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Polarization and Quid Pro Quo: The Role of Party Cohesiveness

When can an interest group exploit ideological and affective polarization between political parties to its advantage? We study a model where an interest group credibly promises payments to legislators conditional on voting for its favored policy. Legislators value voting as their friends within their party, and suffer an ideological-disutility upon voting against their party's ideologically preferred policy. Affective polarization, owing to its interpersonal nature, is modeled by assuming a legislator values distinguishing her voting decision from legislators in the opposite party. Our main finding is that an aggregate measure of relative cohesiveness of social networks in the two parties determines whether the interest group can profitably exploit increasing polarization. However, the significance of relative cohesiveness vanishes if there is no ideological polarization between the two parties.


Introduction
The influence of special interest money on elected officials tops the list when people -regardless of their political leaning -are asked to specify the biggest problem with elected officials in Washington (Pew Research Center, 2015). At the same time, political polarization in the United States has increased dramatically over the last few decades.
Ideological polarization in the general public is reflected in more than 70% of Americans believing that voters in both parties "not only disagree over plans and policies, but also cannot agree on the basic facts" (Pew research Survey, 2019). Affective polarization -ill will, animus, or antipathy towards those that lean towards the other party -is reflected in nearly 55% of Republicans saying that Democrats are "more immoral" when compared with other Americans; 47% of Democrats say the same about Republicans (Pew Research Center, 2019). Despite deep ideological and affective polarization across party lines, voters across the political spectrum remain concerned about the role of money in politics.
Whether material or informational exchange between special interest groups and legislators has an impact on legislative outcomes remains a matter of intense scolarly scrutiny (Battaglini and Pattachini, 2018;Bombardini and Trebbi, 2020). Recent studies have documented increasing ideological, as well as, increasing affective polarization political elites (Enders, 2021). However, the impact of rising polarization between political parties on the ability of an interest group to influence legislation in its favor has not been explored. The present paper aims to fill this gap.
We draw upon the framework offered in Battaglini and Pattachini (2018) and suitably modify it to account for ideological and affective polarization in order to investigate when can an interest group exploit polarization between political parties to its advantage. Our baseline model involves two ideologically polarized parties that hold opposite positions on an issue (e.g., relaxed versus tighter controls on carbon emissions). The interest group strictly favors one policy over the other, and credibly promises resources to legislators if they vote for its favored policy. We refer to the party ideologically inclined towards the interest group's favored policy as the party in "favor" of the interest group; and, the other as the party "against" the interest group. We assume each legislator derives resourceutility from the investment she receives from the interest group. Legislators suffer an ideological-disutility if they vote against their party's ideologically preferred policy. The magnitude of this disutility serves as a measure of ideological polarization between the parties.
A key feature of our baseline model is that legislators within a party are connected via a network of friendships. We assume each legislator in a party derives network utility if her friends within her party vote as she does, regardless of the policy they vote for.
In particular, voting against the ideologically preferred policy of one's party may be costly due to a variety of factors, but such costs may be attenuated if one's friends also vote similarly. The final ingredient of the baseline model is that each legislator obtains a private preference shock prior to voting which provides all additional relevant information.
We assume the interest group maximizes the sum of the probabilities of legislators voting for the policy it favors.
While the two major political parties in the United States currently seem to be as ideologically polarized as ever, political scientists view increasing affective polarization as one of the most striking developments in U.S. politics (Mason, 2015;Rogowski and Sutherland, 2016;Iyengar and Krupenkin, 2018;Iyengar et al., 2019;Enders, 2021).
Affective polarization differs from ideological polarization due to its inter-personal nature as ideological polarization simply refers to opposite policy preferences.
The bulk of existing literature in economics and political science has focused on documenting affective polarization in the general public. Using survey data from delegates to presidential nominating conventions, Enders (2021) finds that even the political elites "exhibit extreme emotional reactions toward political groups just like their mass counterparts". Further, recent studies suggest that affective polarization can be conceptualized as if legislators value distinguishing their voting decisions from legislators in the ideologically opposite party (Druckman et al., 2021). Drawing upon these studies, we formalize affective polarization by assuming a legislator suffers an affective-disutility when legislators in the opposite party vote as she does. The magnitude of this affective-disutility serves as the measure of affective polarization between the parties.
The analysis of both models involves two technical assumptions. The first ensures the probabilities of legislators voting in favor of the interest group are strictly in the interior of the unit interval such that the interest group is sufficiently uncertain about the voting decisions of legislators. The second ensures that investments towards legislators never serve as a "bad" from the perspective of the interest group (i.e., the interest group has an incentive to invest towards legislators).
We show that the relative structure of party networks determines whether growing polarization between parties aids the interest group. Specifically, the marginal impact of increasing polarization (ideological or affective) between the parties on the interest group depends on the relative Bonacich-cohesiveness of the two parties, where Bonacichcohesiveness of a party refers to the sum of Bonacich centralities of all its legislators 1 . For instance, an increase in ideological polarization between parties benefits the interest group if and only if the party in favor of the interest group is relatively more Bonacich-cohesive.
Similarly, an increase in affective polarization increases the marginal returns from increasing ideological polarization if and only if the party in favor of the interest group is relatively more Bonacich-cohesive. However, the significance of relative cohesiveness of party networks vanishes if there is no ideological polarization.
Our paper is most closely related to Battaglini and Pattachini (2018), henceforth BP2018, that examines competition between multiple interest groups in influencing vot-1 Bonacich centrality of an agent is related to the centralities of those with whom she is connected (Bonacich, 1987;Ballester et al., 2006). ing decisions of socially connected legislators while allowing cross-party linkages. The politically salient notions of "party", "ideology", and "polarization" are not central to the theoretical analysis in BP2018 2 . It shows that equilibrium investments by the interest groups are proportional to the Bonacich centralities of legislators in the overall network that allows for cross-party linkages. In our baseline model, we assume there are no cross-party linkages. Our extended model that accounts for affective polarization in addition to ideological polarization permits (negative) cross-party linkages. These modeling choices provide a plausible and tractable way to make the notions of party, ideology, and polarization salient. In both of our models, relative cohesiveness of party networks determines whether the interest group can profitably exploit polarization. This highlights that the significance of relative cohesiveness is robust as it exists with or without cross-party links 3 .
BP2018 supplements equilibrium existence and uniqueness results with empirical analysis. Our focus is on the comparative statics. In particular, we show that a simple aggregate level feature of the network -relative cohesiveness of party networks -determines whether the interest group can exploit increasing polarization to its advantage. BP2018 shows that two competing interest groups with symmetric budgets nullify each other's impact in so far as legislators' voting decisions are concerned. This is not the case with asymmetric budgets. To the extent that some asymmetry between interest groups is a plausible assumption, our choice to focus on one interest group can be viewed as an extreme form of asymmetry that helps obtain the sharpest analytical results.
Finally, our work contributes to the growing literature on affective polarization within economics. This nascent literature has focused on documenting trends in affective polarization in the general public and understanding the reasons behind it (Boxell et al., 2021).
To the best of our knowledge, our paper is the first to formally model the notion of affective polarization and theoretically analyze its impact in a politically and economically relevant context.

