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Incentive Compatibility in Stochastic Dynamic Systems

Ke Ma, P. R. Kumar

TL;DR

This paper tackles incentive-compatible mechanism design for stochastic dynamic systems, showing that static VCG is insufficient in dynamic settings. It develops a layered VCG approach for LQG agents that decouples intertemporal effects and yields dominant truth-telling of dynamic states under known parameters, while introducing a Scaled VCG variant to achieve budget balance and individual rationality under a Market Power Balance condition. The authors prove asymptotic convergence of SVCG payments to Lagrange payments as the market expands and extend results to partially observed states and non-Gaussian noises, with numerical illustrations. They also demonstrate limitations when system parameters are unknown and provide a MinMax approach to approximate Lagrange optimality in large markets, establishing the practical relevance of load aggregators and scalable pricing in dynamic power systems.

Abstract

While the classic Vickrey-Clarke-Groves mechanism ensures incentive compatibility for a static one-shot game, it does not appear to be feasible to construct a dominant truth-telling mechanism for agents that are stochastic dynamic systems. The contribution of this paper is to show that for a set of LQG agents a mechanism consisting of a sequence of layered payments over time decouples the intertemporal effect of current bids on future payoffs and ensures truth-telling of dynamic states by their agents, if system parameters are known and agents are rational. Additionally, it is shown that there is a "Scaled" VCG mechanism that simultaneously satisfies incentive compatibility, social efficiency, budget balance as well as individual rationality under a certain "market power balance" condition where no agent is too negligible or too dominant. A further desirable property is that the SVCG payments converge to the Lagrange payments, the payments corresponding to the true price in the absence of strategic considerations, as the number of agents in the market increases. For LQ but non-Gaussian agents the optimal social welfare over the class of linear control laws is achieved.

Incentive Compatibility in Stochastic Dynamic Systems

TL;DR

This paper tackles incentive-compatible mechanism design for stochastic dynamic systems, showing that static VCG is insufficient in dynamic settings. It develops a layered VCG approach for LQG agents that decouples intertemporal effects and yields dominant truth-telling of dynamic states under known parameters, while introducing a Scaled VCG variant to achieve budget balance and individual rationality under a Market Power Balance condition. The authors prove asymptotic convergence of SVCG payments to Lagrange payments as the market expands and extend results to partially observed states and non-Gaussian noises, with numerical illustrations. They also demonstrate limitations when system parameters are unknown and provide a MinMax approach to approximate Lagrange optimality in large markets, establishing the practical relevance of load aggregators and scalable pricing in dynamic power systems.

Abstract

While the classic Vickrey-Clarke-Groves mechanism ensures incentive compatibility for a static one-shot game, it does not appear to be feasible to construct a dominant truth-telling mechanism for agents that are stochastic dynamic systems. The contribution of this paper is to show that for a set of LQG agents a mechanism consisting of a sequence of layered payments over time decouples the intertemporal effect of current bids on future payoffs and ensures truth-telling of dynamic states by their agents, if system parameters are known and agents are rational. Additionally, it is shown that there is a "Scaled" VCG mechanism that simultaneously satisfies incentive compatibility, social efficiency, budget balance as well as individual rationality under a certain "market power balance" condition where no agent is too negligible or too dominant. A further desirable property is that the SVCG payments converge to the Lagrange payments, the payments corresponding to the true price in the absence of strategic considerations, as the number of agents in the market increases. For LQ but non-Gaussian agents the optimal social welfare over the class of linear control laws is achieved.

Paper Structure

This paper contains 16 sections, 11 theorems, 122 equations, 1 figure, 1 table.

Table of Contents

  1. introduction
  2. Related Works
  3. Static and Dynamic Deterministic Systems
  4. Static Deterministic Systems
  5. Static Deterministic Systems with Quadratic Costs: Asymptotics
  6. Dynamic Deterministic Systems
  7. Deterministic Linear Systems with Quadratic Costs
  8. Dynamic Stochastic VCG
  9. Linear Quadratic Gaussian Systems
  10. LQG systems with unknown system parameters
  11. Budget Balance and Individual Rationality in LQG systems
  12. Lagrange Optimality in LQG Systems
  13. Numerical Example
  14. LQG systems with partially observed states
  15. Linear Non-Gaussian Agents with Quadratic Costs
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\boldsymbol{u}^{*}$ be the optimal solution to utilitymax and suppose that $u^{*}$ is unique. Suppose that $\boldsymbol{u}^{(i)}$ is the unique optimal solution to: Let $H_{i}:=\sum_{j\ne i}F_{j}(\boldsymbol{u}^{(i)})$, and let $H_{max}:=\max_{i}H_{i}$. Let $F_{total}=\sum_{j}F_{j}(\boldsymbol{u}^{*})$. If $F_{total}>0$, $H_{i}>0$ for all $i$, and the following Market Power Balance (MPB) con

Figures (1)

  • Figure 1: Convergence of $c^{N}$ and $d^{N}$

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • ...and 21 more