Cohesion, Ideology, and Tolerance

Factions within political parties, politically active non-governmental organizations, and interest groups share the feature of serving an underlying core of values and ideas —their ideology. These ideologies, regardless of whether they stem from convictions or serve the agenda of a specific group, dictate the goals of such politically active entities. These goals are often conflicting and agents must secure support from allies to achieve their goals at the expense of their opponents.

In contrast to what naive intuition would suggest, seemingly close agents —in terms of their ideology —are often opponents (Boucek, 2009). We tend think of the value of an alliance as independent of other alliances, however, it is sensible to assume that potential supporters (e.g., donors or voters) would perceive it negatively were many allies of an agent in fact opponents. Indeed, Greene and Haber (2015) document the value for political parties to be perceived as cohesive entities with little divide between its factions.¹¹1Their argument is that supporters perceive internal divide as a signal of low competence and support this with evidence of lower vote shares in elections. This interpretation is consistent with the model we will propose, however, other interpretations are possible as well. This interdependence between the benefits created by different alliances gives rise to a complex network describing the spillovers between allies and opponents.

The focus of our paper is to understand how a concern for cohesion shapes the emerging alliances in an environment where ideological agents interact. To do so, we propose a model with ideologically heterogeneous agents who choose the ideological composition of their allies —their tolerance. Two agents are allies when they tolerate each others’ ideologies and ideological proximity reduces the burden of tolerance (Murphy, 1997). Intuitively, interacting with ideologically dissimilar allies entails a reputation loss in the eyes of supporters. The extent of this reputation loss is driven by the most distant allies. Agents who do not tolerate each others’ ideologies are opponents.

Sustaining alliances requires effort, and even more so when an agent wants to sustain many alliances at once. To model this, we let agents exert costly effort to strengthen the connections to their allies, however, agents cannot target these efforts towards any specific ally. Instead, the weight of a connection to an ally is (i) increasing in the effort of both allies, and (ii) decreasing in the efforts of other allies that either of the two agents have.

The tolerance and effort choices determine a weighted network which describes who are allies, who are opponents, as well as the intensity of connections between allies.²²2Since alliances also determine oppositions, the network can be interpreted as a signed graph. We think of the intensity of a connection as the support of agents towards each other. The network fully determines all benefits in this model. In particular, agents benefit from dispute with their opponents to capture the idea of conflicting ideologies and the incompatible goals they dictate. An agent’s benefits from a dispute increase in the number and intensity of their connections relative to their opponent. We interpret this as the expected support an agent secures for potential proposals, which we simply call their strength.³³3This is similar to the idea of probabilistic voting models, e.g., Austen-Smith (1987). We introduce a dispute specific notion of cohesion such that an agent is more cohesive in a dispute when they share fewer connections with their opponent. Cohesion increases the effectiveness of an agent in disputing against a given opponent.

The effectiveness of cohesion determines the structure of the equilibrium network. Generally, allies are not ideologically as similar as possible. If cohesion is sufficiently effective, a segregated equilibrium network arises where opponents share no mutual allies. In other words, the network consists of distinct cliques. Ideologically more diverse cliques require more tolerance from its members. If cohesion is highly effective, agents are willing to be more tolerant and society consists of few cliques where at least some of them are ideologically diverse. Otherwise, the equilibrium network is an overlapping society where opponents share mutual allies and allies are ideologically closer.

Some important insights arise from the equilibrium characterization. First, despite homophilous preferences, i.e., agents find it easier to tolerate ideologically similar ones, agents may ally themselves with ideologically distant types and dispute against those with closer ideologies to increase their cohesion. Second, securing broad support across the ideological spectrum need not make agents more successful in achieving their goals as ideological diversity among allies may decreases their cohesion.

The degree of polarization induced by the endogenous network is naturally of interest in our context. We think of polarization as the total dispute intensity, comprising of the number of disputes (extensive margin) and how intensely disputes are fought (intensive margin), which is captured by the strengths of the opponents.

Agents with more allies exert lower efforts in equilibrium because (i) they are in fewer disputes and investing in their strengths pays off in fewer instances, and (ii) their neighbors contribute to their strength as well, thereby crowding out their own equilibrium efforts.⁴⁴4A similar mechanism is present in other economic contexts, for instance, fighting efforts of armed militias (König et al., 2017) or collective action (Chwe, 1999).

To build stronger intuition on how polarization depends on the network structure, we impose a widely used parametric structure. In particular, we let the canonical Tullock contest capture the benefits from an agent’s relative strength (Tullock, 1980) and treat the benefits from cohesion as an additively separable component.⁵⁵5This is a conservative assumption since we allow agents to perfectly substitute away from investing in their cohesion.

We are interested in how the effectiveness of cohesion affects polarization, and how this may depend on the initial network structure. The effectiveness of cohesion can increase (i) directly, when supporters value cohesion more, or (ii) indirectly, when exerting effort to intensify one’s ties with allies becomes costlier. In some sense, agents substitute away from investing into their strengths and become more concerned with their cohesion instead. For instance, restricting communication between allies may erode their support for each other in critical situations or regulations on campaign spending and content would reduce the possibility of signalling potential alliances to supporters.

The effect of a direct increase in the effectiveness of cohesion depends on the initial network structure. Agents want to have fewer mutual allies with their opponents when cohesion becomes more effective increase. In an initially overlapping society, the network moves towards segregation, resulting in more disputes overall. This also increases aggregate equilibrium efforts and dispute intensity increases in an initially overlapping society.