Baseline model
Suppose n ≥ 2 legislators have to vote for one of two policies: policy-f or policy-a.
Each legislator belongs to a party P ∈ {F, A}. Let n F ≥ 1 and n A ≥ 1 denote the number of legislators in party F and party A, respectively. There exists an interest group that strictly favors policy-f over policy-a, and credibly promises investments towards legislators if they vote for its favored policy, i.e., policy-f . Party F is ideologically inclined towards policy-f and thus favors the interest group, whereas party A is ideologically inclined towards policy-a and thus against the interest group.
Legislators within a party are assumed to be connected via an exogenously given social network. The strength of the mutual friendship between legislators i and j in a party is denoted by g ij ∈ [0, 1], with g ii = 0. We assume that no two legislators belonging to different parties are friends such that g ij = 0 if legislator i belongs to party P ∈ {A, F } and legislators j belong to party P = P . This assumption is not essential for the analysis but helps focus on how differences in the networks across parties impact the interest group.
The n P × n P symmetric matrix G P shall represent the social network in party P ∈ {F, A}. We denote the "overall network" by G, which is a n × n square matrix that can be viewed as a 2 × 2 block matrix as shown below The n F × n F matrix G F and the n A × n A matrix G A are the two blocks on the main diagonal of G, and the two off diagonal blocks are matrices of zeroes.

