---
title: Intended Cooperation Values in Multi-Agent Systems
url: https://www.emergentmind.com/topics/intended-cooperation-values-icvs
type: topic
---

# Intended Cooperation Values in Multi-Agent Systems

Intended Cooperation Values (ICVs) denote a family of formal devices for representing, attributing, or inferring cooperation-relevant influence in multi-agent systems. In the explicit usage introduced for multi-agent reinforcement learning (MARL), ICVs are an information-theoretic, Shapley-value method that quantifies each agent’s causal influence on its co-players’ instrumental empowerment by measuring action effects on policy uncertainty and preference alignment [2508.15652]. Closely related literatures use different names for nearby constructs—such as cooperation-strength parameters, history-conditioned cooperation levels, internal innate values, bargaining weights, and selectively legible compatibility signals—so the term occupies a broader technical field rather than denoting a single universally adopted formalism [2112.10951], [2604.01240], [2606.08274].

## 1. Direct meaning and conceptual scope

The most direct and technically specific definition of ICVs appears in the MARL explainability literature. There, ICVs are a post-hoc, causal action-attribution method intended to work when one has access to policies at inference time but does not want to rely on explicit rewards or learned value functions [2508.15652]. The method asks how an agent’s executed action changes the future decision conditions of agents that have not yet acted, rather than how much return that action eventually earns.

A broader reading is also well supported. Several adjacent papers do not use the term *Intended Cooperation Value* explicitly, but they introduce quantities that play closely related roles: a cooperation-strength parameter in co-utility theory, a local cooperation-dependent payoff field in spatial dilemmas, a time-indexed cooperation level in sequential reciprocity models, and bargaining weights for compromise across differing values [2112.10951], [1802.06745], [2604.01240], [2307.04879]. This suggests that ICVs are best understood as a research area centered on cooperation-relevant valuation and influence, not only as a single estimator.

| Source | Role in the literature | Core construct |
|---|---|---|
| [2508.15652] | Explicit ICV formalism | Information-theoretic Shapley attribution over policy effects |
| [2604.01240] | Sequential ICV-like mapping | Current or next-period cooperation level \(a_i^t\) |
| [2112.10951] | Mathematical precursor | Cooperation-strength or altruism parameter \(\lambda\) |
| [2307.04879] | Normative compromise model | Common weighted sum selected by bargaining |

The field is also marked by the absence of a single human-centered synthesis. A 2026 survey argues that there is no widely adopted framework integrating cognition, culture, values, social behavior, and cooperation into autonomous agents, even though those components are jointly necessary for human-centered multi-agent systems [2606.08274]. In that sense, ICVs are both a defined MARL technique and a candidate umbrella concept for a wider integration problem.

## 2. Information-theoretic ICVs in multi-agent reinforcement learning

The formal ICV construction is defined on a Markov game
$$
\mathcal{G}_{m} \coloneqq (\mathcal{N}, \mathcal{S}, \{\mathcal{A}^i\}_{i=1}^n, P, \rho, \{R^i\}_{i=1}^n, \gamma),
$$
with stochastic policies \(\pi^i(a\mid s_t)\) and joint policy
$$
\boldsymbol{\pi}(\boldsymbol{a}\mid s_t)=\prod_{i=1}^n \pi^i(a\mid s_t).
$$
To isolate action effects, the method replaces the one-step transition with a Sequential Value Markov Game, in which the agents’ actions are processed in an ordering \(\sigma\), producing intermediate states \(S_{t,(k)}\) [2508.15652]. The executed action is treated interventionally as \(\mathrm{do}(A_t^{\sigma(k)} = a_t^{\sigma(k)})\), so attribution is attached to the action itself rather than to its upstream causes.