In an initially segregated network, any agent who is in more disputes due to an increase in the effectiveness of cohesion is so because others no longer tolerate their ideology. The number of disputes in the economy must therefore fall, thereby also decreasing the aggregate equilibrium effort and decreasing dispute intensity in a segregated society.

Equilibrium efforts decrease in the effort cost and thus always depress polarization on the intensive margin. However, the effort cost also determines the alliances that agents form. Again, the initial network structure plays a crucial role for the effect on polarization.

For high initial effort costs, the relative benefits from cohesion are high to begin with and the equilibrium network is segregated. Agents rely even more on cohesion when strengthening their ties to allies becomes costlier, thereby causing (some) cliques to grow more diverse. This results in fewer disputes and dispute intensity decreases.

In an initially overlapping society, increasing the cost of effort moves the network towards segregation. The extensive and intensive margin then go in opposing directions and how strong the crowding out of efforts is relative to the increase in the number of disputes determines the overall effect on polarization. For initially low effort costs, the crowding out of efforts is larger and dominates. This effect slows down as the network approaches segregation as agents are in more disputes. Eventually, the increase in the number of disputes (extensive margin) dominates, which is the case for an intermediate effort cost, i.e., when the network is close to segregation. Then, subsidizing efforts presents a viable option to reduce polarization.

Our model also rationalizes the paradoxical appeal that many extremist candidates have on their allies. Since extreme ideologies are at the far end of the ideological spectrum, agents with extreme ideologies are in many disputes and thus increase the cohesion of their allies by a lot. Attempts to influence extremists may thus have unintended effects and actually lead to more polarization.⁶⁶6Mostagir and Siderius (2023) find a trade off in the same spirit in a model of community formation on social media.

The seminal work of Downs (1957) conceptualizes political actors as economic entities that are responsive to incentives. Scholars have thus become concerned with the incentives of political actors who care about winning office (e.g., Riker (1962)).⁷⁷7See for instance Austen-Smith and Banks (1988); Baron and Ferejohn (1989) for seminal contributions on voter incentives from which we abstract in our paper. Indeed, Kriesi et al. (1996) emphasize the importance of the benefits for political agents when cooperating.

What links this literature to our paper is the idea of cohesion and its role for the benefits of agents. Cohesion contributes to a political agent’s issue ownership, i.e., how competent they are perceived to be in tackling specific issues (Petrocik, 1996; Van der Brug, 2004; Bélanger and Meguid, 2008). In some sense, political agents in our model have a “brand value” to protect (Snyder Jr and Ting, 2002).

Our model rationalizes why ideological differences sometimes lead to the collapse of a political entity (e.g., the Christian Democratic Party in Italy in the early 1990s (Boucek, 2009)) and sometimes not (e.g., the Republican Party (Noel, 2016)). In a changing society, separation into more cohesive entities becomes profitable only if the initial divide in the political landscape is not too large.

Esteban and Ray (1994) establish a measurement of polarization in society with the leading application of the polarization among income groups. Following their notion of polarization, our model studies what drives the emergence of polarization in the context of political factions.

Ideologies differentiate types horizontally and thereby induce homophily —the tendency to interact predominantly with similar individuals (McPherson et al., 2001; Currarini et al., 2009, 2016).

Interactions in our model arise from tolerance as in Genicot (2022), where individuals choose their behavior to establish connections to others and want to comply to their own “ideal behavior”. Each agent tolerates some behaviors and departs from their own ideal behavior to be tolerated by others. Agents thus shirk their type whereas, consistent with the contexts we have in mind, types are common knowledge in our model. Tolerance is thus bilateral, whereas agents never compromise for each other in Genicot (2022). Moreover, cohesion introduces an interdependence between different links, which is, in contrast to friendship networks, key in our application.⁸⁸8Iijima and Kamada (2017) provide a general framework to study how social distance determines the relationships between agents. Social distance also determines the structure of groups whose members contribute the public good of their groups (Baccara and Yariv, 2013, 2016). In our model, distance between agents determines the cost of tolerance.

Our paper follows the network formation protocol of Cabrales et al. (2011), where generic socialization efforts determines the intensity of interactions between agents.⁹⁹9This protocol has been used to study, for instance, job search via friends (Galeotti and Merlino, 2014). A key difference to their protocol is that we include a tolerance decision, i.e., players can exclude certain types from their network. Our network formation model is thus an intermediate case between the canonical pairwise stability criterion of Jackson and Wolinsky (1996) (related to tolerance in our model), and one where generic efforts jointly determine the weighted links between agents as in Cabrales et al. (2011). Due to the complexity of our network formation protocol and in line with our application, we consider bilateral deviations as in Goyal and Vega-Redondo (2007) to account for deviations in tolerance and effort decisions.

We borrow from the well established literature on contests (Tullock, 1967, 1980; Hirshleifer, 1989) and use two well established modelling tools. First, we use weighted links as a notion of strength (König et al., 2017). Second, we study the endogenous emergence of a signed network. Hiller (2017) studies signed and unweighted network formation with homogeneous agents. Our model requires a weighted network to account for the different incentives for ideologically differentiated types. This provides novel insights on the role of cohesion for segregation and polarization, as well as the role of extreme ideologies for polarization.

The remainder of the paper is organized as follows. Section 2 introduces the model. Section 3 characterizes the equilibrium. Section 4 presents how the characteristics of the economy shape dispute intensity. Section 5 discusses the role of extreme ideologies. Section 6 describes various extensions and additional results of the model. Section 7 concludes. All proofs are in the appendix.