Objective of legislators
The total utility of legislator i belonging to party P ∈ {F, A} upon voting for policy-a, i.e., against the interest group, is assumed to be A legislator in party F suffers an ideological-disutility of magnitude σ ≥ 0 from voting against the interest group given the ideological inclination of party F towards the policy favored by the interest group. A legislator in party A does not suffer such a disutility as party A is ideologically inclined against the interest group. The magnitude of σ can be interpreted as the intensity of ideological polarization between the two parties regarding the underlying issue.
The second term captures the network effect due to friendships. The indicator variable v j (a) takes the value 1 when legislator j = i votes against the interest group, and zero otherwise. A legislator in any party who votes against the interest group derives a utility δ ≥ 0 per unit strength of friendship with a legislator in her party that also votes against the interest group.
The final termia -indicates a private exogenous preference shock that is realized after the interest group announces the vector of investments toward legislators but before the legislators vote. It may be interpreted as a private signal that provides a legislator all additional relevant information to assess the utility she will derive if she votes against the interest group. The preference shocks are independently and identically distributed across legislators, and common knowledge. Specifically, they are uniformly distributed over the interval [− 1 2θ , 1 2θ ] with density θ > 0 and mean zero. In specifying the utility derived by legislators upon voting in favor of the interest group, we assume, without loss of generality, that the preference shock can be normalized to zero. The utility of legislator i belonging to party P upon voting in favor of the interest group, i.e, voting for policy-f , is The first term here captures the idea that legislators in the party against the interest group suffer an ideological-disutility of σ upon voting in favor of the interest group.
The second term indicates the resource-utility to a legislator from the investment she would receive from the interest group conditional on voting in favor of the interest group.
We assume resource-utility function is continuous, strictly increasing, twice continuously differentiable, and strictly concave in m i , with u(0) = 0 and marginal utility u i → ∞ as m i → 0. The third term captures the network utility as described earlier.
The network effect in our model thus captures the benefit derived by a legislator when her friends in her party vote as she does, regardless of which policy they vote for. Further, the above formulation amounts to implicitly assuming that legislators care about their actions rather than the outcome of voting process.

Objective of the interest group
The ex-ante probability that legislator i votes in favor of the interest group -after the investments are announced by the interest group but before the private preference shocks are observed -is the expected value of v i (f ) given the distribution of preference shocks.
We henceforth denote this expectation E v i (f ) by q i . We restrict attention to pure investment strategies by the interest group and assume it allocates its resource of size M > 0 among the legislators to maximize the sum of ex-ante probabilities of the legislators voting for its favored policy.