For a given ordering \(\sigma\), the successor coalition of agent \(i\) is
$$
\mathcal{C}_{\sigma}^i \coloneqq \{j : j \succ_{\sigma} i\},
$$
namely the agents whose actions have not yet been processed when \(i\) acts. A characteristic function \(\nu(\mathcal{C}_{\sigma}^i, s_{t,(k)})\) is evaluated before and after \(i\)’s action, and the marginal contribution is
$$
\Delta \nu\left(\mathcal{C}_{\sigma}^i, s_{t, (k)}\right)
\coloneqq
\nu\left(\mathcal{C}_{\sigma}^i, s_{t,(k)}\right)
-
\nu\left(\mathcal{C}_{\sigma}^i, s_{t,(k-1)}\right).
$$
The Intended Cooperation Value of agent \(i\) is then the ordering-averaged time average
$$
\Phi_i\left(\nu\right)
\coloneqq
\frac{1}{T}
\sum_{t=0}^{T-1}
\sum_{\sigma \in \Sigma_+^i}
w \cdot
\Delta \nu\left(\mathcal{C}_{\sigma}^i, s_{t}\right),
$$
with a Monte Carlo estimator
$$
\hat{\Phi}_i\left(\nu\right)
\coloneqq
\frac{1}{T}
\sum_{t=0}^{T-1}
\Delta \nu\left(\mathcal{C}_{\sigma_t}^i, s_{t}\right),
\quad
\sigma_t \sim \mathcal{U}_{\Sigma_+^i}.
$$

This formulation gives ICVs a distinctive status relative to classical credit assignment. Standard reward decomposition asks who contributed to return; ICVs ask who altered others’ decision-making landscape. The method is therefore post-hoc, policy-based, and explicitly local in time and causality [2508.15652].

## 3. Characteristic functions: value, peakness, consensus, and diversity

The ICV framework is driven by the choice of characteristic function \(\nu\). The value-based baseline is
$$
\nu_{v}(\mathcal{C}_{\sigma}^i, s_{t,(k)})
\coloneqq
\sum_{j\in \mathcal{C}_{\sigma}^i} V^j(s_{t,(k)}),
$$
which reduces to the centralized critic \(V(s_{t,(k)})\) in centralized-critic cooperative settings [2508.15652]. Its marginal contribution satisfies
$$
\Delta \nu_{v}
=
\sum_{j \in \mathcal{C}_{\sigma}^i}
V^j(s_{t,(k)}) - V^j(s_{t,(k-1)}),
$$
and is identified with the sum of successor-agent advantages.

The policy-based peakness characteristic replaces explicit value access with decision certainty. The peakedness of policy \(j\) is defined by
$$
\mathrm{H}(A_{t,(k+1)}^j)
\coloneqq
\log|\mathcal{A}^j|
-
H(A_{t,(k+1)}^j \mid s_{t,(k)}),
$$
and
$$
\nu_p(\mathcal{C}_{\sigma}^i, s_{t,(k)})
\coloneqq
\sum_{j\in \mathcal{C}_{\sigma}^i}
\mathrm{H}\left( A_{t,(k+1)}^j \right).
$$
Its marginal contribution is derived as conditional mutual information,
$$
\Delta \nu_{p}
=
\sum_{j \in \mathcal{C}_{\sigma}^i}
\mathrm{I}\left(A_{t,(k)}^j ; a_t^i \mid s_{t,(k-1)} \right),
$$
which the paper interprets as a policy-based proxy for teammates’ instrumental empowerment [2508.15652].

Consensus and diversity are defined through the Jensen–Shannon divergence. With
$$
\mathrm{JSD}(p \| q)
=
\frac{1}{2} D_{KL}(p \| u)
+
\frac{1}{2} D_{KL}(q \| u),
\quad
u=\frac{1}{2}(p+q),
$$
the similarity score is
$$
J(p \| q) \coloneqq 1 - \mathrm{JSD}(p \| q).
$$
Consensus-based ICVs compare what one agent would do in another’s state and vice versa; dissimilarity-based ICVs replace \(J\) with \(\mathrm{JSD}\) [2508.15652]. This permits a finer distinction between helpful alignment and role-preserving diversity.

