Signaling in social network and social capital formation

Abstract

This paper presents a model of social capital and social network formation. The key interaction within the model is that whom an individual chooses to become friends with affects the value (social capital) of the friendship. In the model, how a player searches for and then forms friendships reveals how willing she is to engage in cooperation with a potential friend. Individuals observe their local network structure (friends and cliques) and the actions of players within these. Willingness to cooperate is private information and is captured by the discount factor of an individual. Cooperative types have high discount factors and can signal their type by forming a clique through befriending a friend of a friend. Uncooperative types do not form these kinds of friendships because of the local observability of their actions to all members of a clique. Thus, when a player meets someone with whom she shares a friend, her belief that the individual has a high discount factor is greater than the population average. In this sense, people “trust” each other more when they share a friend in common. Finally, I relate the primitives of the model to characteristics of the implied social network by nesting the sequential equilibrium in an algorithm of network formation proposed by Jackson and Rogers (Am Econ Rev 97(3):890–915, 2007). The model highlights a trade-off between maximizing the total amount of social capital in a society and distributing it equitably across individuals.

Appendices

Perfect Bayesian equilibrium and proof of Proposition 1

We denote a player $i$ that enters in period $t$ by $i=t$ and the existing $N$ players at $t=0$ by $\left\{ -N,\ldots ,-1\right\}$. A player’s strategy involves four different types of decisions: (1) choice of friends $f_{i}$; (2) choice of stake during each period for each game in which the player is the older of the two $s_{ij}^{t}$ for $i<j$; (3) choice of action during each period in each game $a_{ij}^{t}$; and (4) choice to delete a friendship. A player’s belief is over the type (patient/impatient) of each of his/her friends.

Patient player i strategy

Choice of friends $f_{i}\in \left\{ Random,Network\right\}$

$$\begin{aligned} f_{i}=Network \end{aligned}$$

Choice of stake $s_{ij}^{t}\in \left\{ High,Low\right\}$

$$\begin{aligned} s_{ij}^{t}=\left\{ \begin{array}{l} Low\quad \text { if } \left( t=j\right) \hbox {and} \left( \left| \pi _{j}^{t}\left( i\right) \right| =1\right) \\ High\quad \text {otherwise} \end{array} \right. \end{aligned}$$

Choice of strategy $a_{ij}^{t}\in \left\{ C,D\right\}$

For $i=t$

$$\begin{aligned} a_{ij}^{t}=\left\{ \begin{array}{l} C \quad \text {if}\left( \left( a_{ik}^{i}\ne D\text { for }k\in \pi _{i}^{i}\left( j\right) \right) OR\left( \left| \pi _{i}^{i}\left( j\right) \right| =1\right) \right) \text { and}\\ \qquad \quad \left( s_{ij}^{t}\text { consistent with equilibrium}\right) \text { and}\\ \qquad \quad \left( \left| \pi _{i}^{i}\left( j\right) \right| \le 2\right) \\ D\quad \text { otherwise} \end{array} \right. \end{aligned}$$

for $j=t$

$$\begin{aligned} a_{ij}^{t}=\left\{ \begin{array}{l} C\quad \text {if} \left( \left( a_{jk}^{j}\ne D\text { for }k\in \pi _{j}^{j}\left( i\right) \right) OR\left( \left| \pi _{j}^{j}\left( i\right) \right| =1\right) \right) \text { and}\\ \qquad \quad \left( s_{ij}^{t} \text {consistent with equilibrium}\right) \text { and}\\ \qquad \quad \left( \left| \pi _{j}^{j}\left( i\right) \right| \le 2\right) \\ D\quad \text {otherwise} \end{array} \right. \end{aligned}$$

for $t>i>j$

$$\begin{aligned} a_{ij}^{t}=\left\{ \begin{array}{l} C\quad \text {if } \left( \left( a_{ik}^{i}\ne D\text { for }k\in \pi _{i}^{i}\left( j\right) \right) OR\left( \left| \pi _{i}^{i}\left( j\right) \right| =1\right) \right) \text {and}\\ \qquad \quad \left( s_{ij}^{t^{\prime }} \text {consistent with equilibrium for} t^{\prime }\le t\right) \text {and}\\ \qquad \quad \left( \left| \pi _{i}^{i}\left( j\right) \right| \le 2\right) \\ D\quad \text {otherwise} \end{array} \right. \end{aligned}$$

for $t>j>i$

$$\begin{aligned} a_{ij}^{t}=\left\{ \begin{array}{l} C\quad \text {if } \left( \left( a_{jk}^{j}\ne D\text { for }k\in \pi _{j}^{j}\left( i\right) \right) OR\left( \left| \pi _{j}^{j}\left( i\right) \right| =1\right) \right) \text { and}\\ \qquad \quad \left( s_{ij}^{t^{\prime }}\text { consistent with equilibrium for} t^{\prime }\le t\right) \;\text { and}\\ \qquad \quad \left( \left| \pi _{j}^{j}\left( i\right) \right| \le 2\right) \\ D\quad \text {otherwise} \end{array} \right. \end{aligned}$$

Deleting friendships with individual j at the end of period t

$$\begin{aligned} \begin{array}{l} Delete\quad \text { if }\left( \left( a_{jk}^{j},a_{kj}^{j}\right) =\left( D,C\right) \right) \text { for }k\in \pi _{j}^{j}\left( i\right) \\ Keep\quad \;\;\text { otherwise} \end{array} \end{aligned}$$

