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Cited by 45Evolutionary Game Theory and CooperationExperimental Behavioral Economics StudiesGame Theory and Applications

Conditional investment strategy in evolutionary trust games with repeated group interactions

Open access
Liu, Linjie, Chen, Xiaojie·2022·Information Sciences
DOI 10.1016/j.ins.2022.07.073
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Written from the paper’s full text.

TL;DR

The paper introduces a conditional investment strategy into a repeated \(N\)-player trust game, where investors pay an observation cost and invest in the first round but only continue investing if the number of trustworthy trustees in the group reaches a threshold \(M\). Using a Markov decision process in finite well-mixed populations, it finds that an intermediate threshold allows conditional investors and trustworthy trustees to form an alliance that suppresses untrustworthy trustees.

Key findings

  • An intermediate investment threshold promotes high trust and trustworthiness: conditional investors (\(CI\)) and trustworthy agents (\(T\)) form an alliance that crowds out untrustworthy agents (\(U\)). When \(M=0\) or \(M=N\), \(U\) is more prevalent or agents only benefit from the first round, and the \(CI\)–\(T\) advantage disappears.
  • The \(CI\)–\(T\) alliance is robust across many parameters. Average levels of \(CI\), \(T\), and \(U\) remain almost unchanged with increases in the untrustworthy trustee multiply factor \(R_U\) (tested at \(R_U=6.66, 7.98, 9.96\)), investment cost \(tv\), discount factor \(w\), and group size \(N\) (with \(M/N=0.5\)).
  • Increasing observation cost \(\sigma\) decreases the average level of \(CI\) and increases the average level of \(T\), while leaving \(U\) almost unchanged; the \(CI\)–\(T\) alliance still resists \(U\) invasion.
  • When selection intensity \(\beta\) is small (\(\beta<10^{-3}\)) or mutation rate \(\mu\) is close to 1, average levels of all strategies approach \(1/3\). As \(\beta\) increases or \(\mu\) decreases, the \(CI\)–\(T\) alliance emerges and crowds out most \(U\) agents. Average payoffs of all agents first increase then decrease with \(M\); at \(M=0\) or \(M=N\), average payoffs of \(CI\) agents are negative.

Methods

  • Finite well-mixed population of \(Z\) agents playing a repeated \(N\)-player trust game. At each time step, \(N\) agents are sampled; the game repeats with probability \(w\), giving average rounds \(r=1/w\). Three strategies are modeled: conditional investor (\(CI\)), trustworthy trustee (\(T\)), and untrustworthy trustee (\(U\)). \(CI\) pays observation cost \(\sigma\), invests \(tv\) in the first round, and subsequently invests only if the number of \(T\) agents in the group is at least threshold \(M\).
  • Payoffs are computed using multiple hypergeometric sampling of group configurations. Social learning follows a pairwise comparison Fermi rule with selection intensity \(\beta\), plus mutation with probability \(\mu\). Evolutionary dynamics are analyzed as a Markov chain/Markov decision process via the master equation, stationary distribution, and gradient of selection.
  • Main analyses report stationary distributions, gradients of selection, average strategy levels, and average payoffs. Baseline parameters include \(Z=100\), \(N=4\), \(tv=1\), \(R_U=8\), \(R_T=6\), \(\mu=1/Z\), \(w=0.8\), \(\sigma=0.1\), and \(\beta=5\), with robustness checks varying \(R_U\), \(tv\), \(\sigma\), \(w\), \(\beta\), \(\mu\), and \(N\).

Limitations

  • The paper states that future work could explore the repeated \(N\)-player trust game in structured populations, since the MDP approach adopted is suitable for complete graph networks and can be extended to other network structures; it also suggests theoretical approximation and agent-based simulations on any population structure.
  • Evident from the text: the model assumes a finite well-mixed population and complete graph interactions, so results do not directly address structured or networked populations. The findings are based on numerical stationary distributions and gradients of selection under specified parameter ranges rather than empirical data.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

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Crossref · 10.1016/j.ins.2022.07.073

Cites

42 of 42 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 25 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § 4. Results · p. 11

    With an intermediate investment threshold, the population spends most of its time in states with a high prevalence of trustworthy agents and conditional investors.

    “The weakening of investment tendency will lead the population to spending most of the time in states with a high prevalence of T and CI (see Fig. 2(b)).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the model, investors invest deterministically in the first round of the game.
    • The authors state this as an assumption: In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.
  • Findingown data
    § Abstract · p. 1

    Conditional investors can form an effective alliance with trustworthy trustees in the repeated N-player trust game.

