Targeted Lobbying on Council Networks

Abstract

A lobbyist with a limited budget must decide not only how hard to push a bill but whom to push, since influence cascades through a council's network of party, committee, and personal ties. We model this as a static network game in the linear–quadratic tradition of Galeotti, Golub, and Goyal (2020) and embed it in a voting institution that aggregates members' actions into a passage decision. The headline finding is a clean dichotomy: linear voting selects Katz–Bonacich centrality; smooth voting selects slope-weighted Bonacich centrality. Concretely, (i) under a linear passage rule, the cost-minimizing lobbyist concentrates the entire budget on the member with the highest Katz–Bonacich centrality with decay $\beta/c$, refining the eigenvector-centrality result of Galeotti et al. (2020); and (ii) under any smooth strictly increasing vote-aggregation rule, the small-budget optimal target is the member with the largest slope-weighted Bonacich return $$w_j \;=\; \sum_i f'(a_i^0)\, M_{ij}, \qquad a^0 \;=\; M b,$$ where $a^0$ is the no-lobbying equilibrium. When $f$ is linear or baseline actions are zero, $w$ collapses to Katz–Bonacich centrality; when baselines differ and $f$ is concave, slope-weighting can overturn the Katz ranking, so a peripheral fence-sitter can be cheaper to target than a saturated central partisan. Two extensions follow: a bounded-action variant solved exactly by a greedy-with-clawback algorithm (positive externalities only), and a probabilistic-vote companion model in which the swing-voter slope $\sigma'(a^*_i) = a^*_i(1-a^*_i)$ plays the same role as $f'(a^0_i)$.

Introduction

The classical theory of political corruption following Shleifer and Vishny (1993) treats public officials as isolated agents bargaining one at a time. Real legislative bodies are not isolated. Members are connected by parties, committees, whip structures, and personal relationships. If one member declares early support for a bill, an ally often finds it easier to do the same. A lobbyist with a limited budget therefore faces a targeting problem: the question is not only how hard to push, but whom.

We model this as a one-shot game between a lobbyist and an $n$-member council connected by an undirected weighted network $G$. Each member $i \in N := \{1, \ldots, n\}$ chooses a continuous level of support $a_i \in \mathbb{R}$ — speeches, whip activity, committee work — and faces strategic complementarities: the marginal benefit of supporting the bill rises when network neighbors also support it. The lobbyist publicly commits to a non-negative targeting vector $t = (t_1, \ldots, t_n)$ that shifts members' marginal benefits, pays linear cost $\sum_i t_i$, and the institution then aggregates equilibrium actions $a^*$ into passage or failure. The lobbyist wants to clear the passage threshold as cheaply as possible.

Our main result identifies what target is cheapest under a smooth voting rule. Suppose the bill passes when $\sum_i f(a_i^*) \geq K$ for some $C^2$ strictly increasing $f$. For small support gaps, the cheapest target is the member with the largest slope-weighted Bonacich return $w_j = \sum_i f'(a_i^0)\, M_{ij}$, where $a^0 = Mb$ is the no-lobbying equilibrium and $M = (cI - \beta G)^{-1}$. The matrix entry $M_{ij}$ tells us how pressure on member $j$ spreads through the council; the weight $f'(a_i^0)$ tells us how responsive member $i$ is at the no-lobbying point. With a concave $f$ and nonzero baselines, a peripheral fence-sitter can beat a central member whose action is already saturated. The proof reduces to a linear-program perturbation: at leading order, the passage constraint becomes $w^\top t \geq \Delta$, where $\Delta = K - F(a^0)$, so the corner solution spends on $\arg\max_j w_j$.

Two cases collapse this formula to plain Katz–Bonacich centrality. If the passage rule is linear, $f(a) = a$, then $f' \equiv 1$ and $w = M\mathbf{1} =: v$ — the canonical Katz–Bonacich vector with decay $\beta/c$. The lobbyist targets the member with the highest $v_i$ at cost $C^*(\bar T) = (\bar T - v^\top b)/\max_i v_i$. If $b = 0$, every member's no-lobbying action is zero, all slopes equal $f'(0)$, and the target is again the highest-$v_i$ member, with cost rescaled by $1/f'(0)$. Outside these cases, slope weights can change the answer.

