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Aligning Individual and Collective Objectives in Multi-Agent Cooperation

Yang Li, Wenhao Zhang, Jianhong Wang, Shao Zhang, Yali Du, Ying Wen, Wei Pan

TL;DR

A novel optimization method named AgA is introduced that effectively attracts gradients to stable fixed points of the collective objective while considering individual interests and is theoretically prove that AgA effectively attracts gradients to stable fixed points of the collective objective while considering individual interests.

Abstract

Among the research topics in multi-agent learning, mixed-motive cooperation is one of the most prominent challenges, primarily due to the mismatch between individual and collective goals. The cutting-edge research is focused on incorporating domain knowledge into rewards and introducing additional mechanisms to incentivize cooperation. However, these approaches often face shortcomings such as the effort on manual design and the absence of theoretical groundings. To close this gap, we model the mixed-motive game as a differentiable game for the ease of illuminating the learning dynamics towards cooperation. More detailed, we introduce a novel optimization method named \textbf{\textit{A}}ltruistic \textbf{\textit{G}}radient \textbf{\textit{A}}djustment (\textbf{\textit{AgA}}) that employs gradient adjustments to progressively align individual and collective objectives. Furthermore, we theoretically prove that AgA effectively attracts gradients to stable fixed points of the collective objective while considering individual interests, and we validate these claims with empirical evidence. We evaluate the effectiveness of our algorithm AgA through benchmark environments for testing mixed-motive collaboration with small-scale agents such as the two-player public good game and the sequential social dilemma games, Cleanup and Harvest, as well as our self-developed large-scale environment in the game StarCraft II.

Aligning Individual and Collective Objectives in Multi-Agent Cooperation

TL;DR

A novel optimization method named AgA is introduced that effectively attracts gradients to stable fixed points of the collective objective while considering individual interests and is theoretically prove that AgA effectively attracts gradients to stable fixed points of the collective objective while considering individual interests.

Abstract

Among the research topics in multi-agent learning, mixed-motive cooperation is one of the most prominent challenges, primarily due to the mismatch between individual and collective goals. The cutting-edge research is focused on incorporating domain knowledge into rewards and introducing additional mechanisms to incentivize cooperation. However, these approaches often face shortcomings such as the effort on manual design and the absence of theoretical groundings. To close this gap, we model the mixed-motive game as a differentiable game for the ease of illuminating the learning dynamics towards cooperation. More detailed, we introduce a novel optimization method named \textbf{\textit{A}}ltruistic \textbf{\textit{G}}radient \textbf{\textit{A}}djustment (\textbf{\textit{AgA}}) that employs gradient adjustments to progressively align individual and collective objectives. Furthermore, we theoretically prove that AgA effectively attracts gradients to stable fixed points of the collective objective while considering individual interests, and we validate these claims with empirical evidence. We evaluate the effectiveness of our algorithm AgA through benchmark environments for testing mixed-motive collaboration with small-scale agents such as the two-player public good game and the sequential social dilemma games, Cleanup and Harvest, as well as our self-developed large-scale environment in the game StarCraft II.
Paper Structure (25 sections, 3 theorems, 7 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 25 sections, 3 theorems, 7 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Mixed-motive cooperation.
  4. Gradient-based Methods.
  5. Preliminaries
  6. Differential Game
  7. Gradient Adjustment Optimization
  8. Method
  9. Differentiable Mixed-motive Game
  10. Altruistic Gradient Adjustment
  11. Alignment Effectiveness of AgA: A Toy Experiment
  12. Experiments
  13. Two-Player Public Goods Game
  14. Sequential Social Dilemma: Cleanup and Harvest
  15. Selfish-MMM2
  16. ...and 10 more sections

Key Result

Corollary 4.2

In the neighborhood of fixed points of the collective objective, AgA will pull the gradient toward stable fixed points, which means $\theta(\Tilde{{\bm{\xi}}}, \nabla{\mathcal{H}}_{c}) \leq \theta({\bm{\xi}}_{c}, \nabla{\mathcal{H}}_{c})$, and push away from unstable ones, indicated by $\theta(\Tild

Figures (5)

  • Figure 1: Trajectories of optimization in a two-player DMG (as delineated in Example \ref{['game_eg']}). Fig.\ref{['fig:toy_opt1']} displays the trajectories over the collective reward landscape - deeper orange equates to higher rewards. Remarkably, only Simul-Co and AgA make successful strides towards the social optimum. Fig.\ref{['fig:toy_opt2']} and Fig. \ref{['fig:toy_opt3']} delineate trajectories on the individual reward contour, underscoring Simul-Co's neglect for Player 1's interests as it navigates through the crests and troughs of its reward. Conversely, our AgA optimizes along the summit of Player 1's reward while also maximizing the collective reward, demonstrating successful alignment.
  • Figure 2: Left figure a: Illustration of Corollary \ref{['thm:aga']}. In case 1, within an unstable fixed point's neighborhoods, an appropriate selection of the $\lambda$ sign push AgA to evade the unstable fixed point and pull towards a stable fixed point in its neighborhoods as shown in case 2. Right figure b: Alignment Effectiveness of AgA. The comparison between AgA (shown in red) and AgA without sign alignment (AgA-Sign, in purple) trajectories spans 40 steps, marked at every tenth step. Norm gradients are represented with blue arrows. Starting from the 14th step, sign alignment pulls the gradient toward the steepest direction, resulting in AgA reducing the number of steps by approximately 15% compared to AgA-Sign by the end of the trajectory.
  • Figure 3: The first row displays results comparing different values of the alignment parameter $\lambda$ across three environments: Cleanup and Harvest (measurement of social welfare, SW) and Selfish-MMM2 (focusing on win rate). The second row examines the performance differences between the proposed AgA and AgA without sign alignment (AgA-Sign) on the three testbeds. The bold lines indicate the mean social welfare calculated in three seeds, while the surrounding shaded areas represent the 95% confidence interval. The third row compares the AgA method with baseline approaches on these testbeds. Each bar represents the mean collective results of each method and the error bars indicate the 95% confidence interval.
  • Figure 4: The scatter of actions in a two-player public goods game achieved through different optimization methods. Each circle represents the position attained within a maximum of 100 steps, with the color indicating the corresponding method. The 'X' mark represents the mean actions of 50 random runs. With the exception of Simul-Co, the baseline methods converge towards the Nash equilibrium (0,0). Notably, while both AgA and Simul-Co display altruistic behavior, the actions of AgA are more tightly clustered around the (1,1) point compared to Simul-Co.
  • Figure 5: Semantic Diagram of MMM2 map in SMAC

Theorems & Definitions (9)

  • Definition 3.1: Differential Game balduzzi2018mechanicsLetcher2019Diff
  • Definition 3.2
  • Example 4.1
  • Definition 4.2: Altruistic Gradient Adjustment
  • Corollary 4.2
  • Corollary B.0
  • proof
  • Lemma B.1: Sign of $align$.
  • proof