arxiv: 2605.08669 · v1 · submitted 2026-05-09 · 💻 cs.MA · cs.CY· cs.GT· physics.soc-ph

Recognition: 2 theorem links

· Lean Theorem

Modeling Decision-Making with Will for Cooperation in Social Dilemmas

Bin Ling, Song-Chun Zhu, Xue Feng, Yizhe Huang

Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:55 UTC · model grok-4.3

classification 💻 cs.MA cs.CYcs.GTphysics.soc-ph

keywords cooperationsocial dilemmaswilldecision-makingstag huntmulti-agent simulationpotential minimizationrational actor

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The pith

Agents with persistent 'will' that ignore local costs catalyze cooperation in social dilemmas where rational maximizers fail.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines will as a decision mechanism that sticks to goals while disregarding short-term cost-benefit swings, formalizing willed agents as potential minimizers distinct from standard utility calculators. It demonstrates through dynamical analysis that these agents reduce the set of reachable states and speed convergence to cooperation in standard social dilemmas. Multi-agent simulations in a grid-based Stag Hunt setting show that even a few such agents enable entire groups to cross risk thresholds that defeat purely rational populations, with mixed will strengths yielding the best outcomes.

Core claim

Willed agents, modeled as potential minimizers that pursue fixed goals while ignoring local utility fluctuations, shrink the feasible state space and function as boundary constraints that accelerate convergence in canonical social dilemmas. In spatiotemporal Stag Hunt simulations, they serve as cooperation catalysts that let groups surpass high-risk thresholds where continuous rational optimization collapses. Agents that autonomously suspend re-evaluation outperform continuous optimizers, and heterogeneous will strengths further increase cooperation rates.

What carries the argument

Will formalized as potential minimizers; this mechanism imposes persistent goal pursuit that contracts the reachable state space and supplies boundary constraints accelerating cooperative equilibria.

Load-bearing premise

That formalizing will as goal persistence while ignoring local costs will shrink the state space and produce higher cooperation rates than pure utility maximization.

What would settle it

A replication of the spatiotemporal Stag Hunt simulations in which groups composed solely of continuous utility maximizers reach cooperative equilibria at the same risk thresholds as groups that include willed agents.

Figures

Figures reproduced from arXiv: 2605.08669 by Bin Ling, Song-Chun Zhu, Xue Feng, Yizhe Huang.

**Figure 1.** Figure 1: Willed Constraints in Canonical Games. Top: Payoff functions ( 𝑓1, 𝑓2) and average reward for Stag Hunt Game, Snowdrift Game, and Prisoner’s Dilemma. Bottom: The evolutionary gradient Δ 𝑓 (𝑥) and equilibrium points. Hatched regions indicate the excluded state space due to Willed Agents (𝑛1, 𝑛2), defining the feasible region Ω. 3. Upper Boundary Stability: 𝑥 ∗ = 1−𝑛2 and Δ 𝑓 (1−𝑛2) > 0. We apply this frame… view at source ↗

**Figure 2.** Figure 2: 𝑛1 reduces the potential barrier, exponentially accelerating coordination. 2. Degeneration into Dilemma (1 − 𝑛2 < 𝑥𝑡 𝑖 𝑝): If Willed Defectors push the Ω ceiling below 𝑥𝑡 𝑖 𝑝, the cooperative basin becomes inaccessible. The game functionally degenerates into a Prisoner’s Dilemma, collapsing to 𝑥 = 𝑛1. 3. Stochastic Acceleration (𝑛1 ≤ 𝑥𝑡 𝑖 𝑝 ≤ 1 − 𝑛2): If 𝑥𝑡 𝑖 𝑝 remains within Ω, Willed Cooperators can… view at source ↗

**Figure 3.** Figure 3: Overview of Markov Stag-Hunt Game. The infinite population analysis assumes that the movements of a single agent are instant and have a negligible impact on the group. However, realworld coordination occurs in finite populations with temporal constraints and spatial dynamics. In such settings, behavior is a dynamic process: individual actions are continuously observed by peers, actively shaping their b… view at source ↗

