{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T02:40:33Z","timestamp":1790822433185,"version":"4.1.0"},"reference-count":64,"publisher":"AIP Publishing","issue":"6","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12461054"],"award-info":[{"award-number":["12461054"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Key Foundation of Guizhou Province","award":["2025089"],"award-info":[{"award-number":["2025089"]}]},{"name":"Guizhou Province High-Level Talent Cultivation Support Program","award":["KJLYRC-[2026]014"],"award-info":[{"award-number":["KJLYRC-[2026]014"]}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:p>Trust underpins large-scale cooperation, yet it remains fragile among self-interested individuals. Trust games provide a classic framework for studying the evolutionary dynamics of trustworthy and untrustworthy behaviors under asymmetric payoff structures. However, in most existing trust-game models, investment rules are based on equal or linear allocation, failing to incentivize and constrain the behaviors of trustees effectively. To address this gap, we propose a reputation-based investment allocation mechanism in a networked N-player trust game. In this framework, individual behavior shapes reputation, reputation regulates investment allocation through a nonlinear softmax rule, and both payoff and reputation jointly influence strategy evolution, thereby forming a feedback loop. Numerical simulations demonstrate that this mechanism effectively directs resources toward high-reputation trustees, suppresses untrustworthy behavior, and improves both role-based cooperation index and the average payoff. Reputation-based nonlinear allocation is the key component, which converts reputation differences into resource-allocation advantages. A suitable balance between investment sensitivity and reputation preference is required: excessive reputation preference may reduce investor proportions and limit wealth generation. Thus, the optimal reputation preference is one that suppresses untrustworthy behavior while preserving enough investors to sustain capital flow. Tests on regular lattice, small-world, and scale-free networks show that the mechanism is not limited to the lattice, although stationary payoffs remain network-dependent. This study proposes a novel incentive framework in which reputation scoring, nonlinear allocation, payoff generation, and strategy updating are coupled into a feedback loop that sustains trust in networked N-player trust games.<\/jats:p>","DOI":"10.1063\/5.0336959","type":"journal-article","created":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T13:07:52Z","timestamp":1781615272000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Reputation-based nonlinear investment promotes cooperation in networked N-player trust games"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-7298-9487","authenticated-orcid":false,"given":"Xiaoyi","family":"Jiang","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, State Key Laboratory of Public Big Data, Guizhou University 1 , Guiyang 550025,","place":["China"]},{"name":"Guizhou Provincial Key Laboratory of Game Decision and Control System 2 , Guiyang 550025,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5596-5091","authenticated-orcid":false,"given":"Wensheng","family":"Jia","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, State Key Laboratory of Public Big Data, Guizhou University 1 , Guiyang 550025,","place":["China"]},{"name":"Guizhou Provincial Key Laboratory of Game Decision and Control System 2 , Guiyang 550025,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,6,16]]},"reference":[{"key":"2026061609074980000_c1","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1146\/annurev-soc-082120-082850","article-title":"Trust in social relations","volume":"47","year":"2021","journal-title":"Annu. Rev. Sociol."},{"key":"2026061609074980000_c2","volume-title":"Foundations of Social Theory","year":"1990"},{"key":"2026061609074980000_c3","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1086\/228311","article-title":"Economic action and social structure: The problem of embeddedness","volume":"91","year":"1985","journal-title":"Am. J. Sociol."},{"key":"2026061609074980000_c4","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1016\/j.evolhumbehav.2019.04.001","article-title":"The primacy of trust within romantic relationships: Evidence from conjoint analysis of hexaco-derived personality profiles","volume":"40","year":"2019","journal-title":"Evol. Human Behav."},{"key":"2026061609074980000_c5","first-page":"343","article-title":"Gifts and exchanges","volume":"1","year":"1972","journal-title":"Philos. Public Affairs"},{"key":"2026061609074980000_c6","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1257\/0002828041464498","article-title":"The role of social capital in financial development","volume":"94","year":"2004","journal-title":"Am. Econ. Rev."},{"key":"2026061609074980000_c7","doi-asserted-by":"publisher","first-page":"709","DOI":"10.2307\/258792","article-title":"An integrative model of organizational trust","volume":"20","year":"1995","journal-title":"Acad. Manag. Rev."