Equilibrium and comparative statics
As described earlier, the expected value of v j (f ) from the perspective of legislator i = j is q j -the probability that legislator j votes in favor of the interest group -since preference shocks are observed privately. Consequently, the probability that legislator j votes against the interest group will be (1 − q j ). Further, the expected value of the preference shock conditional on observing the preference shock will simply be the observed preference shock. Hence, conditional on her preference shock, the expected utility of legislator i in any party P if she votes against the interest group is Conditional on the private shock, we assume a legislator will vote for the interest group if her expected utility from voting for the interest group is at least as high as her expected utility from voting against the interest group. Consequently, the ex-ante probability that a legislator votes for the interest group -i.e., before she observes her private preference shock -will be the probability that her private preference shock does not exceed a certain threshold. The probability that legislator i in party P ∈ {F, A} votes in favor of the interest group turns out to be where σ P = σ if P = F and −σ if P = A. Letq(P ) and q(P ) denote the highest and lowest probabilities of any legislator in party P voting for policy-f . The highest probability of any legislator in party P , say, legislator i, voting in favor of the interest group is where, ifq(P ) ∈ [0, 1], then it would be given bȳ This is becauseq(P ) would correspond to the scenario where (i) the interest group invests all its resource on legislator i, (ii) g ij = 1 for all j = i in party P , and (iii) q j = 1 for all j = i in party P . Analogously, the lowest probability of a legislator in party P , say, legislator i, voting in favor of the interest group is where q(P ) = 1 2 This is becauseq(P ) would correspond to the scenario where (i) the interest group invests nothing towards legislator i, (ii) g ij = 1 for all j = i in party P , and (iii) q j = 0 for all j = i in party P . Following Battaglini and Pattachini (2018), we assume the following to ensure voting probabilities always lie in the interior of the unit interval.
Assumption A1 effectively says that any parameter in the model, given the values of the other parameters, cannot be too large. It will hold, for instance, if the density of preference shocks, θ, is small enough to ensure the interest group is sufficiently uncertain about preferences of legislators.
Given the interdependence in voting probabilities of legislators, a vector of equilibrium voting probabilities of legislators conditional on a vector of investments by the interest group will be a solution of the system of linear equations generated by (7). This system of linear equations induces a continuous mapping of probability vectors from [0, 1] n to (0, 1) n . The domain of this mapping is a compact and convex set, the co-domain is a subset of the domain, and the mapping is continuous since it is linear. Hence, Brouwer's fixed-point theorem ensures a solution exists. Our next assumption ensures the solution is unique.
When both A1 and A2 hold, the unique equilibrium vector of voting probabilities that solves the system of linear equations generated by (7) conditional on investment vector m is where 1 is the n × 1 vector of ones, I is the n × n identity matrix, G is the overall network specified in (1), u(m) = (u(m 1 ), u(m 2 ), · · · , u(m n )) is the vector of resource-utilities derived by legislators, and σ = (σ, σ, · · · , −σ, −σ) is the vector that accounts for ideological inclinations of legislators (with the positive entries corresponding to legislators in the party in favor of the interest group).
Assumption A2 holds when the matrices (I − 2δθG F ) and (I − 2δθG A ) are invertible. 4 An additional, and crucial, implication of A2 is that every component of the associated vector b(δ, θ, G) = (I − 2δθG) −1 · 1 is strictly positive (Ballester et al., 2006). As we shall shortly describe, this ensures investments towards every legislator serve as a "good" from the perspective of the interest group.
The vector b specifies the Bonacich centralities of the legislators, whereby one's centrality depends on the centralities of those with whom one is connected. In general, given a network G = [g ij ] n×n involving n agents, the Bonacich centrality of agent i is which, in turn, implies the vector of Bonacich centralities is where the non-negative decay factor β ∈ [0, 1) scales down the relative weight of longer paths between agents, 1 is a n × 1 vector of one's, and I is the n × n identity matrix. In the context of our model, the decay factor is given by β = 2δθ.
Assumption A1 implies β < 1 and assumption A2 implies the Bonacich centrality of each legislator is well-defined and strictly positive. Under these assumptions, we obtain the following proposition 5 .

Ideological polarization
An increase in ideological polarization implies legislators in the party against the interest group would suffer greater disutility if they vote in favor of the interest group. On the one hand, this would require the interest group to increase investments toward these legislators to induce them to vote in its favor. On the other hand, given the limited 5 The proof is analogous to the proof of Proposition 1 in BP2018.
budget of the interest group, such tailoring of investments cannot compensate for the greater ideological-disutility once ideological polarization becomes sufficiently large. The following corollary highlights that the relative cohesiveness the two parties determines the impact of an increase in ideological polarization on the interest group.

Corollary 1. An increase in ideological polarization benefits the interest group if and
only if the party in favour of the interest group is relatively more Bonacich-cohesive.
Using (15), the equilibrium value of the interest group's objective is Corollary 1 follows directly from the observation that ∂Q * ∂σ ≥ 0 if and only if B F > B A . It highlights the crucial role of relative cohesiveness of networks in the two parties in determining whether increasing ideological polarization between parties aids the interest group. To further see the import of this proposition, it is helpful to contrast it with a model that ignores the network among legislators. Such a model is a special case of our model, and corresponds to both G F and G A being the empty network. The Bonacich centrality of every legislator will then be unity. The Bonacich-cohesiveness of a partythe sum of Bonacich centralities of all legislators in the party -will thus be the number of legislators in the party. Consequently, if the party against the interest group has relatively fewer legislators, then an increase in ideological polarization can never hurt the interest group. In this sense, Corollary 1 highlights that greater Bonacich-cohesiveness of the party against the interest group can compensate for its potential numerical disadvantage in limiting the influence of the interest group.

