jackson, Francs Bloch and partcpants n the Networks and Development Conference at Lousana State Unversty. The paper has also receved useful suggestons from semnar partcpants at ISI Delh, Unverste Bordeaux, Erasmus Unversty Rotterdam, and the Asan Meetngs of the Econometrc Socety,

2 1 Introducton A growng body of evdence has shown that whle household ncome n developng countres vares greatly, consumpton s remarkably smooth at a communty level (e.g., Townsend, 1994, Paxson, 1992, Jacoby and Skouas, 1997). Gven the lack of formal nsurance especally n the rural areas, ths suggests that nformal nsttutons play a crucal role n helpng households to counter the effects of ncome varaton. In ths paper we study the formaton of these nformal mutual nsurance networks buldng on several stylzed facts. A frst key feature s that nformal nsurance s not a vllage level phenomenon. Indeed, as has been well documented n the lterature on socal networks, n tmes of need ndvduals do not rely on the entre vllage, nstead they seek help prmarly through a network of mutual nsurance relatonshps wth frends and famly (see Fafchamps and Lund, 2003, and Wellman and Currngton, 1988). Another mportant feature s that mutual nsurance networks are not complete wthn the observed set of ndvduals. That s, wthn any communty, ndvduals do not enjoy the benefts of beng nsured by all others ndvduals of the vllage. Fnally, the sharng of resources n tmes of need s not equal (Townsend, 1994). In fact to the best of our knowledge, ths aspect of the formaton of mutual nsurance networks has not been addressed before. Our goal s to provde a pcture of the mutual nsurance arrangements wthn the communty from a networks perspectve. We examne when symmetrc and asymmetrc network archtectures can be stable among ex ante symmetrc agents. We ask whether such arrangements may be locally complete,.e., nvolve every ndvdual n a small group. We also study effcency and the mpact of agent heterogenety n ths problem. In our model mutual nsurance takes place between pars of ndvduals n a vllage or a small communty. A specfc feature s the way agents share ther resources (and hence the rsk): ndvduals who draw hgh resources gve a fxed amount of resources to ndvduals n ther mmedate neghborhood n the network who draw low resources. Thus, agents do not engage n equal sharng of resources. Ths type of sharng mechansm has two realstc features: () t ensures that ndvduals who draw hgh resources can always transfer resources to all ther neghbors who draw low resources, and () ndvduals who draw hgh resources always obtan hgher benefts than ndvduals who draw low resources. 2

3 In our benchmark model, we assume that each agent obtans random resources whch take on two values: hgh or low. If a person draws the hgh endowment state, then she gves an amount δ > 0 of resources to each of her neghbors (agents wth whom she has establshed a blateral rsk-sharng agreement) that has drawn the low endowment state. Conversely, f a person draws the low endowment state, then she obtans an amount δ of resources from each of her neghbors who has drawn the hgh endowment state. 1 Note that such a mutual nsurance network exposes agents to the rsk of ther neghbors. Indeed, f two ndvduals decde to nsure each other, then each of them ncreases her chances of obtanng a satsfyng payoff when her own resources are low, but also ncreases her chances of reducng her payoffs when her own resources are hgh. We also assume that nformal agreements are not bndng and hence to make them work agents need to nvest tme n ther relatonshps. So, n our model, establshng such mutual nsurance agreements s costly. More precsely, the cost of an agreement (a lnk) between two ndvduals depends on the number of agreements establshed by them. In partcular, we assume that the margnal cost of ndvdual, when she forms a lnk wth an ndvdual j, s ncreasng wth the number of lnks formed by. Ths captures the dea that a mutual nsurance agreement between two agents requres the agents to spend a mnmal amount of tme on t. Now the more lnks they have already formed, the less tme they have to spend on any addtonal lnk, and so the hgher s the cost of tme. It follows that the cost of an addtonal lnk ncreases wth the number of lnks. Usng ths framework, we examne the formaton of mutual nsurance lnks and ask what structures wll emerge when agents cannot coordnate lnk formaton across the entre populaton. We use parwse stablty as the equlbrum concept (see Jackson and Wolnsky, 1996). In a parwse stable network no par of unlnked agents has an ncentve to reach a mutual nsurance agreement (add a lnk), and no ndvdual has an ncentve to break a mutual nsurance agreement (remove one of her lnks). We contrast parwse stable networks wth the effcent networks for mutual nsurance agreements. An effcent network s one whch maxmzes the 1 In realty transfers can take a wde range of values dependng on the ncomes of the ndvduals and may be even the needs of the agents n a communty. However, n our model ndvduals have ether hgh or low ncomes. Therefore t s reasonable to assume that n the hgh ncome state agents gve a fxed amount to ther neghbors who have drawn the low ncome state. 3

4 amount of total payoffs obtaned by agents. In the basc model, we have several fndngs. We establsh that there exst parwse stable networks, n whch ndvduals are n asymmetrc stuatons relatve to the rsk they support. Thus, agents have dfferent rsk-sharng outcomes, despte that they have dentcal preferences and ther ncomes are dentcally dstrbuted. Moreover, n parwse stable networks agents who obtan the smallest amount of nsurance are always lnked together. We show that n effcent networks agents always obtan smlar amounts of nsurance. In that case, an effcent network s a parwse stable network (or a network very smlar to a stable network), but the converse s not true. Thus there may exst a conflct between parwse stable networks and effcent networks. We conclude that the mutual nsurance mechansm descrbed here does not always lead to effcent networks. Then we extend the basc model by ntroducng two types of agents heterogenety, an exogenous one and an endogenous one. Frst, we consder stuatons where agents are exogenously heterogeneous: ether they do not obtan the same ncome when they draw the hgh ncome state or they do not have the same preferences wth respect to the generosty of the transfer scheme. In the frst case, we assume that there exst two types of agents concernng the ncome they get when they draw the hgh ncome state. In that case, we show that there exst stuatons where only people who get the hghest ncome when they draw the hgh ncome state, that s the rch people, get access to nsurance, whle people who get the lowest ncome when they draw the hgh ncome state, that s the poor people, wll never be nsured. It follows that the nsurance mechansm ncreases the gap between the expected well-beng of rch people and the expected well-beng of poor people. In the second case, we assume that there exst two types of agents: the generous ones who are more gvng more than the mserly ones. In that case, we establsh that there exst condtons under whch, n a parwse stable network, generous agents are lnked together, mserly agents are lnked together, but there are no lnks between these two types of agents. Ths knd of parwse stable networks are compatble wth a result stressed by several emprcal studes: the majorty of transfers takes place only between sub-groups of agents (see Rosenzweg, 1988, and Udry, 1994). 4