Model

A single lobbyist faces a council of $n \geq 3$ members connected by an undirected weighted graph $G = (g_{ij})_{i,j \in N}$ with $g_{ii} = 0$, $g_{ij} = g_{ji} \geq 0$, and largest eigenvalue $\lambda_{\max}(G)$. The graph is public and exogenous; entries $g_{ij}$ measure the strength of committee, party, or relational ties between members. Each member $i$ chooses an action $a_i \in \mathbb{R}$, interpreted as intensity of support: positive values mean active advocacy, negative values mean active opposition. Effort has no hard ceiling; the quadratic cost below is what keeps actions finite.

Before the council acts, the lobbyist publicly commits to a non-negative targeting vector $t \in \mathbb{R}^n_+$ and pays linear cost $\sum_i t_i$. Targeting enters as an intercept shift in member $i$'s payoff. Given $t$, member $i$'s payoff is the linear–quadratic form $$U_i(a_i, a_{-i}; t_i) \;=\; a_i\!\left(b_i + t_i + \beta \sum_{j \in N} g_{ij}\, a_j\right) - \tfrac{1}{2}\, c\, a_i^2,$$ where $b_i \in \mathbb{R}$ is member $i$'s baseline inclination toward the bill, $\beta > 0$ governs peer effects, and $c > 0$ is the cost parameter. Each member is pushed toward support by their own preference $b_i$, by lobbying effort $t_i$, and by neighbor support $\beta \sum_j g_{ij} a_j$; the term $\tfrac{c}{2} a_i^2$ is the quadratic cost of visible support.

The lobbyist values passage at $V_L > 0$ and zero from failure. Under the linear-passage rule, the bill passes if and only if $\sum_{i \in N} a_i \geq \bar T$. Assuming $V_L$ is large enough that the lobbyist strictly prefers passage to any zero-effort alternative, the lobbyist's strategic problem reduces to minimizing the cost of guaranteeing passage: $$\min_{t \in \mathbb{R}^n_+}\; \mathbf{1}^\top t \quad \text{subject to} \quad \sum_{i \in N} a_i^*(t) \;\geq\; \bar T,$$ where $a^*(t)$ is the unique Nash equilibrium of the council subgame induced by $t$. Two regularity assumptions keep this well posed: the network is symmetric and non-negative ($g_{ij} = g_{ji}$, $g_{ii} = 0$, $g_{ij} \geq 0$); and the spectral condition $c > \beta\,\lambda_{\max}(G)$ holds, equivalently, $cI - \beta G$ is positive definite. The spectral condition is standard for linear–quadratic network games: peer feedback gets amplified through loops in the graph, and $\beta\,\lambda_{\max}(G)$ measures the strongest feedback mode. If private cost $c$ dominates, best responses stay finite and the equilibrium is unique.

Linear passage rule

The council subgame has a closed-form Nash equilibrium. Each member's payoff is strictly concave in $a_i$, so the first-order condition $b_i + t_i + \beta \sum_j g_{ij} a_j = c\, a_i$ stacks into the linear system $(cI - \beta G)\, a = b + t$. Under the spectral condition, $cI - \beta G$ is positive definite and invertible, giving the closed form $$a^*(t) \;=\; (cI - \beta G)^{-1}(b + t) \;=:\; M\,(b + t),$$ where $M$ is symmetric, positive definite, and entrywise non-negative. The Neumann expansion $$M \;=\; \frac{1}{c}\sum_{k=0}^\infty \left(\frac{\beta}{c}\right)^{\!k} G^k$$ converges in operator norm under the spectral condition and admits a transparent walk-counting interpretation: $M_{ij}$ sums every walk from $i$ to $j$, weighted by a factor $(\beta/c)^k$ that decays geometrically in walk length. The decay parameter is $\beta/c$, not $\beta$ alone — longer walks are penalized both because each link transmits a fraction $\beta$ of upstream pressure and because each node absorbs a fraction $1/c$ of pressure received.

The vector summarizing the network's targeting-relevant structure is the Katz–Bonacich centrality with decay $\beta/c$, defined as the row sum $$v \;:=\; M\,\mathbf{1} \;=\; (cI - \beta G)^{-1}\,\mathbf{1} \;\in\; \mathbb{R}^n_{++}.$$ Operationally, $v_i$ is exactly the aggregate council support generated when the lobbyist places one unit of effort on member $i$ and zero elsewhere, holding $b = 0$. Equivalently, $v_i = \partial(\sum_j a_j^*(t))/\partial t_i$ is the marginal effect of an extra dollar of lobbying on member $i$. Substituting the equilibrium into the lobbyist's program collapses it to a linear program: $$\min_{t \in \mathbb{R}^n_+}\; \mathbf{1}^\top t \quad \text{subject to} \quad v^\top t \;\geq\; \Delta,$$ where $\Delta := \bar T - v^\top b$ is the support gap (the residual aggregate support the lobbyist must induce after baseline contributions $S_0 = v^\top b$ are accounted for).