Impact of changes in network structure
We shall say a network G benefits the interest group relative to some other network G if Q * (G ) > Q * (G). We suppress the dependence on other primitives for notational convenience. For any party P ∈ {F, A}, we say network G s P = [g s ij ] is a stronger network among legislators in party P than network G P = [g ij ] if every g s ij ≥ g ij and at least one inequality is strict.

Consider a pair of overall networks G and G s such that the network is party A is relatively stronger under the overall network G s . There exists a critical level of ideological
The formal proof is provided in the Appendix. Intuitively, part (a) follows because a stronger network in the party in favor of the interest group allows the interest group to better exploit the increased network effect under the relatively stronger network. Part (b) highlights that the interest group may be able also to profitably exploit an increase in the strength of the network in the party against it.
Given any voting pattern of legislators in the party against the interest group, a stronger network in this party would increase the utility derived by some legislators due to the increased network effect under the stronger network. Legislators in the party against the interest group suffer an ideological-disutility from voting in favor of the interest group. However, if the ideological-disutility is sufficiently low, then the interest group can tailor higher investments towards at least some legislators that have sufficiently high Bonacich centralities, and thereby increase their probability of voting in favor of the interest group. This, in turn, may increase the probability of their "close" friends voting in favor of the interest group as well. However, no such tailoring of investments can compensate for the greater ideological-disuitlity suffered by these legislators once ideological polarization between the parties becomes sufficiently high. Our next proposition notes two additional implications of the baseline model.

Proposition 3(b). An increase in strength of the network in a party increases total investment towards it.
Part (a) follows from the fact that the optimization problem of the interest group is independent of the level ideological polarization between parties. With respect to part (b), note that a stronger network in party P will weakly or strictly increase the Bonacich centrality of every legislator in party P , and will not alter the Bonacich centrality of any legislator in party P = P . Consequently, the interest group will attach weakly greater weights on legislators in party P in its optimization problem when the network in party P gets stronger; but, the weights on legislators in party P will remain unchanged. Given the fixed budget of the interest group, the equilibrium investment on every legislator in party P will be relatively lower under the relatively stronger network in party P . Hence, total investment towards party P will decrease. Consequently, total investment towards party P will increase.

