{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:13:49Z","timestamp":1760228029780,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["61973007","61633002","4192026"],"award-info":[{"award-number":["61973007","61633002","4192026"]}]},{"name":"Beijing Natural Science Foundation","award":["61973007","61633002","4192026"],"award-info":[{"award-number":["61973007","61633002","4192026"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Division of labor in self-organized groups is a problem of both theoretical significance and application value. Many application problems in the real world require efficient task allocation. We propose a model combining bio-inspiration and evolutionary game theory. This research model theoretically analyzes the problem of target search in unknown areas for multi-robot systems. If the robot\u2019s operating area is underwater, the problem becomes more complicated due to its information sharing restrictions. Additionally, it drives strategy updates and calculates the fixed probability of relevant strategies, using evolutionary game theory and the commonly used Fermi function. Our study estimates the fixed probability under arbitrary selection intensity and the fixed probability and time under weak selection for the two-player game model. In the multi-player game, we get these results for weak selection, which is conducive to the coexistence of the two strategies. Moreover, the conducted simulations confirm our analysis. These results help to understand and design effective mechanisms in which self-organizing collective dynamics appears in the form of maximizing the benefits of multi-agent systems in the case of the asymmetric game.<\/jats:p>","DOI":"10.3390\/sym14050941","type":"journal-article","created":{"date-parts":[[2022,5,6]],"date-time":"2022-05-06T02:46:39Z","timestamp":1651805199000},"page":"941","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Evolutionary Dynamics of Division of Labor Games for Underwater Searching Tasks"],"prefix":"10.3390","volume":"14","author":[{"given":"Minglei","family":"Xiong","sequence":"first","affiliation":[{"name":"State Key Laboratory for Turbulence and Complex Systems, Intelligent Biomimetic Design Lab, College of Engineering, Peking University, Beijing 100871, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guangming","family":"Xie","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Turbulence and Complex Systems, Intelligent Biomimetic Design Lab, College of Engineering, Peking University, Beijing 100871, China"},{"name":"Peng Cheng Laboratory, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1177\/002200278002400101","article-title":"Effective Choice in the Prisoner\u2019s Dilemma","volume":"24","author":"Axelrod","year":"1980","journal-title":"J. Confl. Resolut."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"13327","DOI":"10.1038\/ncomms13327","article-title":"Altruistic punishment does not increase with the severity of norm violations in the field","volume":"7","author":"Balafoutas","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"13273","DOI":"10.1038\/ncomms13273","article-title":"Competition among networks highlights the power of the weak","volume":"7","author":"Iranzo","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1109\/TCSS.2016.2584103","article-title":"A Signaling Game for Uncertain Data Delivery in Selfish Mobile Social Networks","volume":"3","author":"Xia","year":"2016","journal-title":"IEEE Trans. Comput. Soc. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1109\/TAC.2016.2555879","article-title":"Feedback Nash Equilibria in Linear-Quadratic Difference Games With Constraints","volume":"62","author":"Reddy","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"6222","DOI":"10.1109\/TIE.2014.2317133","article-title":"Game-Theoretical Persistent Tracking of a Moving Target Using a Unicycle-Type Mobile Vehicle","volume":"61","author":"Zhang","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"58002","DOI":"10.1209\/0295-5075\/116\/58002","article-title":"Fixation of strategies driven by switching probabilities in evolutionary games","volume":"116","author":"Xu","year":"2016","journal-title":"EPL"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"9098","DOI":"10.1038\/srep09098","article-title":"How insurance affects altruistic provision in threshold public goods games","volume":"5","author":"Zhang","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Gong, Y., and Yu, Q. (2019). Evolution of conformity dynamics in complex social networks. Symmetry, 11.","DOI":"10.3390\/sym11030299"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Musaev, A., and Borovinskaya, E. (2021). Evolutionary Optimization of Case-Based Forecasting Algorithms in Chaotic Environments. Symmetry, 13.","DOI":"10.3390\/sym13020301"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Sz\u00e9p, J., and Forg\u00f3, F. (1985). Two-Person Games, Springer.","DOI":"10.1007\/978-94-009-5193-8_7"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.knosys.2017.02.026","article-title":"Collaborative content dissemination based on game theory in multimedia cloud","volume":"124","author":"Li","year":"2017","journal-title":"Knowl. Based Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0165-4896(83)90012-4","article-title":"Evolutionary stability in extensive two-person games","volume":"5","author":"Selten","year":"1986","journal-title":"Math. Soc. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"E7518","DOI":"10.1073\/pnas.1604096113","article-title":"An oscillating tragedy of the commons in replicator dynamics with game-environment feedback","volume":"113","author":"Weitz","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"12004","DOI":"10.1088\/1742-5468\/2010\/12\/P12004","article-title":"Group penalty on the evolution of cooperation in spatial public goods games","volume":"2010","author":"Zhang","year":"2010","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11721-008-0021-5","article-title":"Glowworm swarm optimization for simultaneous capture of multiple local optima of multimodal functions","volume":"3","author":"Krishnanand","year":"2009","journal-title":"Swarm. Intell."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/s11721-017-0153-6","article-title":"Closed-loop interactions between a shoal of zebrafish and a group of robotic fish in a circular corridor","volume":"12","author":"Bonnet","year":"2018","journal-title":"Swarm. Intell."