5 Second, we consder stuatons where the cost of lnkng to an agent s ncreasng n the number of lnks that ths agent has already formed. Ths captures the fact that nsurance agreements are nformal and are honored f agents nvolved n a relatonshp nvest tme. In such stuaton, t s more dffcult to establsh a relatonshp wth an agent who already has numerous lnks snce she has less tme avalable. Note that n ths stuaton agents heterogenety arses endogenously n the model. As n the basc model, we show that agents who obtan the smallest amount of nsurance are always lnked together. However, by contrast wth the basc model, we show that agents who have the hghest amount of nsurance are never lnked together. Concernng effcency, we show that unlke n the basc model, f a network s effcent, then t s not always parwse stable (or a network very smlar to a parwse stable network). More precsely, an effcent network can be over connected wth respect to stablty. Ths result s due to the fact that when an agent decdes whether to delete a lnk, she does not take nto account the postve externalty that would accrue to agents who wll be stll lnked wth ths agent after the deleton of the lnk. A recent theoretcal lterature about revenue sharng n developng economes examnes the formaton of rsk-sharng networks. Bramoulle and Kranton (2006 and 2007) dscuss the stablty/effcency dlemma of rsk-sharng networks. A dstnctve aspect of ther work s that after the ncome realzaton occurs, lnked pars of agents meet (sequentally and randomly) and share ther current money holdng equally. The authors show that wth many rounds of such meetngs, an ndvduals money holdng converges to the mean of realzed ncome n her group, 2 that s n a group there s always equal revenue sharng among ndvduals. Belhaj and Deroan [1] also examne a stuaton where the blateral partal rsk-sharng rule s such that neghbors share equally a part of ther revenue. However, they focus on the mpact of nformal rsk-sharng on rsk takng ncentves when transfers are organzed through a socal network. By contrast, n our paper we deal wth stuatons where ndvduals do not engage n equal ncome sharng. In partcular, after ncome sharng, an ndvdual who has ntally obtaned hgh ncome always ends up better than an ndvdual who has obtaned low ncome. There s an nterestng dfference between our paper and the Bramoulle and Kranton papers concernng externalty generated by lnks. In our paper, when an agent forms a lnk wth an agent j, ths lnk may have a negatve 2 In ther paper, a group conssts of agents who are both drectly or ndrectly lnked. 5

6 mpact on the utlty of s neghbors (there s a negatve externalty), snce agent wll now have less tme to spend on relatons wth her neghbors (dem for j s neghbors). It follows that when two agents have an ncentve to form a lnk, ths lnk may decrease socal welfare. In the Bramoulle and Kranton model, when an agent forms a lnk wth an agent j, ths lnk has a postve mpact on the expected utlty of s neghbors. Indeed, n ther model there s equal ncome sharng between all the agents of the components. Therefore due to the addtonal lnk between and j, s neghbors wll share ther ncome wth an addtonal agent (agent j) and ther expected utlty wll ncrease. It follows that t can be that two agents have no ncentve to form a lnk, and ths lnk ncreases the socal welfare. Some papers explan partal rsk-sharng by self-enforcng mechansms n networks (Bloch et al., 2008). 3 These models consder that ndvduals can use ther lnks to punsh ndvduals who devate from the nsurance scheme. For nstance, f an agent devates from the nsurance scheme (.e. fals to transfer money to drectly connected agents that have negatve ncome shocks), the vctm wll communcate such behavor to other connected agents who n turn wll termnate the nsurance scheme wth the devatng household as a punshment. In ths paper, we do not deal wth the self-enforcement mechansm problem. Instead, we assume that establshng a relatonshp s costly and t commts the partes to future resource sharng, say, due to a socal norm or a socal punshment n case of non-sharng. 4 More precsely, we assume that the self-enforcement mechansm problem s solved when agents nvest enough tme and resources n the nformal nsurance agreements. 5 The rest of the paper s organzed as follows. In secton 2, we present the defntons and the basc model setup. In secton 3, we examne parwse stable networks and effcent networks n the basc model context. In secton 4, we extend the basc model by ntroducng dfferent types of agents heterogenety. In secton 5, we conclude. 3 Ths lterature extends the lterature about the robustness of mutual nsurance (see for nstance Gencot and Ray, 2003). 4 Ths knd of relaton can be llustrated wth the marrage of daughters n Inda whch are arranged to maxmze gans from rsk sharng, see Rosenzweg and Stark, The tme an agent nvests n the relatonshp and the socal punshment n case of non-sharng are related snce a blateral relaton n whch agents have nvested a lot of tme can be more easly observed by the peers. 6

7 2 Basc model setup Let N = {1,..., n} be a communty of n, n 3, ex ante dentcal agents. Agents receve an endowment and consume resources. Each agent s endowment s a random varable that takes two values. The low endowment state s called state 0 whle the hgh endowment state s called state 1. Each agent obtans an endowment 0 n state 0 whle she obtans Θ > 0 n state 1. State 1 occurs wth probablty p > 0 whle the low endowment state occurs wth probablty 1 p > 0. The realzatons of endowments are ndependent and dentcal across the agents. Networks. To model blateral mutual nsurance agreements n a small populaton, we use tools from the theory of networks. Although the agreements themselves are blateral, the amount of resources consumed by each agent depends on how many other agents she s connected wth, and the endowments of these agents. Hence tools from network theory are useful for modelng such blateral nsurance networks. In the model, we assume that ndvduals and j can have a mutual nsurance agreement by formng a costly lnk between themselves. Ths assumpton reflects the dea that there are always some costs (tme at the least) to buld a relatonshp. We represent lnks and a network of lnks wth the followng notaton: A network g s an n n matrx, where g j = 1 when and j are lnked (.e., have establshed a rsk-sharng agreement) and g j = 0 otherwse. We assume that rsk-sharng relatons are mutual, so that g j = g j. By conventon, g = 0. Let g + g j denote the network obtaned by replacng g j = 0 n g by g j = 1. Smlarly, let g g j denote the network obtaned by replacng g j = 1 n g by g j = 0. We say that there s a chan between two agents and j n the network g f there exsts a sequence of agents 1,..., k such that g 1 = g 1 2 = g 2 3 =... = g k j = 1. A subset of agents s connected f there s a chan between any two agents n the subset. A component of the network g s a maxmal connected subset of agents. Moreover, network g N s a sub-network of g f t conssts n the agents n N N, and N and j N are lnked n g N f and only f they are lnked n g. The empty network s the network where all agents have formed no lnks. The complete network s the network where each agent has formed lnks wth all the agents. A k-regular network s a network where all agents have formed exactly k lnks. A k -regular network s 7