Players: Let $N=\{1,2,...,n\}$ be the set of players, where $i$ is the typical player. To rule out trivial cases, we assume $n\geq 3$. Player $i$ is of type $\theta_{i}\in[0,1]$, which represents their ideological location or simply their ideology. One natural interpretation of $\theta_{i}$ would be the extent to which $i$ favors right-wing positions. Without loss of generality, we choose $\theta_{0}=0$ and $\theta_{1}=1$ as the most extreme types. Ideologies are commonly observable to everyone. Moreover, no two players have precisely the same ideology. Formally, $\theta_{i}$ is drawn from some cumulative distribution $\Phi$ with continuous probability density function $\varphi$. The probability to draw identical types is then zero, i.e., $P(\exists i,j:\theta_{i}=\theta_{j})=0$ for all $i,j\in N$, and the probability that player $i$ is equally distant from some players $j$ and $h$ is zero as well, i.e., $P(\exists i,j,h\in N:|\theta_{i}-\theta_{j}|=|\theta_{i}-\theta_{h}|)=0$ for all $i,j,h\in N$.¹⁰¹⁰10While intuitive, this assumption is also technically convenient since it limits the concern of equilibrium multiplicity. The qualitative results of this paper, however, do not depend on it. If $\theta_{i}>\theta_{j}$, we say $i>j$. If $\max\{1-i,i\}<\max\{1-j,j\}$, $i$ is more moderate than $j$.

Strategies: Strategies in this model determine the interactions among players and comprise of two actions. Each player $i$ nominates an interval of tolerable types, denoted by $\mathbf{t}_{i}=[\underline{t}_{i},\bar{t}_{i}]\in[0,1]$. We refer to this decision as their tolerance. Denote by $\mathbf{t}=(\mathbf{t}_{1},\mathbf{t}_{2},...,\mathbf{t}_{n})$ the vector of tolerance choices. It is costly to tolerate different ideologies and even more so if these ideologies are farther from one’s own ideology. For simplicity, let the total cost of tolerance associated with a given tolerance interval for player $i$ be $\tau[(\underline{t}_{i}-\theta_{i})^{2}+(\bar{t}_{i}-\theta_{i})^{2}]$, where the parameter $\tau>0$ captures the flexibility or stubbornness of agents.¹¹¹¹11Any specification that is increasing and convex in the distance between an agent’s type and the bounds of the tolerance interval delivers the qualitative results presented in this paper. For lower values of $\tau$, agents are more flexible.¹²¹²12In Section 6, we discuss the possibility of $\tau_{i}\neq\tau_{j}$. The tolerance cost can, for instance, represent a reputation loss from interacting with individuals of a different ideology. Such a reputation loss would be substantially larger if an agent interacted with someone far from their own ideology. Moreover, a reputation loss of this form would be driven by the “most distant ideology” one tolerates. We allow players to choose asymmetric tolerance intervals, so generally, $(\underline{t}_{i}-\theta_{i})^{2}\neq(\bar{t}_{i}-\theta_{i})^{2}$. In principle, players may choose to tolerate types far to the right (left) of their own ideology and simultaneously be intolerant towards more similar ideologies to their left (right).¹³¹³13Note, we do not consider the cases $\bar{t}_{i}=\underline{t}_{i}>\theta_{i}$ ($\bar{t}_{i}=\underline{t}_{i}<\theta_{i}$) or $\underline{t}_{i}<0$ ($\bar{t}_{i}>1$) since such strategies would obviously never occur in an equilibrium.

Each player $i$ chooses a generic effort to establish connections to others which we denote by $x_{i}\in X_{i}=[0,+\infty)$. Effort entails a constant marginal cost $c>0$. The parameter $c$ may represent regulations on the communication between agents or their campaigning efforts, e.g., whether they can signal future alliances to supporters. The effort profile is then a vector $x=(x_{1},x_{2},...,x_{n})$.

We write the strategy of player $i$ as $s_{i}\in S_{i}=S\equiv X\times T$, where $\mathbf{t}_{i}\in T=[0,1]$. Denote by $s=(s_{i},s_{-i})$ the strategy profile, where $s_{-i}$ denotes the strategies of all players, other than $i$.

Network formation: Our network formation protocol allows players to choose the composition of their neighborhoods through their tolerance decision and the intensity of their interactions through their efforts. The two choices jointly generate a weighted network, which describes alliances, i.e., links of strictly positive weight, and disputes, i.e., links of weight zero. One can thus interpret the resulting network as a signed graph of positive (alliances) and antagonistic relationships (disputes). We denote this network by an adjacency matrix $g$ and, abusing notation, refer to $g$ simply as the network. Let $K_{i}(g)=\{j\in N:\theta_{j}\in[\underline{t}_{i},\bar{t}_{i}]\}$ denote the set of players, whose types fall into $i$’s tolerance interval. Denote by $k_{i}(g)$ the cardinality of this set.¹⁴¹⁴14Note, we also use this notation to denote the neighborhood of player $i$. These coincide straightforwardly in equilibrium. We sometimes write $K_{i}$ for $K_{i}(g)$ and $k_{i}$ for $k_{i}(g)$ when no confusion arises. Define a weighting function

There is no link between $i$ and $j$ whenever $i$ does not tolerate $j$’s ideology ($\theta_{j}\not\in[\underline{t}_{i},\bar{t}_{i}]$) or vice versa. Similarly, there is no link between $i$ and $j$ when either player exerts no effort to strengthen connections, i.e., $x_{i}=0$ or $x_{j}=0$. In all other cases, links are weighted by the aggregate socialization effort of every potential neighbor of $i$ or $j$ (everyone in their respective tolerance intervals). Congestion effects therefore reduce the intensity of a connection between two nodes when other connections exert high efforts.