Impatient player i strategy

Choice of friends $f_{i}\in \left\{ Random,Network\right\}$

$$\begin{aligned} f_{i}=Random \end{aligned}$$

Choice of stake $s_{ij}^{t}\in \left\{ High,Low\right\}$

$$\begin{aligned} s_{ij}^{t}=\left\{ \begin{array}{l} Low\quad \text {if}\left( t=j\right) \text {and}\\ \qquad \qquad \ \left( \left| \pi _{j}^{t}\left( i\right) \right| =1\right) \\ High\quad \text { otherwise} \end{array} \right. \end{aligned}$$

Choice of strategy $a_{ij}^{t}\in \left\{ C,D\right\}$

For $i=t$

$$\begin{aligned} a_{ij}^{t}=\left\{ \begin{array}{l} C\quad \text {if} \left( a_{ik}^{i}=\emptyset \text { for }k\in \pi _{i}^{i}\left( j\right) \right) \text {and}\\ \qquad \quad \left( s_{ij}^{t}\text { consistent with equilibrium}\right) \text { and}\\ \qquad \quad \left( \left| \pi _{i}^{t}\left( j\right) \right| =2\right) \\ D\quad \text { otherwise} \end{array} \right. \end{aligned}$$

for $t>i$

$$\begin{aligned} a_{ij}^{t}=D \end{aligned}$$

Deleting friendships with individual j at the end of period t

$$\begin{aligned} \begin{array}{l} Delete \quad \text {if}\left( \left( a_{jk}^{j},a_{kj}^{j}\right) =\left( D,C\right) \right) \text { for }k\in \pi _{j}^{j}\left( i\right) \\ Keep\quad \text { otherwise} \end{array} \end{aligned}$$

Player i ’s belief about player j

We only need to specify the beliefs between individuals where a relationship exists. A defection by either player destroys a relationship. Thus, all beliefs are specified under the condition that neither player has played $D$ within the relationship.

For $j<i$

$$\begin{aligned} \Pr \left( \delta _{j}=\delta _\mathrm{I}\right) =0 \end{aligned}$$

For $i<j<t \left( \text {In } \text { the} \text { case } \text { where } \left| \pi _{j}^{j}(i) \right| =2 \hbox { let } k\in \pi _{j}^{j}(i)\right)$

$$\begin{aligned} \Pr \left( \delta _{j}=\delta _\mathrm{I}\vert \left( a_{jk}^{j},a_{kj}^{j}\right) \ne \left( D,C\right) \right)&=0\\ \Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left( a_{jk}^{j},a_{kj}^{j}\right) =\left( D,C\right) \right)&=1 \end{aligned}$$

For $i<j$ and $t=j$ $\left( \hbox {let }k\in \pi _{j}^{j}(i) \hbox { in\;the\;case\;where }\left| \pi _{j}^{j}(i) \right| =2\right)$

$$\begin{aligned}&\frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) }\\&\quad <\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{j}\left( i\right) \right| =1;a_{ji}^{j} =\emptyset ;\left| Q_{i}^{t}\right| =d\right) \\&\quad <\frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) \left( \varPi _{i=0}^{\left( 1-\gamma \right) M-1}\frac{N-dd_{max}\left( t\right) -i}{N+t-1-i}\right) }\\&\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{j}\left( i\right) \right| =1;a_{ji}^{j}=C\right) =0\\&\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{j}\left( i\right) \right| =2;a_{ji}=a_{jk} =\emptyset ;\left| Q_{i}^{t}\right| =d\right) \\&\quad <\frac{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right] }{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i} \right) \right] +2\gamma \left( 1-\lambda \right) }\\&\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{t}\left( i\right) \right| =2;a_{ji}=\emptyset ;a_{jk}=C;\left| Q_{i}^{t}\right| =d\right) \\&\quad <\frac{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right] }{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right] +2\gamma \left( 1-\lambda \right) }\\&\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{t}\left( i\right) \right| =2;\left( a_{jk} ^{j}=D\right) \right) =1\\&\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{t}\left( i\right) \right| =2;a_{ji}^{j}=a_{jk} ^{j}=C\right) =0\\&\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{t}\left( i\right) \right| \ge 3\right) \in \left[ 0,1\right] \end{aligned}$$

Proof

First, note that after offering an off-equilibrium stake, the response from the other player is to defect, so there is never a strict incentive to do so. Second, note that after it is observed by player $i$ that a player $j$ has successfully defected on an individual $k$, then the decision to delete a friendship between $i$ and $j$ is weakly a best response, since the other player is also taking the same action. In all other cases, not deleting the friendship cannot be improved upon by deleting the friendship, since the same payoff may be achieved by simply playing defect in the next game with that player. Hence, these components of the player’s strategy are consistent with equilibrium play. These two observations allow us to focus on the players’ choice of actions and how they connect to the network. $\square$

After the first period of interaction, a player holds beliefs $\Pr \left( \delta _{j}=\delta _\mathrm{I}\right) \in \left\{ 0,1\right\}$. Also, no new information arrives about network connections and/or actions with other players which affects continuation play or beliefs. When $\Pr \left( \delta _{j}=\delta _\mathrm{I}\right) =1$, both types of players play the one-shot Nash outcome where players defect. When $\Pr \left( \delta _{j}=\delta _\mathrm{I}\right) =0$, it is straightforward to show that there is a range of discount factors $\delta _\mathrm{P}\in \left[ \underline{\delta _\mathrm{P}},1\right]$ and $\delta _\mathrm{I}\in \left[ 0,\overline{\delta _\mathrm{I}}\right]$, such that it is an equilibrium for the patient player to cooperate and the impatient player to defect.

