A Reduction from Multi-Parameter to Single-Parameter Bayesian Contract Design
arxiv(2024)
摘要
The main result of this paper is an almost approximation-preserving
polynomial-time reduction from the most general multi-parameter Bayesian
contract design (BCD) to single-parameter BCD. That is, for any multi-parameter
BCD instance I^M, we construct a single-parameter instance I^S such that
any β-approximate contract (resp. menu of contracts) of I^S can in turn
be converted to a (β -ϵ)-approximate contract (resp. menu of
contracts) of I^M. The reduction is in time polynomial in the input size and
log(1/ϵ); moreover, when β = 1 (i.e., the given
single-parameter solution is exactly optimal), the dependence on
1/ϵ can be removed, leading to a polynomial-time exact
reduction. This efficient reduction is somewhat surprising because in the
closely related problem of Bayesian mechanism design, a polynomial-time
reduction from multi-parameter to single-parameter setting is believed to not
exist. Our result demonstrates the intrinsic difficulty of addressing moral
hazard in Bayesian contract design, regardless of being single-parameter or
multi-parameter.
As byproducts, our reduction answers two open questions in recent literature
of algorithmic contract design: (a) it implies that optimal contract design in
single-parameter BCD is not in APX unless P=NP even when the agent's type
distribution is regular, answering the open question of [Alon et al. 2021] in
the negative; (b) it implies that the principal's (order-wise) tight utility
gap between using a menu of contracts and a single contract is Θ(n)
where n is the number of actions, answering the major open question of
[Guruganesh et al. 2021] for the single-parameter case.
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