2. The model

We consider a finite population of agents $1,\ldots ,n$ . For each agent $i$ there is a collection of other agents $N(i)\subset \{1,\ldots ,n\}\setminus \{i\}$ , that we call the friends or neighbors of $i$ . Friendship is reciprocal, so $j\in N(i)$ implies $i \in N(j)$ . This information defines a network with nodes the agents $i=1,\ldots ,n$ , and a link from $i$ to $j$ if $j\in N(i)$ (Jackson, Reference Jackson2008; Newman, Reference Newman2018).

At each time step $t=1,2,3,\ldots$ each agent $i$ chooses an action $\sigma _i\in \{x,y\}$ (we suppress the dependence on time). An agent’s payoff at each time depends on their own action and those of their neighbors as specified by the payoff function $\pi (\sigma _i,\sigma _{N(i)})$ . For an agent $i$ , let $k_x(i)$ and $k_y(i)$ denote the number of neighbors playing $x$ and $y$ , respectively. When the focal agent is clear or unimportant, we drop the dependence on $i$ , and simply write $k_x$ or $k_y$ . We assume that payoffs are anonymous in the sense that they depend only on the number of neighbors playing each of the two strategies, not on their specific identities. In this case, the payoffs can be specified by two payoff functions: $\pi _x(k_x,k_y)$ and $\pi _y(k_x,k_y)$ , the payoffs to playing strategy $x$ or $y$ , respectively, when an agent has $k_x$ neighbors playing $x$ and $k_y$ neighbors playing $y$ .

To simplify the analysis, assume that each agent has a fixed number of neighbors, $k$ (or equivalently, each agent plays with the expectation of meeting a fixed number of agents on average).Footnote ³ Then we can write the payoffs as functions of the number of neighbors playing strategy $x$ : $\pi _{x}(k_x)$ and $\pi _{y}(k_x)$ .

Agents in the model update their actions using a best response rule similar to the dynamics studied by Blume (Reference Blume1995). At each time step, an agent is chosen uniformly at random to update their strategy. We assume that agents are myopic and only attempt to maximize their next period payoffs. Thus, because only one agent updates at a time, it is rational for the agent to choose the best response to the play of their neighbors in the previous period. Since payoffs are determined entirely by the number of neighbors playing $x$ , we can define the best response function, $BR:\{0,\ldots ,n\}\to \{x,y\}$ so that $BR(k_x)$ is the best response for an agent with $k_x$ neighbors playing $x$ . When the payoffs to both strategies are equal, we arbitrarily designate $x$ as the best response.

Example 1. Pairwise games

One example that fits within this framework is a situation in which agents play a symmetric binary game with each of their neighbors in which they are constrained to play the same strategy with all of their neighbors at any given time, and they receive the sum of the payoffs from each of these pairwise interactions. For example, suppose that for each of an agent $i$ ’s neighbors $j\in N_i$ , $i$ receives payoffs depending on $\sigma _i$ and $\sigma _j$ as specified in Table 1.

Table 1.

Payoffs to agent $i$ from agent $j$

Then $\pi _x(k_x,k_y)=k_xa+k_yb$ and $\pi _y(k_x,k_y)=k_xc+k_yd$ . In this case, the best response function follows a threshold rule:

(1) \begin{equation} BR(k_x,k_y)=\begin{cases} x & k_x(a-c)\geq k_y(d-b)\\[5pt] y & k_x(a-c)\lt k_y(d-b) \end{cases} \end{equation}

If $a-b-c+d\gt 0$ , the action $x$ becomes more attractive as more of an agent’s neighbors play $x$ and, as we will show below, agents tend to coordinate locally on a given strategy. When $a-b-c+d\lt 0$ , agents prefer to play $x$ unless too many of their neighbors are also playing $x$ . In this situation we find that agents’ strategies tend to be locally negatively correlated.

Example 2. $\boldsymbol{k}$ -person stag hunt

The framework also allows for situations in which an agent’s payoffs depend on the play of their entire network neighborhood such as the $k$ -person stag hunt (Skyrms, Reference Skyrms2003). In this game, an agent receives a payoff of 1 from playing $x$ if more than $\tau$ of their neighbors also play $x$ , but receives 0 from playing $x$ otherwise. Playing $y$ yields a payoff $\alpha \lt 1$ independent of the play of the agents’ neighbors. Here again the best response function follows a simple threshold rule: if $k_x\gt \tau$ play $x$ , otherwise play $y$ .