8 a network where all agents but one have formed k lnks; the agent who s the excepton has formed k 1 lnks. An k + -regular network s a network where all agents but one have formed k lnks; the agent who s the excepton has formed k + 1 lnks. An almost-k-regular-networks s ether a k -regular network, or k + -regular network. We llustrate the notons of k-regular network, and k + -regular network n Fgure regular network 3 + -regular network Fgure 1: Networks archtectures In the followng, the neghbors of agent, that s agents wth whom has formed a lnk, wll play a crucal role. Hence we defne N (g) = {j N g j = 1} as the set of the neghbors of. Let n (g) = N (g) be the degree of agent. Payoffs. Havng descrbed the set of players and ther strateges, we now ask: Gven a network g, how are expected payoffs determned under dfferent endowment realzatons n the network? We consder a benchmark model where ex ante agents are dentcal: they get the same resources Θ when they draw state 1 and the same resources 0 when they draw state 0. Moreover, f an agent draws state 1, then she always gves δ (0, 1) to each of her neghbors who has drawn state 0. 6 Conversely, f an draws state 0, then each of her neghbors who has drawn state 1 always gves her δ. Note that n our model, agents may receve dfferent amounts of transfers dependng on the network archtecture. Moreover, we assume that Θ > (n 1)δ. In ths paper we assume that the payoff obtaned by an agent, say, can be dvded nto two parts. 1. The benefts part whch nvolves uncertanty captures the fact that each addtonal lnk 6 Here, we assume that the transfer amount δ comes from some knd of socal norm. Our goal n ths paper s to study what s the archtecture of the mutual nsurance network wthn a communty. Hence we do not explctly model where δ comes from. 8

9 formed by allows her to obtan addtonal nsurance when she draws the bad state (0), and the fact that has to nsure more agents when she draws the good state (1). 2. The costs part whch nvolves no uncertanty captures the fact that lnks are costly, wth addtonal lnks beng more costly. We now present these two parts of an agent s payoff functon. Benefts. Workng wth a general payoff functon n the context of a network formaton problem poses tractablty ssues. Hence from now on we deal wth the exponental utlty functon: u (x) = 1 exp[ ρx], where x s the ncome of agent, and ρ s a postve parameter. Consequently, our model exhbts Constant Absolute Rsk Averson. 7 It follows that f agent draws state 0 and k agents n her neghborhood draw state 1, then she obtans a beneft equal to b g (0, k) = u (kδ) = 1 exp[ ρkδ]. Conversely, f agent draws state 1 and k agents n her neghborhood draw state 0, then she obtans a beneft equal to b g (1, k) = u (Θ (n (g) k)δ) = 1 exp[ ρ(θ (n (g) k)δ)]. We now defne the expected neghborhood benefts (ENB) functon, B (g), whch captures the expected benefts obtaned by an agent gven her neghborhood n (g). We have: B (g) = φ(n (g)) = p n (g) k=0 +(1 p) n (g) k=0 ( n (g)) k p k (1 p) n(g) k b g (1, k) ( n (g)) k p k (1 p) n(g) k b g (0, k), (1) where ( x y) s just the probablty of y hgh resources out of x draws. In the followng, for each functon f we set f(x) = f(x) f(x 1). Moreover, snce we use non-contnuous functon, we use a slghtly modfed verson of convexty and concavty. We say that f s concave f for all x, f(x + 1) f(x) 0 and f s convex f for all x, f(x + 1) f(x) 0. Costs of lnks. In the frst part of the paper, we assume that the costs of formng lnks of agent depends on the number of lnks formed by, but does not depend on the degree of the agents wth whom frm forms lnks. More precsely, we assume that the cost of lnks 7 Here we use an exponental functon, but we obtan the same qualtatve results for some other CARA functons. 9

10 depends only on the number of lnks formed by : C (g) = f 1 (n (g)), where f 1 s a strctly ncreasng and convex functon. In addton, we assume that f 1 (0) = 0. Expected payoffs functon. The expected payoff functon, U (g), of each agent, gven the network g, s the dfference between the ENB functon and the cost functon of formng lnks: U (g) = B (g) C (g) = Φ(n (g)) = φ(n (g)) f 1 (n (g)). (2) Parwse stable networks and effcent networks. A network g s parwse stable f no par of unlnked agents would beneft by formng a lnk n g and f no agent would beneft from severng one of her exstng lnks n g. Formally, followng Jackson and Wolnsky (1996) we have () for all g j = 1, U (g) U (g g j ) and U j (g) U j (g g j ); and () for all g j = 0, f U (g) < U (g + g j ), then U j (g) > U j (g + g j ). An effcent network s one that maxmzes the sum of the expected payoffs of the agents. Let W (g) = N U (g) be the total expected payoffs obtaned n a network g. A network g e s effcent f W (g e ) W (g) for all networks g. 3 Parwse stable and effcent networks analyss Frst, we analyze the ENB functon. Second, we study parwse stable networks. Thrd, we turn to effcent networks. Our frst proposton provdes some useful propertes of the ENB functon. We provde a sketch of proof of ths proposton n the appendx. Proposton 1 Suppose that the ENB s gven by equaton 1. Then, the expected neghborhood benefts functon of agent s strctly ncreasng and strctly concave wth the number of lnks she has formed. Proposton 1 states that the ENB obtaned by agent s ncreasng. In other words, each agent prefers to be more nsured than less nsured when the cost of nsurance (the cost of formng lnks) s suffcently low. Moreover, Proposton 1 mples that Φ s concave: the margnal ENB 10

