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The Conditional Value of Costly Information in Collective-Risk Dilemmas

Elias Fernández Domingos, Genki Ichinose, Wakaba Tateishi, Tom Lenaerts

arXiv:2610.33895v1cs.GTcs.CY

TL;DR

Collective-risk dilemmas require a group to meet a threshold to avoid loss, yet that threshold may be uncertain and misinformation can complicate coordination. The paper models costly information acquisition and sharing in an evolutionary collective-risk dilemma. It finds that information’s effects depend on its cost and distribution, while buyers can also limit disinformation’s spread.

  • Problem

    Collective-risk dilemmas require cooperation around a possibly uncertain threshold, while misinformation can sustain conflicting information that complicates decisions.

    Source: §1 · pp. 1–2

  • Method

    The authors develop an evolutionary model of costly threshold information, group sharing, and network diffusion, including noisy and false relays.

    Source: §1 · p. 2, §3.4 · p. 5, §5 · p. 8

  • Results

    Costly information improves collective outcomes when sufficiently cheap; sharing shifts the cooperation tipping point to lower risk, and network diffusion is robust to noise while false information harms success.

    Source: §5 · pp. 8–9

  • Takeaways & Limitations

    A few information buyers can benefit others through sharing and limit false information’s spread, though diffusion also encourages free-riding.

    Source: §5 · p. 9

    The network model uses 100-node networks, fixed noncontributing attackers and sources, and receivers that do not learn to distrust messages.

    Source: S8 Limitations of the network model · p. 14

Abstract

from arXiv · show

Many high-stakes collective problems, e.g., climate mitigation and pandemic response, share a structure in which risk materializes unless a group threshold is met. This threshold is often uncertain, but uncertainty can be reduced at a cost. We study this information cost tradeoff in an evolutionary game theoretic model of social learning in the collective-risk dilemma, where individuals must cooperate to reach a possibly uncertain threshold to prevent collective loss. Individuals may pay a fixed cost to observe the threshold's true value and condition their cooperation on it. Across four threshold priors, we characterize risk levels at which cooperative equilibria emerge and show that low cost information increases group success. However, this effect is reversed at higher costs. We define a private \emph{value of information} and show that information is worth buying only when it is considerably cheaper than contributing. When purchased information is shared with the group, the cooperative tipping point shifts to lower risk levels. Despite bearing a private cost, above this tipping point buyers earn more in expectation than information free-riders. We then study these dynamics when information diffuses through a network, allowing nodes to relay it. Although fewer individuals pay for information, relaying increases group success, and noisy relaying does not reduce it. By contrast, false information does, especially when it originates at the hubs of heterogeneous networks, and large attacks can bring collective action close to collapse. Buyers act as a firewall against disinformation, confining it to source neighbourhoods, and once disinformation sources are removed, the population recovers. The value of costly information thus depends on its price, the level of risk, and how it is shared, and, in our setting, the effort of acquiring it also protects the population against disinformation.

1 Introduction

Collective-risk dilemmas require coordination around a threshold, but uncertainty and misinformation can complicate cooperation. This paper introduces an evolutionary model to examine whether costly threshold information and its sharing help groups avoid disaster and resist disinformation.

1 Introduction

  • Threshold uncertainty removes a focal point for coordination, while misinformation can sustain conflicting accounts that complicate protective decisions.The introduction distinguishes uncertainty inherent in a system from manufactured uncertainty sustained by actors who benefit from public confusion.Source: §1 · pp. 1–2
  • The authors isolate information cost and threshold uncertainty in a simplified evolutionary model, then extend it to group sharing and network diffusion.The model uses a homogeneous population, four stylized threshold priors, and strategies that act unconditionally or pay to observe the threshold and condition their behavior on it.Source: §1 · p. 2
  • The model examines whether costly threshold information improves collective success and whether buyers who relay truthful information can increase resilience to disinformation.Prior evolutionary CRD models had not examined agents acquiring and relaying information, to the authors’ knowledge.Source: §1 · p. 2
  • The paper’s stated contributions include a value-of-information account, group sharing that lowers the cooperation tipping point, and network relaying that raises success despite fewer buyers.It also reports that noisy relaying does not harm collective success, whereas false information does, especially when originating at hubs.Source: §1 · p. 2