Example 3. Best-shot public goods game

In the $k$ -person stag hunt there is pressure for agents to conform with the actions of their neighbors. In the best-shot public goods game, agents would always prefer for their neighbors to provide the public good while they free ride, but they would rather provide the public good themselves if none of their neighbors is willing to do so. Specifically, playing $x$ yields a payoff of $1-c$ for some $c\in (0,1)$ , while playing $y$ has a payoff of 1 as long as at least one of the agent’s neighbors plays $x$ and a payoff of 0 otherwise. This game captures the incentives in situations like volunteering to be the “room parent” for a child’s class, where only one agent needs to pay the cost of providing the public good in order for all of their neighbors to reap the benefits.

3. Information

Even in small populations with simple network structures this model often admits a multitude of equilibria. To help solve the equilibrium selection problem, we assume agents only have partial information about the play of their neighbors. Galeotti et al. (Reference Galeotti, Goyal, Jackson, Vega-Redondo and Yariv2010) use a similar strategy by assuming that agents know the probability that their neighbors play a given strategy conditional on their own degree. Effectively, the agents know how many people they expect to interact with but not their specific identities. Here we employ an alternative strategy that allows us to retain local correlation in neighbors’ actions. Specifically, agents know the probability that their neighbors play a given strategy conditional on their own action.

One way to understand the difference between these two approaches is to think of them as compartmental models (similar to standard models of disease spread such as the SI, SIS, and SIR models (Bailey, Reference Bailey1975). Jackson and Yariv (Reference Jackson and Yariv2006); Jackson and Rogers (Reference Jackson and Rogers2007), and López-Pintado (Reference López-Pintado2008) apply direct analogs of these models in economics). The approach employed by Galeotti et al. (Reference Galeotti, Goyal, Jackson, Vega-Redondo and Yariv2010) has compartments for each strategy for each degree type and models the flows between these types (more specifically, they consider equilibrium states of such a flow). Our model has compartments for each pair type – e.g. $xx$ , $xy$ and $yy$ connected pairs—and keeps track of transitions between these pair types. For this reason, our approach is often called a pair approximation.

We use the first three Greek letters as variables to denote either of the two strategies: $\alpha ,\beta ,\gamma \in \{x,y\}$ . Let $p_{\alpha }$ denote the proportion of agents in the population playing strategy $\alpha$ . Let $p_{\alpha \beta }$ denote the proportion of connected ordered pairs of agents in the network with the first agent playing $\alpha$ and the second agent playing $\beta$ . Because we use ordered pairs, $p_{xy}=p_{yx}$ . Let $p_{\alpha |\beta }$ denote the probability that a random neighbor of an agent playing strategy $\beta$ is playing strategy $\alpha$ . Note that the following relationships hold among these variables:

(2) \begin{align} p_x+p_{y}&=1 \end{align}

(3) \begin{align} p_{xx}+2p_{xy}+p_{yy}&=1 \end{align}

(4) \begin{align} p_{x|\alpha }+p_{y|\alpha }&=1 \end{align}

(5) \begin{align} p_{\alpha \beta }&=p_{\alpha |\beta }p_{\beta }. \end{align}

The last equation follows from Bayes’ rule.

In a given time step only one agent updates their strategy, so if $p_x$ changes it increases or decreases by $1/N$ . The probability that $p_x$ increases equals the probability that the agent selected to update their strategy is currently playing $y$ , $p_{y}$ , times the probability that the best response for that agent is $x$ , $P[k_x\in BR^{-1}(x)]$ . We need to calculate the probability that a random agent playing $y$ has $k_x$ neighbors playing $x$ . Let $i$ be the agent chosen to update their strategy and suppose that $i$ is playing strategy $y$ . Then the probability that a random neighbor of $i$ is playing $x$ is $p_{x|y}$ , and the probability that a random neighbor of $i$ plays $y$ is $p_{y|y}$ . If (conditional on $i$ playing $y$ ) the strategies of $i$ ’s neighbors are independent then the probability that $i$ has $k_x$ neighbors playing $x$ and $k_y=k-k_x$ neighbors playing $y$ is

(6) \begin{equation} \frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y}. \end{equation}