We obtain the network $g$ by setting

We simply write $g_{ij}$ when doing so creates no confusion. Ultimately, we want to interpret $g_{ij}$ as the expected support for $i$ from their neighbor $j$. For this reason, we impose $g_{ij}\leq 1$ for all $i,j\in N$. Connections strengthen as an agent invests more into their connections, all else equal, and even more so if their neighbors invest more into their connections as well. We capture this idea by letting the weight of a link depend on the product of the socialization efforts.

Following this protocol, $\rho_{ij}(x,\mathbf{t})=\rho_{ji}(x,\mathbf{t})$ and links are symmetric, i.e., $g_{ij}=g_{ji}$ for all $i,j\in N$. Moreover, self-loops are a genuine feature of this model, so $g_{ii}\in[0,1]$. Self-loops allow agents to pursue their own agenda independently from others.

Network definitions: A player $i$ is isolated if $g_{ij}=0$ for all $j\in N\setminus\{i\}$. The network $g$ is empty if every agent agent is isolated.

A player $j$ is a neighbor of $i$ whenever $g_{ij}>0$. Player $j$ is an opponent of $i$ whenever $i$ and $j$ are not neighbors, i.e., $g_{ij}=0$. In this case, $i$ and $j$ are in dispute.

A player’s degree is the number of their connections, which we call $k_{i}$.

A subset of players is a clique $C(g)$, if for all $i\in C(g)\subseteq N$, $g_{ij}>0$ if $j\in C(g)$ and $g_{ij}=0$ otherwise.

A network $g$ is ordered with respect to types, if for any two nodes $i$ and $j$, with $i>j$, $\bar{t}_{i}\geq\bar{t}_{j}$ and $\underline{t}_{i}\geq\underline{t}_{j}$.

A network $g$ is segregated if it consists of cliques. A segregated network exhibits strong structural balance if it consists of exactly two cliques and weak structural balance otherwise (Cartwright and Harary, 1956).

A network is an overlapping society if it is not segregated.

Benefits: The network determines all disputes and the resulting benefits from them. Connections influence a player’s strength and how effectively they campaign against their opponents. Establishing links to others yields only indirect benefits through altering the benefits from disputes. In a model of competing ideological agendas, it is natural to model dispute as a contest between two players. An agent’s strength, captured by the sum of the weights of all their links, i.e., $\lambda_{i}(g)=\sum_{j\in N}g_{ij}$, uniformly increases their benefits in all disputes.¹⁵¹⁵15We can study alternative versions of this notion where higher order neighbors also contribute to an agent’s strength or where common neighbors do not contribute to the strength of players in dispute (see Section 6). This is in line with interpreting strengths as the expected support from allies.

Another ingredient of our model is how effectively opponents campaign against each other. We posit that this depends on the cohesion of an agent. An agent is more cohesive in a dispute if they share many connections with the opponents of their opponent. Formally, let $\lambda_{ij}(g)=\sum_{j\in N}\operatorname{sgn}(g_{ih})(1-\operatorname{sgn}(g_{hj}))$ denote the number of $i$’s neighbors who are in dispute with $i$’s opponent $j$. The function $\operatorname{sgn}$ denotes the sign function, which equals one if $g_{ih}>0$ and zero otherwise. The dispute technology thus composes of a global component (the strengths of players) and a dispute-specific component (the dispute-specific cohesion).

Denote the benefits for player $i$ from a dispute with player $j$ by $f(\lambda_{i}(g),\lambda_{j}(g),\lambda_{ij}(g))$. Call $f(\cdot,\cdot,\cdot)$ the contest success function (henceforth: CSF). For convenience, we write $\lambda_{i}$ for $\lambda_{i}(g)$ and $f(\lambda_{i},\lambda_{j},\lambda_{ij})$ for $f(\lambda_{i}(g),\lambda_{j}(g),\lambda_{ij}(g))$ when no confusion arises. The function $f(\cdot,\cdot,\cdot)$ is strictly increasing in its first argument, the strength of the player, and concave. Moreover, $f(\cdot,\cdot,\cdot)$ is strictly decreasing and convex in its second argument, the strength of the opponent. We normalize $f(\lambda_{i},\lambda_{j},\lambda_{ij})=0$ for all $\lambda_{i},\lambda_{j}$ such that $\lambda_{i}=\lambda_{j}\text{ and }\lambda_{ij}=0$. Let $f(\cdot,\cdot,\cdot)$ be increasing in the third argument.

Define a function $\delta(y)=f(\lambda_{i},\lambda_{j},y)-f(\lambda_{i},\lambda_{j},y-1)$ for all $y\in\{1,...,n-2\}$.¹⁶¹⁶16Note, $n-2$ is the maximum number of agents, an individual can have a mutual opponent with. The function $\delta(y)$ captures the additional benefits for player $i$ from dispute with player $j$ when adding the $y^{th}$ opponent of $j$ to their neighborhood, all else equal. Define $\underline{\delta}=\min_{y\leq n-2}\delta(y)$ as the lowest value this function can possibly take. The value $\underline{\delta}$ establishes a lower bound on how effectively a player can campaign against others. We say cohesion is more effective when $\underline{\delta}$ is larger.

The utility of player $i$ is given by

The utility of an agent comprises of the benefits from each dispute (first term), the cost of their effort (second term) and their cost of tolerance (third term).

To build stronger intuition, we will use the normalized Tullock CSF, augmented by cohesion, later on to establish comparative statics.