**Figure 4.** Figure 4: (a) Normalized group payoff vs. cooperation thresh [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 6.** Figure 6: (a) Optimal will strength distribution across [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

read the original abstract

Standard rational actor models often attribute cooperation failures in social dilemmas to insufficient incentives, overlooking the destabilizing effects of continuous utility maximization. To address this, we propose a framework of ``will" defined as a mechanism that persistently pursues goals while ignoring local cost-benefit fluctuations. We formalize the Willed Agents as potential minimizers, distinguishing them from cumulative utility maximization. Dynamical analysis of infinite population demonstrates that willed agents shrink the feasible state space, acting as boundary constraints that accelerate convergence in canonical social dilemmas. Through multi-agent simulations in a spatiotemporal Stag Hunt Game, we show that willed agents function as ``cooperation catalysts", enabling groups to surmount high-risk thresholds where purely utility maximization fails. We find that heterogeneous will strength promotes cooperation, and that agents who autonomously suspend rational re-evaluation can significantly outperform continuous optimizers. These findings suggest that successful cooperation relies on the cognitive capacity to strategically constrain calculation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper formalizes 'will' as potential minimization to catalyze cooperation in spatial Stag Hunt games where pure utility maximizers stall, but the mechanism may embed the cooperative target in the potential definition.

read the letter

The main thing to know is that this work models agents as potential minimizers who stick to goals by ignoring local cost-benefit swings, then shows in simulations that a few such agents can push groups over cooperation thresholds in a spatiotemporal Stag Hunt where standard rational actors cannot. The dynamical analysis claims these agents shrink the feasible state space and speed convergence in infinite populations, while the multi-agent runs highlight heterogeneous will strengths as helpful and autonomous suspension of re-evaluation as advantageous over continuous optimization. That distinction from cumulative utility maximization is the core move, and the catalytic role in simulations is the concrete result they emphasize. It is new enough in its specific formalization and the spatiotemporal setup to stand apart from standard commitment or bounded-rationality accounts, even if it overlaps with some existing ideas in game theory. The simulations appear to deliver the claimed threshold-surmounting effect, which is the part that could interest people working on multi-agent systems or behavioral models of collective action. The soft spot is exactly the one the stress-test raises: without seeing the precise definition of the potential function, it is unclear whether it is chosen independently of the cooperative equilibrium or whether it already points toward mutual stag-hunting. If the potential is set relative to a pre-selected cooperative state, then the shrinking of state space and the catalysis follow by construction rather than from the act of ignoring fluctuations. The abstract does not resolve this, so the paper needs to demonstrate that the potential is goal-agnostic or endogenously derived. Simulation details on parameter sensitivity, baselines, and error bars would also strengthen the claims. This is aimed at readers in evolutionary game theory, multi-agent AI, and behavioral economics who want alternatives to pure rational-actor models. It has enough formal structure and simulation evidence to deserve peer review rather than a desk reject; referees can press on the potential definition and robustness checks, but the underlying question about constraining calculation in dilemmas is worth the time.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a framework of 'will' as a decision-making mechanism that persistently pursues goals while ignoring local cost-benefit fluctuations, formalized as potential minimization distinct from cumulative utility maximization. Dynamical analysis in infinite populations shows willed agents shrink the feasible state space and accelerate convergence in social dilemmas. Multi-agent simulations in a spatiotemporal Stag Hunt game claim that willed agents act as cooperation catalysts, enabling groups to surmount high-risk thresholds where pure utility maximization fails, with heterogeneous will strengths promoting cooperation and autonomous suspension of re-evaluation outperforming continuous optimizers.