In general, $i$ ’s neighbors will not be (conditionally) independent. In empirical social networks, two individuals with a common friend are much more likely to be friends with each other than two randomly chosen individuals (Newman and Park, Reference Newman and Park2003). This feature, known as clustering or transitivity (Jackson, Reference Jackson2008), implies that often the state of two neighbors of a given node will affect one another directly. Thus, calculating the probability that $i$ has $k_x$ neighbors playing $x$ and $k_y$ neighbors playing $y$ requires knowing the probability of larger configurations such as $p_{x|yx}$ —the probability that an agent plays $x$ given that they neighbor an agent playing $y$ that has another neighbor playing $x$ . To keep track of these probabilities requires tracking the frequency of triple configurations in the network, which in turn requires knowing the frequency of quadruples and so on. Rather than continue this expansion, we stop here and make the assumption that the agents behave as if the actions of their neighbors are independent conditional on their own actions. Thus, $p_{\alpha |\beta \gamma }$ is simply $p_{\alpha |\beta }$ . We call this the conditionally independent neighbors (CIN) assumption.

If the network is a tree (i.e. has no closed loops), this assumption follows from the fact that the only path between two neighbors of an agent $i$ goes through $i$ . The assumption is also approximately true in large Poisson random networks, because these networks have few short loops (Newman, Reference Newman2018).Footnote ⁴ In general, however, this will not be the case. Nevertheless, the CIN assumption has proven to lead to highly accurate approximations in other models, even in clustered networks (Morris, Reference Morris1997; Newman, Reference Newman2018). Under this assumption, equation (6) gives the probability that an agent playing $y$ has $k_x$ neighbors playing $x$ and $k_y$ neighbors playing $y$ . In Section 6, we relax the CIN assumption and examine dependence of equilibria on network clustering.

The probability that $p_x$ decreases equals the probability that the agent selected to update their strategy is currently playing $x$ , $p_x$ , times the probability that the best response for that agent is $y$ , $P[k_x\in BR^{-1}(y)]$ . The probability that a random agent playing $x$ has $k_x$ neighbors playing $x$ and $k_y$ neighbors playing $y$ is

(7) \begin{equation} \frac {k!}{k_x!k_y!}p_{x|x}^{k_x}p_{y|x}^{k_y}. \end{equation}

Thus, the net rate of change for $p_x$ is

(8) \begin{equation} \dot {p}_x=\sum _{k_x\in BR^{-1}(x)} \frac {p_{y}}{N} \frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y} - \sum _{k_x\in BR^{-1}(y)} \frac {p_x}{N} \frac {k!}{k_x!k_y!}p_{x|x}^{k_x}p_{y|x}^{k_y}. \end{equation}

Equation (8) does not fully specify the game dynamics because the right-hand side has the terms $p_{\alpha |\beta }$ , which by equation (5) depend on the pair frequencies $p_{\alpha \beta }$ .

We can write down an equation describing the rate of change of the pairs $p_{\alpha \beta }$ in the same way that we did for the singletons $p_x$ . The identities in equations (2) through (5) imply that it is sufficient to consider any one of the four pair types. Consider the $xx$ pairs. Let $i$ be the agent randomly selected to update their strategy. If $i$ is playing strategy $y$ and has $k_x$ neighbors playing strategy $x$ , where $k_x\in BR^{-1}(x)$ , then $p_{xx}$ increases by $\frac {2k_x}{kN}$ (there are $kN$ total pairs and the change from playing $y$ to $x$ creates $2k_x$ $xx$ pairs since pairs are counted in both directions). Making use of the CIN assumption, the probability that a randomly chosen agent is of type $y$ and has $k_x$ neighbors playing $x$ equals the probability that a random agent plays $y$ times the probability that an agent playing $y$ has $k_x$ neighbors playing $x$ :

(9) \begin{equation} p_{y}\frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y}. \end{equation}

Thus, the rate of increase of $p_{xx}$ is

(10) \begin{equation} \frac {2k_x}{kN}p_{y}\frac {k!}{k_x!k_y}p_{x|y}^{k_{x}}p_{y|y}^{k_y}. \end{equation}

A similar calculation gives the rate of decrease of $p_{xx}$ leading to the net rate of change

(11) \begin{equation} \dot {p}_{xx}=\sum _{k_x\in BR^{-1}(x)} \frac {2k_x}{kN}p_{y} \frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y} - \sum _{k_x\in BR^{-1}(y)} \frac {2k_x}{kN}p_x \frac {k!}{k_x!k_y!}p_{x|x}^{k_x}p_{y|x}^{k_y}. \end{equation}

Equations (8) and (11) completely describe the strategy dynamics of the game. To distinguish this model from models without local correlation, we refer to this model as a locally correlated model and models without local correlation, such as the one developed by Galeotti et al. (Reference Galeotti, Goyal, Jackson, Vega-Redondo and Yariv2010), as independent models.