The values of $\phi\in[0,1]$¹⁷¹⁷17In principle, we could generalize the model to $\phi>1$., $\beta\geq 0$ and $\alpha\geq 0$ parameterize the CSF. Higher values of $\phi$ favor stronger agents. The parameter $\beta$ scales the value of cohesion and $\alpha$ captures the marginal changes in the value of cohesion.¹⁸¹⁸18While the Tullock contest poses a natural framework to think about disputes, other forms of modelling disputes are possible. We refer the reader to Appendix B.

A strategy profile $s^{\ast}$ is a bilateral equilibrium (BE) (Goyal and Vega-Redondo, 2007), if

1. (i)

   $u_{i}(s^{\ast}_{i},s^{\ast}_{-i})\geq u_{i}(s^{\ast}_{i},s^{\prime}_{-i})\;\forall i\in N\text{ and }s^{\prime}\in S$;

2. (ii)

   for each player pair $i,j\in N$, and strategy pair $(s_{i},s_{j})$,
   $u_{i}(s_{i},s_{j},s^{\ast}_{-i-j})>u_{i}(s^{\ast}_{i},s^{\ast}_{j},s^{\ast}_{-i-j})$ implies $u_{j}(s_{i},s_{j},s^{\ast}_{-i-j})<u_{j}(s^{\ast}_{i},s^{\ast}_{j},s^{\ast}_{-i-j})$.

Our solution concept requires that no agent can increase their utility by changing their strategy (condition (i)) and that any conceivable bilateral deviation of agents $i$ and $j$ reduces the utility of at least on of the two agents (condition (ii)).

Total effort is the sum of individual efforts, $\sum_{i\in N}x_{i}$.

We require a measure of divide in society. A natural candidate is polarization captured by the aggregate intensity of disputes (Esteban and Ray, 1994). Formally, dispute intensity is $\iota(s)=\sum_{i\in N}\sum_{j\neq i}\lambda_{i}(1-\operatorname{sgn}(g_{ij}))\lambda_{j}$. Dispute intensity increases on the extensive margin when there are more disputes. Dispute intensity increases on the intensive margin if players’ strengths increase.

We now proceed to characterizing the equilibrium of the game. We present conditions under which a bilateral equilibrium network exhibits (strong) structural balance, or is an overlapping society. First, we establish a powerful preliminary result.

According to Lemma 1, we can infer the ordering of the bounds of the tolerance intervals from the types. Players in our model are ex ante identical apart from their ideologies. Ideologies only affect the costs of tolerating and being tolerated by others. Agents want to achieve a certain level of cohesion at the lowest possible tolerance cost. Orderedness arises since all agents behave this way.

Orderedness is a powerful result since it restricts the deviations we have to consider when characterizing the BE network. However, let us first note existence and uniqueness.

We use a constructive argument to prove existence. Clearly, players only tolerate those who tolerate them as well. Otherwise, those players are in dispute and one agent has a profitable deviation in narrowing their tolerance interval (since Lemma 1 establishes orderedness). We then proceed to let $1$ and $0$ choose their tolerance interval. Players $1$ and $0$ never tolerate each other, since they would be in no disputes and receive zero benefits. Either $1$, $0$, or both, would then receive a strictly greater utility from being in dispute. Since agents always bilaterally agree to tolerate each other in equilibrium, only players who tolerate $1$ ($0$) are also in $1$’s ($0$’s) tolerance interval. Among all players who are not connected to $1$ or $0$, we can take the most extreme type. If no such player exists, we reach an equilibrium by letting everyone else choose their tolerance intervals since this would not induce any profitable deviation for anyone who has already made their tolerance decision. Otherwise, we can let this player choose their tolerance interval and repeat this process until no players remain. By construction, no profitable deviation emerges. This process eventually yields a BE of the game, thereby proving existence.

The endogenous tolerance decisions define who are allies and opponents and thus uniquely pin down the equilibrium effort profile. Hence, if there existed multiple BE networks, there would exist multiple equilibrium tolerance profiles. At least one of two things must then be true. Either, (i) at least one player is indifferent between tolerating and not tolerating some other type, or (ii), some player is indifferent between tolerating two distinct types. In case (i), the benefits of a connection must precisely coincide with the additional cost of tolerance. Since agents derive benefits from each dispute, the sum of all benefits is not a continuous function. Given our assumptions on the type distribution, such instances have zero probability. In case (ii), establishing either of two connections yields exactly the same net benefits. Since agents always differ somewhat in their ideology, this indifference occurs with probability zero as well and the BE network is indeed unique.¹⁹¹⁹19While bilateral equilibrium is a natural solution concept in this context, employing weaker concepts would also be possible. However, issues of equilibrium multiplicity would arise.

Having established existence and uniqueness, we next turn to characterizing the BE network. The dispute technology and players’ ideologies jointly determine the incentives for interactions in this model. Dispute technology comprises of an agent’s strength, i.e., the number of allies and the intensity of connections to them, and how cohesive the agents are in disputes, i.e., the number allies who are opponent of their opponent as well. The benefits from cohesion push towards segregation, since players want to coordinate their neighborhoods to share many opponents with their allies. Homophily pushes towards overlaps in the neighborhoods of players in dispute. The structure of the BE network therefore depends critically on how effective cohesion is, and hence $\delta(y)$.

The next Proposition provides a full characterization of the unique BE network subject to the effectiveness of cohesion.

Proposition 2 establishes the link from cohesion to the BE network structure. It is beneficial to have similar allies as one’s allies if cohesion is highly effective. Agents are then willing to tolerate even distant ideologies as long as this increases their cohesion in disputes with opponents who are possibly ideologically more similar. The resulting network comprises of two cliques, where agents are allies with everyone in their clique and opponents of everyone else. At least one of the cliques contains agents of greatly different ideologies and sustaining it requires high tolerance. This is beneficial only when cohesion generates sufficiently high benefits. For intermediate effectiveness of cohesion, agents are not willing to tolerate as far. Cliques are consequently smaller and more homogeneous and the network comprises of more than two cliques.