    “We find that conditional investors can form an effective alliance with trustworthy trustees, hence they can sweep out untrustworthy trustees.”
    • Also stated in: § 4. Results · p. 12 · boosted
      “With the increase of threshold, CI and T agents can form a strong alliance, preventing the invasion of U agents.”
    • Also stated in: § 4. Results · p. 12 · plain
      “The intermediate investment threshold is conducive to the formation of CI and T alliance, and then effectively improves the level of trust.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the model, investors invest deterministically in the first round of the game.
    • The authors state this as an assumption: In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.
  • Findingreasoning
    § 1. Introduction · p. 3

    The introduction of the conditional investment strategy can lead to the emergence of high trust and high trustworthiness in repeated group interactions.

    “We find that the introduction of the conditional investment strategy can lead to the emergence of high trust and high trustworthiness in repeated group interactions.”
    • Also stated in: § 4. Results · p. 18 · plain
      “we show that the introduction of the conditional investment strategy can lead to the emergence of high trust and high trustworthiness in repeated group interactions”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the model, investors invest deterministically in the first round of the game.
    • The authors state this as an assumption: In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.
    • The authors state this as an assumption: Investors pay a permanent observation cost σ to collect information about the trustworthiness of the trustee, and then make decisions in subsequent possible group interactions based on what they observe.
  • Findingown data
    § Abstract · p. 1

    The alliance between conditional investors and trustworthy trustees can exist in a wide range of model parameters.

    “Moreover, we verify that such alliance can exist in a wide range of model parameters.”
    • Also stated in: § 1. Introduction · p. 3 · plain
      “Furthermore, we verify that such evolutionary outcomes are robust against model parameters.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the model, investors invest deterministically in the first round of the game.
    • The authors state this as an assumption: In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.
  • Findingown data
    § 5. Conclusions and Discussion · p. 19

    In our model, even though the social dilemma is extremely difficult (R_U/R_T > 1.66), a high level of trust can always be achieved.

    “In our model, even though the social dilemma is extremely difficult (i.e. the ratio of temptation to defect R U R T &gt; 1 . 66), a high level of trust can always be achieved (see Fig. 5(a)).”
    Quote in the paperSubstantial claim
  • May combine two claims. 13
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 7

  • May combine two claims. The quote only partly states this claim.

    Findingown data
    § 4. Results · p. 11

    When the investment threshold M is zero, untrustworthy agents can be more prevalent than conditional investors and trustworthy agents, and the population spends significant time in states where trustworthy and untrustworthy agents coexist with untrustworthy agents highly prevalent.

    “We can notice that the population will spend a significant time on configurations near the TU -edge ( T means trustworthy agents and U means untrustworthy agents) of the triangle simplex, as shown in Fig. 2(a). Besides, most of the arrows flow to the coexistence states of T and U where U is highly prevalent.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the model, investors invest deterministically in the first round of the game.
    • The authors state this as an assumption: In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.

  • May combine two claims.

    Findingown data
    § Abstract · p. 1

    The alliance between conditional investors and trustworthy trustees can sweep out or prevent the invasion of untrustworthy trustees.

    “We find that conditional investors can form an effective alliance with trustworthy trustees, hence they can sweep out untrustworthy trustees.”
    • Also stated in: § 4. Results · p. 12 · boosted
      “With the increase of threshold, CI and T agents can form a strong alliance, preventing the invasion of U agents.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the model, investors invest deterministically in the first round of the game.
    • The authors state this as an assumption: In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.

  • May combine two claims.

    Findingown data
    § 4. Results · p. 11

    When the investment threshold M equals N, the system spends most of its time in states where trustworthy and untrustworthy agents coexist, and untrustworthy agents have a greater advantage than trustworthy and conditional investor agents.

    “The evolutionary outcomes in finite populations are similar to the result reported in Fig. 2(a), that is, the system spends most of the time in states where T and U agents coexist (see Fig. 2(c)).”
    • Also stated in: § 4. Results · p. 12 · plain
      “In this case, the advantage of U agents is greater than that of T and CI agents.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the model, investors invest deterministically in the first round of the game.
    • The authors state this as an assumption: In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.

  • May combine two claims.

    Findingown data
    § 4. Results · p. 18

    When N changes appropriately, the population spends most time in the intermediate region of the CI-T edge, and most arrows point to states where CI and T agents coexist.

    “the population spends most time in the intermediate region of CI -T edge, and most arrows point to the states where CI and T agents coexist when N changes appropriately (see Fig. 11(b)-(d)).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4. Results · p. 18

    When β is small (β < 10^-3) or when μ is significantly large (close to 1), the evolutionary process is mainly affected by imitation error or behavioral mutation, leading to average levels of all strategies close to 1/3.

