When Competing Incentive Systems Can Reach Equilibrium

When rivals change what is feasible

A principal–agent problem arises when a decision-maker designs incentives for someone with private information or actions the decision-maker cannot directly control. Roberson’s Interacting Principal-Agent Games studies multiple such relationships operating together: each principal chooses a mechanism for its own team while competing or otherwise interacting with other teams.

A mechanism specifies how reported information translates into decisions and recommendations. It is incentive compatible when agents cannot improve their expected payoff by misreporting their information or disobeying recommendations, assuming everyone else reports honestly and obeys.

The complication is that competitors’ mechanisms can change those incentives. A mechanism that encourages honesty against one rival’s choices may stop doing so after an arbitrarily small adjustment by that rival. The principal’s feasible choices therefore move with its opponents’ choices. Economists call this a generalized game.

The paper revisits Roger Myerson’s 1982 example, where this dependence prevents an exact principals’ equilibrium. Here, equilibrium means that every principal chooses its best mechanism among those satisfying its own agents’ incentive constraints, given the other principals’ mechanisms. It does not mean that the outcome maximizes social welfare.

A small tolerance on the agents’ side

The first result changes which mechanisms count as feasible. Rather than requiring perfect incentive compatibility, it permits each agent an expected gain from its best unilateral deviation of at most a positive tolerance, epsilon.

Under the paper’s assumptions, an equilibrium exists for every positive epsilon, however small. Principals still optimize exactly over their permitted mechanisms. The approximation concerns agents’ incentives, not principals’ willingness to accept less profitable choices.

Crucially, the tolerance is an average across an agent’s possible private types, weighted by their probabilities. It does not guarantee that every type faces only a tiny temptation. An unlikely type can have a relatively large deviation gain while contributing little to the average.

This also distinguishes the result from Myerson’s quasi-equilibrium. That concept keeps incentive compatibility exact but weakens the principals’ optimality requirement, allowing certain profitable deviations to be disregarded because small changes in opponents’ mechanisms would make them infeasible.

Neither approach automatically produces an exact equilibrium. The paper’s worked example shows that relaxed equilibria can converge, as epsilon approaches zero, to a profile that is exactly incentive compatible but leaves a principal with a profitable deviation.

How the existence proof works

The research is mathematical, not an experiment or an analysis of observed organizations. Its general model accommodates multiple agents per team, multidimensional information and decisions, and both reporting and subsequent action deviations.

The proof represents mechanisms through probability distributions over types and team decisions. This provides a compact, convex setting for analyzing possible choices. A baseline mechanism that remains exactly incentive compatible against every opponents’ profile supplies a dependable starting point.

Mixing a feasible mechanism slightly toward that baseline creates room beneath the positive regret limit. That room can survive small changes in competitors’ mechanisms, preventing feasible choices from disappearing abruptly. Established maximum and fixed-point theorems then deliver an equilibrium.

The assumptions matter: spaces must be compact, payoffs bounded and continuous, type distributions satisfy specified density conditions, and the universally incentive-compatible baseline must exist. Participation constraints are not included in this general theorem.

Exact equilibrium in a narrower contest

The second result restores exact incentive compatibility in a specialized contest with one privately informed agent per principal. Principals assign outputs and shares of contest winnings. Agents have type-dependent production costs, can misreport their types, and can decline participation for a payoff of zero.

Contest shares must satisfy monotonicity, concavity, and regularity conditions. There must also be a common output level that can cover the highest-cost type’s production cost with money left over, even against maximal rival outputs.

Here, truthful reporting is characterized by output increasing with type and an envelope formula linking output to expected rewards. Concavity allows stochastic output choices to be replaced by conditional-mean outputs without reducing the principal’s payoff. The profitable common-output mechanism supplies budget slack, replacing the incentive slack used in the first theorem. The resulting exact equilibrium withstands all admissible stochastic mechanism deviations.

What the results leave open

The exact theorem does not cover teams with several agents. Their promised rewards must be delivered jointly from one realized prize; separate average-payment checks are insufficient. Neither theorem establishes that real organizations will discover or converge to equilibrium, and the relaxed theorem does not establish actual truthful behavior when deviations remain profitable.

This article draws on the supplied full-paper text of arXiv version 4, dated November 8, 2026. It is a theoretical preprint; the supplied material does not establish its peer-review status.