This LP has a sharp corner solution. For any feasible $t \geq 0$, the constraint $v^\top t \geq \Delta$ combined with $v^\top t \leq \max_i v_i \cdot \mathbf{1}^\top t$ forces $\mathbf{1}^\top t \geq \Delta/\max_i v_i$. The vertex that achieves this bound concentrates the entire budget on a single coordinate: setting $t^*_{i^*} = \Delta/v_{i^*}$ for $i^* \in \arg\max_i v_i$ and zero elsewhere is feasible and optimal. The minimum lobbying cost is $$C^*(\bar T) \;=\; \frac{\bar T - v^\top b}{\max_{i} v_i}.$$ If $\arg\max_i v_i$ is a singleton, the optimum is unique. The optimal target $i^*$ does not depend on the baseline-belief vector $b$: a friendlier council lowers total cost via a smaller $\Delta$ but does not change whom to target. This is because $a^*(t) = M(b+t)$ is linear in $t$, so the marginal product of targeting a member is a property of network and cost parameters alone.

To make this concrete, consider the three-member path: members 1, 2, 3 with $g_{12} = g_{23} = 1$, $b = 0$, $c = 1$, $\beta = 0.4$, $\bar T = 1$. The eigenvalues of $G$ are $\{-\sqrt{2}, 0, \sqrt{2}\}$, so the spectral condition $0.4\sqrt{2} \approx 0.566 < 1$ holds. Direct calculation gives Katz–Bonacich vector $v \approx (2.059,\, 2.647,\, 2.059)^\top$. The center node strictly dominates: every walk between the endpoints must pass through it, so it accumulates extra walk weight. The unique optimal target is $i^* = 2$ at cost $C^* \approx 0.378$. The peripheral counterfactual — loading the budget on member 1 — costs $\approx 0.486$, a $28.6\%$ centrality premium for mis-targeting that comes purely from network geometry.

Theorem 1 is the LP version of the eigenvector-alignment theorem in Galeotti, Golub, and Goyal (2020). The difference is the centrality measure. Their work characterizes interventions using the top principal components of $G$; here the relevant object is Katz–Bonacich centrality with decay $\beta/c$. The two coincide only in the near-critical limit $\beta\,\lambda_{\max}(G) \to c^-$ when the leading eigenvalue is simple. Away from that limit, the ranking depends on both $\beta$ and $c$, and eigenvector centrality can pick the wrong target.

Smooth passage rule

Real institutions do not aggregate effort linearly. Members caucus, whip, and eventually cast discrete votes, so the mapping from effort to votes is usually curved. We replace the linear rule with a smooth strictly increasing aggregator $F(a) = \sum_i f(a_i) \geq K$, where $f : \mathbb{R} \to \mathbb{R}$ is $C^2$, $f'(\xi) > 0$ everywhere, and $f''$ is bounded on compact intervals. The identity $f(a) = a$ recovers the linear case; a sigmoid $f(a) = \sigma(a)$ gives a soft step. The corner-targeting result survives at leading order, but the relevant vector is now slope-weighted Bonacich centrality.

Define the slope-weighted centrality vector $w := M\, \nabla f(a^0)$, with components $w_j = \sum_{i \in N} f'(a^0_i)\, M_{ij}$, where $a^0 = Mb$ is the no-lobbying equilibrium. Under the maintained assumptions and assuming the constraint binds ($F(a^0) < K$), the cost-minimizing lobbying cost satisfies $$C^*(K) \;=\; \frac{\Delta}{w_{j^*}} \;+\; O(\Delta^2), \qquad j^* \in \arg\max_j w_j,$$ where $\Delta := K - F(a^0)$ is the smooth support gap. As $\Delta \downarrow 0$, the cost-minimizing strategy converges in direction to single-coordinate concentration on $j^*$. The proof linearizes $F$ via second-order Taylor expansion: $F(a^*(t)) = F(a^0) + w^\top t + R(t)$ with $|R(t)| \leq \tfrac{1}{2}\, B\, \|M\|_2^2\, \|t\|_2^2$ and $B$ a bound on $|f''|$. Dropping the second-order remainder reduces the lobbyist's program to the leading-order LP $\min_{t \geq 0} \mathbf{1}^\top t$ subject to $w^\top t \geq \Delta$ — exactly the linear-rule LP with $w$ in place of $v$. As $\Delta \downarrow 0$, the optimal $\|t\|_2$ shrinks linearly, the remainder is $O(\Delta^2)$, and the leading-order analysis is asymptotically valid.