},{"key":"2026061609074980000_c8","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1287\/orsc.9.2.141","article-title":"Does trust matter? exploring the effects of interorganizational and interpersonal trust on performance","volume":"9","year":"1998","journal-title":"Organ. Sci."},{"key":"2026061609074980000_c9","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1146\/annurev.polisci.3.1.475","article-title":"Political trust and trustworthiness","volume":"3","year":"2000","journal-title":"Annu. Rev. Pol. Sci."},{"key":"2026061609074980000_c10","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1353\/wp.2006.0022","article-title":"All for all: Equality, corruption, and social trust","volume":"58","year":"2005","journal-title":"World Pol."},{"key":"2026061609074980000_c11","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1006\/game.1995.1027","article-title":"Trust, reciprocity, and social history","volume":"10","year":"1995","journal-title":"Games Econ. Behav."},{"key":"2026061609074980000_c12","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1016\/j.jtbi.2005.06.007","article-title":"The evolution of n-player cooperation\u2014threshold games and ess bifurcations","volume":"238","year":"2006","journal-title":"J. Theor. Biol."},{"key":"2026061609074980000_c13","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1109\/TEVC.2015.2484840","article-title":"The n-player trust game and its replicator dynamics","volume":"20","year":"2015","journal-title":"IEEE Trans. Evol. Comput."},{"key":"2026061609074980000_c14","doi-asserted-by":"publisher","first-page":"866","DOI":"10.1109\/TEVC.2017.2769081","article-title":"A networked n-player trust game and its evolutionary dynamics","volume":"22","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"2026061609074980000_c15","doi-asserted-by":"publisher","first-page":"20200491","DOI":"10.1098\/rsif.2020.0491","article-title":"The evolution of trust and trustworthiness","volume":"17","year":"2020","journal-title":"J. R. Soc. Interface"},{"key":"2026061609074980000_c16","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1038\/31225","article-title":"Evolution of indirect reciprocity by image scoring","volume":"393","year":"1998","journal-title":"Nature"},{"key":"2026061609074980000_c17","doi-asserted-by":"publisher","first-page":"176","DOI":"10.1140\/epjb\/s10051-021-00185-w","article-title":"Evolutionary dynamics of trust in the N-player trust game with individual reward and punishment","volume":"94","year":"2021","journal-title":"Eur. Phys. J. B"},{"key":"2026061609074980000_c18","doi-asserted-by":"publisher","first-page":"114031","DOI":"10.1016\/j.chaos.2023.114031","article-title":"N-player repeated evolutionary trust game under government management","volume":"175","year":"2023","journal-title":"Chaos, Solitons Fractals"},{"key":"2026061609074980000_c19","doi-asserted-by":"publisher","first-page":"1694","DOI":"10.1016\/j.ins.2022.07.073","article-title":"Conditional investment strategy in evolutionary trust games with repeated group interactions","volume":"609","year":"2022","journal-title":"Inf. Sci."},{"key":"2026061609074980000_c20","doi-asserted-by":"publisher","first-page":"114078","DOI":"10.1016\/j.chaos.2023.114078","article-title":"Evolution of trust in a hierarchical population with different investors based on investment behavioral theory","volume":"176","year":"2023","journal-title":"Chaos, Solitons Fractals"},{"key":"2026061609074980000_c21","doi-asserted-by":"publisher","first-page":"129177","DOI":"10.1016\/j.amc.2024.129177","article-title":"Time delay effects on evolutionary dynamics of n-player trust game with diverse investment behaviors","volume":"489","year":"2025","journal-title":"Appl. Math. Comput."},{"key":"2026061609074980000_c22","doi-asserted-by":"publisher","first-page":"130898","DOI":"10.1016\/j.physa.2025.130898","article-title":"The influence of heterogeneous investment on the evolution of trust in complex networks","volume":"677","year":"2025","journal-title":"Phys. A"},{"key":"2026061609074980000_c23","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.jtbi.2004.06.005","article-title":"How should we define goodness?\u2014Reputation dynamics in indirect reciprocity","volume":"231","year":"2004","journal-title":"J. Theor. Biol."},{"key":"2026061609074980000_c24","doi-asserted-by":"publisher","first-page":"e40218","DOI":"10.1371\/journal.pone.0040218","article-title":"Inferring reputation promotes the evolution of cooperation in spatial social dilemma games","volume":"7","year":"2012","journal-title":"PLoS One"},{"key":"2026061609074980000_c25","doi-asserted-by":"publisher","first-page":"868","DOI":"10.1177\/147470491201000509","article-title":"Harnessing the power of reputation: Strengths and limits for promoting cooperative behaviors","volume":"10","year":"2012","journal-title":"Evol. Psychol."