11 that an agent obtans from an addtonal lnk decreases wth the number of lnks she has formed. Consequently, f the cost of formng lnks was constant, then the ncentve of an agent to form an addtonal lnk would decrease wth the number of lnks she has formed. In the followng proposton, we examne the ncentve of agents to form an addtonal lnk when ther ncome ncreases. Proposton 2 Suppose that the ENB s gven by equaton 1. Then, the margnal expected neghborhood benefts functon of agent ncreases wth Θ. Proof By nspectng the proof of Proposton 1, we know that B (g, j) s equal to B (g, j) = p(1 p)(1 exp[ ρδ])(1 + p(exp[ ρδ] 1)) n (g) p(1 p)(exp[ρδ] 1) exp[ ρθ](exp[ρδ] + p(1 exp[ρδ])) n (g). (3) From ths we have: B (g, j) Θ = ρp(1 p)(exp[ρδ] 1) exp[ ρθ](exp[ρδ] + p(1 exp[ρδ])) n (g) 1 > 0. Let us explan the above result. Obvously, when agent draws the low ncome state, then her ENB s not affected by her ncome. Suppose now that agent draws the hgh ncome state. Due to the concavty of the utlty functon, when the ncome ncreases, the utlty functon of agent s less affected by the loss of money she ncurs when she has to help one of her neghbors. It follows that the margnal expected neghborhood benefts functon of agent ncreases wth Θ. We now deal wth parwse stable networks. We prove the exstence and we characterze parwse stable networks. To ensure the exstence of parwse stable networks, we use a theorem establshed by Erdös and Galla. We need the followng defnton to present ths theorem. Defnton 1 A fnte sequence s = (d 1, d 2,..., d n ) of nonnegatve ntegers s graphcal f there exsts a network g whose nodes have degrees d 1, d 2,..., d n. 11

12 Theorem 1 (Erdös and Galla, 1960) A sequence s = (d 1, d 2,..., d n ) of nonnegatve ntegers, such that d 1 d 2... d n, and whose sum s even s graphcal f and only f r n d r(r 1) + mn{d, r}, for every r, 1 r < n. 8 (4) =1 =r+1 In the followng lemma we provde condtons that ensure the exstence of three knds of networks that turn out to be qute useful subsequently. Lemma 1 Let n and k be nonnegatve ntegers wth n > k. 1. Let n or k be even. Then, the sequence s = (k,..., k) s graphcal. 2. Let n and k be odds. Then, the sequences s = (k, k,..., k, k + 1), s = (k, k,..., k, k 1) are graphcal. Proof See Appendx. Snce Φ s strctly concave, there exsts ˆk such that Φ(ˆk) > Φ(k) for all k ˆk. To present the next proposton, we need the addtonal defnton: let M(g) = { N : n (g) ˆk} be the set of players who have formed ˆk lnks n g. Proposton 3 Suppose that the benefts functon s gven by equaton 1. Network g s parwse stable f and only f for every agent M(g), n (g) < ˆk, and g M(g) s complete. Moreover, (a) f n or ˆk are even, then ˆk-regular networks are always parwse stable; (b) If n and ˆk are odd then, ˆk -regular networks are always parwse stable. Proof Let g be a parwse stable network. If an agent has formed more than ˆk lnks n g, then g s not a parwse stable snce ˆk maxmzes {Φ(k) : k {0,... n 1}}, and agent has an ncentve to remove a lnk. Smlarly, f there exst two unlnked agents and j who have formed less than ˆk lnks n g, then g s not a parwse stable network snce and j have an ncentve to form a lnk together. It follows that for M(g), n (g) < ˆk, and g M(g) s complete. Let us now assume that n or ˆk are even. By Lemma 1, the sequence of degrees s = (ˆk, ˆk..., ˆk) s graphcal: we can buld a ˆk-regular-network where no agent has an ncentve to remove lnks and no couple of agents have an ncentve to form an addtonal lnk. The 8 The theorem can also be found n Harary, 1969, Chapter 6 pp and the statement here s based on hs presentaton. 12

13 result follows. Let us now assume that n and ˆk are odd. By Lemma 1, the sequence of degrees s = (ˆk, ˆk..., ˆk 1) s graphcal: we can buld a ˆk -regular-network where no agent has an ncentve to remove lnks and where no couple of agents have an ncentve to form an addtonal lnk. The result follows. Suppose ˆk = 3. Network g drawn n Fgure 2 satsfes propertes gven n Proposton 9. We have M(g) = {1, 2, 3}. It s worth notng that snce the players only take nto account Fgure 2: Network g ther degree when they form lnks, the dentty of neghbors of players n M(g) cannot, n general, be known. The followng corollary establshes that when the costs of formng lnks are zero, the agents have to form some lnks n order to get some nsurance from the others. Corollary 1 Suppose that the ENB s gven by equaton 1. If the costs of formng lnks are null, then the the parwse equlbrum network s the complete network. Proof We know by Proposton 1 that the ENB s ncreasng n n (g). By nspectng the proof of Proposton 1, we know that the ENB s postve for n (g) = n 2. It follows that the complete network s the only parwse equlbrum network when the costs of formng lnks are null. Corollary 1 mples that f lnks had no costs, then all the agents would form nsurance lnks wth all other agents. By contnuty, ths result s true for small costs of formng lnks. We now focus on effcent networks. Proposton 4 Suppose that the ENB s gven by equaton 1. Suppose n or ˆk are even, then ˆk-regular networks are the unque effcent networks. Suppose n and ˆk are odd. If Φ(ˆk + 1) < Φ(ˆk 1), then ˆk -regular networks are the unque effcent networks. If Φ(ˆk + 1) > Φ(ˆk 1), then ˆk + -regular networks are the unque effcent networks. 13