4. Strategic complements and substitutes

In this section, we examine two specific classes of games: games of strategic complements and strategic substitutes. These two classes encompass any game in which agents play independently with each of their neighbors and receive the sum of those individual payoffs.

4.1 Strategic complements

In a game of strategic complements the payoff functions satisfy

(12) \begin{equation} \pi _x(k_x)-\pi _y(k_x)\geq \pi _x(k_x')-\pi _y(k_x') \end{equation}

for any $k_x$ and $k_x'$ with $k_x\geq k_x'$ . In other words, as the number of neighbors playing $x$ increases, the change in benefits to playing $x$ exceeds the change in benefits to playing $y$ .

4.1.1 Example: standards competition with positive externalities

Games of strategic complements provide a simple model of agents choosing between two standards with positive externalities. For example, suppose the population of agents represents the population of network scientists worldwide and each of the two strategies corresponds to the choice of a statistical software package. Links join network scientists who collaborate with one another. Assume that network scientists can only use one software package at a time and because they share files with their collaborators the benefits to using a particular package increase with the number of colleagues that use the software. Specifically, we might have

(13) \begin{equation} \pi _x(k_x)=f(k_x)-c_x \end{equation}

and

(14) \begin{equation} \pi _y(k_x)=f(k-k_x)-c_y, \end{equation}

where $c_x$ and $c_y$ are the costs of using $x$ and $y$ , respectively, and $f$ is a non-decreasing function capturing the increasing benefits of using the same package as your collaborators.

4.1.2 Dynamics with strategic complements

In games of strategic complements, the best response function follows a threshold rule: if $k_x\geq \tau$ play $x$ , if $k_x\lt \tau$ play $y$ . Equations (8) and (11) reduce to

(15) \begin{equation} \dot {p}_x=\sum _{k_x=\tau }^k \frac {p_{y}}{N} \frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y} - \sum _{k_x=0}^{\tau -1} \frac {p_x}{N} \frac {k!}{k_x!k_y!}p_{x|x}^{k_x}p_{y|x}^{k_y}, \end{equation}

and

(16) \begin{equation} \dot {p}_{xx}=\sum _{k_x=\tau }^k \frac {2k_x}{kN}p_{y} \frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y} - \sum _{k_x=0}^{\tau -1} \frac {2k_x}{kN}p_x \frac {k!}{k_x!k_y!}p_{x|x}^{k_x}p_{y|x}^{k_y}. \end{equation}

Figure 1.

The evolution of the fraction of agents playing the preferred strategy over time in a game of strategic complements under a range of initial conditions.

From equations (15) and (16), we can immediately see two possible equilibria: all agents play $x$ , or all agents play $y$ . If the network has multiple connected components, then any situation in which all the agents within any given component take the same action is also an equilibrium, because in any such state $p_{x|y}=p_{y|x}=0$ . It is not obvious however whether any unsegregated equilibria exist in which some agents play $x$ while others choose to play $y$ .

For a given set of initial conditions, equations (15) and (16) can be numerically integrated to give the predicted evolution of play in the population over time. (Analytic solutions under an alternative updating dynamic are described in Appendix A. In Appendix B we compare the numeric solutions in the main text with simulation results.) For example, Figure 1 plots the fraction of agents playing strategy $x$ over time in a network with 1000 nodes each of degree ten.Footnote ⁵ In the figure, the best response threshold for playing strategy $x$ is $\tau =4$ . In other words, $x$ is the best response for any agent with four or more neighbors playing $x$ .

Figure 2.

Comparison of the correlation and independent models for a game of strategic complements.