},{"key":"2026061609074980000_c26","doi-asserted-by":"publisher","first-page":"8","DOI":"10.3390\/g8010008","article-title":"The evolution of reputation-based cooperation in regular networks","volume":"8","year":"2017","journal-title":"Games"},{"key":"2026061609074980000_c27","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1038\/nature25763","article-title":"Social norm complexity and past reputations in the evolution of cooperation","volume":"555","year":"2018","journal-title":"Nature"},{"key":"2026061609074980000_c28","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1016\/S0167-2681(99)00045-1","article-title":"Collective action as a social exchange","volume":"39","year":"1999","journal-title":"J. Econ. Behav. Org."},{"key":"2026061609074980000_c29","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.jtbi.2014.01.037","article-title":"Rewarding evolutionary fitness with links between populations promotes cooperation","volume":"349","year":"2014","journal-title":"J. Theor. Biol."},{"key":"2026061609074980000_c30","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1016\/j.chaos.2019.01.006","article-title":"Reciprocal reward promotes the evolution of cooperation in structured populations","volume":"119","year":"2019","journal-title":"Chaos, Solitons Fractals"},{"key":"2026061609074980000_c31","doi-asserted-by":"publisher","first-page":"127108","DOI":"10.1016\/j.physleta.2020.127108","article-title":"Reciprocal rewards promote the evolution of cooperation in spatial prisoner\u2019s dilemma game","volume":"390","year":"2021","journal-title":"Phys. Lett. A"},{"key":"2026061609074980000_c32","doi-asserted-by":"publisher","first-page":"5928","DOI":"10.1038\/s41467-022-33551-y","article-title":"Reputation effects drive the joint evolution of cooperation and social rewarding","volume":"13","year":"2022","journal-title":"Nat. Commun."},{"key":"2026061609074980000_c33","doi-asserted-by":"publisher","first-page":"103147","DOI":"10.1063\/5.0233220","article-title":"Threshold incentive mechanisms for the sustainable management of public resources","volume":"34","year":"2024","journal-title":"Chaos"},{"key":"2026061609074980000_c34","doi-asserted-by":"publisher","first-page":"129951","DOI":"10.1016\/j.physleta.2024.129951","article-title":"Evolution of cooperation under pure institutional reward and punishment in collective-risk social dilemma games","volume":"526","year":"2024","journal-title":"Phys. Lett. A"},{"key":"2026061609074980000_c35","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1038\/415137a","article-title":"Altruistic punishment in humans","volume":"415","year":"2002","journal-title":"Nature"},{"key":"2026061609074980000_c36","doi-asserted-by":"publisher","first-page":"083005","DOI":"10.1088\/1367-2630\/12\/8\/083005","article-title":"Punish, but not too hard: How costly punishment spreads in the spatial public goods game","volume":"12","year":"2010","journal-title":"New J. Phys."},{"key":"2026061609074980000_c37","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/j.chaos.2018.12.040","article-title":"Spontaneous punishment promotes cooperation in public good game","volume":"120","year":"2019","journal-title":"Chaos, Solitons Fractals"},{"key":"2026061609074980000_c38","doi-asserted-by":"publisher","first-page":"113669","DOI":"10.1016\/j.chaos.2023.113669","article-title":"Facilitating the evolution of cooperation through altruistic punishment with adaptive feedback","volume":"173","year":"2023","journal-title":"Chaos, Solitons Fractals"},{"key":"2026061609074980000_c39","doi-asserted-by":"publisher","first-page":"128879","DOI":"10.1016\/j.physleta.2023.128879","article-title":"Evolution of cooperation driven by sampling punishment","volume":"475","year":"2023","journal-title":"Phys. Lett. A"},{"key":"2026061609074980000_c40","doi-asserted-by":"publisher","first-page":"073123","DOI":"10.1063\/5.0215702","article-title":"Emergence of cooperation under punishment: A reinforcement learning perspective","volume":"34","year":"2024","journal-title":"Chaos"},{"key":"2026061609074980000_c41","doi-asserted-by":"publisher","first-page":"886","DOI":"10.1016\/j.physa.2019.04.190","article-title":"Reputation-based investment strategy promotes cooperation in public goods games","volume":"523","year":"2019","journal-title":"Phys. A"},{"key":"2026061609074980000_c42","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1016\/j.jtbi.2005.08.008","article-title":"The leading eight: Social norms that can maintain cooperation by indirect reciprocity","volume":"239","year":"2006","journal-title":"J. Theor. Biol."},{"key":"2026061609074980000_c43","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1038\/nature02978","article-title":"Indirect reciprocity can stabilize cooperation without the second-order free rider problem","volume":"432","year":"2004","journal-title":"Nature"},{"key":"2026061609074980000_c44","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1098\/rspb.2000.1573","article-title":"Evolution of cooperation through indirect reciprocity","volume":"268","year":"2001","journal-title":"Proc. R. Soc. London., Ser. B."},{"key":"2026061609074980000_c45","doi-asserted-by":"publisher","first-page":"2666","DOI":"10.1073\/pnas.0407370102","article-title":"Indirect reciprocity, image scoring, and moral hazard","volume":"102","year":"2005","journal-title":"Proc. Natl. Acad. Sci."