Risk Aversion and Social Networks

A Perfect rationality for m=3

There is an important issue concerning the rationality of players for m>2. In particular, the clustering of the network can lead to a situation, such that anytime a node i links through network there is a positive probability that one or more of newly observed neighbors of neighbor have already been observed and not chosen in previous linking stages. A completely rational individual should take this into account. Since the entering node has not linked up to such node(s), i has information about them. This affects the mean field analysis, because the expected payoff from linking to such neighbors of neighbors is lower than the expected payoff i gets from observing and linking to someone i has no information about. In this section, we illustrate this argument formally.

Denote the nodes that t+1 connects in each linking stage as j₁, j₂ and j₃. First, note that this issue never concerns the first and last linking decision, since the first is always random, while in the last linking decision the entering nodes do not care about who they observe afterwards. Hence, for m=3 the only linking decision, in which he may observe someone, whom he has already observed, is the second one. Suppose that t+1 decides to link to a node j₂∈N_j1(t+1) such that j2∈arg maxj∈Nj1(t+1)uij. If so, then there is a positive probability that j₁ is connected to a neighbor of j₂. It this occurs there exist a node l, an out-degree neighbor of both j₁ and j₂, who t+1 observes after linking to j₁ and will observe after linking to j₂. Furthermore, there is an important information in the fact that t+1 observed l, but has not connected to him.

Formally, t+1 links to a j2∈arg maxj∈Nj1(t+1) uij if

(7)maxj∈Nj1(t+1)uij−∑s=1m−1{C(g)s(m−s)m[u¯−m−smu¯−sm∫amaxj∈Nj1(t+1)uijudF(u)]}>u¯−rt+1 (7)

where C(g) is the fraction of transitive triples in the population and measures the average probability that a triangle exists. C(g)=pθ(m−1)m2−pθm for m=3 and reflects the average probability that t+1 observes m–1 new individuals and 1 individual t+1 has already been observed and have not chosen because the utility he would reported to t+1 was lower that maxj∈Nj1(t+1)uij. The second expression, u¯−m−1mu¯−1m∫amaxj∈Nj1(t+1)uijudF(u), reflects the expected utility loss due the fact that t+1 observes only m–1 new individuals (instead of m), taking into account the expected utility from the individual observed and unchosen in the previous linking stage.

After some simplification of (7), we get

(8)maxj∈Nj1(t+1)uij−∑s=1m−1C(g)ss2ms[u¯−∫amaxj∈Nj1(t+1)uijudF(u)]>u¯−rt+1 (8)

(9)maxj∈Nj1(t+1)uij−[∑s=1m−1C(g)ss2ms][∫maxj∈Nj1(t+1)uijbudF(u)]>u¯−rt+1. (9)

As a result, the probability that t+1 links through network search in its second linking decision is

p2nd(rt+1)=1−F[u¯−rt+1+∑s=1m−1C(g)ss2ms∫maxj∈Nj1(t+1)uijbudF(u)m]>p(rt+1).

Then, the expected probability of node i<t+1 to receive a new link in t+1, analogous to expression (15), is

(10)ddi(t)dt=1t+[1−p2nd(rt+1)t+p2nd(rt+1)di(t)t1m]+[1−p(rt+1)t+p(rt+1)di(t)t1m]. (10)

The only difference between (15) and (10) is the intermediate term. The effect of perfect rationality is to enhance global search. In (10), it increases the probability of receiving a random link and decreases the probability of receiving a link from a neighbor of a neighbor. The overall effect, hence, depends on the current in-degree of each node. In particular, nodes with large in-degrees will be negatively affected by perfect rationality, because a large fraction of nodes they receive is through local linking. Nodes with low connectivity, on the other hand, benefit from the form of rationality we model here, since they receive almost no links through network anyway. Formally, ∂ddi(t)dtp(rt+12nd)=di(t)t1m−1t>0 if d_i(t)>m. Hence, the effect of the rationality discussed here is the following:

• If d_i(t)>m, i receives a link with lower probability than in the original specification,

• If d_i(t)=m, i is unaffected by the new specification,

• If d_i(t)<m, i receives a link with higher probability in the new specification.