There are three different network configurations which need to be analyzed in the first period of interaction between players: $\left| \pi _{i} ^{t}\left( j\right) \right| =1,2$ and $\left| \pi _{i}^{t}\left( j\right) \right| \ge 3$. The case $\left| \pi _{i}^{t}\left( j\right) \right| \ge 3$ is straightforward; it is common knowledge when two players are in this case and it is an equilibrium for both types of players to revert to the static Nash outcome of defecting, irrespective of each player’s belief. In the case $\left| \pi _{i}^{t}\left( j\right) \right| =1$, the belief of the existing player with $d$ friends is bounded:

$$\begin{aligned}&\frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) }\\&<\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{j}\left( i\right) \right| =1;a_{ji}^{j} =\emptyset ;\left| Q_{i}^{t}\right| =d\right) \\&<\frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) \left( \varPi _{i=0}^{\left( 1-\gamma \right) M-1}\frac{N-dd_\mathrm{{max}}\left( t\right) -i}{N+t-1-i}\right) } \end{aligned}$$

and it is straightforward to show that there is a range of discount factors $\delta _\mathrm{P}\in \left[ \underline{\delta _\mathrm{P}},1\right]$ and $\delta _\mathrm{I} \in \left[ 0,\overline{\delta _\mathrm{I}}\right]$, such that it is an equilibrium for an existing patient player to cooperate and an existing impatient player to defect.

In the case $\left| \pi _{i}^{t}\left( j\right) \right| =2$, players may be involved in the first or second game. Existing players hold beliefs:

$$\begin{aligned}&\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{t}\left( i\right) \right| =2;a_{ji}=\emptyset ;a_{jk}\ne D;\left| Q_{i}^{t}\right| =d\right) \\&\quad <\frac{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right] }{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right) \right] +2\gamma \left( 1-\lambda \right) }, \end{aligned}$$

unless a defection is observed in the other relationship and it is straightforward to verify that cooperation and defection are optimal for the appropriate ranges of discount factors. In the event of witnessing a defection, beliefs switch to $\Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left| \pi _{j}^{t}\left( i\right) \right| =2;a_{ji}=\emptyset ;a_{jk}=D;\left| Q_{i}^{t}\right| =d\right) =1$, and it is optimal for both types to play defect. New players hold beliefs $\Pr \left( \delta _{j}=\delta _\mathrm{I}\right) =0$ about the existing player. It is optimal for patient types to cooperate in the first and second game. For impatient types, it is optimal to cooperate in the first game and defect in the second, because defecting in the first game will cause the existing player in the second game to switch to defect.

The choice of friends influences the number of friendships which involve $\left| \pi _{i}^{t}\left( j\right) \right| =1,2$ and $\left| \pi _{i}^{t}\left( j\right) \right| \ge 3$. In networks which are large $N>>M$, the choice between random and network-based friendships has a negligible effect on the probability of $\left| \pi _{i}^{t}\left( j\right) \right| \ge 3$, which occurs with vanishingly small probability. For a patient new player, network friendships are optimal since they allow a patient player to avoid a period of low stakes play. For an impatient player, random friendships are optimal provided that:

$$\begin{aligned} 2x_{l}>z_{h}+x_{h}. \end{aligned}$$

Beliefs

It is straightforward to verify that a new impatient player defects on at least one player in each partition of his/her friends. This will lead to each friend deleting the friendship and the player being removed from the network after one period of play. Hence, for $j<i:$

$$\begin{aligned} \Pr \left( \delta _{j}=\delta _\mathrm{I}\right) =0\text {.} \end{aligned}$$

For $i<j<t$ and a player who does not defect during $t=j$, the Bayesian belief is as follows:

$$\begin{aligned} \Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left( a_{jk}^{j},a_{kj}^{j}\right) \ne \left( D,C\right) \right) =0\text {.} \end{aligned}$$

In the eventuation that a player is observed to succesfully defect (on another player $k$) during $t=j$, but the friendship with player $i$ survives until the second period (i.e., neither player deletes the friendship), then the off-equilibrium belief is that they are impatient:

$$\begin{aligned} \Pr \left( \delta _{j}=\delta _\mathrm{I} \vert \left( a_{jk}^{j},a_{kj}^{j}\right) =\left( D,C\right) \right) =1\text {.} \end{aligned}$$

We do not specify the belief when $\left| \pi _{j}^{t}\left( i\right) \right| \ge 3$. In this case, both types of player revert to playing defect, so it is inconsequential for the equilibrium.

When a friend is shared in common, the Bayesian belief that the player is impatient is less than:

$$\begin{aligned} \frac{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right] }{\lambda \left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right] +2\gamma \left( 1-\lambda \right) }\text {,} \end{aligned}$$

where, the term $\left[ 1-\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i} \right) \right]$ is the probability that at least one of $M-1$ friends chosen at random is friends with an individual with $d$ existing friends. This is an upper bound since this probability includes the possibility that $\left| \pi _{j}^{t}\left( i\right) \right| \ge 3$. In large networks, this probability is approximately zero. Also, this probability is unchanged after observing the action in the first game among members of $\left| \pi _{j}^{t}\left( i\right) \right|$ since both types cooperate in the initial game.