As shown in Figure 1, there is a “critical mass” of initial strategy $x$ players required in order for strategy $x$ to diffuse throughout the population. The existence of such a threshold is a general feature of the model for games of strategic complements. Figure 2 compares predictions from the correlation model and the independent model in a game of strategic complements for the same parameters. In this example, the critical mass is approximately .26. The middle upward bending purple curve starts with $p_x=.265$ , which is just enough above this tipping point for the $x$ strategy to diffuse and eventually converge to the all $x$ equilibrium. However, the green line for the independent model that starts at this same fraction of initial strategy $x$ players trends downward to the all $y$ equilibrium. In other words, the critical mass required for strategy $x$ to spread is lower in the correlation model than in the analogous model without local correlation. Because $\tau /k\lt \frac {1}{2}$ , the all $x$ equilibrium is socially preferred to the all $y$ equilibrium. As in this example, in general, the basin of attraction for the socially preferred outcome is larger in the correlation model developed here than in the independent model. In other words, the additional individual information leads to better collective decision-making.

4.2 Strategic substitutes

We also consider games of strategic substitutes, which are particularly interesting for understanding the provision of public goods. In many cases, it may only be necessary that a public good be provided locally. For example, in the case of information provision if one person in a network neighborhood does the work of gathering and disseminating important information, then there is no incentive for others in that neighborhood to duplicate their effort. Similarly, for a parent it may only be important that parents of children at the same school volunteer to lead field trips or after school activities, and it may only be important to us that people in our neighborhood make the effort to pick up liter.

The payoff inequalities for games of strategic substitutes follow the opposite relationship as for games of strategic complements studied in the previous section:

(17) \begin{equation} \pi _x(k_x)-\pi _y(k_x)\leq \pi _x(k_x')-\pi _y(k_x') \end{equation}

for any $k_x$ and $k_x'$ with $k_x\geq k_x'$ .

The best shot public goods game described above is a game of strategic substitutes.

4.2.1 Dynamics with strategic substitutes

With strategic substitutes the best response function again satisfies a threshold rule, but in this case the inequality is reversed: if $k_x\leq \tau$ play $x$ , if $k_x\gt \tau$ play $y$ . The analogous versions of equations (15) and (16) are

(18) \begin{equation} \dot {p}_x=\sum _{k_x=0}^\tau \frac {p_{y}}{N} \frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y} - \sum _{k_x=\tau +1}^{k} \frac {p_x}{N} \frac {k!}{k_x!k_y!}p_{x|x}^{k_x}p_{y|x}^{k_y}, \end{equation}

and

(19) \begin{equation} \dot {p}_{xx}=\sum _{k_x=0}^\tau \frac {2k_x}{kN}p_{y} \frac {k!}{k_x!k_y!}p_{x|y}^{k_x}p_{y|y}^{k_y} - \sum _{k_x=\tau +1}^{k} \frac {2k_x}{kN}p_x \frac {k!}{k_x!k_y!}p_{x|x}^{k_x}p_{y|x}^{k_y}. \end{equation}

Figure 3.

The evolution of the fraction of agents playing $x$ over time in a game of strategic substitutes under a range of initial conditions.

As with strategic complements, equations (18) and (19) can be numerically integrated to determine the evolution of play in the population for any given initial conditions. Figure 3 plots the fraction of strategy $x$ players over time for a game of strategic substitutes with $\tau =4$ for a range of initial conditions. Unlike the strategic complements case, with strategic substitutes the long run outcome is the same regardless of the initial conditions. In general, if $\tau \lt k/2$ , then the predicted equilibrium fraction of agents playing $x$ in the correlation model is lower than that predicted by the independent model, a point we return to in Section 5.

5. Local correlation

Besides the strategy frequencies over time, the model provides additional information about the correlation in neighbors’ behavior. For example, Figure 4 plots the conditional probability $p_{x|x}$ that a given neighbor of an agent playing $x$ is also playing $x$ in the model for strategic substitutes (solid orange) and strategic complements (solid blue) ( $\tau =4$ in both cases).

Figure 4.

The conditional probability $p_{x|x}$ from the correlation model (solid lines), and the fraction of agents playing strategy $x$ from an independent model (dashed lines). The blue lines are for a game of strategic complements and the orange lines are for a game of strategic substitutes.

For comparison, we also examine the analogous independent network game, which corresponds to the framework developed by Galeotti et al. (Reference Galeotti, Goyal, Jackson, Vega-Redondo and Yariv2010).Footnote ⁶ Since the independent model assumes that neighboring agents’ strategies are independent, $p_x$ is the best estimate for $p_{x|x}$ available. The dashed lines in Figure 4 display the predicted fraction of agents playing strategy $x$ from the independent network game.