14 Proof Suppose that n or ˆk are even. By Lemma 1, ˆk-regular networks exst. In a ˆk-regular network g, each agent maxmzes ts expected payoffs. It follows that g s effcent. Suppose that n and ˆk are odd. By Theorem 1, t s not possble to buld a ˆk-regular network. By Lemma 1, ˆk + -regular networks and ˆk -regular exst. In an almost-ˆk-regular network g, each agent except one, say, maxmzes ts expected payoffs. Snce cannot form ˆk lnks and Φ s concave, she maxmzes her payoffs when she forms k + 1 or k 1 lnks. If Φ(ˆk + 1) < Φ(ˆk 1), then the agent, who has not formed ˆk lnks, forms ˆk 1 lnks n an effcent network; ths agent forms ˆk + 1 lnks n an effcent network f Φ(ˆk + 1) > Φ(ˆk 1). We note that f n or ˆk are even then an effcent network s always parwse stable whle there exst parwse stable networks that are not effcent. Suppose that n and ˆk are odd. If Φ(ˆk + 1) < Φ(ˆk 1), then ˆk -regular network are effcent and parwse stable networks. If Φ(ˆk + 1) > Φ(ˆk 1), then ˆk + -regular networks are effcent. These networks are very smlar to parwse stable ˆk -regular networks. 4 Parwse stable networks wth heterogeneous agents In ths secton, we assume that agents are heterogeneous. Although the followng frameworks are harder to analyze than the prevous one, we are able to obtan some nsghts by makng some smplfyng assumptons. We restrct attenton to the lnear cost functon and assume that agent N ncurs a cost F > 0 for each lnk she forms. 9 Frst, we examne a stuaton where agents do not obtan the same ncome when they draw state 1. Agents n N Θ obtan an ncome equal to Θ f they draw state 1 and agents n N Θ obtan an ncome equal to Θ < Θ f they draw state 1. In the followng, we assume that N Θ and N Θ are even. We denote by Φ Θ (n (g)) the expected payoff of agent N Θ when she has formed n (g) lnks and Φ Θ (n (g)) the expected payoff of agent N Θ when she has formed n (g) lnks. By Proposton 1, we know that the ENB s strctly concave. It follows that there exst ˆk Θ whch maxmzes {Φ Θ (k) : k {0,... n 1}} and ˆk Θ whch maxmzes {Φ Θ (k) : k {0,... n 1}}. By Proposton 2, we know that B (g, j)/ Θ > 0, so ˆk Θ ˆk Θ. For the followng proposton we need some addtonal defntons. Let 9 It s possble to do the analyss wth the cost functon assumed n secton 2, but ths would be an harder task that would not provde any addtonal nsghts. 14

15 N Θ (k Θ, g) = { N Θ : n (g) = k Θ }, and N Θ (k Θ, g) = { N Θ : n (g) = k Θ }. Fnally, we set M (g) = N \ (N Θ (g) N Θ (g)). Proposton 5 Network g s parwse stable f and only f for every agent M (g), n (g) < ˆk x, x {Θ, Θ }, and g M(g) s complete. Proof Suppose that g s parwse stable. Then there s no agent who has an ncentve to remove one of ther lnks and no couple of unlnked agents (, j) who have smultaneously an ncentve to form a lnk together. It follows that no agent N x forms more than ˆk x lnks, and agents j M (g) are all lnked together. Suppose that for every agent M (g), n (g) < ˆk x, x {Θ, Θ }, and g M (g) s complete. By constructon, agents n N \ M (g) have no ncentve to modfy ther strategy. Smlarly, agent M (g) does not have any ncentve to remove a lnk; and can form addtonal lnks only wth agents n N \ M (g) who wll not accept any addtonal lnks. Consequently, g s parwse stable. We now llustrate, through an example, that ˆk Θ > ˆk Θ does not mply that for N Θ and N Θ, we have n (g) n (g) n a parwse stable network g. Suppose that N Θ = {1, 2}, N Θ = {3, 4}, ˆk Θ = 2, and ˆk Θ = 1. Then network g drawn n Fgure 3 s parwse stable. We observe that n 2 (g ) < n 3 (g ) whle agent 2 wants to form more lnks than agent Fgure 3: Networks g Fnally, there exst some costs of formng lnks such that no agent n N Θ has formed lnks and agents n N Θ have formed lnks. Corollary 2 There exsts F such that only agents n N Θ have formed lnks. 15

16 Proof By Proposton 2, we know that agents n N Θ have more ncentve to form lnks than agents n N Θ. We set ι(x) = (exp[ρx] exp[ρδ])(exp[ρδ] 1) exp[ρδ] exp[ρx]. By Corollary 1 and the concavty of the ENB, f F {ι(θ ), ι(θ)}, then agents n N Θ snce F < ι(θ), agents n N Θ have an ncentve to form lnks. have no ncentve to form any lnks. Smlarly, There exst stuatons where the ncomes of the agents and the costs of formng lnks are such that all rch agents have formed lnks together whle poor agents have formed no lnks, that s only rch people get access to nsurance. In that case, the nsurance mechansm ncreases the gap between the expected well-beng of rch people and the expected well-beng of poor people. We now propose an example where ths happens. Example 1 We assume N = {1,..., 6}, Θ = 7, Θ = 6, N Θ = {1, 2, 3}, N Θ = {4, 5, 6}, ρ = 0.2, δ = 0.5, and F = Then network g, where g N Θ s complete and g N Θ s empty, s the unque parwse stable network. Tll now, we have assumed that all agents who draw state 1 gve the same amount of resources, δ, to ther neghbors who draw state 0. We now consder an extenson of our benchmark model where ths assumpton s relaxed. In partcular, one can magne that agents do not have the same preferences wth respect to the generosty of the transfer scheme. Some agents may be wllng to gve a large amount of resources to ther unlucky neghbors, whle some others may be wllng to gve them small amount of resources. We wll assume that the populaton s parttoned nto two sets: the set of generous agents, N G, and the set of mserly agents, N M. If a generous agent draws state 1, then she gves δ G to each of her neghbors who draws state 0. Smlarly, f a mserly agent draws state 1, then she gves δ M to each of her neghbors who draws state 0. Obvously, δ G > δ M. We now assume that each agent has the same probablty of drawng state 0 or state 1: t does not depend on her preference regardng the transfer scheme. Moreover, f there are α 0 agents who belong to N G n the neghborhood of and k agents n N (g) who draw state 1, then there s a probablty gven by [( )( )] [ k ( )( ) ] α n (g) α α n (g) α Q(α, x, k) = / x k x l k l that x, x α, of these k agents belong to N G. We now defne b g (0, α, k), whch descrbes the expected benefts obtaned by n g when she 16 l=0