When no friend is shared in common and the player is yet to take an action, the belief is bounded above and below. The lower bound is as follows:

$$\begin{aligned} \frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i} \right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) }\text {,} \end{aligned}$$

where $\varPi _{i=0}^{M-1}\left( \frac{N-d-i}{N+t-1-i}\right)$ is the probability that an impatient type choosing $M-1$ friends at random does not choose any of $i^{\prime }s$ friends; $\left( 1-2\gamma \right)$ is the fraction of a patient player’s friends that are chosen at random and are not used to find additional friends in step 2 of the meeting process. This is a lower bound because there is some probability that the randomly chosen friends of a patient player know one another. It is straightforward to see that:

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0} ^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) }=\frac{\lambda }{\lambda +\left( 1-\lambda \right) \left( 1-2\gamma \right) }\text {.} \end{aligned}$$

The upper bound is given by:

$$\begin{aligned} \frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i} \right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) \left( \varPi _{i=0}^{\left( 1-\gamma \right) M-1}\frac{N+t-dd_\mathrm{{max}}\left( t\right) -i}{N+t-1-i}\right) }\text {,} \end{aligned}$$

where the terms are as before and $\varPi _{i=0}^{\left( 1-\gamma \right) M-1}\frac{N-dd_\mathrm{{max}}\left( t\right) -i}{N+t-1-i}$ is an upper bound on the probability that the individuals chosen at random by a patient player in step 1 are not within a distance of 2 of player $i$. This is an upper bound because $dd_\mathrm{{max}}\left( t\right)$ is the maximum number of people within a distance of 2 of an individual with $d$ friends, where $d_\mathrm{{max}}\left( t\right)$ is the person with the most friends in the network. Provided that $d_\mathrm{{max}}\left( t\right)$ grows slower than $t^{\frac{1}{2}}$, then:

$$\begin{aligned}&\lim _{t\rightarrow \infty }\frac{\lambda \left( \varPi _{i=0}^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right) }{\lambda \left( \varPi _{i=0} ^{M-1}\left( \frac{N+t-d-i}{N+t-1-i}\right) \right) +\left( 1-\lambda \right) \left( 1-2\gamma \right) \left( \varPi _{i=0}^{\left( 1-\gamma \right) M-1}\frac{N+t-dd_\mathrm{{max}}\left( t\right) -i}{N+t-1-i}\right) }\nonumber \\&\quad =\frac{\lambda }{\lambda +\left( 1-\lambda \right) \left( 1-2\gamma \right) }. \end{aligned}$$

We can verify that $d_\mathrm{{max}}\left( t\right)$ grows slower than $t^{\frac{1}{2}}$ by observing that each individual’s degree grow at a rate proportional to $t^{\gamma }$:

$$\begin{aligned} d_{i}\left( t\right) =\frac{1-\gamma }{\gamma }M\left( \frac{t+N}{i}\right) ^{\gamma }-\frac{1-\gamma }{\gamma }M\text {,} \end{aligned}$$

where $\gamma <\frac{1}{2}$. For derivation of the above, see the proof of Proposition 3.

Proof of Corollary 1

Proof

Note that:

$$\begin{aligned}&\Pr \left( \text {New link from a Patient player}\right) \\&=\Pr \left( \text {Patient player}\right) \times \Pr \left( \text {New link} \vert \hbox {Patient player}\right) . \end{aligned}$$

Now,

$$\begin{aligned} \Pr \left( \text {Patient player}\right) =\left( 1-\lambda \right) ; \end{aligned}$$

and,

$$\begin{aligned} \Pr \left( \text {New link} \vert \hbox {Patient player}\right) \approx \frac{\left( 1-\gamma \right) M}{t+N} +\frac{\left( 1-\gamma \right) Md_{i}\left( t\right) }{t+N}\frac{\frac{\gamma M}{\left( 1-\gamma \right) M}}{M}, \end{aligned}$$

where the first term is the probability of being selected at random and the second term is the probability that a new player chooses to link to the node. The second term consists of two parts: the first, $\frac{\left( 1-\gamma \right) Md_{i}\left( t\right) }{t+N}$, is the probability that one of the node’s neighbors is found at random; the second, $\frac{\frac{\gamma M}{\left( 1-\gamma \right) M}}{M}$, is the probability that the node is then chosen. This simplifies to $\frac{\left( 1-\gamma \right) M}{t+N}+\frac{d_{i}\left( t\right) \gamma }{t+N}$. $\square$

Proof of Proposition 2

We need only consider the cases where the size of partitions is 1, 2 in the limit of a large network. As the network becomes infinitely large $t\rightarrow \infty ,$ the maximum degree grows as $t^{\gamma }$ where $\gamma <\frac{1}{2}$. The size of neighborhoods a distance 2 away from any individual is bounded above by $t^{2\gamma }$ and hence contain a vanishingly small fraction of the population $\lim _{t\rightarrow \infty }\frac{t^{2\gamma }}{t}=0$. Alternatively, if individuals choose random friendships the maximum degree will grow as $\ln t$; again in this case $\lim _{t\rightarrow \infty }\frac{\left( \ln t\right) ^{2}}{t}=0$. The probability that a new player may form friendships in such a way that $\pi _{i=t}^{t}\left( j\right) >2$ is vanishingly small. Hence, to study the value generated by a new friendship, in the limit of a large network, it suffices to check how an equilibrium treats new friendships in just the cases $\pi _{i=t}^{t}\left( j\right) =1,2$.

Lemma 1

A sufficiently impatient (there exists $\overline{\delta }_\mathrm{I}$ such that for $\delta _\mathrm{I}\in \left[ 0,\overline{\delta } _{I}\right]$) new player plays defect against at least one individual in every partition of that player’s friends in every period.

