Who Controls the Rules When Nobody Controls All of Them?
The game theory of distributed institutional control
Most economic models assume somebody owns the rules. A regulator designs a market. A platform owner writes the terms of service. A principal designs a contract. The entire apparatus of mechanism design—the single most productive branch of economic theory in the last forty years—rests on one structural premise: a single actor controls the full design space.
That premise fails in most institutional settings that matter.
Consider any environment where different actors control different components of the rules governing economic activity, and where those same actors participate in the economic activity those rules define. Protocol governance works this way. Regulatory architecture works this way. Platform ecosystems with distributed authority work this way. In each case, no single actor sets the full rule vector. Different actors set different dimensions. And the actors who set the rules also compete, trade, invest, and extract value under the rules they set.
The question is what happens to the equilibrium when rule-setting authority splits across participants in the downstream game.
The structure
Define the setting precisely. A rule vector has multiple dimensions. Each dimension maps to a distinct rule-setter. A timing structure determines the order in which rule-setters move. Downstream agents observe the complete rule vector and then invest. Every rule-setter’s payoff depends on that downstream investment. Rule-setters do not stand outside the game. They participate in the economic environment the rules define.
Two features distinguish this structure from everything adjacent in the literature. First, distributed control: different actors control different rule dimensions, and no actor controls all of them. Second, participation: every rule-setter’s payoff depends on the downstream investment that the rules jointly govern. These two features operate simultaneously. Removing either one collapses the structure to a known special case.
Without distributed control, the problem reduces to standard mechanism design. A single principal controls all design variables and optimises subject to incentive constraints. The classical results of Myerson, Laffont, and Tirole apply. The equilibrium rule vector reflects the principal’s objective, distorted only by information rents.
Without participation, the rule-setters resemble common-agency principals who influence a shared agent but do not compete downstream. The results of Bernheim and Whinston, Martimort and Stole, and the multi-principal mechanism design literature apply. The principals offer contracts; they do not invest.
Without downstream investment, there is no third party whose sunk capital responds to the rule vector. The investment response ratio—the object that drives the entire equilibrium characterisation—does not exist.
The conjunction of all three features produces a distinct strategic environment. The question is whether that environment admits tractable equilibrium analysis.
The factorisation
It does, and the reason is structural.
For the two-dimensional sequential case—two rule-setters, one downstream investor, strictly convex investment cost—the leading rule-setter’s reduced-form problem admits a factorisation. The derivative of the leader’s objective decomposes as the product of the choice variable and a bracket containing three terms: a share-weighted investment response ratio, a rent response ratio, and a linear cost.
The investment response ratio encodes how the leader’s rule choice affects downstream investment through the follower’s equilibrium response. It combines three objects: the curvature of the investment cost function, the scale of equilibrium investment, and the sensitivity of that investment to the leader’s choice. The rent response ratio encodes how the leader’s rule choice affects the follower’s equilibrium rent extraction. The linear cost is the leader’s direct cost of setting a higher rule level.
This factorisation holds for any strictly convex investment cost function with non-decreasing marginal cost, any smooth displacement technology satisfying regularity conditions, and any degree of rent alignment between the two rule-setters. It extends to non-separable rent functions. The factorisation does not depend on functional form. It depends on the structure of the game: distributed control, participation, and downstream investment response.
The factorisation converts a potentially intractable multi-stage distributed-control problem into a one-dimensional condition on whether the bracket is positive or negative at the social optimum.
The threshold
Under a monotonicity condition—that the combined investment and rent response ratios are strictly decreasing in the leader’s choice variable—the equilibrium has a clean characterisation. The leader’s equilibrium is unique. A closed-form threshold separates two regimes.
Below the threshold: the leader underprovides. The rule level sits below the social optimum. The leader does not internalise enough of the downstream surplus to justify the cost of setting a high rule level. This is the standard hold-up logic of Grossman–Hart and Hart–Moore, operating through institutional rule dimensions rather than through bilateral contracts.
Above the threshold: the leader overprovides. The rule level sits above the social optimum. The leader overinvests in native capability to suppress the follower’s rent extraction—a strategic commitment effect. In the taxonomy of Fudenberg and Tirole, this is a top-dog strategy: the leader makes an aggressive commitment that reduces the follower’s marginal return to gatekeeping.
The threshold itself takes a closed form. It equals the cost parameter divided by the investment response ratio evaluated at the social optimum, adjusted for rent alignment. When the leader partially internalises the follower’s rents (positive rent alignment), the threshold rises: the leader needs a larger share of downstream surplus before over-provision becomes individually rational.
The comparative statics follow from the factorisation. The leader’s equilibrium choice strictly increases in the share parameter and strictly decreases in the cost parameter. These results require the monotonicity condition. They do not hold in general.
The timing result
The factorisation and the threshold characterise the sequential game: the leader moves first, the follower observes and responds, the investor observes both and invests. Sequential timing is not merely a modelling convenience. It is a substantive assumption, and the equilibrium depends on it.
Under simultaneous rule-setting—both rule-setters choose simultaneously, the investor observes both and invests—the leader’s problem changes. With linear productivity and sufficient curvature in the investment cost function, the leader’s simultaneous-play problem is globally concave. Over-provision does not occur. At every simultaneous Nash equilibrium, the leader underprovides relative to the social optimum.
This result identifies over-provision as a commitment effect. The leader over-invests in native capability only when the leader can commit to a high rule level before the follower responds. When the leader cannot commit—because the timing is simultaneous—the strategic incentive to suppress the follower’s rents vanishes.
