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Achieving PAC Guarantees in Mechanism Design through Multi-Armed Bandits

Takayuki Osogami, Hirota Kinoshita, Segev Wasserkrug

Abstract

We analytically derive a class of optimal solutions to a linear program (LP) for automated mechanism design that satisfies efficiency, incentive compatibility, strong budget balance (SBB), and individual rationality (IR), where SBB and IR are enforced in expectation. These solutions can be expressed using a set of essential variables whose cardinality is exponentially smaller than the total number of variables in the original formulation. However, evaluating a key term in the solutions requires exponentially many optimization steps as the number of players $N$ increases. We address this by translating the evaluation of this term into a multi-armed bandit (MAB) problem and develop a probably approximately correct (PAC) estimator with asymptotically optimal sample complexity. This MAB-based approach reduces the optimization complexity from exponential to $O(N\log N)$. Numerical experiments confirm that our method efficiently computes mechanisms with the target properties, scaling to problems with up to $N=128$ players -- substantially improving over prior work.

Achieving PAC Guarantees in Mechanism Design through Multi-Armed Bandits

Abstract

We analytically derive a class of optimal solutions to a linear program (LP) for automated mechanism design that satisfies efficiency, incentive compatibility, strong budget balance (SBB), and individual rationality (IR), where SBB and IR are enforced in expectation. These solutions can be expressed using a set of essential variables whose cardinality is exponentially smaller than the total number of variables in the original formulation. However, evaluating a key term in the solutions requires exponentially many optimization steps as the number of players increases. We address this by translating the evaluation of this term into a multi-armed bandit (MAB) problem and develop a probably approximately correct (PAC) estimator with asymptotically optimal sample complexity. This MAB-based approach reduces the optimization complexity from exponential to . Numerical experiments confirm that our method efficiently computes mechanisms with the target properties, scaling to problems with up to players -- substantially improving over prior work.

Paper Structure

This paper contains 30 sections, 14 theorems, 62 equations, 9 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Settings
  4. Optimization problem for automated mechanism design
  5. Analytical solutions to the optimization problem
  6. Evaluating the analytical solutions with a PAC guarantee
  7. Estimating the best mean reward in multi-armed bandits
  8. Numerical experiments
  9. Conclusion
  10. Details
  11. Details of Section \ref{['sec:LP']}
  12. Details of Section \ref{['sec:solution']}
  13. Solutions that guarantee WBB
  14. On dependent types
  15. Details of Section \ref{['sec:bandit2']}
  16. ...and 15 more sections

Key Result

Lemma 1

The LP given by eq:LP-obj-eq:LP-WBB is feasible if When types are independent ($t_m$ and $t_n$ are independent for any $m\neq n$ under $\mathbb P$), the LP feasible if and only if eq:condition holds. When types are dependent, the LP may be feasible even if eq:condition is violated.

Figures (9)

  • Figure 1: (a) Representative sample path that shows the estimated values of $\mathbb E[w^\star(t)\mid t_n]$ for $t_n\in\mathcal{T}_n$ against sample size used by the $(0.25, 0.1)$-PAC SE-BME for 8 players, each with 8 types. (b)-(c) The unique sample size (the number of unique $t$ which $w^\star(t)$ is evaluated with) required by the exact computation of $\min_{t_n\in\mathcal{T}_n}\mathbb E[w^\star(t)\mid t_n], \forall n\in\mathcal{N}$ (dashed) and by SE-BME (solid).
  • Figure 2: The red dot shows expected utility in (a) and (c) and expected revenue in (b) and (d), when analytical solutions are evaluated exactly (horizontal) or estimated with $(0.25,0.1)$-PAC SE-BME (vertical) for environments with $8$ players, each having $8$ possible types. The analytical solution guarantees IR in (a) and (b) and SBB in (c) and (d). Results are plotted for 10 random seeds.
  • Figure 3: The total sample size required by $(\varepsilon,\delta)$-PAC BME (Mean; Algorithm \ref{['alg:SE-BME']}) and by $(\varepsilon,\delta)$-PAC BAI (Arm; Algorithm \ref{['alg:SE-BAI']}) when arms have Bernoulli rewards with equally separated means for varying values of $\varepsilon$ and $\delta$.
  • Figure 4: Representative sample paths that show the estimated values of $\mathbb E[w^\star(t)\mid t_n]$ for $t_n\in\mathcal{T}_n$ against sample size used by $(0.25,0.1)$-PAC SE-BME.
  • Figure 5: The unique sample size (the number of unique $t$ which $w^\star(t)$ is evaluated with) required by the exact computation of $\min_{t_n\in\mathcal{T}_n}\mathbb E[w^\star(t)\mid t_n], \forall n\in\mathcal{N}$ (dashed curves) and by $(0.25,0.1)$-PAC SE BME (solid curves).
  • ...and 4 more figures

Theorems & Definitions (29)

  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Lemma 4
  • Lemma 5
  • Proposition 2
  • ...and 19 more