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Tightness of the two-outcome approximation ratio for linear contracts

Determine whether the worst-case approximation ratio between the principal’s expected utility under the optimal linear contract and under the optimal (general) contract equals 2 in settings with exactly two outcomes when both outcomes have positive rewards.

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Background

The survey derives tight worst-case approximation bounds for linear contracts across several parameters but notes a specific gap for the two-outcome case. When one outcome has zero reward, linear contracts are optimal; when both outcomes have positive rewards, an example shows a lower bound of 2 on the ratio. It remains unknown whether this lower bound is tight.

Resolving this would complete the performance landscape of linear contracts in the smallest nontrivial outcome space and inform the choice of simple contracts in practice.

References

If both outcomes have positive rewards then Example~\ref{ex:equal-rev} with $n=m=2$ actions and outcomes shows that the approximation ratio is at least $2$; it is unknown whether the approximation ratio of~$2$ is tight.

— Algorithmic Contract Theory: A Survey (2412.16384 - Duetting et al., 20 Dec 2024) in Table 1 (Approximation guarantees of linear contracts) — caption