Affective polarization
We model "pure" affective polarization as an affective-disutility that a legislator suffers when a legislator in the opposite party votes as she does, regardless of the policy they vote for. This is equivalent to assuming each legislator in one party values distinguishing her vote from legislators in the other party. The size of the affective-disutility serves as a measure of affective polarization between the parties. Building up on the baseline model, we assume that utility of legislator i belonging to party P ∈ {F, A} upon voting against the interest group (i.e., voting for policy-a) is where the indicator variable v j (a) takes the value 1 when legislator j = i votes against the interest group, and zero otherwise. The second term captures the network effect and its interpretation is as in the baseline model. The third term captures the idea that a legislator in any party who votes for policy-a suffers an affective-disutility α > 0 per legislator of the other party that also votes for policy-a. Finally, as in the baseline model, the preference shock ia is assumed to be uniformly distributed over the interval [− 1 2θ , 1 2θ ] with density θ > 0 and mean zero. Turning to the utility of legislator i in party P upon voting in favor of the interest group, we assume where the indicator variable v j (f ) is 1 when legislator j votes in favor of the interest group, and zero otherwise. The third term captures the idea that a legislator in any party who votes for policy-f suffers an affective-disutility α > 0 per legislator of the other party that also votes for policy-f .
Conditional on the private shock, we assume a legislator will vote for the interest group if her expected utility from voting for the interest group is at least as high as her expected utility from voting against the interest group. Consequently, as in the baseline model, the ex-ante probability that a legislator votes for the interest group -i.e., before she observes her private preference shock -will be the probability that her private preference shock does not exceed a certain threshold. The probability that legislator i in party P ∈ {F, A} votes in favor of the interest group turns out to be Conditional on an investment vector m, the probability that any legislator i in party P ∈ F, A votes in favor of the interest group is where σ P = σ if P = F and −σ if P = A. As in the baseline model, letq(P ) and q(P ) denote the highest and lowest probabilities of any legislator in party P voting for policyf . As in the baseline model, the following assumption ensures the voting probabilities lie in the interior of the unit interval.
A vector of equilibrium voting probabilities of legislators conditional on a vector of investments by the interest group is a solution to the system of linear equations generated by (19). As in the baseline model, Brouwer's fixed-point theorem ensures a solution exists.
In order to ensure uniqueness, we need an assumption analogous to assumption A2 in the baseline model. It is helpful to first define the matrix where the blocks on the main diagonal correspond to the networks in the two parties, and the matrices 1 in the off diagonal blocks are matrices of ones having the appropriate order such thatG is a symmetric n × n square matrix.
where the vector u(m) = u(m 1 ), u(m 2 ), · · · , u(m n ) specifies the resource-utility of legislators, the vector σ = (σ, σ, · · · , −σ, −σ) corresponds to the ideological inclinations of legislators, and the associated vector provides the modified Bonacich centralities of legislators. Algebraic manipulation of the above equation reveals that, in the presence of affective polarization, the modified Bonacich centrality of legislator i in party P ∈ {F, A} is related to the modified Bonacich centralities of all other legislators as follows: where β = 2δθ andα = 2θα. The modified Bonacich centrality of a legislator is thus positively related to the modified Bonacich centralities of legislators in ones own party, and negatively related to the modified Bonacich centralities of legislators in the opposite party. As we detail in the Appendix, (22) implies the modified Bonacich centrality of where ω P is a factor that modifies the "standard" Bonacich centrality b i (P ) of each legislator i in party P in the presence of affective polarization. B P and B P are the "standard" Bonacich-cohesiveness of party P and party P as in the baseline model.
Affective polarization modifies the standard Bonacich centralities of legislators in two ways. The denominator of ω P highlights that the mutuality inherent in affective polarization amplifies the standard Bonacich of every legislator in both parties. Higher affectivedisutility and greater cohesiveness of either party increase this amplification factor. The term in the numerator of ω P dampens the standard Bonacich centrality, where the dampening factor increases with increasing affective polarization and the cohesiveness of the opposite party. Further, the amplifying factor is identical for every legislator in both parties and decreases non-linearly with an increase in affective polarization. In contrast, the dampening factor is relatively stronger for the party that is relatively less cohesive and declines linearly with an increase in affective polarization.
Lemma 1. Suppose B P ≥ B P such that party P is relatively more Bonacich-cohesive.
(a) ω P (0) = 1, lim α→α ω P (α) = 0, and ∂ω P ∂α < 0 for α ∈ [0,α]. (b) ω P (0) = 1, lim α→α ω P (α) = 1, and Under our assumptions, the dampening effect outweighs the amplifying effect such that the Bonacich centralities of legislators in the presence of affective polarization are lower than in the absence of affective polarization. However, as lemma highlights, the impact of increasing affective polarization on Bonacich centralities the can be monotonic or non-monotonic depending on whether a legislator belongs to the party that is relatively less or relatively more cohesive.
Using (21), the value of the objective of the interest group as a function of its invest-ments is Equation (25) There exists a critical α c ∈ (0,α) such that Part ( which is the difference in the equilibrium values of the interest group's objective at the highest and lowest feasible levesl of affective polarization. Using Lemma 1 and equation (25), if party P is more Bonacich-cohesive than party P , then where σ P = σ if P = F and −σ if P = A. Suppose the party in favor of the interest group is relatively more Bonacich-cohesive. Since investments by the interest group are independent of the level of ideological polarization, equation (27) suggests that higher ideological polarization makes it more likely that the interest group benefits from suf-ficiently high affective polarization relative to the absence of affective polarization. In contrast, if the party against the interest group is relatively more Bonacich-cohesive, then this implication is more likely when ideological polarization is relatively lower.
In general, the impact of affective polarization on investments by the interest group towards legislators is difficult to pin down without further assumptions. Assuming logarithmic resource-utility, increasing affective polarization increases total investment towards the party in favor of the interest group if and only if it is more Bonacich-cohesive than the party against the interest group (see Proposition 6 in Appendix B).