In sum, the effect of the perfect rationality considered here is to enhance random search. This will affect the in-degree of each agent as a function of his connectivity. From the global point of view, the tails of the degree distribution shift down, more frequent global search lowers the clustering coefficient, and the distances would shrink.

B Generated Network Architectures

In this section, we show theoretically and via simulations that the network formation process proposed in Section 3 exhibits typical features of empirical social networks. Given the complexity of the problem (especially due to the dependence of meetings on the network structure), we rely on mean-field analysis of the model. The mean-field approach approximates the complex evolution of a stochastic system by a simpler deterministic system, in which the evolution is determined by the expected change. The first results is theoretical: ^[20]

Theorem 6Under mean field approximation,

1. if m>1 and p(r_L)>0, the (complementary) cumulative distribution function of in-degrees in period t can be characterized as

   • (11)1−Ft(d)=(m(m+pθ−mpθ)(m−1)pθd+m(m+pθ−mpθ)(m−1)pθ)m(m−1)pθ (11)

2. the average clustering coefficient in the network satisfies

   • (12)C(g)≥pθm2(m−1). (12)

Proof of Theorem 6. Let us first prove part (i) of the theorem. For an entering individual t+1 after linking to j_s, p(r) equals the probability that at least one of m friends of j_s is more attractive than the benefits of linking randomly, that is

(13)p(r)=1−F(u¯−r)m. (13)

This probability naturally depends on the risk aversion of agent t+1, such that p(r_H)>p(r_L), and the expected probability that a random agent finds it optimal to follow a network-based meeting is p_θ=θp(r_H)+(1–θ)p(r_L).

At entrance, the linking process of t+1 is as follows: She first links up randomly. Thus, for a particular agent i<t+1 the probability of receiving this link is 1t. Once this link has been created, t+1 faces m–1 decisions between linking locally through the network (by observing the neighbors of his neighbors), or linking to a randomly chosen agent from the population. In this case, the probability of i to increase its degree in one of these decisions is approximately

(14)1−p(rt+1)t+p(rt+1)di(t)t1m, (14)

where the first part corresponds to the probability that t+1 decides for a random search and links up to i. The second part of the expression is the joint probability of three events: (i) t+1 finds it attractive to connect through the network structure, p(r_t+1), (ii) she has connected to one of the d_i (in-degree) neighbors j of i in the previous decision, d_i(t)/t, ^[21] and (iii) i has the largest gain for t+1 out of the (out-degree) neighbors of j, 1/m.

Given that each link i<t+1 can receive at most one link in each period, that is, multiple links are ruled out, we can write the deterministic change of i’s in-degree in period t as

(15)ddi(t)dt=1t+(m−1)[1−pθt+pθdi(t)t1m]. (15)

Note that (15) can be rewritten as ddi(t)dt=adi(t)t+bt+c, where a=(m−1)pθm,b=[1+(m–1)(1–p_θ)], and c=0. Given that m>1 and that p(r_L)>0 ensures p_θ>0, the first part of Lemma 1 in Jackson and Rogers (2007) applies.

As for the clustering coefficient, consider an agent i. Each agent initially creates 1 random link and afterwards faces m–1 decisions to either link locally or search randomly. The first case occurs with probability p(r_i). Thus the agent has on average p(r_i)×(m–1) links that are based on network search. If k is found through network search, then it must be through j to whom i is also linked. So we have g_ij=g_jk=g_ik=1. Each such network-searched link creates at least one transitive triple. Given that the amount of triples for which g_ij=g_jk=1 equals m² and E[p(r_i)]=p_θ, we obtain (12).    ■

Theorem 6 shows that – as long as there is a positive probability of low-risk individuals to find an attractive agent through network – f(d)∝d−(m(m−1)pθ+1) for large d; that is, the in-degree distribution of agents for large d has a power-law distribution in the tail, and the average clustering coefficient will be strictly positive independently of other characteristics of the model.

