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Stackelberg POMDP: A Reinforcement Learning Approach for Economic Design

Gianluca Brero, Alon Eden, Darshan Chakrabarti, Matthias Gerstgrasser, Amy Greenwald, Vincent Li, David C. Parkes

TL;DR

This work formalizes indirect mechanism design as a Stackelberg Partially Observable Markov Game (Stackelberg POMG) and introduces the Stackelberg POMDP as a single-agent RL formulation that captures follower adaptation to a leader’s rules. It proves that, within a restricted leader policy class, the optimal Stackelberg leader policy corresponds to the optimal policy of the Stackelberg POMDP, and solves this using centralized training with decentralized execution (MAPPO) paired with policy-interactive follower models. The framework accommodates diverse follower behaviors, including no-regret learning leading to Bayesian Coarse Correlated Equilibria (BCCE), and is validated on increasingly complex settings like Simple Allocation Mechanisms and Sequential Price Mechanisms with Messages (MSPMs), achieving welfare gains with limited communication. The results suggest this approach can automate and improve the design of indirect mechanisms in multi-round, uncertainty-rich markets, while highlighting practical considerations around data representativeness and deployment objectives.

Abstract

We introduce a reinforcement learning framework for economic design where the interaction between the environment designer and the participants is modeled as a Stackelberg game. In this game, the designer (leader) sets up the rules of the economic system, while the participants (followers) respond strategically. We integrate algorithms for determining followers' response strategies into the leader's learning environment, providing a formulation of the leader's learning problem as a POMDP that we call the Stackelberg POMDP. We prove that the optimal leader's strategy in the Stackelberg game is the optimal policy in our Stackelberg POMDP under a limited set of possible policies, establishing a connection between solving POMDPs and Stackelberg games. We solve our POMDP under a limited set of policy options via the centralized training with decentralized execution framework. For the specific case of followers that are modeled as no-regret learners, we solve an array of increasingly complex settings, including problems of indirect mechanism design where there is turn-taking and limited communication by agents. We demonstrate the effectiveness of our training framework through ablation studies. We also give convergence results for no-regret learners to a Bayesian version of a coarse-correlated equilibrium, extending known results to correlated types.

Stackelberg POMDP: A Reinforcement Learning Approach for Economic Design

TL;DR

This work formalizes indirect mechanism design as a Stackelberg Partially Observable Markov Game (Stackelberg POMG) and introduces the Stackelberg POMDP as a single-agent RL formulation that captures follower adaptation to a leader’s rules. It proves that, within a restricted leader policy class, the optimal Stackelberg leader policy corresponds to the optimal policy of the Stackelberg POMDP, and solves this using centralized training with decentralized execution (MAPPO) paired with policy-interactive follower models. The framework accommodates diverse follower behaviors, including no-regret learning leading to Bayesian Coarse Correlated Equilibria (BCCE), and is validated on increasingly complex settings like Simple Allocation Mechanisms and Sequential Price Mechanisms with Messages (MSPMs), achieving welfare gains with limited communication. The results suggest this approach can automate and improve the design of indirect mechanisms in multi-round, uncertainty-rich markets, while highlighting practical considerations around data representativeness and deployment objectives.

Abstract

We introduce a reinforcement learning framework for economic design where the interaction between the environment designer and the participants is modeled as a Stackelberg game. In this game, the designer (leader) sets up the rules of the economic system, while the participants (followers) respond strategically. We integrate algorithms for determining followers' response strategies into the leader's learning environment, providing a formulation of the leader's learning problem as a POMDP that we call the Stackelberg POMDP. We prove that the optimal leader's strategy in the Stackelberg game is the optimal policy in our Stackelberg POMDP under a limited set of possible policies, establishing a connection between solving POMDPs and Stackelberg games. We solve our POMDP under a limited set of policy options via the centralized training with decentralized execution framework. For the specific case of followers that are modeled as no-regret learners, we solve an array of increasingly complex settings, including problems of indirect mechanism design where there is turn-taking and limited communication by agents. We demonstrate the effectiveness of our training framework through ablation studies. We also give convergence results for no-regret learners to a Bayesian version of a coarse-correlated equilibrium, extending known results to correlated types.
Paper Structure (24 sections, 3 theorems, 19 equations, 6 figures, 7 tables, 2 algorithms)

This paper contains 24 sections, 3 theorems, 19 equations, 6 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Further related work.
  3. Preliminaries
  4. The Leader's Problem
  5. Learning an Optimal Leader Policy
  6. Applications of Stackelberg POMDP
  7. Simple Allocation Mechanisms
  8. Sequential Price Mechanisms with Messages
  9. Conclusion
  10. Broader Impact and Limitations.
  11. Formal Definition of Subgame of a POMG
  12. Optimal Policy in Stackelberg POMDP
  13. Learning an Optimal Policy in Stackelberg POMDP
  14. No-Regret Followers
  15. Multiplicative-Weights Algorithm
  16. ...and 9 more sections

Key Result

Theorem 4.1

Let $\pi_{\mathcal{S}}^*$ be the optimal policy in our Stackelberg POMDP among the set of leader's policies $\Pi_\ell$. Then, $\pi_{\mathcal{S}}^*$ also solves the leader's Stackelberg problem of Definition def:leaderProblem.

Figures (6)

  • Figure 1: A Stackelberg POMDP episode. The follower response phase starts with an initial PI response algorithm state $s_{b,0}$ and corresponding followers' policies $\pi_{-\ell,0}$. These policies interact with the leader policy $\pi_{\ell}$ in game $\mathcal{G}$, leading to a new PI response algorithm state $s_{b,1}$ and corresponding followers' policies $\pi_{-\ell,1}$. This process continues until completing $E$ plays of the game with final state $s_{b,E}$, which includes the behavior policies $\pi^*_{-\ell}$. Subsequently, in the reward phase, the game is played one more time against the behavior strategies $\pi^*_{-\ell}$, generating reward $R_\ell$.
  • Figure 2: Training curves for Simple Allocation Mechanism, averaged over 25 runs. Standard errors are shown in lighter shades. Optimal welfare corresponds to a leader reward of 1.
  • Figure 3: Training curves for Escape averaged over 25 runs. Standard errors are displayed in lighter shades. The leader policy is trained using MAPPO or PPO in the Stackelberg POMDP. Rewards are normalized between 0 and 1.
  • Figure 4: Training curves for Maintain averaged over 25 runs. Standard errors are displayed in lighter shades. Results show leader policy training using MAPPO in the Stackelberg POMDP under deterministic and randomized leader policy. Rewards are normalized between 0 and 1.
  • Figure 5: The training curves for the Matrix Design Game, introduced in monderer2003k, illustrate the mean outcomes across 25 simulations. The standard errors are indicated by the lighter shaded areas. The designer’s reward has been normalized to a range of 0 to 1.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 2.1: Partially Observable Markov game
  • Definition 2.2: Subgame of a POMG (Informal Definition, Formal in Appendix)
  • Definition 2.3: Stackelberg Partially Observable Markov Game
  • Definition 3.1: Policy-Interactive Response Algorithm
  • Definition 3.2: Optimal leader strategy
  • Definition 4.1: Stackelberg POMDP
  • Theorem 4.1
  • Definition A.1: Subgame of a POMG
  • Theorem B.1
  • Definition D.1: Bayesian coarse correlated equilibrium
  • ...and 1 more