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Willy Wonka Mechanisms

Thomas Archbold, Bart de Keijzer, Carmine Ventre

TL;DR

This work addresses incentive compatibility under bounded rationality by introducing not obviously manipulable (NOM) mechanisms and two design primitives, golden tickets and wooden spoons, which together guarantee NOM behavior. It develops Willy Wonka mechanisms for revenue maximisation and frugal procurement, showing that with α-approximate (or maximal-in-range) allocations one can achieve strong prior-free guarantees; specifically, BNOM/WA-consistency yields revenue within $\alpha(1-1/\tau)$ of the optimum in binary and general outcome spaces, and procurement mechanisms achieve frugality within a factor of two of the second-best solution. The core contributions are the NOM characterisations for binary and general allocations, the explicit Willy Wonka construction with simple, single-line payment rules, and the frugality refinement that bounds overpayment even in procurement settings. These results offer practical, robust mechanisms for settings where bidders are cognitively limited, with clear theoretical guarantees and directions for extending allocation design and payment schemes.

Abstract

Bounded rationality in mechanism design aims to ensure incentive-compatibility for agents who are cognitively limited. These agents lack the contingent reasoning skills that traditional mechanism design assumes, and depending on how these cognitive limitations are modelled this alters the class of incentive-compatible mechanisms. In this work we design mechanisms without any "obvious" manipulations for several auction settings that aim to either maximise revenue or minimise the compensation paid to the agents. A mechanism without obvious manipulations is said to be "not obviously manipulable" (NOM), and assumes agents act truthfully as long as the maximum and minimum utilities from doing so are no worse than the maximum and minimum utilities from lying, with the extremes taken over all possible actions of the other agents. We exploit the definition of NOM by introducing the concept of "golden tickets" and "wooden spoons", which designate bid profiles ensuring the mechanism's incentive-compatibility for each agent. We then characterise these "Willy Wonka" mechanisms, and by carefully choosing the golden tickets and wooden spoons we use this to design revenue-maximising auctions and frugal procurement auctions.

Willy Wonka Mechanisms

TL;DR

This work addresses incentive compatibility under bounded rationality by introducing not obviously manipulable (NOM) mechanisms and two design primitives, golden tickets and wooden spoons, which together guarantee NOM behavior. It develops Willy Wonka mechanisms for revenue maximisation and frugal procurement, showing that with α-approximate (or maximal-in-range) allocations one can achieve strong prior-free guarantees; specifically, BNOM/WA-consistency yields revenue within of the optimum in binary and general outcome spaces, and procurement mechanisms achieve frugality within a factor of two of the second-best solution. The core contributions are the NOM characterisations for binary and general allocations, the explicit Willy Wonka construction with simple, single-line payment rules, and the frugality refinement that bounds overpayment even in procurement settings. These results offer practical, robust mechanisms for settings where bidders are cognitively limited, with clear theoretical guarantees and directions for extending allocation design and payment schemes.

Abstract

Bounded rationality in mechanism design aims to ensure incentive-compatibility for agents who are cognitively limited. These agents lack the contingent reasoning skills that traditional mechanism design assumes, and depending on how these cognitive limitations are modelled this alters the class of incentive-compatible mechanisms. In this work we design mechanisms without any "obvious" manipulations for several auction settings that aim to either maximise revenue or minimise the compensation paid to the agents. A mechanism without obvious manipulations is said to be "not obviously manipulable" (NOM), and assumes agents act truthfully as long as the maximum and minimum utilities from doing so are no worse than the maximum and minimum utilities from lying, with the extremes taken over all possible actions of the other agents. We exploit the definition of NOM by introducing the concept of "golden tickets" and "wooden spoons", which designate bid profiles ensuring the mechanism's incentive-compatibility for each agent. We then characterise these "Willy Wonka" mechanisms, and by carefully choosing the golden tickets and wooden spoons we use this to design revenue-maximising auctions and frugal procurement auctions.
Paper Structure (7 sections, 8 theorems, 7 equations, 3 algorithms)

This paper contains 7 sections, 8 theorems, 7 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Revenue Maximisation
  4. Binary allocation
  5. General outcome spaces
  6. Frugal Procurement
  7. Conclusion

Key Result

Lemma 1

Let $M = (a,p)$ be an auction where the allocation function has any positive approximation $\alpha > 0$ to the optimal social welfare. Then $M$ is BNOM if and only if for every agent $i \in [n]$ with valuation $v_i$ there exists a bid profile such that $i$ wins the auction for free.

Theorems & Definitions (18)

  • Lemma 1: Golden ticket for binary allocation
  • proof
  • Lemma 2: Wooden spoon for binary allocation
  • proof
  • Definition 1: Willy Wonka mechanism for binary allocation
  • Theorem 1
  • proof
  • Lemma 3: Golden ticket for general outcome spaces
  • proof
  • Lemma 4: Wooden spoon for general outcome spaces
  • ...and 8 more