Conclusion
The influence of special interest money on elected officials and growing political polarization have been salient themes in public discourse and academic inquiry. We investigate when can an interest group exploit increasing polarization between political parties to its advantage by accounting for both ideological and affective polarization between parties, and the social connections within parties.
Our analysis underscores that the impact of rising ideological or affective polarization between political parties on the interest group depends on the relative cohesiveness of party networks. Higher ideological polarization benefits the interest group if and only if the party in favor of the interest group is relatively more cohesive. An increase in affective polarization increases the returns to the interest group from increasing ideological polarization if and only if the party in favor of the interest group is relatively more cohesive. If the party in favor of the interest group is relatively more cohesive, then the higher the ideological polarization between the parties, the more likely it is that the interest group benefits from affective polarization relative to the absence of affective polarization. However, the crucial role of relative cohesiveness cannot arise unless there exists ideological polarization between the parties.
Our analysis can be extended to account for other relevant considerations (e.g., withinparty ideological heterogeneity). Exploring the co-evolution of political polarization and influence activities may be particularly insightful. Our model may also be modified to investigate other questions related to polarization. For instance, there is growing suspicion that social media platforms facilitate ideological and affective polarization among the general public. Our model can serve as the starting point to investigate these issues by replacing the interest group with a firm, and appropriately modeling its action space and objective function.

APPENDIX Appendix A: Proofs for the baseline model
The arguments behind the proofs of Proposition 1, Corollary 1, and Proposition 3 appear in the main text. In the following we provide the proof of Proposition 2. Consider a pair of overall networks G and G s such that the network is party A is relatively stronger under the overall network G s . There exists a critical level of ideological Consider a pair of overall networks G and G s such that the network is party P is relatively stronger under the overall network G s , while the network in party P = P is identical under G and G s . Let the Bonacich centrality vectors of the legislators under G and G s be b and b s . Since G s is stronger than G, it follows that b s ≥ b, where b s iP ≥ b iP for all i ∈ P with the inequality being strict for at least two legislators in party P , and b s jP = b jP for all j ∈ P . Consequently, the Bonacich-cohesiveness of the parties under the overall networks G and G s will be such that Let m * and m * s denote the equilibrium investment vectors under G and G s . Fix any network G A and consider a pair of overall network G and G s such that the network in party F is stronger in G s . The value of the interest group's objective under G and G s will be as follows.
We first note that where inequality (33) follows from b s i ≥ b i for every i ∈ F, A. For the proof of part (a), fix any G A and consider a pair of overall network G and G s such that the network in party F is stronger in G s than in G. The interest group benefits from the stronger network in party F if Q * (m * , G) ≤ Q * (m * s , G s ) (34) It follows from (28) and (33) that σ(G, G s ) > 0. Hence, (35) holds for every σ ≥ 0, thereby implying that a stronger network in party F always benefits the interest group regardless of the level of ideological polarization between the parties.
For part (b), fix any network G F in party F and consider a pair of overall network G and G s such that the network in party A is stronger under G s than under G. The interest group benefits from the stronger network in party A if Thus, there exists a critical σ(G, G s ) > 0 such that a stronger network in party A benefits the interest group if and only if σ ≤ σ(G, G s ).
where b P is the n P × 1 vector of the "standard" Bonacich centralities of legislators in party P ∈ {F, A}. Also,α ∈ [0,α max ), whereα max = min{ 1 B F , 1 B A }, ensures the modified Bonacich centralities of all legislators are non-negative.
The technical arguments behind the proof of Proposition 4 appear in the main text.
The proofs of Lemma 1, Proposition 5 (a), and Proposition 5 (b) are straightforward. In the following, we provide proofs of Proposition 5 (c) and Proposition (6).
Using (44) and Lemma 1, dQ * dα < 0 at α = 0. If B P > B P , then ω P (α) = 0. This, in turn, implies m * j = 0 for all j ∈ P when α =α. Hence, Proposition 5 (c) follows from the observation that when P = F , both terms on the RHS in (45) are positive. Also note that when P = A, the first term is positive but the second term is negative. Hence, the marginal impact of increasing polarization on the interest group is ambiguous when the party against the interest group is relatively more Bonacich-cohesive. The equilibrium investments by the interest group turn out to be which, in turn, imply