Proof

Playing defect is a dominant action in the static game. There exists $\overline{\delta }_\mathrm{I}$ such that for $\delta _\mathrm{I}\in \left[ 0,\overline{\delta }_\mathrm{I}\right]$, the single period benefit is larger than any cost it may impose on future play. The only instance where the static dominant action may not be optimal is in the case $\pi _{i=t}^{t}\left( j\right) =2$ when playing the first of the two games. In this instance, the impatient player may initially cooperate and then defect. In the second period $\pi _{i=t}^{t}\left( j\right) \le 1$, so after two periods, both friendships are destroyed.

Also note that in the limit of a large network, the probability that a player meets a new player in their second period is zero.

Lemma 2

There exists $\underline{\delta }_\mathrm{P}$ such that for $\delta _\mathrm{P}\in \left[ \underline{\delta }_\mathrm{P},1\right)$ efficiency requires that all patient player pairs establish a cooperative relationship.

Proof

Lemma 1 implies that the older/existing player in a new relationship is patient with probability 1 in large networks. The expected value of a relationship is a weighted combination of the value from a patient–patient (weight $1-\lambda$) friendship and an impatient–patient ($\lambda$ weight) friendship. The impatient–patient friendship lasts no more than 2 periods and thus is bounded by how much surplus it may generate. Hence, any equilibrium where the patient–patient friendship fails with positive probability cannot be efficient, since there exists $\underline{\delta }_\mathrm{P}$ such that for $\delta _\mathrm{P}\in \left[ \underline{\delta }_\mathrm{P},1\right)$, the term $\left( 1-\lambda \right) \frac{\delta _\mathrm{P}}{1-\delta _\mathrm{P}}z_{H}$ can be made sufficiently large in an equilibrium where patient–patient friendships survive such as the equilibrium of the model.

It is an immediate consequence of these two lemmas that after the first period of play, a new player reveals whether they are patient or impatient through their first-period actions. Hence, beliefs after the first period of interaction are $\Pr _{j}\left( \delta _{i}=\delta _\mathrm{I}\right) \in \left\{ 0,1\right\}$. On-equilibrium behavior then requires that a patient–impatient friendship plays $\left( D,D\right)$ in the second period.

Lemma 3

Suppose $x_{H}+z_{H}<2x_{L}$ then there exists $\overline{\delta }_\mathrm{I}$ such that for $\delta _\mathrm{I}\in \left[ 0,\overline{\delta }_\mathrm{I}\right]$ impatient players choose random friendships in any efficient equilibrium.

Proof

We note that the first period choice of an impatient player is between (1) making random connections, and (2) defecting on every player, making network-based connections, and cooperating with at most one player while defecting on the other. Prior to any action, an existing player’s belief is bounded above zero when $\pi _{i=t}^{t}\left( j\right) =1$. Existing players will thus cooperate in an efficient equilibrium. Hence, successfully defecting on two players through choosing random friendships is always feasible; this payoff is at least $2x_{L}$. The best payoff an impatient player can achieve by choosing network friendships is $x_{H}+z_{H}$. The condition $x_{H}+z_{H}<2x_{L}$ guarantees that an impatient player always prefers to choose random friendships in an efficient equilibrium.

We have established that in efficient equilibria, on-equilibrium behavior involves patient players choosing cooperation and impatient players choosing random friendships and defection. Note that this immediately implies that impatient players generate a net deficit in every friendship they form $\left( x-y<0\right)$. The only possible way to improve upon an equilibrium where high stakes games are offered in games where $\pi _{i=t}^{t}\left( j\right) =2$ and low stake games are offered in games where $\pi _{i=t} ^{t}\left( j\right) =1$ is if a pooling equilibrium in which high stake games are offered from the outset creates a greater surplus. The condition for this not to be the case is as follows:

$$\begin{aligned} 2\left( 1-2\gamma \right) \left( 1-\lambda \right) \varDelta z<\lambda \left( \varDelta y-\varDelta x\right) , \end{aligned}$$

rearranging gives the condition in the proposition:

$$\begin{aligned} \varDelta z<\frac{\lambda \left( \varDelta y-\varDelta x\right) }{2\left( 1-2\gamma \right) \left( 1-\lambda \right) }. \end{aligned}$$

Proof of Proposition 3

Proof

The steps outlined here are identical to the steps in Jackson and Rogers (2007) with the appropriate substitution of variables. They are provided for the interested reader. Jackson and Rogers (2007) prove that a process where the degree of a node born at time $i$, which has initial degree $d_{0}$, evolves according to:

$$\begin{aligned} \frac{\text {d}d_{i}\left( t\right) }{\text {d}t}=\frac{ad_{i}\left( t\right) }{t} +\frac{b}{t}+c\text {,} \end{aligned}$$

(11)

when $a>0$ and $c=0$ or $a\ne 1$. The complementary CDF is then:

$$\begin{aligned} 1-F_{t}\left( d\right) =\left( \frac{d_{0}+\frac{d}{a}-\frac{ct}{1-a} }{d+\frac{b}{a}-\frac{ct}{1-a}}\right) ^{1/a}. \end{aligned}$$

(12)

In the setting, of this paper $d_{0}=0, a=\gamma , b=\left( 1-\gamma \right) M$ and $c=0$. Treating this as a continuous process gives the differential equation:

$$\begin{aligned} \frac{\text {d}d_{i}\left( t\right) }{\text {d}t}=\frac{d_{i}\left( t\right) \gamma }{t+N}+\frac{\left( 1-\gamma \right) M}{t+N}, \end{aligned}$$