In the game of strategic substitutes (orange), the independent model predicts that any given agent will play strategy $x$ with probability .465, and regardless of their strategy, any neighbor of a given agent will also play $x$ with probability .465. The resulting probability that an agent playing $x$ has more than four of their neighbors playing $x$ is

(20) \begin{equation} \sum _{k_x=5}^{10} \frac {10!}{k_x!(10-k_x)!}.465^{k_x}(1-.465)^{10-k_x} = .534. \end{equation}

Thus, at any given time more than half of the agents playing $x$ would be better off playing $y$ . If we think of playing $x$ as providing a public good, then the good is over provided in the independent model. The over provision results from agents’ limited information. While agents are responding optimally to the average play in the population, by chance an agent playing $x$ will frequently be matched with more than four other agents playing $x$ .

Figure 5.

The expected number of neighbors playing $x$ for an agent playing $x$ in a game of strategic substitutes with $\tau =4$ under the correlation model (purple) and the independent model (green).

As the figure shows, the correlation model predicts a dissociative relationship in neighboring agents’ strategies (solid orange). The probability that a random agent plays $x$ is .473, but the probability that the neighbor of an agent playing $x$ also plays $x$ at equilibrium is only .230. The probability that an agent playing $x$ has more than four neighbors playing $x$ is only

(21) \begin{equation} \sum _{k_x=5}^{10} \frac {10!}{k_x!(10-k_x)!}.230^{k_x}(1-.230)^{10-k_x} = .057. \end{equation}

Agents in the correlation model fare better because they take their own behavior into account when calculating the expected behavior of their neighbors. For example, consider the case of information provision. If I have provided information to my friends in the past, then I know that in the future my friends are less likely to seek out information because they may count on free riding off of me. Therefore, I should be more likely to continue to provide information because I know that my friends are unlikely to provide it. Figure 5 provides a second illustration of this point by plotting the expected number of neighbors playing strategy $x$ for an agent playing strategy $x$ under both the correlation (purple) and independent (green) models. The dashed horizontal line at three indicates the optimal number of neighbors. On average, in the independent model agents playing $x$ can expect to have two more neighbors playing $x$ than they would prefer, while the equilibrium in the correlation model is very near the optimum.

With strategic complements (blue lines in Figure 4), players coordinate locally. Ultimately, the independent model and our model make similar predictions: both models predict that nearly all of the agents play $x$ and therefore the probability that a neighbor of an agent playing $x$ also plays $x$ approaches one. However, our model results in rapid local coordination as clusters of agents playing $x$ form and then expand.

6. Clustering

Up to this point we have always made the conditionally independent neighbors assumption; however, many empirical social networks exhibit significant clustering (Newman and Park, Reference Newman and Park2003; Watts, Reference Watts2004), casting doubt on this assumption. In this section, we describe a refinement to the model that incorporates clustering. We measure clustering using the well-known clustering coefficient:

(22) \begin{equation} C=\frac {(\textrm {number of closed paths of length two})}{(\textrm {number of paths of length two})} \end{equation}

(p. 184 Newman, Reference Newman2018). The clustering coefficient captures the probability that any two neighbors of a given node are themselves neighbors.

When $C$ is nonzero, we expect that some neighboring nodes are neighbors of each other, and thus their strategies are not independent. Consider three nodes, a central node $i$ with two neighbors, $j$ and $k$ . Let $p_{\alpha |\beta \gamma }$ denote the probability that $j$ uses strategy $\alpha$ when $i$ and $k$ play strategies $\beta$ and $\gamma$ , respectively. We can split this probability into two components, the probability when $j$ and $k$ are connected forming a closed triangle, $p_{\alpha |\beta \gamma }^{\bigtriangledown }$ , and the probability when $j$ and $k$ are not connected, $p_{\alpha |\beta \gamma }^{\vee }$ . Then

(23) \begin{equation} p_{\alpha |\beta \gamma }=C p_{\alpha |\beta \gamma }^{\bigtriangledown } + (1-C) p_{\alpha |\beta \gamma }^{\vee }. \end{equation}