2 Related Work

  • Prior CRD research studied uncertainty’s effects, but costly endogenous acquisition and network diffusion of threshold information remained unexamined in evolutionary CRD models.Related work reports that threshold uncertainty generally hurts cooperation, and earlier experiments found cooperation could be restored by narrowing uncertainty around the true threshold.Source: §2 · pp. 2–3

3 Model

The model examines costly threshold information in a one-shot collective-risk dilemma, using four threshold priors and strategies that either act unconditionally or condition cooperation on observed thresholds. It analyzes evolutionary dynamics and extends the setup to group sharing and network diffusion.

3.1 The collective-risk dilemma with costly information

  • Groups succeed when at least M members cooperate; otherwise each player loses the endowment with probability r, while cooperators pay their contribution cost even if the group fails.The threshold is independently drawn from one of four priors: uniform, gaussian, low-threshold, or high-threshold.Source: §3.1 · p. 3
  • Conditional strategies pay to observe M and compare it with a fixed reference threshold, while unconditional strategies neither buy information nor condition their actions on M.The conditional strategies represent simplified decisionmakers with an aspiration about the collective target.Source: §3.1 · p. 3

3.2 Evolutionary dynamics

  • The model defines private information value as the informed individual’s gain from choosing an action by threshold rather than the best unconditional action, holding the population state fixed.Information is valuable only if it changes the individual’s decision; its value cannot exceed the cost of cooperation, and acquisition pays when c_I < φc.Source: §3.2 · p. 4

3.3 Sharing information within groups

  • With group sharing, buyers, free-riders, and providers separate information acquisition from contribution; on networks, buyers and informed nodes can relay threshold information before players act.The network analysis also considers noisy relaying, malicious relays, external sources, and intervention-and-recovery phases.Source: §3.3 · pp. 4–5, §3.4 · p. 5

4 Results

Across threshold priors, risk shapes when cooperation emerges and how costly information affects group success. Sharing can move the cooperative tipping point to lower risk, while network relaying spreads information but leaves groups vulnerable to disinformation.

4 Results

  • Across the full risk range, payoff-maximizing cooperation and replicator equilibria remain prior-dependent, with each prior reaching cooperation at a different critical risk.At risk 0.9, the low-threshold prior has unstable full defection, gaussian and high-threshold priors are bistable, and the uniform prior has a critical risk of 0.9 in the stated setting.Source: §4 · p. 5, Figure 1 · p. 6
  • With information cost cI=0.01, costly information raises group success versus no information across nearly the entire risk range for every prior, but this reverses at cI=0.10.At the lower cost, information-acquiring strategies reach about 30–50% of the population depending on the prior; few buy at the higher cost.Source: §4 · p. 6, Figure 2 · p. 6
  • Buying information is privately beneficial only at sufficiently high risk, when pivotal contributions can be worth their cost; this threshold is reached earliest for low-threshold and latest for high-threshold priors.The studied conditional heuristics do not capture the full value of information because they do not account for whether a contribution is likely to be pivotal.Source: §4 · p. 6, Figure 2 · p. 6
  • Sharing information shifts the cooperative tipping point to lower risk, with the largest success gain at that transition and for the high-threshold prior.The gain declines at high risk, when groups also succeed without shared information; one informed member can coordinate the group's conditional contributions.Source: §4 · p. 7, Figure 3 · p. 7
  • With sharing, buyers and information free-riders coexist; under uniform and high-threshold priors, buyers increase with risk and can earn more than free-riders relying on them.Sharing lowers the private value of buying under uniform, gaussian, and low-threshold priors, although buying remains privately beneficial over most of the risk range.Source: §4 · p. 7, Figure 3 · p. 7
  • Network relaying increases group success while reducing the share of buyers under uniform and high-threshold priors; noisy relays do not significantly change results.Without relaying, success is only slightly lower except under uniform priors on random-regular networks and high-threshold priors on scale-free networks.Source: §4 · p. 7, Figure 4 · p. 8
  • Malicious relays and biased sources reduce group success while present, with larger damage from sources at scale-free hubs; success recovers after their removal.On degree-heterogeneous networks, a few biased hubs can suppress success almost entirely under the high-threshold prior.Source: §4 · pp. 7–8, Figure 4 · p. 8
  • Buyers constrain false information largely to source neighbours because true-information cascades compete with it; fewer buyers allow disinformation to spread farther.False information changes the actions of information free-riders, whose contributions depend on what they hear.Source: §4 · pp. 7–8