    “When β is small ( β &lt; 10 -3 ) or when µ is significantly large (close to 1), the evolutionary process is mainly affected by imitation error or behavioral mutation, which leads to the average levels of all strategies close to 1 / 3 (see Figs. 9(a) and 10(a)).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4. Results · p. 18

    The average level of T is the highest, the second largest frequency is CI, and U makes up the smallest fraction.

    “In particular, the average level of T is the highest, the second largest frequency is formed by CI , while U makes up the smallest fraction.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 5. Conclusions and Discussion · p. 19

    An intermediate threshold leads to the evolution of high trust and high trustworthiness, while too low or too high threshold levels cannot promote the evolution of trust.

    “We find that an intermediate threshold can lead to the evolution of high trust and high trustworthiness, while too low threshold level or too high threshold level cannot promote the evolution of trust.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Investors pay a permanent observation cost σ to collect information about the trustworthiness of the trustee, and then make decisions in subsequent possible group interactions based on what they observe.

  • May combine two claims.

    Findingown data
    § 4. Results · p. 17

    Changing the tv values does not influence the gradient of selection and the stationary distribution of the system.

    “In addition, we find that changing the tv values does not influence the gradient of selection and the stationary distribution of the system (see Fig. 6(b)-(d)).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data
    § 4. Results · p. 13

    An intermediate investment threshold ensures a more evident advantage of the alliance formed by CI and T agents over U agents, although this advantage can shrink as M increases.

    “In general, an intermediate investment threshold can ensure a more evident advantage of alliance formed by CI and T agents over U agents, even if this advantage can shrink with the increase of M .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § 4. Results · p. 18

    As β increases or μ decreases, a strategy's performance becomes increasingly important for its survival, eventually favoring the alliance of CI and T strategies, which crowds out most U agents.

    “As β increases or µ decreases, a strategy's performance becomes increasingly important for the strategy's survival, eventually favoring the emerging alliance of the CI and T strategies, which crowds out the most U agents (see Fig. 9 (b)-(d) and Fig. 10 (b)-(d)).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5. Conclusions and Discussion · p. 19

    The alliance formed by CI and T agents can effectively resist the invasion of U agents, and this result is robust to the observation cost.

    “the result that the alliance formed by CI and T agents can effectively resist the invasion of U agents is robust to the observation cost.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Investors pay a permanent observation cost σ to collect information about the trustworthiness of the trustee, and then make decisions in subsequent possible group interactions based on what they observe.

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Premisenone given
    § 2. Related Works · p. 3

    The analysis of replicator dynamics requires that the network structure is a complete graph, and thus cannot be used to analyze evolutionary dynamics when agents interact with local neighborhoods in social networks.

    “Nevertheless, the analysis of replicator dynamics in the previous literature requires that the network structure is a complete graph, and thus this approach cannot be used to analyze the evolutionary dynamics when agents interact with local neighborhoods in social networks.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § 5. Conclusions and Discussion · p. 19

    Analysis of evolutionary dynamics of NTG has been mostly dealt with either in the limit of rare mutations or in the limit of weak selection.

    “analysis of evolutionary dynamics of NTG has been mostly dealt with either in the limit of rare mutations [26] or in the limit of weak selection [44].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [26] [26] I. S. Lim, Stochastic evolutionary dynamics of trust games with asymmetric parameters, Phys. Rev. E 102(6) (2020) 062419. -

      Not checked in the free run.

    • cites [44] [44] C. E. Tarnita, Fairness and trust in structured populations, Games 6(3) (2015) 214-230. -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 5. Conclusions and Discussion · p. 19

    Previous work on networked NTG has revealed that the level of trust is correlated with how 'difficult' the game is.

    “Previous work on networked NTG has revealed that the level of trust is correlated with how 'difficult' the game is [8].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [8] [8] M. Chica, R. Chiong, M. Kirley, H. Ishibuchi, A networked N -player trust game and its evolutionary dynamics, IEEE Trans. Evol. Comput. 22(6) (2018) 866-878. -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § 5. Conclusions and Discussion · p. 19

    The MDP approach adopted in our work is suitable for complete graph networks and can also be extended to other network structures.

    “the MDP approach adopted in our work is suitable for complete graph networks and can also be extended to other network structures [7].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [7] [7] Y. Chen, K. R. Liu, Indirect reciprocity game modelling for cooperation stimulation in cognitive networks, IEEE Trans. Commun. 59(1) (2011) 159-168. -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 2. Related Works · p. 4

    The weak-mutation limit assumption is not suitable for modeling social learning.