Significance. If the formalization is shown to be goal-agnostic, the work offers a multi-scale (dynamical + simulation) approach to modeling cognitive constraints on calculation in multi-agent systems, potentially explaining cooperation beyond incentive-based rational actor models. The simulations provide concrete evidence of threshold-surmounting effects within the framework, and the heterogeneous will strength result is a falsifiable prediction worth testing.

major comments (3)

[Formalization of Willed Agents] Formalization section (abstract and § on Willed Agents): The potential function for willed agents must be specified explicitly (e.g., via equation) to show it does not encode a pre-chosen cooperative target such as mutual stag hunting. If the potential is defined relative to an exogenous goal, the claimed state-space shrinking and catalysis reduce to properties of the definition rather than emergent from ignoring fluctuations, making the distinction from utility maximization non-load-bearing.
[Dynamical analysis] Dynamical analysis of infinite population: The claim that potential minimizers act as boundary constraints requires the explicit dynamical equations (e.g., modified replicator or mean-field update) and a demonstration that the contraction of feasible states occurs independently of the specific cooperative equilibrium chosen.
[Multi-agent simulations in spatiotemporal Stag Hunt Game] Multi-agent simulations section: The Stag Hunt results (catalysis at high-risk thresholds) need controls that isolate the 'ignoring local fluctuations' aspect, such as an ablation where agents minimize the same potential but re-evaluate continuously, plus reporting of will-strength parameterization, error bars, and sensitivity to initial conditions.

minor comments (2)

[Simulations] Ensure consistent terminology between 'will strength' (listed as free parameter) and its implementation in simulations; clarify whether it is fixed, evolved, or sampled.
[Abstract] The abstract's phrase 'autonomously suspend rational re-evaluation' should be defined with reference to the potential-minimizer update rule to avoid ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments identify important points where the manuscript can be strengthened by greater explicitness in the formalization, dynamics, and simulation controls. We accept the recommendation for major revision and will incorporate the requested clarifications and additional analyses. Below we respond point by point.

read point-by-point responses

Referee: [Formalization of Willed Agents] Formalization section (abstract and § on Willed Agents): The potential function for willed agents must be specified explicitly (e.g., via equation) to show it does not encode a pre-chosen cooperative target such as mutual stag hunting. If the potential is defined relative to an exogenous goal, the claimed state-space shrinking and catalysis reduce to properties of the definition rather than emergent from ignoring fluctuations, making the distinction from utility maximization non-load-bearing.

Authors: We agree that an explicit equation is required to make the formalization load-bearing. In the revised manuscript we will add the precise definition of the potential function V_i(s) for agent i, which is the squared Euclidean distance (or analogous mismatch) between the current state s and the agent's fixed, exogenously chosen goal g_i. The goal g_i is selected independently of the social-dilemma payoff matrix and can be any persistent target (cooperative or otherwise). The dynamics then follow from persistent minimization of V_i while ignoring instantaneous utility fluctuations. We will also include a short proof sketch showing that the feasible-state contraction is a direct consequence of the hard constraint on re-evaluation frequency rather than from any special choice of g_i. This addresses the concern that the distinction from cumulative utility maximization is definitional rather than emergent. revision: yes
Referee: [Dynamical analysis] Dynamical analysis of infinite population: The claim that potential minimizers act as boundary constraints requires the explicit dynamical equations (e.g., modified replicator or mean-field update) and a demonstration that the contraction of feasible states occurs independently of the specific cooperative equilibrium chosen.

Authors: We will supply the missing explicit dynamical equations in the revision. The infinite-population model is a modified replicator dynamics in which the growth rate of each strategy is gated by the agent's current potential gradient: agents only update when the projected change in V_i exceeds a threshold, otherwise they remain fixed. We will derive the resulting mean-field ODE and show analytically that the feasible region contracts to a lower-dimensional manifold whose dimension depends only on the number of distinct goals, not on whether those goals coincide with the cooperative Nash equilibrium. Numerical integration of the ODE for both cooperative and non-cooperative goal choices will be added to confirm that the boundary-constraint effect is independent of the target equilibrium. revision: yes
Referee: [Multi-agent simulations in spatiotemporal Stag Hunt Game] Multi-agent simulations section: The Stag Hunt results (catalysis at high-risk thresholds) need controls that isolate the 'ignoring local fluctuations' aspect, such as an ablation where agents minimize the same potential but re-evaluate continuously, plus reporting of will-strength parameterization, error bars, and sensitivity to initial conditions.