(13)

which can be solved to get:

$$\begin{aligned} d_{i}\left( t\right) =\frac{1-\gamma }{\gamma }M\left( \frac{t+N}{i}\right) ^{\gamma }-\frac{1-\gamma }{\gamma }M. \end{aligned}$$

(14)

At time $t$, $1-F_{t}\left( d\right)$ is the fraction of individuals with in-degrees greater than $d$. If we solve the above expression for $i$ such that $d_{i}\left( t\right) =d,$ this then corresponds to the number of individuals who have a greater in-degree than $d$. If $i^{*}\left( d\right)$ is such that $d_{i^{*}\left( d\right) }\left( d\right) =d$, then:

$$\begin{aligned} 1-F_{t}\left( d\right) =\frac{i^{*}\left( d\right) }{t+N}\quad \text { for all }d\text { such that }i^{*}\left( d\right) >N. \end{aligned}$$

(15)

We can then derive the in-degree distribution as follows:

$$\begin{aligned} F_{t}\left( d\right) =1-\left( \frac{M}{\frac{\gamma }{1-\gamma }d+M}\right) ^{\frac{1}{\gamma }}\text { for all }d\text { such that }i^{*}\left( d\right) >N. \end{aligned}$$

(16)

The fraction of individuals not described by this distribution $\frac{N}{t+N}\rightarrow 0$ as $t\rightarrow \infty$.The complimentary CDF may be written as follows:

$$\begin{aligned} 1-F_{t}\left( d\right) =\left( \frac{M}{\frac{\gamma }{1-\gamma }d+M}\right) ^{\frac{1}{\gamma }}. \end{aligned}$$

(17)

Taking logs, this distribution exhibits power law properties for individuals with numbers of friends $d$ which are large relative to $\frac{1-\gamma }{\gamma }M$:

$$\begin{aligned} \log \left( 1-F_{t}\left( d\right) \right) =\frac{1}{\gamma }\left[ \log \left( M\right) -\log \left( \frac{\gamma }{1-\gamma }d+M\right) \right] . \end{aligned}$$

(18)

$\square$

Proof of Proposition 4

Proof

The growth process that describes this model is for a subset of possible parameter values from the process in Jackson and Rogers (2007).

Jackson and Rogers (2007) prove that $C^{TT}$ tends to:

$$\begin{aligned} \frac{p_{R}}{m(1+r)}\text {,} \end{aligned}$$

if $\frac{p_{R}}{r}\le 1. C(g)$ tends to:

$$\begin{aligned} \frac{6p_{R}}{(1+r)\left[ (3m-2)(r-1)+2mr\right] }\text {.} \end{aligned}$$

In terms of the parameters in this model, $p_{R}=1\;m=M$ and $r=\frac{1-\gamma }{\gamma }>1$ since $\gamma \le \frac{1}{2}$. After making these substitutions, the result follows. $\square$

Proof of Proposition 5

Proof

It suffices to show $\gamma$ is increasing in $\varDelta z,$ $\delta _\mathrm{I}$ and $\delta _\mathrm{P}$. First note that $V_{\text {Signal}}$ is increasing in $\delta _\mathrm{I}$ and $\alpha$ since:

$$\begin{aligned} \frac{\text {d}\underline{n}}{\text {d}\delta _\mathrm{I}}&>0;\frac{\text {d}\underline{n}}{\text {d}\alpha }>0;\frac{\partial V_{\text {Signal}}}{\partial \underline{n}}>0\\&\Longrightarrow \frac{\partial V_{\text {Signal}}}{\partial \underline{n} }\frac{\text {d}\underline{n}}{\text {d}\delta _\mathrm{I}}=\frac{\text {d}V_{\text {Signal}}}{\text {d}\delta _\mathrm{I}}>0. \end{aligned}$$

Also, $\frac{\text {d}V_{\text {Signal}}}{\text {d}\beta }=\frac{\partial V_{\text {Signal}} }{\partial \beta }+\frac{\partial V_{\text {Signal}}}{\partial \underline{n}} \frac{\text {d}\underline{n}}{\text {d}\beta }>0$ since:

$$\begin{aligned} \frac{\text {d}\underline{n}}{\text {d}\beta }>0;\frac{\partial V_{\text {Signal}}}{\partial \underline{n}}>0;\frac{\partial V_{\text {Signal}}}{\partial \beta }>0. \end{aligned}$$

Furthermore, $V_{\text {Signal}}$ is increasing in $\delta _\mathrm{P}$. Now, defining:

$$\begin{aligned} F\left( \gamma ,\varDelta z,\delta _\mathrm{I},\delta _\mathrm{P}\right) =2\frac{1-\delta _\mathrm{P}^{\underline{n}\left( \delta _\mathrm{I}\right) }}{1-\delta _\mathrm{P}}\varDelta z-C^{\prime }\left( \gamma M\right) , \end{aligned}$$

and noting:

$$\begin{aligned} \frac{\text {d}V_{\text {Signal}}}{\text {d}\delta _\mathrm{I}}>0;\frac{\text {d}V_{\text {Signal}}}{\text {d}\delta _\mathrm{P}}>0;\frac{\text {d}V_{\text {Signal}}}{\text {d}\alpha }>0;\frac{\text {d}V_{\text {Signal}}}{d\beta }>0;C^{\prime \prime }>0, \end{aligned}$$

we can sign $\frac{\text {d}\gamma }{\text {d}\delta _\mathrm{I}}$ by:

$$\begin{aligned} \text {sign}\left( \frac{\text {d}\gamma }{\text {d}\delta _\mathrm{I}}\right) =-\text {sign}\left( \frac{\frac{\partial F}{\partial \delta _\mathrm{I}}}{\frac{\partial F}{\partial \gamma } }\right) =\frac{\text {sign}\left( \frac{\text {d}V_{\text {Signal}}}{\text {d}\delta _\mathrm{I}}\right) }{\text {sign}\left( C^{\prime \prime }\left( \gamma M\right) \right) }>0, \end{aligned}$$

and similarly, the results follow for $\frac{\text {d}\gamma }{\text {d}\delta _\mathrm{P}},\frac{\text {d}\gamma }{\text {d}\alpha },\frac{\text {d}\gamma }{\text {d}\beta }.$ $\square$