To check the precision of the mean-field approximations and to check whether the generated architectures exhibit other stylized facts of empirical social networks, we also run simulations of the model and match them with the approximations. The simulation assumes that u_ij is drawn from a standard uniform distribution. Agents have a risk premium of either r_L=0 and r_H=0.25 with equal probability (θ=0.5). We initially set m=2 and we generate a network of 5000 nodes. Figures 4 and 5 contain various plots for four of the five stylized facts of observed social networks that our model predicts. The distances are only discussed at the end of this section.

Figure 4

(A) The Predicted (solid line) and Simulated (crosses and circles) Degree Distributions. (B) The Average Clustering Coefficient.

The solid line is the predicted lower bound from proposition 4.

Figure 5

(A) Relation Between Degree and Clustering. (B) Assortativity.

The simulations confirm the findings from Theorem 6 and Proposition 1. Figure 4A contrasts the predicted in-degree distribution with the simulated one. We can conclude that the mean field approach approximates very well the degree distribution generated by the model. Moreover, we distinguish between the high (crosses) and low (circles) risk premium types and find no systematic difference. Figure 4B plots the average clustering coefficient (in terms of fraction of transitive triples) for several values of m. The figure shows that the clustering coefficient is indeed positive and lies above the lower bound derived in Proposition 6. In fact, the simulated values of the average clustering coefficient are well above and increase over m, suggesting that the more connections the agents of our model form, the more clustered the network becomes.

In addition, the generated networks exhibit short network distances, assortativity, and the negative clustering/degree correlation. The first states that the average network distances and the largest distance between two (reachable) nodes in real-life networks are in general low in relation to the size of the network. The second property, assortativity, is a tendency such that high (low) degree nodes are more likely linked to high (low) degree nodes. Last, negative clustering/degree correlation simply suggests that the larger the degree of the node the lower its clustering coefficient. The reader is referred to Goyal (2007) or Jackson (2008) for formal definitions and evidence.

Figure 5A shows the negative clustering/degree correlation. (Here the clustering coefficient is measured ignoring the direction of the links.) The x- and y-axes plot the degree and clustering, respectively, and there is an obvious negative relation between the two variables in the graph. Moreover, in this plot we also make a distinction between the clustering coefficient of high risk-averse agents and low risk-averse agents. The plot shows that the clustering coefficient is substantially higher for high-risk averse agents, in particular for low degree values, where the majority of the nodes lies. To check for assortativity, in Figure 5B we draw a plot with the degree of a node on the x-axis and the average degree of an out-neighboring node on the y-axis. This plot shows a positive correlation. Nodes with high degree have also high degree neighbors, indicating positive assortativity. We also compute the degree correlation, which is 0.260, well above zero.

To check for network distances, we compute the average networks distance and the largest distance between two nodes in the resulting simulated network, again ignoring directions. The obtained values are 5.74, and 13, respectively, thus of the order of ln(n).

C Proofs

Proof of Proposition 1. Using that the initial in-degree of entering agents is 0, solving (15) leads to the in-degree of an agent i at period t:

(16)di(t)=[m(m+pθ−mpθ)(m−1)pθ](ti)(m−1)pθm−m(m+pθ−mpθ)(m−1)pθ. (16)

Given that it is independent of r_i, the first part of the proposition directly follows. The second follows from that p(r_H)>p(r_L), thus high types are more likely to search through the network. Each time an agent i decides to link through the network at least one transitive triple is created in its neighborhood, whereas the probability that a transitive triple is created after a random linking decision converges to 0 for large t. The proposition directly follows.       ■

Proof of Proposition 2. Note that

E[∑j∈Niuij|ri=r]=u¯+(m−1){[1−p(r)]u¯+p(r)E[maxj∈Niuij|maxj∈Niuij>u¯−r]}.