5 Discussion and Conclusion

Costly information helps collective action only under specific conditions: it must be sufficiently cheap, and its effects depend on how information is shared and transmitted. Buyers can also limit disinformation, while populations recover after false sources are removed.

5 Discussion and Conclusion

  • Costly information improves collective outcomes when its cost is sufficiently low, but its value to individuals is bounded by the cost of cooperation.The paper gives the condition c_I < φc ≤ c and notes that private incentives to acquire information can diverge from its collective value.Source: §5 · p. 8
  • Sharing purchased information within groups shifts the cooperation tipping point to lower risk, while network diffusion reduces the number of buyers.A single informed group member can coordinate contributions across the group; diffusion extends information beyond groups and encourages free-riding.Source: §5 · p. 8
  • Noisy relaying does not reduce success in the model, whereas false information harms collective success, especially when introduced at hubs.Large attacks can bring collective action close to collapse; false information injected at hubs was up to three times more damaging than at random nodes.Source: §5 · pp. 8–9
  • Figure 5 maps false-information reach, strategy types, and source locations across attacks that vary buyer prevalence and source placement.Its panels include attacks on random regular networks and attacks involving a hub or random sources on scale-free networks.Source: Figure 5 · p. 9
  • Buyers' true-information cascades act as a firewall, limiting false information mainly to source neighbours and affecting information free-riders.After false sources are removed, success is restored within a few thousand generations, even after an attack that nearly suppressed it.Source: §5 · p. 9

6 Contributions

The author-contribution statement assigns conceptualization to all authors, with distinct roles in methods, supervision, analysis, investigation, and writing.

6 Contributions

  • E.F.D. is credited with methodology, software, formal analysis, investigation, and the original draft, alongside conceptualization.T.L. is credited with supervision and review and editing; G.I. and W.T. are credited with conceptualization.Source: §6 · p. 9
  • All authors are credited with reading and approving the final manuscript.Source: §6 · p. 9

Supporting Information for:

The supporting-information heading identifies the paper and its four authors.

Supporting Information for:

S1 Notation

The notation section specifies fixed group and population sizes, a fixed cooperation cost, and a reference threshold shared across prior settings.

S1 Notation

  • Group size is fixed at N = 8 and population size at Z = 100 throughout.Source: S1 Notation · p. 12
  • The cooperation cost is fixed at c = 0.1 throughout.Source: S1 Notation · p. 12
  • The reference threshold M_f = 4 equals the mean for uniform and Gaussian priors, but not for low- and high-threshold priors, whose means are 2.38 and 5.62.Source: S1 Notation · p. 12

S2 Payoffs, fitness, and value of information

The section derives expected payoff differences and population-level measures for the costly-information game. It also defines an individual's value of information by comparing informed and uninformed choices across uncertain thresholds.

Two-strategy reduction

  • With only UD and UC, a contribution changes payoff only when exactly M−1 co-players contribute, yielding the two-strategy replicator analysis.The resulting analysis is reported in Figure 1.Source: S2 Payoffs, fitness, and value of… · p. 12

Finite-population fitness and population-level variables

  • In a finite population, co-players are sampled without replacement, and group success and cooperation and information-purchase frequencies are defined from population strategy counts.The group-success measure is averaged over the stationary distribution.Source: S2 Payoffs, fitness, and value of… · p. 12

Value of information

  • An individual's value of information is the smaller of the expected gains from acting when the threshold favors contributing versus defecting, and is bounded by the contribution cost.The informed individual chooses separately for each threshold, unlike an uninformed individual; the values are averaged across individuals and the stationary distribution.Source: S2 Payoffs, fitness, and value of… · p. 12

S3 No-information baseline

The baseline comparison accounts for mutation acting on strategy labels: duplicating payoff-identical strategies changes the chance that a mutating defector becomes a cooperator.