In all other cases, at least some players ally themselves with opponents of an ally. This requires less tolerance and players forgo some benefits from cohesion to reduce the burden of tolerating other ideologies.

Figures 1 and 2 sketch equilibria for different dispute technologies. Circles represent agents who share allies with their opponents, whereas other shapes and colors are clique specific. Increasing the benefits of cohesion transforms an overlapping society (Panel (a) of Figure 1) into a segregated society (Panel (b) of Figure 1) by encouraging agents to choose more mutual opponents with their allies. For high effectiveness of cohesion, we obtain a society consisting of two cliques as displayed in Panel (a) of Figure 2. Reducing the benefits from cohesion also decreases the incentives to choose similar allies as one’s allies. The equilibrium network then consists of more than two cliques, which are also more homogeneous cliques (Panel (b) of Figure 2).

Our simple equilibrium characterization uncovers important economic implications. Tolerance intervals are generally asymmetric in the BE. The value of a connection depends on the entire network in our model because cohesion depends on the allies of allies. Agents thus have an incentive to choose, at least to some extent, similar allies as their allies, thereby increasing their cohesion and their benefits from disputes with opponents. Proposition 2 illustrates why there is no one-to-one relationship between agents’ ideological proximity and their probability of being allies. In fact, agents with almost identical ideologies may be opponents and at the same time ally themselves with individuals whose ideologies are much farther from their own.

This result rationalizes several seemingly paradoxical, yet common phenomena, for instance, (i) why extremist figures are able to secure support from moderates within their organization, (ii) why seemingly small ideological differences between organizations or factions prevent cooperation between them, or (iii), why seemingly small ideological differences within a political organization can cause its split. To go beyond those insights, the next section investigates how the equilibrium structure and the associated outcomes are affected in a changing economy.

The equilibrium characterization uncovers several important features about interactions among ideologically differentiated agents. A natural next step is to investigate which changes we might expect in the alliance network in a changing economy and the implication for polarization. To do so, we study the effect of an increase in the effectiveness of cohesion. Cohesion may become more effective either directly, for instance because potential supporters value cohesion more, or indirectly, because the cost of effort determines how much benefits agents can extract from disputes through their strengths. Whenever it is costly to strengthen the ties to allies, agents become more homogeneous in their strengths and the relative benefits from cohesion increase. Indeed, many countries regulate campaigning expenditures or the contents that campaigns may contain to prevent coordination between political agents. Moreover, anti-lobbying efforts increase the hurdles for exchanges between politics and interest groups.

Dispute intensity (our measure of polarization) depends on two characteristics of the network: (i) the number of disputes in society, i.e., the extensive margin of dispute intensity, and (ii) the strength of players in dispute captured by the sum of the weights of all their connections, i.e., the intensive margin of dispute intensity. It is thus useful to first establish a general result on the relationship between equilibrium efforts and equilibrium degrees.

Lemma 2 links a player’s equilibrium degree to their equilibrium effort. Players with more allies exert lower efforts in equilibrium for two reasons. First, the network determines who are allies and opponents, so agents with more allies are necessarily in fewer disputes. Their efforts thus generate benefits in fewer instances, thereby reducing the incentives to exert effort. Second, efforts of neighbors also contribute to the strength of an agent. High degree players enjoy more spillovers from their neighbors’ efforts and need not exert as much effort themselves.

Lemma 2 Lemma is useful to analyze the comparative statics of our model because it tells us when the extensive and intensive margin go in the same direction.

From now on, let us consider the parametric dispute technology of Definition 1, i.e., the augmented Tullock contest. We are interested two key comparative statics. First, how does a direct increase in the value of cohesion affect polarization? Second, how does an increase in the cost of exerting effort affect polarization?

Increasing the effectiveness of cohesion has ambiguous effects on polarization depending on whether the initial network is segregated or not. In an overlapping society, higher effectiveness of cohesion gives incentives for agents to coordinate on more similar neighborhoods with their allies. There are consequently fewer overlaps in opponents’ neighborhoods which leads to an increase in the number of disputes in the economy.²⁰²⁰20This is because tolerance is increasingly costly in the distance between the own and others’ ideologies. Hence, agents choose narrower tolerance intervals in an overlapping society when cohesion becomes more effective. An increase in polarization follows immediately because efforts are increasing in the number of disputes they are in (Lemma 2). Polarization increases unambiguously on the extensive and intensive margin and the first statement of Proposition 3 follows.

The reverse is true in an initially segregated society. As players derive greater benefits from cohesion, at least some cliques must become larger. The players in large cliques are in fewer disputes and exert lower efforts. This results in a decrease in dispute intensity on the extensive and intensive margin, thereby unambiguously decreasing polarization.

The results of Proposition 3 illustrate the important role of the initial network structure for the effect of interventions in this economy. In general, there is no universally best tool to tackle polarization in ideologically heterogeneous societies. Instead, a similar intervention has drastically different effects in an overlapping society compared to a segregated society.

An interesting insight of Proposition 3 is that small groups of ideologically extreme types may be a “necessary evil” for a society. Attempts to dissolve such extremist group may in fact backfire and result in more polarization. This is because reducing the effectiveness of cohesion would at first reduce the incentive to be part of the “silent majority”, and encourage agents to join an ideologically extreme group.