The resulting interpretive claim is deliberately non-monotone. An action can be beneficial either by making teammates more deterministic or by preserving flexibility for future choices. The paper therefore rejects the simple rule that “more certainty” is always equivalent to “more cooperation” [2508.15652].

## 4. Empirical behavior and explainability claims

In Level-Based Foraging, an agent moving into a useful coordination position increased its teammates’ policy certainty and all agents’ value. Across 500 episodes, the reported \(\hat{\Phi}(\nu_v)\) and \(\hat{\Phi}(\nu_p)\) were positive, but the acting agents showed decreased self-consensus \(\hat{\Phi}(\nu_{cs})\), while others-consensus effects \(\hat{\Phi}(\nu_{co})\) were minor [2508.15652]. The interpretation is that positive cooperation can coexist with differentiated roles rather than policy homogeneity.

In Multi-Particle Environment Tag, the prey’s value aligned reasonably with action certainty, and the prey showed high dissimilarity relative to predators. Over 500 episodes, prey contributions aligned with value and determinism, while predators displayed the opposite pattern: beneficial behavior often involved reducing determinism, keeping options open, increasing alignment among predators, and promoting diversity versus prey [2508.15652]. This is the clearest empirical case against treating low entropy as a universal signature of cooperative value.

In Google Research Football, the analysis used TiZero with a shared policy \(\pi_{tz}\), a shared centralized critic, and a fixed order
$$
\sigma = (\boldsymbol{p}_{o}, \boldsymbol{p}_{t}, p_{s}).
$$
The paper defines a strategy-change proxy
$$
\bar{\nu}_d
=
\mathrm{JSD}(\pi_{tz}(\cdot \mid s_{t,(k+1)}) \,\|\, \pi_{tz}(\cdot \mid s_{t,(k)})),
$$
and reports that losing or regaining possession near the penalty area caused uncertainty and policy adaptation, while opponent scoring caused a drop in value-based contribution and a rise in strategy adaptation [2508.15652]. Just before a goal, opponents negatively affected the striker or ball player’s value and determinism, whereas teammates positively increased decisiveness to shoot.

The empirical record is therefore not merely confirmatory. It establishes that policy-based ICVs often track value-based assessments, but only in a task-dependent and role-dependent way. The same action effect can be interpreted differently for prey, predator, teammate, or opponent, and the paper explicitly notes that there is no formal ablation table isolating every component [2508.15652].

## 5. Formal precursors and ICV-like quantities in adjacent theories

Several earlier or parallel theories define quantities that function much like ICVs without using the name. In the microlevel theory of joint intentionality and group-mindedness, the central object is the co-utility
$$
u_{\lambda} = (1-\lambda)u(a) + \lambda u(b),
$$
with \(\lambda \in [0,1]\) interpreted as the strength of cooperation or degree of altruism [2112.10951]. The parameter \(\lambda\) changes action ranking by placing positive weight on the partner’s payoff, and the paper treats it as the closest formal analogue to a measurable cooperation-valuing quantity.

In sequential coopetition, the cleanest ICV-like mapping is
$$
\mathrm{ICV}_i^t \equiv a_i^t,
$$
where \(a_i^t\) is the actor’s current cooperation level, or alternatively the history-conditioned equilibrium choice \(\sigma_i^*(h^{t-1})\) [2604.01240]. The dynamics are shaped by bounded reciprocity response functions, memory-windowed history, structural reciprocity sensitivity \(\rho_{ij} = \rho_0 D_{ij}^\eta\), and trust-gated reciprocity. In the case-study simulation, the next cooperation level is updated explicitly by a reciprocity term, a baseline-reversion term, and noise. This gives ICVs a path-dependent and trust-sensitive reading rather than a purely informational one.