The theorem includes the linear-rule corner solution as a special case ($f(a) = a$ gives $\nabla f \equiv \mathbf{1}$ and $w = v$ exactly). It also shows where baselines and curvature enter. Three cases are worth highlighting. First, if $b = 0$, then $a^0 = 0$ and $f'(a^0_i) = f'(0)$ for every $i$. Slope weights are uniform across members, so $w = f'(0)\, v$ and the optimal target is again $\arg\max_j v_j$, with costs rescaled by $1/f'(0)$. Second, if $b_i = b > 0$ for all $i$ and $f$ is strictly concave, the slopes $f'(b v_i)$ are strictly decreasing in $v_i$. More central members are more saturated and less responsive at the margin — slope weights penalize saturated rows of $M$, and $\arg\max_j w_j$ need not equal $\arg\max_j v_j$. Third, intermediate cases interpolate continuously: the targeting rule deforms smoothly as the baseline shifts.

To make the curvature effect concrete, return to the three-member path with $c = 1$, $\beta = 0.4$, the centered sigmoid $f(a) = \sigma(a) - \tfrac{1}{2}$, but raise the symmetric baseline to $b_i = 2$ for every member. Then $a^0 = 2v \approx (4.118,\, 5.294,\, 4.118)$, so the central member begins highly saturated. Sigmoid slopes evaluated at $a^0$ are $\nabla f(a^0) \approx (0.01576,\, 0.00497,\, 0.01576)$, and the slope-weighted centrality becomes $w \approx (0.02611,\, 0.02585,\, 0.02611)$. The leaves now beat the center: $w_1 = w_3 \approx 0.02611 > w_2 \approx 0.02585$, even though $v_2 \approx 2.647$ strictly exceeds $v_1 = v_3 \approx 2.059$. The center loses because its high Katz centrality pushes $a^0_2$ into the sigmoid's saturated tail, where $f'$ is small.

A companion model adds the same idea from a different angle. Instead of unbounded actions, let $a_i \in (0,1)$ be the probability member $i$ votes yes, and replace the quadratic penalty with a binary-Shannon entropy cost. The first-order condition becomes a sigmoid best response, $$a_i^* \;=\; \sigma\!\left(\frac{b_i + t_i + \beta \sum_{j \in N} g_{ij}\, a_j^*}{\kappa}\right), \qquad \sigma(x) \;:=\; \frac{1}{1 + e^{-x}},$$ with noise parameter $\kappa > 0$. Differentiating implicitly in $t$ and aggregating gives a marginal targeting return that factors as $\tilde v_j = a_j^*(1 - a_j^*) \cdot [\text{modified Bonacich centrality}]_j$. The first factor is the swing-voter slope $\sigma'(a_j^*) = a_j^*(1 - a_j^*)$: large when $j$ is on the fence, small when $j$ is locked in. The second factor sums over walks reweighted by the swing slopes of members along each walk. The product can flip the centrality ranking — a peripheral leaf on the fence beats a hostile central member — confirming that slope weighting is not an artifact of the Taylor expansion but a structural feature of any model where vote responsiveness varies across members.

Figure 3 makes this concrete on the same 3-node path with $\beta/\kappa = 0.4$ and hostile center $b = (0,\, -2,\, 0)$. Numerical root-finding of the sigmoid system gives $a^* \approx (0.517,\, 0.170,\, 0.517)$ and swing slopes $\sigma'(a^*) \approx (0.250,\, 0.141,\, 0.250)$. The product with the modified Bonacich column sums yields $\tilde v \approx (0.267,\, 0.171,\, 0.267)$: each peripheral leaf delivers roughly 56% more expected vote-share per lobbying dollar than the center, even though the center still has the network advantage in raw walks.

Bounded actions

Both prior results treat each member's action as an unbounded real variable disciplined by a quadratic cost. A natural relaxation keeps the linear passage rule but caps each member's action at $a_i \in [0,1]$. Substantively, this enforces that no amount of effort can push a single member past a hard ceiling — a representative can speak in favor of the bill, whip aggressively, and offer every plausible amendment, but cannot do more than one ``maximally supportive vote.'' Mathematically, the ceiling shifts the lobbyist's problem from a single-corner LP to a piecewise-linear convex program with multiple active-set regimes. We characterize the optimum in this regime and identify exactly where a naive ``greedy by Katz–Bonacich centrality'' heuristic breaks down.