The probability $p_{\alpha |\beta \gamma }^{\vee }$ is the easier of the two to deal with. Since in this case $j$ and $k$ are not connected, we assume that their strategies are independent conditional on the strategy of $i$ , and thus $p_{\alpha |\beta \gamma }^{\vee }=p_{\alpha |\beta }$ . To estimate $p_{\alpha |\beta \gamma }^{\bigtriangledown }$ , we use a strategy suggested by Morris (Reference Morris1997) (see also Rand, Reference Rand and McGlade1999). First, we assume that

(24) \begin{equation} \frac {p_{\alpha |\beta \gamma }^{\bigtriangledown }}{p_{\alpha |\cdot \gamma }^{\bigtriangledown }}\cong \frac {p_{\alpha |\beta \gamma }^{\vee }}{p_{\alpha |\cdot \gamma }^{\vee }}. \end{equation}

The idea behind this assumption is that the strategy of $i$ , given the strategies of $j$ and $k$ , should not be affected much by whether $j$ and $k$ are also neighbors. In the right-hand side denominator, since $j$ and $k$ are not connected, we should have $p_{\alpha |\cdot \gamma }^{\vee }=p_{\alpha |\cdot }=p_{\alpha }.$ For the left-hand side denominator, the probability that $j$ uses strategy $\alpha$ given that $j$ is connected to $i$ using an unknown strategy and $k$ using strategy $\gamma$ is simply $p_{\alpha |\gamma }$ . Substituting into equation (24) we obtain the approximation

(25) \begin{equation} p_{\alpha |\beta \gamma }^{\bigtriangledown } \cong \frac {p_{\alpha |\beta }p_{\alpha |\gamma }}{p_\alpha }. \end{equation}

Finally, substituting back into equation (23) gives

(26) \begin{equation} p_{\alpha |\beta \gamma }\cong C\left (\frac {p_{\alpha |\beta }p_{\alpha |\gamma }}{p_\alpha }\right ) + (1-C) p_{\alpha |\beta }. \end{equation}

To see how we incorporate this new conditional probability into the dynamic equations (8) and (11), it helps to rewrite the equations slightly. We start with equation (11) for $\dot {p}_{xx}$ . Rather than thinking of randomly selecting a node to update, we could randomly select an edge and then choose a node at one end of that edge (since all nodes have the same degree, these are equivalent). The probability $p_{xx}$ can only increase if we select the $y$ end of either an $xy$ edge or a $yx$ edge, which occurs with probability $\frac {1}{2}(p_{xy}+p_{yx})=\frac {1}{2}(2p_{xy})=p_{xy}.$

Call the selected agent $i$ and the opposite end of the selected edge $j$ . Agent $i$ will switch to playing $x$ if $k_x\in BR^{-1}(x)$ . Since we already know that $j$ is playing $x$ , the probability that $i$ has exactly $k_x$ neighbors playing $x$ is equal to the probability that $i$ has $k_x-1$ additional neighbors playing $x$ . The probability that any one of $i$ ’s neighbors (besides $j$ ) plays $x$ is $p_{x|yx}$ . Now, rather than the CIN assumption, we assume that $i$ ’s neighbors are independent conditional on $i$ and $j$ ’s strategies. Thus, the probability that $i$ has $k_x$ neighbors playing $x$ and $k_y=k-k_x$ neighbors playing $y$ is

(27) \begin{equation} \frac {(k-1)!}{(k_x-1)!k_y!}p_{x|yx}^{k_x-1}p_{y|yx}^{k_y}. \end{equation}

If $i$ changes from $y$ to $x$ then $p_{xx}$ increases by $2/N$ . Performing a similar calculation for decreases in $p_{xx}$ we obtain

(28) \begin{align} \dot {p}_{xx}&=\sum _{k_x\in BR^{-1}(x)}\frac {2}{N}p_{xy}\frac {(k-1)!}{(k_x-1)!k_y!}p_{x|yx}^{k_x-1}p_{y|yx}^{k_y}-\sum _{k_x\in BR^{-1}(y)}\frac {2}{N}p_{xx}\frac {(k-1)!}{(k_x-1)!k_y!}p_{x|xx}^{k_x-1}p_{y|xx}^{k_y}\\[-6pt]\nonumber \end{align}