17 draws state 0, α agents n her neghborhood are generous, and k agents n her neghborhood draw state 1, as follows: α bg (0, α, k) = Q(α, y, k)(1 exp [ ρ(yδ G + (k y)δ M ) ] ). y=0 It s worth notng that f there exsts only one type of agents, then both defntons of b g (0, ) and b g (0,, ) are equvalent. Indeed, gven that the sze of the neghborhood of agent s n (g), f we assume that all agents n the neghborhood of agent are generous and gve δ = δ G to ther neghbors, then we have b g (0, n (g), k) = b g (0, k). Smlarly, f we assume that all agents n the neghborhood of agent are mserly and gve δ = δ M to ther neghbors, then bg (0, 0, k) = b g (0, k). We now defne the expected neghborhood benefts functon of agent N x, x {G, M}, gven that her neghborhood contans n (g) agents and α among them are generous agents. B (g) = φ(n (g), α) = p n (g) k=0 +(1 p) n (g) k=0 ( n (g)) k p k (1 p) n(g) k b g x(1, k) ( n (g)) k p k (1 p) n(g) k bg (0, α, k), (5) where b g x(1, k) s equal to b g (1, k) when agent gves δ = δ x, x {G, M}, to each of her neghbors who draw state 0. In the followng, we are nterested n the ncentves of agent N x to form a lnk wth an agent j N y, y {G, M}, gven that the cost assocated wth each lnk s F. We denote by Z x,y (g) = B (g + g,j ) B (g) the margnal expected neghborhood benefts obtaned by agent N x, x {G, M}, when she forms a lnk wth an agent whch belongs to N y, y {G, M}, n g. For the next results, we always use the expected neghborhood benefts functon gven by equaton 5. The next proposton shows that generous and mserly agents would prefer to be lnked wth generous agents. Proposton 6 We have Z G,G Proof We only prove that Z G,G (g) > Z G,M (g) > Z G,M (g), and Z M,G (g), snce Z M,G (g) > Z M,M (g). (g) > Z M,M (g) s establshed usng the same arguments. Let g G be the network whch s dentcal to g except that agent N G has formed an addtonal lnk wth agent j N G, and let g M be the network whch 17

18 s dentcal to g except that agent N G has formed an addtonal lnk wth agent j N M. Let α G = N (g G ) N G and α M = N (g M ) N G. We have α G = α M + 1. Moreover, we have Z G,G (g) Z G,M (g) = (1 p) n (g) ) k=0 p k (1 p) n(g) k ( b gg (0, α G, k) b gm (0, α M, k)). ( n (g) k Suppose that k agents respectvely n N (g G ) and n N (g M ) draw state 1. Then, we defne [ (α H(x) = Q(α G, x, k)/q(α M G)(, x, k) = n (g)+1 α ) ] [ G (α M)( / n (g)+1 α ) ] M whch provdes x the rato of the weghts (when they are defned) assocated wth 1 exp [ ρ(xδ G + (k x)δ M ) ]. H(x) s an ncreasng functon wth the number of agents who draw state 1 n the neghborhood of agent. Consequently, the weghts assocated wth the most valuable ncome are hgher n g G than the weghts assocated wth the most valuable ncome n g M. k x x k x It follows that Z G,G (g) > Z G,M (g). In the followng corollary, we establsh that there exst parameters such that a network where agents are lnked only wth other agents of the same type s a parwse stable network. We denote by g G/M the network such that g G/M,j = 1 f and only f, j N x, x {G, M}. The network g G/M can be seen as two dfferent networks: the complete network g G/ 1 whch contans all agents n N G and where g G/,j = gg/m,j contans all agents n N M and where g /M,j for all, j N G, and the complete network g /M whch = g G/M,j. The ENB of each player s not the same n g G/ and n g /M snce agents do not gve and receve the same amount of money n each of these networks. For ths corollary, we frst observe that f δ M = 0, then 0 = Z M,M The strct nequalty holds when δ M s suffcently small. (g) > Z G,M (g). Corollary 3 Suppose that for N M, N G, we have B (g G/M ) B (g G/M g G/M ) > Z G,M (g G/M ). Then, there exsts F > 0 such that g G/M s a parwse stable network. Proof We know that the ENB s concave and for all g, Z G,G N G, A 2 = B (g G/M ) B (g G/M g G/M,j (g) > Z G,M (g). It follows that for ) > Z G,M (g G/M ) = A 1. Moreover, by assumpton, for, j N M, we have A 3 = B (g G/M ) B (g G/M g G/M,j ) > A 1. We set F < mn{a 2, A 3 } and F > A 1. Frst, snce F > A 1, no agent n N G wll accept to form a lnk wth an agent n N M. Second, snce F < mn{a 2, A 3 } no agent N has an ncentve to remove one of her lnks n g G/M. The result follows. Corollary 3 shows that there exst stuatons where n a parwse stable network agents are parttoned accordng to ther type, that s generous agents are lnked together, mserly agents 18

19 are lnked together, but there are no lnks between these two types of agents. It s worth notng that for suffcently small costs of formng lnks, and close enough δ G and δ M, generous agents wll consent to form lnks wth mserly agents. Indeed, generous agents are rsk averse and prefer to be lnked wth mserly agents to be less nsured. 5 Costs of formng lnks depend on the neghborhood of agents 5.1 Cost functon and ENB Informal nsurance arrangements are potentally lmted by the presence of varous ncentve constrants. As a frst cut, t appears that the most mportant constrant arses from the fact that these arrangements are nformal,.e., not wrtten on legal paper. It follows that they wll be honored only f agents nvolved n such a relatonshp nvest tme. Snce each agent has a lmted amount of tme, the costs for agent of formng an addtonal lnk wth some agent j should ncrease wth the number of lnks formed by agent. Moreover, these costs should also ncrease wth the number of lnks formed by agent j. Indeed, t s more dffcult to establsh a relatonshp wth an agent who already has numerous lnks snce she has less tme avalable. 10 We capture these deas through the followng cost functon for lnk formaton: C (g) = f 1 (n (g)) + f 2 (n l (g)), l N (g) where f 2 s strctly ncreasng and convex. In addton, f 1 (0) = 0. Gven ths cost functon, an addtonal lnk formed wth agent j nduces a cost for agent equal to C (g + j) C (g) = f 1 (n (g) + 1) + f 2 (n j (g) + 1). Snce f 1 s strctly ncreasng and f 2 > 0, C (g + j) C (g) > 0. The expected payoff functon, U (g), of each agent, gven the network g, s the dfference between the ENB functon and the cost functon of formng lnks. We assume the same ENB 10 Another opton that makes such nformal arrangements feasble s the threat of punshment as n Bloch, Gencot and Ray,