The bounded-action best response is the truncation of the unbounded one: $$a_i^*(t) \;=\; \max\!\Bigl\{0,\, \min\!\Bigl\{1,\, \tfrac{1}{c}\bigl(b_i + t_i + \beta\textstyle\sum_{j} g_{ij}\, a_j^*\bigr)\Bigr\}\Bigr\},$$ and the projection map $T = \mathrm{proj}_{[0,1]^n} \circ \phi$ is a strict contraction whenever $\beta\,\lambda_{\max}(G)/c < 1$, so the bounded equilibrium $a^*(t)$ is unique. Parametrizing the lobbyist's program by the desired action profile $a$ rather than by the transfer vector $t$, the minimum transfer to sustain $a$ is $t_i(a) = \max\{0, c\, a_i - b_i - \beta \sum_j g_{ij}\, a_j\}$. Substituting gives the equivalent convex piecewise-linear program $$\min_{a \in [0,1]^n}\; \sum_i \max\!\bigl\{0,\, c\, a_i - b_i - \beta\textstyle\sum_j g_{ij}\, a_j\bigr\} \quad\text{s.t.}\quad \sum_i a_i \;\geq\; \bar T,$$ whose objective is convex piecewise-linear and whose feasible set is the intersection of the unit hypercube with a single half-space. Any KKT point of this program is therefore a global optimum, and the only question is algorithmic: how does the lobbyist find one efficiently?

A natural first attempt is to mimic the linear-rule corner solution: target the highest-Katz member, push them to saturation $a_i = 1$, then move on to the next-most-central, and so on. This procedure overpays. On the three-node path with $\beta = 0.5$, $b = 0$, $c = 1$, and $\bar T = 2.5$, naive greedy pushes member 2 to $a_2 = 1$ at cost $t_2 = 1$ (which raises the leaves to $a_1 = a_3 = 0.5$ via peer pressure), then pushes member 1 to $a_1 = 1$ at cost $t_1 = 0.5$, for total cost $1.5$. But once $a_1 = 1$, the peer pressure into member 2 has risen to $\beta(a_1 + a_3) = 0.75$, so $0.75$ of the original bribe to member 2 is now redundant — member 2 was already saturated by the cap, and the additional peer pressure just buys nothing.

The fix is a clawback step. As targeting raises neighbor actions, reduce previous bribes one-for-one with the new incoming peer pressure, keeping the equality $t_i + b_i + \beta \sum_k g_{ik} a_k = c$ for every saturated member $i$. Re-running the example: when the algorithm raises $t_1$ from $0$ toward $0.5$, member 1's action moves from $0.5$ to $1$, sending an extra $\beta \cdot 0.5 = 0.25$ of peer pressure to member 2; the clawback drops $t_2$ from $1$ to $0.75 - 0.5 = 0.25$ in lockstep. The final allocation is $t = (0.5, 0.25, 0)$ with actions $a = (1, 1, 0.5)$ and aggregate support $2.5$, for total cost $0.75$ — half the naive greedy cost.

Greedy-with-clawback solves the bounded-action program exactly when externalities are non-negative ($g_{ij} \geq 0$). The proof lifts the piecewise-linear program to the linear program $$\min_{a \in [0,1]^n,\, z \in \mathbb{R}^n_+}\; \mathbf{1}^\top z \quad\text{s.t.}\quad z_i \;\geq\; c\, a_i - b_i - \beta \textstyle\sum_j g_{ij} a_j, \quad \sum_i a_i \;\geq\; \bar T,$$ and observes that the clawback step maintains $z_i = c - b_i - \beta(Ga)_i$ throughout the saturated set $S$ — exactly complementary slackness for the constraints binding at $z_i$. Because $g_{ij} \geq 0$, peer pressure into every member of $S$ is weakly increasing along the algorithm's trajectory, so the saturated set grows monotonically and the procedure terminates at a basic feasible solution of the lifted LP, which is a global optimum.