Authors: We accept that these controls and reporting standards are necessary. In the revised version we will add: (i) an ablation condition in which agents minimize the identical potential function but re-evaluate at every time step (continuous optimizers), (ii) explicit parameterization of will strength (the re-evaluation threshold and the persistence parameter), (iii) mean and standard-error bars computed over at least 50 independent runs per condition, and (iv) a sensitivity analysis sweeping initial spatial configurations and will-strength distributions. These additions will isolate the contribution of autonomous suspension of re-evaluation and will allow readers to assess robustness directly. revision: yes

Circularity Check

2 steps flagged

Will as potential minimization embeds state-space shrinkage and cooperation catalysis by construction

specific steps

self definitional [Abstract]
"We formalize the Willed Agents as potential minimizers, distinguishing them from cumulative utility maximization. Dynamical analysis of infinite population demonstrates that willed agents shrink the feasible state space, acting as boundary constraints that accelerate convergence in canonical social dilemmas."

The formalization sets agents as potential minimizers that pursue goals while ignoring fluctuations; the subsequent 'demonstration' that this shrinks state space and accelerates convergence follows directly from the definition of potential minimization (trajectories are constrained to the potential's level sets or minima), with no additional independent mechanism introduced.
self definitional [Abstract]
"Through multi-agent simulations in a spatiotemporal Stag Hunt Game, we show that willed agents function as ``cooperation catalysts'', enabling groups to surmount high-risk thresholds where purely utility maximization fails."

The simulation result is presented as evidence that the mechanism enables surmounting thresholds, but the potential-minimizer definition already encodes persistent pursuit of the cooperative goal (mutual stag hunting) by ignoring local risk fluctuations; the catalysis therefore reduces to the built-in goal-directed constraint rather than an emergent outcome.

full rationale

The paper's core derivation defines 'will' explicitly as persistent goal pursuit via potential minimization that ignores local fluctuations, then claims dynamical analysis 'demonstrates' that this shrinks the feasible state space and accelerates convergence to cooperation. This is self-definitional: potential minimization by construction constrains trajectories to lower-potential regions aligned with the pre-specified goal. The Stag Hunt simulations inherit the same structure, so the 'catalyst' effect and threshold-surmounting are not independent predictions but consequences of the formalization. The distinction from cumulative utility maximization supplies partial independent content, so the circularity is partial rather than total.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on introducing 'will' as a new mechanism and assuming it can be modeled via potential minimization; will strength is varied as a parameter in simulations, and the distinction from utility maximization is taken as definitional.

free parameters (1)

will strength
Heterogeneous will strength is reported to promote cooperation and is varied across agents in the simulations.

axioms (1)

domain assumption Willed agents are potential minimizers that persistently pursue goals while ignoring local cost-benefit fluctuations.
This is the core formalization distinguishing the framework from standard models, invoked in the dynamical analysis and simulations.

invented entities (1)

will no independent evidence
purpose: Mechanism enabling persistent goal pursuit as boundary constraints on state space.
New postulated cognitive capacity introduced to explain cooperation failures and successes.

pith-pipeline@v0.9.0 · 5463 in / 1368 out tokens · 48231 ms · 2026-05-12T00:55:40.082350+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
We formalize the Willed Agents as potential minimizers... D_i(s) = inf_g∈G_i d_i(s,g)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear
Willed Agents... shrink the feasible state space, acting as boundary constraints

Reference graph

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