20 as n secton 2. We have: U (g) = B (g) C (g) = φ(n (g)) f 1 (n (g)) + f 2 (n l (g)). (6) l N (g) Proposton 1 allows us to characterze some propertes of the margnal payoffs, U (g, j) = B (g + j) C (g + j) (B (g) C (g)), obtaned by agent n a network g when she forms an addtonal lnk wth agent j. We have U (g, j) = γ(n (g) + 1, n j (g) + 1) and γ(n (g) + 1, n j (g) + 1) = φ(n (g) + 1) f 1 (n (g) + 1) f 2 (n j (g) + 1), Clearly, γ s strctly decreasng n ts frst argument snce φ s strctly decreasng by Proposton 1 and f 1 s strctly ncreasng. Smlarly, γ s strctly decreasng n ts second argument snce f 2 s strctly ncreasng. T. Morrll (2011) has defned the class of network formaton game wth degree-base utlty functon under negatve externaltes. In the followng sectons, we complete ts results snce he manly focuses hs attenton on stuatons where transfers between agents are allowed Parwse stable networks Frst, we present an exstence result n ths context. Let k, k {0,..., n 1}, satsfy the two followng condtons: (1) γ(k, k ) 0, and (2) there s no k, k > k, such that γ(k, k) 0. In other words, k s the number of lnks whch allows an agent who has formed k lnks to obtan a postve margnal expected payoff from a lnk wth an agent j who has formed k lnks. Furthermore, t s a threshold snce no k hgher than k satsfes ths property. Recall that γ(, ) s strctly decreasng n ts two arguments. Hence, we have γ(k, k) > 0 for all k < k. For the next results n ths secton, we always use the payoff functon gven by equaton 6. Proposton 7 A parwse stable network always exsts. Proof See Appendx. Proposton 7 shows that our settng s consstent wth our steady state soluton of parwse stablty. The next proposton mposes condtons that a parwse stable network must satsfy 11 In hs paper, the author does not deal wth the exstence of parwse stable networks. Moreover, he does not provde a full characterzaton of parwse stable networks when transfers are not allowed. 20

21 and sheds lght on when nsurance arrangements wll exhbt symmetrc and asymmetrc structures. In partcular, ths proposton does not exclude from the set of parwse stable networks those networks where agents are n asymmetrc postons relatve to the amount of nsurance they receve. To show ths result, we need the followng defnton. Let S l (g) = {j N n j (g) = l} be the set of agents who have formed l lnks n g. It s worth notng that the set S l (g) s empty f no agents form l lnks n g. 12 Proposton 8 If a network g s parwse stable, then the two followng condtons are satsfed: 1. (Q1) If l, l > k, then there s no lnk between an agent who belongs to S l and an agent who belongs to S l n g. If l, l < k, then there s a lnk between an agent who belongs to S l and an agent who belongs to S l n g. 2. (Q2) Suppose l l < k and k < k k. If there s a lnk between an agent who belongs to S l and an agent who belongs to S k n g, then there s a lnk between an agent who belongs to S l and an agent who belongs to S k n g. 3. (Q3) If γ(k, k + 1) < 0, then there s no agent who have formed more than k lnks. Proof Let g be a parwse stable network. To ntroduce a contradcton, suppose that g does not satsfy property Q1. In partcular, suppose that there s a lnk between an agent, say, who belongs to S l and an agent, say j, who belongs to S l, wth l, l > k, n g. Then, the margnal payoff of agent j from the lnk wth agent s equal to γ(n j (g), n (g)) wth (n j (g), n (g)) (k + 1, k + 1). Snce γ s strctly decreasng n ts two arguments, we have γ(n j (g), n (g)) γ(k + 1, k + 1) < 0. It follows that the lnk between agent and j does not belong to g, a contradcton. Smlarly, suppose that there s no lnk between an agent, say, who belongs to S l and a agent, say j, who belongs to S l, wth l, l < k, n g. Then, the margnal payoff assocated wth the lnk between agents and j for agent j s equal to γ(n j (g), n (g)) wth (n j (g), n (g)) < (k, k ). Snce γ s strctly decreasng n ts two arguments, we have γ(n j (g), n (g)) > γ(k, k ) 0. Smlarly, for agent we have γ(n (g), n j (g)) > γ(k, k ) 0. It follows that agents and j have an ncentve to be lnked n g, a contradcton. 12 We use S l (g) = S l when there s no doubt about the network beng studed. 21

22 Next suppose that g does not satsfy property Q2. In partcular, consder l, l, k, and k such that l l < k and k < k k and suppose that there s a lnk between an agent, say, who belongs to S l and an agent, say j, who belongs to S k n g and there s no lnk between an agent, say, who belongs to S l and an agent, say j, who belongs to S k n g. Snce there s a lnk between agents and j we have γ(n (g), n j (g)) 0 and γ(n j (g), n (g)) 0. Moreover, snce there s no lnk between agents and j we have γ(n (g), n j (g)) < 0 or γ(n j (g), n (g)) < 0 wth n (g) n (g) and n j (g) n j (g). These nequaltes are not compatble wth the fact that γ s strctly decreasng n ts two arguments, a contradcton. Fnally, suppose that γ(k, k +1) < 0, then agents who have formed k lnks wll never form a lnk wth an agent who has k > k neghbors n g snce γ s decreasng n ts second argument. Snce agents who have formed more than k lnks do not formed lnks together, the neghbors of agent who has formed k > k lnks have formed k < k lnks. Agents, who have formed k < k lnks, form lnks together. It follows that agents who have formed k < k are less than k. Consequently the number of neghbors of agent s k < k < k, a contradcton. We now graphcally llustrate Proposton 8. In Fgure 4, network g satsfes the propertes gven n Q1. Indeed, f we assume that k = 4, we observe that agents 1 and 2 are nvolved n k + 1 lnks, agents 3, 4 and 5 are nvolved n k lnks and agents 6 and 7 are nvolved n k 1 lnks, and they are lnked Fgure 4: Network g satsfyng Q1 Note that n Proposton 8 we do not examne agents who have formed exactly k lnks. Indeed, these agents can form lnks both wth agents who have formed x > k lnks and wth agents who have formed y < k lnks. Proposton 8 hghlghts several propertes of parwse stable networks n a communty of ex ante dentcal agents. Frst, agents who obtan nsurance from more than k agents (they are 22