S3 No-information baseline

S4 Collective pivotality

Information-dependent contributions can be jointly decisive even when no individual is pivotal. The section also describes how buying value is evaluated when information is shared and how risk curves are constructed.

S4 Collective pivotality

  • At information cost 0.01, information is jointly decisive in up to 48–96% of successful groups, two to three times more often than an informed contributor is individually pivotal.With sharing, joint decisiveness reaches up to 88% under the high-threshold prior.Source: S4 Collective pivotality · p. 12

S5 Sharing information within groups

S6 Network simulations

Network simulations examine information diffusion and attacks on random regular and Barabási–Albert graphs. False information can sharply reduce success, particularly when sources occupy hubs, while greater buyer presence limits its reach.

S6 Network simulations

  • The simulations use 100-node random 7-regular and Barabási–Albert graphs with mean degree 7.7, and scale group thresholds to keep the required contributor share constant.Source: S6 Network simulations · p. 13
  • Buyers and false sources initiate cascades; informed buyers and free-riders relay messages to uninformed neighbours with probability λ, and nodes keep the first message received.Noisy transmissions add Gaussian noise; malicious relays and biased sources report threshold 0, while noisy sources add noise to the true threshold.Source: S6 Network simulations · p. 13
  • Stationary averages use 30,000 generations after 5,000 burn-in generations, while shock ensembles compare attacks with no-attack counterfactuals across replicates.Experiments also use burn-in and recovery periods, with 95% intervals across chains or replicates.Source: S6 Network simulations · p. 14

S7 Additional network results

  • On Barabási–Albert networks, malicious relays and biased sources at hubs lower success 1.5 to 3 times as much as at random nodes.Five biased sources at hubs reduce high-threshold success from 0.27 to 0.02.Source: S7 Additional network results · p. 14
  • Under the high-threshold prior on random regular networks, 20 biased sources lower success from 0.81 to 0.12, versus 0.41 for 20 silent sources.After removal, success remains 0.35–0.50 below the counterfactual for about 10^3 generations and recovers within 5 × 10^3 generations.Source: S7 Additional network results · p. 14
  • False information reaches 30–66% of sources’ direct neighbours and at most 17% of individuals two steps away; its reach falls as buyer share rises.Relay noise produces errors in both directions, so success does not fall.Source: S7 Additional network results · p. 14
  • Unlike Santos and Pacheco (2011), this study averages payoffs over groups, removing the well-connected-cooperator advantage attributed to accumulated fitness.Source: S7 Additional network results · p. 14

S8 Limitations of the network model

The network model fixes attackers and information sources as noncontributing nodes and does not allow receivers to learn distrust. The accompanying figures compare threshold priors and examine information spread, relay noise, payoffs, and costly-information outcomes.

S8 Limitations of the network model

  • The network model fixes attackers and sources as noncontributing nodes, excludes receiver distrust learning and other-regarding preferences or trust signals, and uses 100-node networks that cannot show a clear power-law degree distribution.The authors leave these limitations for future work.Source: S8 Limitations of the network model · p. 14

S9 Additional figures

  • Figure S1 displays cumulative distributions for four threshold priors alongside the certain threshold M=4.Source: Figure S1 · p. 15
  • Figure S2 tracks disinformation and misinformation reach and contribution changes; its additional panels encode distance to false sources and buyer share against disinformation reach.Panel A uses dark shading for reach and light shading for contribution changes; panels B–C add distance and buyer-share views.Source: Figure S2 · p. 16, S9 Additional figures · p. 16
  • Figure S3 plots group success on a random 7-regular network against relay noise.The caption reports 95% intervals across chains.Source: Figure S3 · p. 16
  • Figure S4 presents the average payoff difference between buyers and information free-riders when information is shared.Source: Figure S4 · p. 16
  • Figure S5 compares cooperation frequency and population-average payoff relative to the NI baseline at information costs 0.01 and 0.10, marking regions where CI performs worse.Red shading marks worse CI performance; solid and dashed lines distinguish the two costs.Source: Figure S5 · p. 17
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