(29) \begin{align} &=\sum _{k_x\in BR^{-1}(x)}\frac {2k_x}{kN}p_{xy}\frac {k!}{k_x!k_y!}p_{x|yx}^{k_x-1}p_{y|yx}^{k_y}-\sum _{k_x\in BR^{-1}(y)}\frac {2k_x}{kN}p_{xx}\frac {k!}{k_x!k_y!}p_{x|xx}^{k_x-1}p_{y|xx}^{k_y}. \end{align}

Under the CIN assumption this equation is equivalent to (11). To see this, note that under CIN $p_{\alpha |\beta \gamma }$ is replaced by $p_{\alpha |\beta }$ giving

(30) \begin{equation} \dot {p}_{xx}=\sum _{k_x\in BR^{-1}(x)}\frac {2k_x}{kN}p_{xy}\frac {k!}{k_x!k_y!}p_{x|y}^{k_x-1}p_{y|y}^{k_y}-\sum _{k_x\in BR^{-1}(y)}\frac {2k_x}{kN}p_{xx}\frac {k!}{k_x!k_y!}p_{x|x}^{k_x-1}p_{y|x}^{k_y}. \end{equation}

Substituting $p_{xy}=p_{x|y}p_y$ and $p_{xx}=p_{x|x}p_x$ , we recover (11).

Now, instead of replacing $p_{\alpha |\beta \gamma }$ by $p_{\alpha |\beta }$ , we use (26) for $p_{\alpha |\beta \gamma }$ in equation (29). A similar argument applies to $\dot {p}_x$ giving

(31) \begin{align} \dot {p}_x&=\sum _{k_x\in BR^{-1}(x)}\frac {k_yp_{yy}}{kN}\frac {k!}{k_x!k_y!}p_{x|yy}^{k_x}p_{y|yy}^{k_y-1}\\[-6pt]\nonumber \end{align}

(32) \begin{align} &-\sum _{k_x\in BR^{-1}(y)}\frac {k_yp_{xy}}{kN}\frac {k!}{k_x!k_y!}p_{x|xy}^{k_x}p_{y|xy}^{k_y-1}\\[-6pt]\nonumber \end{align}

(33) \begin{align} &+\sum _{k_x\in BR^{-1}(x)}\frac {k_xp_{xy}}{kN}\frac {k!}{k_x!k_y!}p_{x|yx}^{k_x-1}p_{y|yx}^{k_y}\\[-6pt]\nonumber \end{align}

(34) \begin{align} &-\sum _{k_x\in BR^{-1}(y)}\frac {k_xp_{xx}}{kN}\frac {k!}{k_x!k_y!}p_{x|xx}^{k_x-1}p_{y|xx}^{k_y}, \end{align}

where again (26) is used for the $p_{\alpha |\beta \gamma }$ terms. When $C=0$ , the equations again collapse to recover the original differential equation for $\dot {p}_x$ shown in equation (8).

Figure 6.

The effect of clustering under strategic substitutes.

Figure 7.

The effect of clustering under strategic complements.

Figure 6 depicts the effect of clustering under strategic substitutes for a variety of initial conditions. In all cases, when the network is highly clustered, fewer agents play strategy $x$ . In the case of public goods provision, this means that more agents play the role of the public good provider in an unclustered network than in a highly clustered network. Clustering allows agents to coordinate locally, ensuring that no more individuals provide the public good than necessary.

Figure 7 depicts the effect of clustering under strategic complements. Here, we see that clustering allows for stable “shared market” equilibria where some agents play $x$ while others play $y$ that do not exist for the same game of strategic complements played in an unclustered network. Here again, clustering allows agents to coordinate locally—so there can be pockets of mostly $y$ players even when outside of these pockets most of the network plays $x$ . In this case since the best response threshold $\tau =4$ is less than half of each agents’ total number of neighbors, $k=10$ , the all $x$ equilibrium is socially preferred to the all $y$ equilibrium. These shared market solutions reduce overall welfare compared to the all $x$ equilibrium because the people in these $y$ -playing clusters would be better off if everyone switched to playing $x$ . If we think of strategy $x$ as a new technology spreading through the population, even with high initial levels of technology $x$ adoption, network clustering prevents that beneficial innovation from reaching some niche populations. On the other hand, for low initial levels of technology $x$ adoption there are benefits to clustering because it allows pockets of technology $x$ users to survive despite the dominance of the inferior incumbent technology $y$ . By maintaining diversity, these clustered niches could serve as incubators for technologies to improve without being wiped out by more popular competitors.