Abstract
Lagrangian duality underlies both classical and modern mechanism design. In particular, the dual perspective often permits simple and detail-free characterizations of optimal and approximately optimal mechanisms. This paper applies this same methodology to a close cousin of traditional mechanism design, one which shares conceptual and technical elements with its more mature relative: the burgeoning field of persuasion. The dual perspective permits us to analyze optimal persuasion schemes both in settings which have been analyzed in prior work, as well as for natural generalizations which we are the first to explore in depth. Most notably, we permit combining persuasion policies with payments, which serve to augment the persuasion power of the scheme. In both single and multi-receiver settings, as well as under a variety of constraints on payments, we employ duality to obtain structural insights, as well as tractable and simple characterizations of optimal policies.
The full paper can be found at https://arxiv.org/abs/1909.10584.
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Notes
- 1.
We use the words “payoff” and “reward” interchangeably in this paper.
- 2.
We refer the reader to [11] for a list examples of Bayesian persuasions and how types are defined in those.
- 3.
It is also easy to see how to implement payments \(p(i)\) from P(i) and an implementation of \(\phi _\theta \): for a given state \(\theta \in \varTheta \), sample a payment \(\tfrac{P(i)}{\phi _\theta (i)}\) whenever signal \(i\sim \phi _\theta \) is recommended.
- 4.
Recall that \(P(i) = \sum _{\theta \in \varTheta } \mu _\theta \phi _{\theta }(i) p(i)\).
- 5.
We will not actually make use of complementary slackness in this paper, but include it here for completeness.
- 6.
Note that this does not hold generally, and absolutely requires the symmetry assumptions.
- 7.
Further recall that the sender/receiver payoffs for action i are completely determined by action i’s type. So this can also be phrased as recommending a uniformly random action with type k, where k maximizes \(\xi _k + \frac{n}{n-1}\rho _k\) over all present types k.
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Acknowledgements
Shaddin Dughmi is supported by NSF CAREER Award CCF-1350900. S. Matthew Weinberg is supported by NSF CCF-1717899.
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Dughmi, S., Niazadeh, R., Psomas, A., Weinberg, S.M. (2019). Persuasion and Incentives Through the Lens of Duality. In: Caragiannis, I., Mirrokni, V., Nikolova, E. (eds) Web and Internet Economics. WINE 2019. Lecture Notes in Computer Science(), vol 11920. Springer, Cham. https://doi.org/10.1007/978-3-030-35389-6_11
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