The timing result holds for any strictly convex investment cost satisfying a curvature bound. Under quadratic costs, the curvature bound reduces to a parameter restriction that the model’s assumptions already impose. Under superquadratic costs (where marginal cost is non-decreasing), the result holds automatically. The timing result extends to concave productivity functions under a weaker parameter restriction, but may fail under convex productivity where the complementarity between quality and investment can overwhelm the concavity.
What existing frameworks miss
Mechanism design misses this because the designer stands outside the game. The designer’s payoff does not depend on downstream investment. The designer controls all rule dimensions. The internally distorted equilibrium—one rule dimension too high, the other too high for different reasons—cannot arise when a single principal optimises the full rule vector.
Common agency misses this because the principals do not participate. In common agency, multiple principals influence a shared agent, but the principals’ payoffs do not depend on the agent’s investment response to the contract vector. The leader’s incentive to distort rules for private competitive advantage has no analogue in common agency.
Standard Stackelberg models miss this because the leader and follower control quantities of the same good. In Stackelberg quantity competition, the leader commits to a high output level and the follower accommodates. The distortion arises from quantity competition in a single dimension. In the distributed-control setting, the leader and follower control different dimensions of a shared institutional environment, and a third party’s investment response closes the equilibrium.
The distinction is not cosmetic. Removing any one of the three features—distributed control, participation, downstream investment—collapses the factorisation to a known special case. The factorisation arises from the conjunction. It does not arise from any one feature operating alone.
Competition and aggregation
The two-player canonical case extends in two directions that preserve the factorisation structure.
First, competition among followers. When multiple followers set the same rule dimension simultaneously, they compete in rent extraction. Each follower’s individual rent declines because the other followers crowd the same extraction channel. But aggregate rent extraction increases: the “tragedy of intermediation” operates through the same logic as the tragedy of the commons. More followers extract more aggregate rent, which depresses downstream investment and dampens the leader’s investment response ratio. The over-provision threshold strictly increases in the number of followers: competition among followers makes over-provision harder.
At the baseline calibration, over-provision at full surplus internalisation occurs against a monopoly follower but vanishes when three or more followers compete. The threshold rises monotonically with the number of followers. The “top-dog” commitment effect—where the leader overinvests in native capability to suppress intermediary rents—requires a concentrated intermediary sector. A competitive intermediation market eliminates over-provision entirely, leaving only underprovision.
Second, heterogeneity among leaders. When multiple leaders each control a share of the leading rule dimension, the equilibrium depends on a single sufficient statistic: the sum of each leader’s surplus share divided by cost. This aggregation result holds for any strictly convex investment cost function. The identity of individual leaders does not matter for the aggregate equilibrium—only the sufficient statistic matters.
This aggregation invariance breaks under hierarchical governance. When one leader moves before others within the leading coalition, the sequential leader restrains her choice, inducing the follower to invest heavily. At the baseline calibration, a hierarchical structure produces approximately 39 percent more aggregate capability than flat governance with the same sufficient statistic. The governance structure among rule-setters—not merely the payoff parameters—determines the equilibrium.
Welfare decomposition
The welfare gap between the decentralised equilibrium and the social optimum decomposes into two components: the cost of the follower’s rent extraction (gatekeeping loss) and the cost of the leader’s distortion (development-cost loss from either underprovision or over-provision).
At the baseline calibration under partial surplus internalisation, gatekeeping accounts for approximately 83 percent of the total welfare gap. The leader’s distortion accounts for the remainder. Under over-provision, the composition shifts: excess development cost accounts for 36 percent, and residual gatekeeping accounts for 64 percent. These numbers are calibration-specific. The qualitative pattern—that intermediary rent extraction dominates the welfare gap under underprovision, while the composition rebalances under over-provision—follows from the model’s structure.
The pressure points
The global characterisation—uniqueness, threshold, comparative statics—depends on the monotonicity condition. That condition is a derived property of the equilibrium objects, not a primitive restriction on the model’s fundamentals. It holds at a quadratic baseline (verified exactly by Sturm’s theorem), extends to all investment cost functions with non-decreasing curvature when infrastructure costs exceed a computable threshold, and holds across a parametric grid covering a range of displacement parameters. A closed-form sufficient condition on primitives that eliminates the need for case-by-case verification remains open.
The timing result requires linear or concave productivity. Under convex productivity, the complementarity between quality and investment can make the leader’s simultaneous-play problem non-concave, and over-provision may survive even without commitment.
The welfare benchmark assumes that the follower’s activity is pure rent extraction: the social optimum sets the follower’s rule level to zero. When the follower’s activity provides productive value—curation, matching, quality certification—the social optimum may involve a positive follower rule level. The distortion result continues to hold when the follower’s private rent exceeds the productive contribution, but the welfare decomposition changes.
These are the boundaries of the analysis. The factorisation and the timing result hold unconditionally within the model class. The global characterisation holds where the monotonicity condition holds. The welfare decomposition holds at the baseline. Each claim sits within its exact conditions.
What this means
Institutional environments where different actors control different rule dimensions and participate in the downstream economic activity produce equilibrium distortions that no single existing framework captures. The leader of a distributed rule-setting game faces a trade-off between native capability investment and intermediary rent suppression. That trade-off admits a clean factorisation, a closed-form threshold, and a timing dependence that identifies over-provision as a commitment effect.
The analysis applies wherever rule-setting authority distributes across participants: protocol governance, platform ecosystems, regulatory architecture, standard-setting bodies with fragmented authority. The specific equilibrium objects depend on functional form and parameters. The structural result—that distributed control plus participation plus downstream investment produces a factorised leader’s problem with a sharp threshold and timing dependence—does not.