},{"key":"ref_18","first-page":"126586","article-title":"Particle swarm intelligence and the evolution of cooperation in the spatial public goods game with punishment","volume":"412","author":"Lv","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"38005","DOI":"10.1209\/0295-5075\/116\/38005","article-title":"A social dilemma structure in diffusible public goods","volume":"116","author":"Matsuzawa","year":"2016","journal-title":"EPL"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"12985","DOI":"10.1073\/pnas.1610244113","article-title":"Networks of conforming or nonconforming individuals tend to reach satisfactory decisions","volume":"113","author":"Ramazi","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1621","DOI":"10.1016\/j.bulm.2004.03.004","article-title":"Evolutionary game dynamics in finite populations","volume":"66","author":"Taylor","year":"2004","journal-title":"Bull. Math. Biol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1038\/nature02414","article-title":"Emergence of cooperation and evolutionary stability in finite populations","volume":"428","author":"Nowak","year":"2004","journal-title":"Nature"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1934","DOI":"10.1080\/00207721.2018.1479469","article-title":"Effects of strategy switching and network topology on decision-making in multi-agent systems","volume":"49","author":"Zhang","year":"2018","journal-title":"Int. J. Syst. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"046004","DOI":"10.1088\/1748-3190\/ab8706","article-title":"Strategies to modulate zebrafish collective dynamics with a closed-loop biomimetic robotic system","volume":"15","author":"Chemtob","year":"2020","journal-title":"Bioinspir. Biomim."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"066008","DOI":"10.1088\/1748-3190\/ab3d05","article-title":"Bio-inspired robotic dog paddling: Kinematic and hydro-dynamic analysis","volume":"14","author":"Li","year":"2019","journal-title":"Bioinspir. Biomim."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1007\/s00285-005-0369-8","article-title":"Evolutionary game dynamics in a Wright-Fisher process","volume":"52","author":"Imhof","year":"2006","journal-title":"J. Math. Biol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.95.238701","article-title":"Coevolutionary dynamics: From finite to infinite populations","volume":"95","author":"Traulsen","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1016\/j.jtbi.2007.01.002","article-title":"Pairwise comparison and selection temperature in evolutionary game dynamics","volume":"246","author":"Traulsen","year":"2007","journal-title":"J. Theor. Biol."},{"key":"ref_29","first-page":"40","article-title":"Fixation Probabilities in Evolutionary Dynamics with a Wright-Fisher Process in Finite Diploid Populations","volume":"33","author":"Chen","year":"2011","journal-title":"J. Southwest Univ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"100565","DOI":"10.1016\/j.swevo.2019.100565","article-title":"Review of methodologies and tasks in swarm robotics towards standardization","volume":"50","author":"Nedjah","year":"2019","journal-title":"Swarm Evol. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"112945","DOI":"10.1016\/j.eswa.2019.112945","article-title":"A multirobot target searching method based on bat algorithm in unknown environments","volume":"141","author":"Tang","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"101103","DOI":"10.1016\/j.phycom.2020.101103","article-title":"Immune genetic algorithm based multi-UAV cooperative target search with event-triggered mechanism","volume":"41","author":"Zhou","year":"2020","journal-title":"Phys. Commun."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"106209","DOI":"10.1016\/j.knosys.2020.106209","article-title":"A constrained differential evolution algorithm to solve UAV path planning in disaster scenarios","volume":"204","author":"Yu","year":"2020","journal-title":"Knowl. Based Syst."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Alhaqbani, A., Kurdi, H., and Youcef-Toumi, K. (2020). Fish-Inspired Task Allocation Algorithm for Multiple Unmanned Aerial Vehicles in Search and Rescue Missions. Remote Sens., 13.","DOI":"10.3390\/rs13010027"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"108713","DOI":"10.1016\/j.knosys.2022.108713","article-title":"UCAV path planning for avoiding obstacles using cooperative co-evolution Spider Monkey Optimization","volume":"246","author":"Zhu","year":"2022","journal-title":"Knowl. Based Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2603","DOI":"10.1007\/s10489-018-1390-0","article-title":"A novel hybrid algorithm based on PSO and FOA for target searching in unknown environments","volume":"49","author":"Tang","year":"2019","journal-title":"Appl. Intell."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"47787","DOI":"10.1109\/ACCESS.2020.2978853","article-title":"An improved potential game theory based method for multi-UAV cooperative search","volume":"8","author":"Ni","year":"2020","journal-title":"IEEE Access"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1109\/TRO.2011.2121170","article-title":"Multirobot tree and graph exploration","volume":"27","author":"Brass","year":"2011","journal-title":"IEEE Trans. Robot."},{"key":"ref_39","unstructured":"Zeng, N., Wang, Z., Liu, W., Zhang, H., Hone, K., and Liu, X. (2020). A dynamic neighborhood-based switching particle swarm optimization algorithm. IEEE Trans. Cybern., 1\u201312."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.neucom.2020.12.065","article-title":"A competitive mechanism integrated multi-objective whale optimization algorithm with differential evolution","volume":"432","author":"Zeng","year":"2021","journal-title":"Neurocomputing"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/941\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:06:29Z","timestamp":1760137589000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/941"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,5]]},"references-count":40,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050941"],"URL":"https:\/\/doi.org\/10.3390\/sym14050941","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,5]]}}}