The monotonicity property is exactly what fails with signed networks. Consider a three-node council with $g_{12} = g_{13} = -1$, $g_{23} = +1$, baselines $b = (0.8, 0.2, 0.2)$, $c = 1$, $\beta = 0.5$, and threshold $\bar T = 2$. A greedy lobbyist sees member 1 as nearly a free vote: a small bribe $t_1 = 0.2$ pushes $a_1$ to the ceiling. But once $a_1 = 1$, the negative externalities collapse the peer-pressure floor at members 2 and 3 by $\beta \cdot 1 = 0.5$, pinning $a_2 = a_3 = 0$. The lobbyist is now stuck fighting $-0.5$ of peer pressure on the only remaining members. The global optimum is qualitatively different: skip member 1 entirely (leave $a_1 = 0.8$), and target the reinforcing pair $\{2,3\}$, who pull each other up. With $g_{23} = +1$, each of $\{2, 3\}$ needs only $t_2 = t_3 = 0.3$ to reach saturation, total cost $0.6$. Greedy with clawback never reaches this solution: it follows the steepest initial gradient at member 1 into a local trap. The bounded-action program is genuinely non-convex once $G$ admits negative entries; this signed-network case is left to future work.

Discussion

The two main results can be summarized in one line. Linear voting selects Katz–Bonacich centrality, while smooth voting selects slope-weighted Bonacich centrality. Theorem 1 is the LP-corner identification under linear passage; the slope-weighted theorem extends it to any smooth strictly increasing aggregator $f$ at small support gaps. These rules agree in the linear and zero-baseline cases, and otherwise can disagree because curvature makes some members more responsive at the margin than others. The probabilistic-vote companion model shows the same idea through the swing-voter slope $\sigma'(a^*) = a^*(1 - a^*)$.

Several limitations are worth flagging. The slope-weighted theorem is local — it applies when the support gap is small enough that equilibrium actions stay near the no-lobbying equilibrium. For larger gaps, curvature of $f$ matters more, and the LP corner may dissolve into an interior optimum spread across several members. The bounded-action result is exact rather than asymptotic, but relies on non-negative externalities; the signed-network counterexample above shows that the problem becomes genuinely non-convex once $g_{ij}$ can flip sign. The model is also static and one-shot, so it leaves out the reputational and re-election dynamics behind Shleifer and Vishny (1993). The network $G$ is symmetric and undirected, ruling out hierarchy or one-way influence such as a committee chair influencing others without being much influenced herself. And there is no opposing lobbyist; a competing interest could change the targeting rule.

Several extensions are natural. A polynomial-time algorithm for the sigmoid-on-line case would be a swing-aware analog of the linear concentration theorem: with sigmoid best responses on a path network, the equilibrium-support map has a recursive one-dimensional structure that we conjecture admits an efficient exact optimum. Dropping symmetry $G = G^\top$ separates two notions of centrality that coincide in the symmetric case: aggregate support becomes $\mathbf{1}^\top a^* = \mathbf{1}^\top M b + s^\top t$ with $s := M^\top \mathbf{1}$ measuring downstream-influence centrality, so a directed version would predict concentration on $\arg\max_j s_j$ — distinguishing who receives spillovers from who sends them. Embedding the static LP in a strategic model with multiple competing lobbyists is a further direction: with pro- and anti-passage interests $k = 1, \ldots, M$, the council observes the net targeting vector $\tau_i = \sum_k \mathrm{sgn}(V_k) m_i^{(k)}$ and the targeting statistics reappear inside each lobbyist's best response, but the equilibrium concept becomes Nash in transfer vectors and the new tension is co-targeting — multiple lobbyists may concentrate on the same Bonacich-maximizer, spending each other's budgets in offsetting transfers. Finally, the bounded-action analysis under mixed-sign networks remains an open combinatorial problem that may require explicit search over active sub-coalitions.

Relative to the existing literature, the contribution is to put a lobbying objective inside a network-game framework. Galeotti, Golub, and Goyal (2020) characterize welfare-maximizing interventions; we replace their planner with a budget-constrained lobbyist trying to clear a passage threshold. That switch turns the intervention problem into an LP and identifies Katz–Bonacich centrality, rather than eigenvector centrality, as the relevant network statistic — eigenvector targeting is recovered only in the near-critical limit $\beta \lambda_{\max}(G) \to c^-$. Relative to Shleifer and Vishny (1993), the change is the network: side payments still shift incentives, but now the cost of corruption depends on which member is captured, not just on the total rent being bought. The voting-rule extension is the substantive twist: once passage is curved, centrality alone is not enough — the lobbyist also has to care where the chamber sits before lobbying begins.