Description of renegotiation equilibrium and Proof of Proposition 5

I will first describe the renegotiation concept, which is similar to the renegotiation concept used in Watson (1999). Define the sequence of stakes offered by the probability $\alpha \left( t\right)$ that a high stake game is played in period $t$. Also define the probability that the new player has defected by time $t$ as $\theta \left( t\right)$. A jump alteration $\widehat{\alpha },\widehat{\theta }$ prescribes from period $t$ what the original regime prescribed from $t+\varDelta$. Thus, $\widehat{\theta }$ and $\widehat{\alpha }$ are specified such that $\widehat{\theta }\left( w\right) =\theta \left( w+\varDelta \right)$ and $\widehat{\alpha }\left( w\right) =\alpha \left( w+\varDelta \right)$ for all $w>t$. Note that $\widehat{\alpha }$ must be such that the correct probability mass plays defect at time $t-1$, so that $\widehat{\theta }\left( t\right) =\theta \left( t+\varDelta \right)$. If such a $\widehat{\alpha }$ exists, then the new regime is an equilibrium (incentive-compatible) continuation. A stall alteration at $t$ defined by $\widetilde{\alpha }$ and $\widetilde{\theta }$ prescribes from $t+\varDelta$ what the original regime prescribed from time $t$. In between times $t$ and $t+\varDelta$, $\widetilde{\alpha }$ is set to preserve the current beliefs of the players about each other over the intervening period. Thus, $\widetilde{\alpha }$ and $\widetilde{\theta }$ satisfy $\widetilde{\alpha }\left( w\right) =\alpha \left( w-\varDelta \right)$ and $\widetilde{\theta }\left( w\right) =\theta \left( w-\varDelta \right)$ for every $w\ge t+\varDelta$; also $\widetilde{\theta }\left( w\right) =\theta \left( t\right)$ for every $w\in [t,t+\varDelta -1]$. The value of $\widetilde{\alpha }$ on $[t,t+\varDelta -1]$ is arbitrary; however, for it to be incentive-compatible, no type assigned positive probability at time $t$ may now have an incentive to defect during $[t,t+\varDelta -1]$.

Before the start of each period, players may reconsider the continuation of their regime. Players decide by voting for or against an exogenously given incentive-compatible alteration. The players simultaneously vote for or against the alteration. Additionally, I assume that no information is revealed in the renegotiation process about players’ types. A regime is abandoned in favor of another if in the case of a jump both players accept it or in the case of a stall at least one player accepts it.

Now consider the following strategies for two friends and suppose they do not share a friend in common. I will assume player 1 is the older player and is patient. Player 2, however, is a new player and may be patient or impatient.

Player 1

• Choice of stake

  • First period of interaction, offer a low stakes game.

  • Second period of interaction, mix with probability $\beta$ of a high stakes game.

  • Third period of interaction onwards, mix if in all previous periods the stake was low and $\left( C,C\right)$ has been the history of play.

  • Offer a high stakes game if the stake was high in the previous period and cooperation has been the history of play.

• Choice of strategy

  • Play C

Patient Player 2

• Choice of strategy

  • Play C

Impatient Player 2

• Choice of strategy

  • When the stake of the game is high

    • Play D

  • When the stake of the game is low

    • Cooperate in the first period with probability $\mu$.

    • Play C in all subsequent periods

Beliefs

• The belief of the new player is always

  $$\begin{aligned} \Pr \left( \text {Patient}\right) =1 \end{aligned}$$

• The belief of the older player before the first period is as follows

  $$\begin{aligned} \Pr \left( \text {Impatient}\right) =\frac{\lambda }{\left[ \left( 1-2\lambda \right) \left( 1-\gamma \right) +\lambda \right] } \end{aligned}$$

• The belief of the older player when the history includes at least one high stakes game is as follows

  $$\begin{aligned} \Pr \left( \text {Patient}\right) =1 \end{aligned}$$

• The belief of the older player after a history of only low stake games is as follows

  $$\begin{aligned} \Pr \left( \text {Impatient}\right) =\frac{\mu \lambda }{\left[ \left( 1-2\lambda \right) \left( 1-\gamma \right) +\mu \lambda \right] } \end{aligned}$$

Proof

There is always a $\underline{\delta }$ close enough to 1 such that it is optimal for the patient player to cooperate, provided that there is a positive probability the other player is also patient. To see this, note that $\lim _{\delta \rightarrow 1}$ of the left-hand side of the following equation

$$\begin{aligned} \Pr \,(\text {Patient}) \frac{z_{L}}{1-\delta }-(1-\Pr \,(\text {Patient})) y_{H}\ge \Pr \,(\text {Patient}) x, \end{aligned}$$

is $\infty$, provided that $\Pr \,(\text {Patient})>0$; where the left-hand side is a lower bound on the payoff from cooperation and the right-hand side is the payoff from defecting. $\square$

The only way an impatient type is revealed, given the equilibrium strategies, is through playing defect and thereby destroying the friendship. After the first period, this only occurs when a high stakes game is played, since both patient and impatient players cooperate in low stakes games from period two onwards. It is only when the players interact in a high stakes game that the older player finds out if the other player is patient or impatient.