Various characteristics of a society, e.g., the media landscape, the general public, or even the characteristics of officials in parties or organizations, influence how effective cohesion is for political agents. A more tangible policy lever is to influence the cost of exerting effort. To serve, exposition, we employ the notion of a dense society. Formally, define a dense society such that $n\to\infty$ and $P(\exists i\in N:\theta_{i}\in[\theta-\epsilon,\theta+\epsilon])\to 1$.

Increasing the cost of effort always reduces the total effort exerted by agents in equilibrium. While intuitive, this result is not entirely trivial. We must consider how potential changes in the network structure affect individual equilibrium efforts since changes in the effort cost also influence the indirect benefits from cohesion. The returns to exerting effort are diminishing, so even agents who engage in more disputes need not exert as much effort when their opponents’ efforts tend to be small, for instance, due to an increase in $c$. Total efforts therefore decrease as exerting effort becomes costlier. This result establishes the effect of an increase in the effort cost on the intensive margin of dispute intensity, which is negative. The total effect on dispute intensity is ex ante ambiguous since increasing the effort cost also alters agents’ tolerance choices.

Whenever an increase in the effort cost decreases the number of disputes, dispute intensity unambiguously decreases. Proposition 4 identifies precisely when this is the case. For sufficiently high initial costs of exerting effort, agents are of relatively similar strength, even when their degrees differ. Poorly connected agents are then (almost) on par with better connected agents in terms of their strengths. While it becomes costlier for agents to establish reliable alliances, cohesive players can still extract high benefits. In other words, agents substitute away from investing in their strengths towards investing in their cohesion. An increase in the effort cost would then imply that at least one clique becomes larger. The agents in larger cliques are in fewer disputes, thereby reducing the number of disputes in the economy. Dispute intensity decreases on the extensive margin and the total effect is negative.

Whenever the initial cost of exerting effort is not too high, the network is an overlapping society. Agents are sufficiently dissimilar in their strengths to ensure enough benefits stem from investing in their connections relative to investing in their cohesion. An increase in the cost of exerting effort reduces equilibrium efforts and thus makes agents more similar in terms of their strengths. The equilibrium network moves towards segregation. This increases the number of disputes. The effects on the extensive and intensive margin go in opposing directions. When the network is initially close to segregation, i.e., the cost of exerting effort is intermediate, increasing the cost of effort crowds out efforts only by little. This is because agents are in many disputes already and engage in even more due to the increase in the effort cost. Since agents in more disputes exert higher efforts (Lemma 2), the effect on the intensive margin becomes smaller as the network approaches segregation. The extensive margin eventually outweighs the intensive margin and dispute intensity increases in the cost of effort for an initially intermediate effort cost.²¹²¹21We assume a dense society in order to deliver a clean characterization of these effects. In a sparse society, similar results would occur, however, the agents would not alter their tolerance choices for marginal increases in the cost of exerting effort. Assuming a dense society ensures the existence of an interval corresponding to “intermediate effort cost”.

Cases where the network is far from segregation correspond to “low costs” of exerting effort. In this case, the extensive and intensive margin of dispute intensity also go in opposing directions when the cost of exerting effort increases. However, the intensive margin of dispute intensity outweighs the extensive margin. In some sense, the intermediate cost case emerges once the effect on the number of disputes dominates the crowding out of efforts.

Overlaps in the neighborhoods of allies are informative about the state of the world, i.e., whether the effort cost is low, high, or intermediate. This is important when trial and error strategies to elicit whether the economy is initially in the low cost or intermediate cost case are unfeasible.

Figure 3 sketches the transformations of the BE network when effort costs increase from intermediate to high. Nodes $i$ and $j$ connect otherwise segregated players in an overlapping society (Panel (a)). An increase in the effort cost increases the relative benefits from cohesion. Player $i$ ($j$) thus benefits more from being in dispute with everyone to their left (right). This is because all players to their right (left) are in dispute with the same players. The cohesion of players $i$ and $j$ is higher in Panel (b) compared to Panel (a) and so is the cost of tolerance. A further increase in the effort cost increases the size of the largest clique and we ultimately arrive in the situation displayed in Panel (c).

Heterogeneity in ideologies always raises the question of the role of “extreme ideologies”. In our model, extreme ideologies are types close to zero or one. On the one hand, homophily makes extreme ideologies unappealing, since allying oneself with extreme ideologies induces a high reputation loss for most types. On the other hand, agents with extreme ideologies are in many disputes, thereby substantially increasing the cohesion of their allies. Extreme types are thus appealing to those who are sufficiently close to them. How these agents respond to changes to the relative benefits from cohesion is crucial to understand the overall effects of attempting to influence extreme types. The next corollary addresses this.

The behavior of players with extreme ideologies is generally ambiguous when cohesion becomes more effective in segregated societies (Proposition 3 and 4). In principle, as one clique grows, extreme types may lose connections and increase their efforts to dispute against others. In an overlapping society, however, those extreme types become increasingly appealing as connections since they are in many disputes and cohesion becomes more effective. Then, players with extreme allies dispute effectively against their opponents and extreme types form more connections. Those types also reduce their efforts and engage in fewer disputes, while overall polarization increases.

The results of Corollary 1 rationalize why relatively moderate types may be drawn towards the ideologically extreme and away from ideologically more similar moderates. Extreme allies serve to show a clear edge towards others and do so more effectively compared to moderate allies.

Corollary 1 also uncovers a trade-off between “more polarization” and “influencing extremists”. Against common intuition, interventions tailored to reduce the influence of few “extremists” on polarization may in fact end up increasing polarization in society. This is because such extremists continue to polarize by dragging more moderate types into their tolerance window, who in turn contribute more to overall polarization.²²²²22This result exhibits similarities to the case of intolerant extremists in Genicot (2022). When extremists are less tolerant, moderate types compromise for extremists and consequently choose extreme actions.