Since

1−p(rH)]u¯+p(rH)E[maxj∈Niuij|maxj∈Niuij>u¯−rH]=∫au¯−rHu¯dF(u)m+∫u¯−rHbudF(u)m=∫au¯−rHu¯dF(u)m+∫u¯−rHu¯−rLudF(u)m+∫u¯−rLbudF(u)m<∫au¯−rHu¯dF(u)m+∫u¯−rHu¯−rLu¯dF(u)m+∫u¯−rLbudF(u)m=[1−p(rL)]u¯+p(rL)E[maxj∈Niuij|maxj∈Niuij>u¯−rL],

it directly follows that E[∑j∈Niuij|ri=rL]>E[∑j∈Niuij|ri=rH].      ■

Proof of Theorem 3. For any existing link ij, define L_ij an indicator that is 1 if i found j through local network search, and 0 if found by random search. For any existing link ij, let

u¯≡E[uij|Lij=0]=∫abudF(u)

and

u˜≡E[uij|Lij=1]=P[ri=rH|Lij=1]E[uij|Lij=1, ri=rH]+P[ri=rL|Lij=1]E[uij|Lij=1, ri=rL]=P[ri=rH|Lij=1]∫u¯−rHbudF(u)m+P[ri=rL|Lij=1]∫u¯−rLbudF(u)m

denote the expected payoff of linking up to a random individual and a neighbor of a neighbor, respectively. Let Li=∑j∈NiLij. Then

E[∑j∈Niuij|Ci=c]=u¯E[m−Li|Ci=c]+u˜E[Li|Ci=c].

Naturally, u˜>u¯. Hence, nodes who search more often locally will tend to earn higher payoffs.

To complete the proof, we now show that E[L_i|C_i=c] is weakly increasing in c for m<5. Note that each node i can close at most ∑j=1m−1j=m(m−1)2 triples in period i since links are directed and entering nodes can only link up to older nodes.

Suppose m=2. Then C_i is (approximately) 0 or positive, depending on whether the second link was random or via a friend of friend. Hence, E[L_i|C_i=0]=0 and E[L_i|C_i>0]=1.

Next, suppose m=3 and denote the out-degree neighbors of i as j₁, j₂, j₃. Then L_i is at most 2, and at most 3 triples can be closed. The first link does not close triples, the second link closes one triple if Lij2=1, and the third link closes one or two triples if Lij3=1. Hence, E[L_i|C_i=0]=0, E[L_i|C_i=1/9]=1, E[L_i|C_i≥2/9]=2.

Finally, suppose m=4 and let i have out-degree neighbors j₁, …, j₄. Then L_i is at most 3, and at most 6 triples can be closed. We have, E[L_i|C_i=0]=0, E[L_i|C_i=1/16]=1, E[L_i|C_i=2/16]=2,, E[L_i|C_i=3/16]∈(2, 3), and E[L_i|C_i=4/16]=3.     ■

Proof of Proposition 4. With utility (5), the certainty equivalent y of linking to a random agent solves

Ui(y)=EF[U(x)].

Hence, with normal distribution of payoffs, y is given by

y=u¯−ρi2σ2

The risk premium is defined as r=u¯−y=ρi2σ2, leading to

p(ρi2σ2)=1−F(u¯−ρi2σ2)m.

With a normal distribution, F(u¯−ρi2σ2)=Φ(−ρi2σ) where Φ(.) is the cumulative distribution function of standard normal distribution. Since it is decreasing with σ² and independent of u̅, the proposition directly follows.      ■

Proof of Theorem. Since p_θ=θ_p(r_H)+(1–θ)p(r_L), it follows from Proposition 4 that ratio of the global and local search probabilities 1+(1−pθ)(m−1)pθ(m−1) decreases with σ² and is independent of u̅_F. It then directly follows from Jackson and Rogers (2007), Theorem 6, that the degree distribution of g′ second order stochastically dominates the degree distribution of g whenever σ²>σ′², independently of u̅ and u̅̅′. Moreover, since p_θ increases with σ² and remains constant with u̅, the results on the clustering coefficient directly follow.      ■