},{"key":"2026061609074980000_c46","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1111\/j.1539-6924.1993.tb01329.x","article-title":"Perceived risk, trust, and democracy","volume":"13","year":"1993","journal-title":"Risk Anal."},{"key":"2026061609074980000_c47","doi-asserted-by":"publisher","first-page":"3087","DOI":"10.1109\/TNSE.2021.3103771","article-title":"Adaptive reputation promotes trust in social networks","volume":"8","year":"2021","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"2026061609074980000_c48","doi-asserted-by":"publisher","first-page":"033105","DOI":"10.1063\/5.0256342","article-title":"Dynamic evolution in multi-player networked trust games with graded punishment","volume":"35","year":"2025","journal-title":"Chaos"},{"key":"2026061609074980000_c49","doi-asserted-by":"publisher","first-page":"826","DOI":"10.1038\/359826a0","article-title":"Evolutionary games and spatial chaos","volume":"359","year":"1992","journal-title":"Nature"},{"key":"2026061609074980000_c50","volume-title":"Evolutionary Dynamics: Exploring the Equations of Life","year":"2006"},{"key":"2026061609074980000_c51","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1145\/355112.355122","article-title":"Reputation systems","volume":"43","year":"2000","journal-title":"Commun. ACM"},{"key":"2026061609074980000_c52","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1016\/j.plrev.2023.05.002","article-title":"Reputation and reciprocity","volume":"46","year":"2023","journal-title":"Phys. Life Rev."},{"key":"2026061609074980000_c53","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/S0144-8188(00)00024-7","article-title":"Punishing repeat offenders more severely","volume":"20","year":"2000","journal-title":"Int. Rev. Law Econ."},{"key":"2026061609074980000_c54","doi-asserted-by":"publisher","first-page":"646","DOI":"10.1038\/nature02414","article-title":"Emergence of cooperation and evolutionary stability in finite populations","volume":"428","year":"2004","journal-title":"Nature"},{"key":"2026061609074980000_c55","volume-title":"Evolutionary Games and Population Dynamics","year":"1998"},{"key":"2026061609074980000_c56","doi-asserted-by":"publisher","first-page":"238701","DOI":"10.1103\/PhysRevLett.95.238701","volume":"95","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"2026061609074980000_c57","doi-asserted-by":"publisher","first-page":"33\u201360","DOI":"10.1016\/j.plrev.2025.11.006","article-title":"Cooperation versus social welfare","volume":"56","year":"2025","journal-title":"Phys. Life Rev."},{"key":"2026061609074980000_c58","doi-asserted-by":"publisher","first-page":"115688","DOI":"10.1016\/j.chaos.2024.115688","article-title":"Cooperation dynamics of reputation-based manhattan distance social circle in spatial prisoner\u2019s dilemma game in evolutionary game theory","volume":"189","year":"2024","journal-title":"Chaos, Solitons Fractals"},{"key":"2026061609074980000_c59","doi-asserted-by":"publisher","first-page":"114668","DOI":"10.1016\/j.chaos.2024.114668","article-title":"The impact of reputation-based heterogeneous evaluation and learning on cooperation in spatial public goods game","volume":"181","year":"2024","journal-title":"Chaos, Solitons Fractals"},{"key":"2026061609074980000_c60","doi-asserted-by":"publisher","first-page":"014216","DOI":"10.1103\/PhysRevE.107.014216","article-title":"Turing patterns in simplicial complexes","volume":"107","year":"2023","journal-title":"Phys. Rev. E"},{"key":"2026061609074980000_c61","doi-asserted-by":"publisher","first-page":"7391","DOI":"10.1038\/s41467-024-50918-5","article-title":"The temporal dynamics of group interactions in higher-order social networks","volume":"15","year":"2024","journal-title":"Nat. Commun."},{"key":"2026061609074980000_c62","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41562-025-02373-5","article-title":"Higher-order interactions shape collective human behaviour","volume":"9","year":"2025","journal-title":"Nat. Human Behav."},{"key":"2026061609074980000_c63","doi-asserted-by":"publisher","first-page":"20190211","DOI":"10.1098\/rsif.2019.0211","article-title":"The evolution of lying in well-mixed populations","volume":"16","year":"2019","journal-title":"J. R. Soc. Interface"},{"key":"2026061609074980000_c64","doi-asserted-by":"publisher","first-page":"20200880","DOI":"10.1098\/rsif.2020.0880","article-title":"Mathematical foundations of moral preferences","volume":"18","year":"2021","journal-title":"J. R. Soc. Interface"}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0336959\/21044376\/063130_1_5.0336959.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0336959\/21044376\/063130_1_5.0336959.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T13:07:54Z","timestamp":1781615274000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/36\/6\/063130\/3395633\/Reputation-based-nonlinear-investment-promotes"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,1]]},"references-count":64,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2026,6,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0336959","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,6]]},"published":{"date-parts":[[2026,6,1]]},"article-number":"063130"}}