    “The above assumption has been justified in population genetics [12, 14], but this approximate does not seem to be suitable for modeling social learning [47].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [47] [47] V. V. Vasconcelos, F. C. Santos, J. M. Pacheco, A bottom-up institutional approach to cooperative governance of risky commons, Nat. Clim. Change 3(9) (2013) 797-801. -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 4. Results · p. 13

    The values R_U = 6.66, 7.98, and 9.96 correspond respectively to mild, moderate, and harsh interaction environments in the trust game.

    “we set R U = 6 . 66, 7.98, and 9 . 96, which can respectively correspond to the mild, moderate, and harsh interaction environment in the trust game (see Ref. [8]).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [8] [8] M. Chica, R. Chiong, M. Kirley, H. Ishibuchi, A networked N -player trust game and its evolutionary dynamics, IEEE Trans. Evol. Comput. 22(6) (2018) 866-878. -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § 2. Related Works · p. 4

    The literature mentioned above does not consider the case in which agents' decision-making may be affected by random factors, including imitation error and behavioral mutation.

    “The literature mentioned above does not consider the case in which agents' decision-making may be affected by random factors, including imitation error and behavioral mutation.”
    Quote in the paperNot a substantial claim
18 of 18

  • The authors state this as an assumption.

    In the model, investors invest deterministically in the first round of the game.

    “Different from previous works [22, 24], we assume investors will invest deterministically in the first round of the game, but in the subsequent game rounds their decision to invest or not depends on the number of the trustworthy trustee in the group.”

  • The authors state this as an assumption.

    In the model, in subsequent game rounds, investors' decision to invest depends on the number of trustworthy trustees in the group.

    “Different from previous works [22, 24], we assume investors will invest deterministically in the first round of the game, but in the subsequent game rounds their decision to invest or not depends on the number of the trustworthy trustee in the group.”

  • The authors state this as an assumption.

    Investors pay a permanent observation cost σ to collect information about the trustworthiness of the trustee, and then make decisions in subsequent possible group interactions based on what they observe.

    “The key assumption of conditional investment is that investors pay a permanent observation cost σ to collect information about the trustworthiness of the trustee, and then make decisions in the subsequent possible group interactions based on what they observe.”

  • The paper relies on this without support.

    Investors generally make decisions based on the current investment environment rather than investing blindly.

    “our model setup is well aligned with reality in which investors generally make decisions based on the current investment environment rather than investing blindly.”
4 of 4

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Cooperation tends to increase when players can vote for exclusion of free riders, observe who is trustworthy, or enter into contracts that enforce minimum reciprocation levels.

    “Cooperation tends to increase when players can vote for exclusion of free riders19, observe who is trustworthy20, or enter into contracts that enforce minimum reciprocation levels21,22.”

    From Deep reinforcement learning can promote sustainable human behaviour in a common-pool resource problem (2025) · cites this paper as [20]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Expected number of rounds conflicts with the repetition probability

    Major

    If a game continues to another round with probability w, its expected number of rounds is 1/(1−w), not 1/w. This changes the expected duration of the repeated interactions used in the model.

  • High thresholds do not prevent investment in the first round

    Minor

    The claim that investors always choose not to invest when M > N−1 conflicts with the stated rule that conditional investors invest in the first round regardless of the threshold.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. Deutsch, 1962, Cooperation and trust: Some theoretical notes, Nebr. Sym. Motiv., 275
  2. Hu, 2021, Adaptive reputation promotes trust in social networks, IEEE Trans. Net. Sci. Eng., 40, 2327
  3. Perc, 2017, The evolution of trust and cooperation between strangers: A computational model, Phys. Rep., 687, 1, 10.1016/j.physrep.2017.05.004 · DOI
  4. Szolnoki, 2013, Correlation of positive and negative reciprocity fails to confer an evolutionary advantage: Phase transitions to elementary strategies, Phys. Rev. X, 3

Cited by

Sorted by citations

Authors
Liu, L., & Chen, X.
Affiliations
University of Electronic Science and Technology of China (CN)
Corresponding author
Chen, Xiaojie
ORCID
Venue
Information Sciences
Publisher
Elsevier BV
Published
2022-09
Volume
609
Pages
1694-1705
ISSN
0020-0255 (ISSN-L)
DOI
10.1016/j.ins.2022.07.073
OpenAlex
W4286255601
Cited by
45
References
42
Citations by year
2026: 8 · 2025: 11 · 2024: 10 · 2023: 15 · 2022: 1
FWCI
13.31
Type
journal article
Language
en
Open access
Licence
Locations
  • Information Sciences (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Social Sciences › Sociology and Political Science › Evolutionary Game Theory and Cooperation
  • Social Sciences › Safety Research › Experimental Behavioral Economics Studies
  • Social Sciences › Decision Sciences › Management Science and Operations Research › Game Theory and Applications
Keywords
trust game, trustworthiness, Markov decision process, coalition formation, artificial agents

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