23 the most nsured agents) are never lnked together. Ths property llustrates the fact that some agents play a specfc role n the provson of mutual nsurance n the parwse stable networks: these agents nsure (and are nsured by) a large part of the populaton. But an agent of ths type does not nteract wth other agents of ths type. In some sense there may exst some nsurance leaders but these leaders themselves are not lnked by mutual nsurance agreements. Secondly, we fnd that there s a knd of soldarty among the less nsured agents: agents who obtan the smallest amount of nsurance are always lnked together. Ths property llustrates the fact that agents, who do not have a suffcent amount of nsurance, wll always reach mutual nsurance agreements. Fnally, Q3 establshes that the asymmetrc poston of agents wth regard to the nsurance, can be reduced f the cost of formng lnks becomes suffcently hgh when the neghbors has more than k neghbors. From these two propertes and the pgeon hole prncple 13, we obtan the followng corollary. Corollary 4 Agents who have formed a number of lnks hgher than k (and obtan the hghest amount of nsurance) are always fewer than the rest of the populaton. Lastly, parwse stable networks may dvde the populaton nto sets of agents who are n asymmetrc postons relatve to ther rsk exposure. In other words, n a parwse stable network, some agents are better off snce they obtan nsurance from others agents, who are nvolved n few mutual nsurance agreements themselves. Note that n our model, agents can have dfferent numbers of blateral nsurance agreements, but ths dfference s bounded. Indeed, let n(g) = max N {n (g)} be the number of lnks formed by the agents who have formed the hghest number of lnks and let n(g) = mn N {n (g)} be the number of lnks formed by the agents who have formed the smallest number of lnks. Corollary 5 Let g be a parwse stable network. Then (a) n(g ) l k S l(g ) and (b) n(g ) l<k S l(g ). The above corollary allows us to bound the dfference of lnks n whch the hghest nsured agents and the lowest ones are nvolved. We have n n S k (g ). 13 Ths prncple states the followng. Let n and k be postve ntegers, and let n > k. Suppose we have to place n dentcal balls nto k dentcal boxes, where n > k. Then there wll be at least one box n whch we place at least two balls. 23

24 Moreover, Proposton 8 establshes that networks where agents are n symmetrc postons and networks where agents are n asymmetrc postons are canddates to be parwse stable. In the next corollary, we show that there always exst some symmetrc networks n the set of parwse stable networks. In other words, there always exst parwse stable networks where ether all agents obtan an dentcal, or a very smlar amount of nsurance. Corollary 6 If n or k are even, then there exsts a k-regular network whch s parwse stable. If n and k are odd, then, there exsts an almost-k-regular network whch s parwse stable. Proof Ths result follows the proof of Proposton 7. Corollary 6 establshes that the set of parwse stable networks always contans ether a k - regular network, or an almost-k -regular network. The fact that agents are n symmetrc postons n a parwse stable mutual nsurance network does not mean that these agents obtan the same amount of benefts. Indeed, n our model agents who draw state 0 always obtan a lower amount of benefts than the benefts obtaned by agents who draw state 1. Fnally, we hghlght the fact that n our settng agents can be parttoned nto dstnct components n a parwse stable network. Moreover some of these stable nsurance networks may also be locally complete as the network between agents 1, 2 and 3 n the followng example. Example 2 Suppose N = {1,..., 12} and k = 2. Then, network g shown n Fgure 5 s a parwse stable network Fgure 5: Network g 24

25 5.3 Effcent Networks We now deal wth the effcent networks. In ths secton, we always use the payoff functon gven by equaton 6. Let η(k) = φ(k) f 1 (k) kf 2 (k). By assumpton, φ s concave, and f 1 s convex. Moreover, kf 2 (k) s convex, snce f 2 s ncreasng and convex. It follows that η s concave and admts a unque maxmum. Moreover, note that the neghbors of agent are connected to an agent wth n (g) neghbors. Consequently, we have W (g) = N [φ(n (g)) f 1 (n (g)) n (g)f 2 (n (g))] = N η(n (g)). We know that there exsts k e {0,..., n 1} such that η(k e ) s maxmal. Therefore N η(ke ) N η(n (g)) for all n (g) {0,..., n 1}. Moreover, we have arg max k k e η(k) {k e 1, k e + 1}, snce η s concave and maxmum for k = k e. Therefore, we have n 1 =1 η(ke ) + max{η(k e 1), η(k e + 1)} n 1 =1 η(ke ) + η(n n (g)) for n n (g) k e. These observatons are summarzed n the next proposton. Proposton 9 Let g e be an effcent network. Then, g e s ether a k e -regular network, or an almost-k e -regular network. Proof We use Lemma 1 to obtan the exstence of one of these three networks. It follows from Corollary 6 that f k or n are even, then only k -regular networks can be both parwse stable and effcent. However, n the followng corollary we show that k -regular parwse stable networks and effcent networks do not always concde. More precsely, we establsh that n the k -regular stable network, agents wll form at least the same number of lnks as n the effcent network. Observe that n ths case, a non-effcent parwse stable network s always over-connected from the effcency perspectve. To smplfy the presentaton, we assume that n s even. Corollary 7 Suppose n s even. Then n the k -regular parwse stable network agents wll form at least the same number of lnks as n the effcent network. Proof Suppose n s even. Then by Lemma 1, network g where all agents have formed k lnks, and network g e where all agents have formed k e lnks, exst. By Corollary 6, we know that g s parwse stable, and by Proposton 9, we know that g e s effcent. Moreover, we have γ(k, k) (η(k) η(k 1)) = (k 1)(f 2 (k) f 2 (k 1)) 0, for 0 < k < n snce f 2 s strctly ncreasng. Now, by defnton of k e, we have η(k e ) η(k e 1) 0. It follows that we 25