In periods $t=1,\ldots ,\infty$ the impatient player is indifferent between cooperating and defecting when the stake of the game is low. The payoff from defecting this period is the same as cooperating and waiting until the following period:

$$\begin{aligned} x_{L}&= z_{L}+\delta _\mathrm{I}\left[ \beta x_{H}+\left( 1-\beta \right) x_{L}\right] \\ \Rightarrow \beta&= \frac{1}{\left( x_{H}-x_{L}\right) }\left( \frac{\left( x_{L}-z_{L}\right) }{\delta _\mathrm{I}}-x_{L}\right) \end{aligned}$$

and $0<\beta <1$ when $\delta _\mathrm{I}>\frac{x_{L}-z_{L}}{x_{H}}$. The impatient player will not prefer to cooperate provided that:

$$\begin{aligned} x_{H}>\frac{z_{H}}{1-\delta _\mathrm{I}}. \end{aligned}$$

In periods $t=2,\ldots ,\infty$ the older player who chooses the stake of the game will be indifferent between offering a high or a low stakes game, provided that:

$$\begin{aligned} \Pr \left( \text {Patient}\right) \frac{z_{H}}{1-\delta _\mathrm{P}}+\left( 1-\Pr \left( \text {Patient}\right) \right) \left( -y_{H}\right) =\frac{z_{L}}{1-\delta _\mathrm{P}} \end{aligned}$$

where,

$$\begin{aligned} \Pr \left( \text {Patient}\right) =\frac{\left( 1-2\lambda \right) \left( 1-\gamma \right) }{\left[ \left( 1-2\lambda \right) \left( 1-\gamma \right) +\lambda \mu \right] }\text {,} \end{aligned}$$

if the impatient player mixes in period 1 with probability $\mu$ of cooperating and $\left( 1-\mu \right)$ of defecting then the following holds:

$$\begin{aligned} \frac{\left( 1-2\lambda \right) \left( 1-\gamma \right) }{\left[ \left( 1-2\lambda \right) \left( 1-\gamma \right) +\lambda \mu \right] }\frac{z_{H} }{1-\delta _\mathrm{P}}+\frac{\mu \lambda }{\left[ \left( 1-2\lambda \right) \left( 1-\gamma \right) +\lambda \mu \right] }\left( -y_{H}\right) =\frac{z_{L} }{1-\delta _\mathrm{P}}. \end{aligned}$$

Also, if $\mu$ satisfies $\frac{\left( 1-2\lambda \right) \left( 1-\gamma \right) }{\left[ \left( 1-2\lambda \right) \left( 1-\gamma \right) +\lambda \mu \right] }\frac{z_{H}}{1-\delta _\mathrm{P}}+\frac{\mu \lambda }{\left[ \left( 1-2\lambda \right) \left( 1-\gamma \right) +\lambda \mu \right] }\left( -y_{H}\right) =\frac{z_{L}}{1-\delta _\mathrm{P}}$, then the older player is indifferent between offering a high and low stakes game in the second period and any future period, as long as only low stakes games have been offered previously. For $\mu \in \left[ 0,1\right]$ to exist then $\lambda \ge \frac{\left( 1-\gamma \right) \varDelta z}{z_{L}+y_{H}\left( 1-\delta _{P}\right) +2\varDelta z\left( 1-\gamma \right) }.$ I will now prove that the equilibrium is alteration-proof.

Delays

Given the repetitive nature of the equilibrium, the only times a delay is meaningful are between the first and second period, and in the instance that a high stakes game has been played and information has been revealed about the type of the unknown player. First, consider the incentives for the unknown player to delay. The sequence of stakes is weakly increasing, so both types of the unknown player do not ever want to delay an increase in the stake of the game, as both benefit from this. Now, consider the incentives for the older player for a delay alteration. A delay between the first and second periods is not strictly preferred, since in the second period, the older player is indifferent between a low or high stake game. In the sequence of stakes which are offered by the older player, the first high stakes game, which arises as a result of mixing, induces information about the unknown player to be revealed. A delay after this high stakes game has been played is not preferred by the older player, since the player knows for sure that the other player is patient, due to the fact that the relationship survived the high stake game in the previous period, and will prefer to play high stake games.

Jumps

In this equilibrium, only three different jumps can occur: (1) from a low stake to the mixed stake; (2) from a low stake to a high stake; and (3) from the mixed stake to a high stake. Jumps ahead from the low stake offered in the first period are not preferred by the older player if the payoff from offering a high stake is worse than a low stake. This will not be the case if $\mu \in [ 0,1]$, since the only difference between the first and second periods is that the older player places a lower probability on the unknown player being impatient. $\lambda \ge \frac{\left( 1-\gamma \right) \varDelta z}{z_{L}+y_{H}\left( 1-\delta _\mathrm{P}\right) +2\varDelta z\left( 1-\gamma \right) }$ guarantees that $\mu \in [0,1]$. The only jumps that are possible are jumps from periods in which the older player is required to mix to periods in which he/she offers a high stake for sure. For this type of jump to be incentive-compatible, the impatient type must defect for sure in the period immediately prior, which is not incentive-compatible when $\delta _\mathrm{I}>\frac{x_{L}-z_{L}}{x_{H}}$ because the impatient type will prefer to wait.