In this section, we discuss some extensions and implications of the model. The appropriate proofs and other extensions can be found in Appendix B.

Initiation of dispute: In the baseline model, tolerance decisions also directly determine disputes. While natural for the contexts we have in mind, this assumption does not qualitatively affect the results of the paper. Suppose agents pay a finite cost for initiating a dispute with someone, denoted by $D$. Let $g_{ij}=-1$ denote a dispute between $i$ and $j$, whereas $g_{ij}=0$ indicates a neutral relationship. It is natural to think of dispute initiation as a one-sided decision, so let $\bar{g}_{ij}=-1$ if $\min\{g_{ij},g_{ji}\}=-1$.

A dispute is initiated if the benefits outweigh the costs. The game is economically equivalent to the baseline model.²³²³23However, some technical inconveniences arise. First, agents would want to fend off (some) disputes by investing a lot into their strength, i.e., choosing high efforts. However, given disputes against them are not initiated, agents would want to reduce their efforts ex post. Non-existence of an equilibrium may arise. Sufficiently low costs ensure the initiation of each dispute.

Heterogeneous flexibility: The baseline model assumes uniform flexibility $\tau$. Allowing for any possible $\tau_{i}\neq\tau_{j}$ is difficult to characterize due to its many degrees of freedom. Suppose instead flexibility correlates with “extremism”. We study two versions of the model with either stubborn or flexible extremists.²⁴²⁴24Extremists hold strong views and consequently insist on them, making them less willing to tolerate other ideologies. Indeed, van Prooijen and Krouwel (2017) document dogmatic intolerance of individuals with extreme political beliefs. On the other hand, extremists may be more grounded in their ideology and therefore appreciate interactions with those of different ideologies.²⁵²⁵25Genicot (2022) uses a similar approach to impose structure on agents’ willingness to compromise for others.

To introduce stubborn extremists formally, let $\tau_{i}=\underline{\tau}+\kappa|\theta_{i}-1/2|$, i.e., more extreme players pay a higher cost for tolerance.

The model with stubborn extremists preserves orderedness of the BE network. Moderates are less constraint in their tolerance decision, since they may tolerate ideologies to their right and their left. Stubborn extremists would therefore never tolerate other extremists, but only more flexible moderates. The flexibility of agents with extreme ideology is thus decisive for whether they connect to moderates and the equilibrium network is ordered. The model with stubborn extremists thus produces qualitatively similar results to the baseline model, however, we would expect segregation to arise for lower benefits from cohesion.

Whereas stubborn extremists seem to be the natural assumption, sometimes extremists are more flexible in whom they want to interact with. Consider the specification $\tau_{i}=\max\{\bar{\tau}-\kappa|\theta_{i}-1/2|,0\}$, which is the model with flexible extremists.

Since the stubborn moderates tend to be intolerant towards flexible extremists, there may arise situations where extremists tolerate extremists from the other end of the ideological spectrum. The BE network with flexible extremists need no longer be ordered and other equilibrium structures may exist. The characterization of these structures is beyond the scope of this paper. However, we could imagine, for appropriate values of $\kappa$, $\alpha$ and $\beta$, a segregated equilibrium where agents with extreme ideology tolerate those of opposing end of the ideology space while no one else tolerates them. Stubborn moderates leave extreme types with too few allies who then resort to becoming tolerant towards extremists instead.

The network in Figure 4 sketches an equilibrium consisting of two cliques in dispute. One clique contains the flexible extremists $1$ and $0$ whereas the other clique contains the remaining stubborn moderates. The tolerance cost for $1$ and $0$ is low enough to tolerate each other whereas moderates benefit enough from cohesion in order not to tolerate either extremist.

This paper studies how the concern for cohesion impacts the arising ally and opponent relationships among ideologically differentiated political actors. Agents choose the ideological composition of their allies by their tolerance decision and the strength of their connections by a generic effort. The number of allies and the intensity of connections to them captures the strength of an agent and uniformly increases their benefits from a dispute with everyone who is not their ally. Moreover, an agent is more cohesive in a dispute and derives greater benefits from it when they share fewer allies with their opponent.

The equilibrium network structure depends on how effective cohesion is. If cohesion is sufficiently effective, the equilibrium network is segregated into cliques. Otherwise, the network is an overlapping society where opponents have mutual allies. Agents are generally not allies with those who are ideologically closest. Instead, agents of relatively similar ideologies may be opponents and have allies who are ideologically farther.

An increase in the effectiveness of cohesion has ambiguous effects on the division in society. In an initially segregated network, at least some cliques become larger and thus ideologically diverse, thereby reducing polarization in society. For an initially overlapping society, on the other hand, higher concerns for cohesion lead to more segregation and polarization.

Altering the cost of exerting effort is a tool for a policy maker to influence the relative benefits from cohesion. For higher effort costs, agents are more concerned with their cohesion and high enough effort costs induce a segregated equilibrium network. Then, higher relative concerns for cohesion would lead more to diverse cliques and reduce polarization.

For low and intermediate initial effort costs, an increase in the effort cost encourages segregation. Only for an intermediate effort cost, the network is initially close to segregation and the increase in the number of disputes outweighs polarizes more than the reduction of efforts decreases polarization.

Our results illustrate the complexity behind the incentives of choosing allies and opponents when pursuing an ideological agenda. In general, ideological proximity does not imply cooperation between agents. This sheds light on why some ideological organizations rally behind extremist candidates within or why seemingly close ideological agents and organizations are sometimes bitter rivals, even when they originate from the same ideas and convictions.