In spatial evolutionary games with environment–strategy feedback, the local cooperation level \(n_C/G\) or its mean-field analogue \(x\) functions as a cooperation-conditioned value signal. It rescales the reward and temptation payoffs as
$$
R = R_0\left(1+\alpha \frac{n_C}{G}\right),
\qquad
T = T_0\left(1+\alpha \frac{n_C}{G}\right),
$$
so cooperation becomes a determinant of the incentive landscape rather than merely an outcome [1802.06745]. The paper reports that this rule promotes cooperation in structured populations, but also produces logarithmically slow relaxation because isolated defectors survive for long periods.

A different precursor is covert signaling. There the relevant object is not a scalar internal value but a selectively legible signal of cooperative compatibility. Overt signals can produce liking among similar agents and dislike among dissimilar agents; covert signals are effectively received by similar individuals with probability \(\gamma r\) and do not generate active dislike from dissimilar individuals [1511.04788]. In the forced-choice case, covert signaling is favored when
$$
\frac{1-s}{s} > \beta(1-\gamma).
$$
This line of work implies that an intended cooperation signal need not be globally transparent; it may instead be audience-dependent and strategically ambiguous.

## 6. Human-centered, normative, and unresolved dimensions

The most expansive survey treatment places cooperation alongside cognition, culture, values, and social behavior, and argues that future AI agents acting on behalf of humans must move beyond task competence toward culturally aware, value-aligned, cognitively grounded, and cooperative multi-agent systems [2606.08274]. It reviews mechanisms such as inverse reinforcement learning, reinforcement learning from human feedback, direct preference optimization, constitutional methods, value–belief–norm reasoning, shared mental models, Theory of Mind, persona modeling, cognitive architectures, and agent-based modeling. It also proposes evaluation dimensions extending beyond task accuracy, including cultural appropriateness, value conformity under pluralistic settings, explanation usefulness, collaboration quality, persona consistency, robustness of social reasoning, and long-term trustworthiness.

A reward-design interpretation appears in innate-values-driven reinforcement learning. There, “innate values” are internal reward-generating preference structures over need satisfaction and team success, operationalized through a critic function \(\mathcal{C}^i\), internal state, and a weight matrix \(\mathbb{W}_{iv}\) over achievement, safety, and basic needs [2401.05572]. The framework is relevant to ICVs because cooperative behavior is shaped by internal value systems rather than a single external team reward, but it stops short of introducing an explicit intention variable or a learned representation of cooperative commitment.

The strongest normative formalization comes from evidential cooperation in large worlds. That work models cooperation across differing values as an incomplete-information bargaining problem and shows that Pareto-optimal cooperation requires all cooperators to maximize the same weighted sum of utility functions [2307.04879]. The paper rejects selecting compromise outcomes merely by utility normalization, argues that the Nash bargaining solution is a plausible Schelling point, and uses dependency equilibria rather than Bayesian Nash equilibria to capture evidential dependence. It also shows that coalition stability is nontrivial: the Nash bargaining solution may fail to lie in the core, and disagreement-point choice remains an unresolved design variable.

Taken together, these strands show that ICVs are not reducible to a single scalar reward, a single explainability score, or a single public declaration of cooperative intent. The explicit MARL formalism defines ICVs as causal policy-based attributions [2508.15652]. The surrounding literature suggests additional interpretations: partner-payoff weighting, trust-gated cooperation levels, endogenous cooperation-conditioned incentive fields, internal reward-generating value systems, bargaining-based compromise utilities, and selectively legible compatibility signals [2112.10951], [2604.01240], [1802.06745], [2401.05572], [2307.04879], [1511.04788]. A plausible implication is that any mature theory of ICVs will need to integrate attribution, representation, compromise selection, and human-centered evaluation rather than treating cooperation as a single undifferentiated objective.

Source: https://www.emergentmind.com/topics/intended-cooperation-values-icvs