{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T17:54:37Z","timestamp":1740160477778,"version":"3.37.3"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T00:00:00Z","timestamp":1643760000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T00:00:00Z","timestamp":1643760000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2022,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>During epidemics, the population is asked to socially distance, with pairs of individuals keeping two meters apart. We model this as a new optimization problem by considering a team of agents placed on the nodes of a network. Their common aim is to achieve pairwise graph distances of at least <jats:italic>D<\/jats:italic>,\u00a0 a state we call <jats:italic>socially distanced<\/jats:italic>. (If <jats:inline-formula><jats:alternatives><jats:tex-math>$$D=1,$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>D<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> they want to be at distinct nodes; if <jats:inline-formula><jats:alternatives><jats:tex-math>$$D=2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>D<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> they want to be non-adjacent.) We allow only a simple type of motion called a lazy random walk: with probability <jats:italic>p<\/jats:italic> (called the <jats:italic>laziness<\/jats:italic> parameter), they remain at their current node next period; with complementary probability <jats:inline-formula><jats:alternatives><jats:tex-math>$$1-p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mi>p<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, they move to a random adjacent node. The team seeks the common value of <jats:italic>p<\/jats:italic> which achieves social distance in the least expected time, which is the absorption time of a Markov chain. We observe that the same Markov chain, with different goals (absorbing states), models the gathering, or multi-rendezvous problem (all agents at the same node). Allowing distinct laziness for two types of agents (searchers and hider) extends the existing literature on predator\u2013prey search games to multiple searchers. We consider only special networks: line, cycle and grid.<\/jats:p>","DOI":"10.1007\/s13235-022-00427-1","type":"journal-article","created":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T14:21:55Z","timestamp":1643811715000},"page":"288-311","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Social Distancing, Gathering, Search Games: Mobile Agents on Simple Networks"],"prefix":"10.1007","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0095-4299","authenticated-orcid":false,"given":"Steve","family":"Alpern","sequence":"first","affiliation":[]},{"given":"Li","family":"Zeng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,2]]},"reference":[{"key":"427_CR1","unstructured":"Alpern S (1974) The search game with mobile hider on the circle. Differential games and control theory, 181-200"},{"key":"427_CR2","doi-asserted-by":"crossref","unstructured":"Alpern S (1995) The Rendezvous Search Problem. SIAM J Control Optim 33(3):673\u2013683","DOI":"10.1137\/S0363012993249195"},{"key":"427_CR3","doi-asserted-by":"crossref","unstructured":"Alpern S (2002) Rendezvous search: a personal perspective. Oper Res 50(5):772\u2013795","DOI":"10.1287\/opre.50.5.772.363"},{"issue":"3","key":"427_CR4","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1002\/nav.10011","volume":"49","author":"S Alpern","year":"2002","unstructured":"Alpern S (2002) Rendezvous search on labeled networks. Nav Res Logist 49(3):256\u2013274","journal-title":"Nav Res Logist"},{"issue":"2","key":"427_CR5","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/s13235-011-0009-4","volume":"1","author":"Steve Alpern","year":"2011","unstructured":"Alpern Steve (2011) A new approach to Gal\u2019s theory of search games on weakly Eulerian networks. Dynam Games Appl 1(2):209\u2013219","journal-title":"Dynam Games Appl"},{"issue":"1","key":"427_CR6","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1239\/jap\/1032374243","volume":"36","author":"S Alpern","year":"1999","unstructured":"Alpern S, Baston VJ, Essegaier S (1999) Rendezvous search on a graph. J Appl Probab 36(1):223\u2013231","journal-title":"J Appl Probab"},{"key":"427_CR7","unstructured":"Alpern S, Gal S (2003) The Theory of Search Games and Rendezvous. Kluwer"},{"issue":"1","key":"427_CR8","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1023\/A:1020891631738","volume":"53","author":"S Alpern","year":"2002","unstructured":"Alpern S, Reyniers DJ (2002) Spatial dispersion as a dynamic coordination problem. Theor Decis 53(1):29\u201359","journal-title":"Theor Decis"},{"key":"427_CR9","doi-asserted-by":"publisher","first-page":"839","DOI":"10.2307\/3214827","volume":"28","author":"EJ Anderson","year":"1990","unstructured":"Anderson EJ, Weber RR (1990) The rendezvous problem on discrete locations. J Appl Probab 28:839\u2013851","journal-title":"J Appl Probab"},{"key":"427_CR10","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1002\/(SICI)1520-6750(199904)46:3<335::AID-NAV6>3.0.CO;2-Q","volume":"46","author":"V Baston","year":"1999","unstructured":"Baston V (1999) Two rendezvous search problems on the line. Nav Res Logist 46:335\u2013340","journal-title":"Nav Res Logist"},{"issue":"1","key":"427_CR11","doi-asserted-by":"publisher","first-page":"504","DOI":"10.1016\/j.jet.2006.01.005","volume":"133","author":"A Blume","year":"2007","unstructured":"Blume A, Franco AM (2007) Decentralized learning from failure. J Econom Theor 133(1):504\u2013523","journal-title":"J Econom Theor"},{"key":"427_CR12","unstructured":"Duvocelle B, Flesch J, Staudigl M, Vermeulen D (2020) A competitive search game with a moving target. arXiv:2008.12032"},{"key":"427_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jet.2021.105293","author":"M Maryam Farboodi","year":"2021","unstructured":"Maryam Farboodi M, Jarosch G, Shimer R (2021) Internal and external effects of social distancing in a pandemic. J Econom Theor. https:\/\/doi.org\/10.1016\/j.jet.2021.105293","journal-title":"J Econom Theor"},{"key":"427_CR14","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1137\/0317009","volume":"17","author":"S Gal","year":"1979","unstructured":"Gal S (1979) Search games with mobile and immobile hiders. SIAM J Control Opt 17:99\u2013122","journal-title":"SIAM J Control Opt"},{"key":"427_CR15","volume-title":"Search Games","author":"S Gal","year":"1980","unstructured":"Gal S (1980) Search Games. Academic Press, New York"},{"key":"427_CR16","doi-asserted-by":"publisher","first-page":"974","DOI":"10.1287\/opre.47.6.974","volume":"47","author":"S Gal","year":"1999","unstructured":"Gal S (1999) Rendezvous search on the line. Oper Res 47:974\u2013976","journal-title":"Oper Res"},{"key":"427_CR17","doi-asserted-by":"crossref","unstructured":"Greenstone M, Nigam V (2020) Does Social Distancing Matter? University of Chicago, Becker Friedman Institute for Economics. Working Paper No. 2020-2620","DOI":"10.2139\/ssrn.3561244"},{"key":"427_CR18","unstructured":"Grenager T, Powers R, Shoham Y (2002) Dispersion Games: General Definitions and Some Specific Learning Results. Proc. AAAI"},{"issue":"4","key":"427_CR19","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1287\/opre.47.4.550","volume":"47","author":"JV Howard","year":"1999","unstructured":"Howard JV (1999) Rendezvous search on the interval and circle. Oper Res 47(4):550\u2013558","journal-title":"Oper Res"},{"key":"427_CR20","volume-title":"Differential Games","author":"R Isaacs","year":"1965","unstructured":"Isaacs R (1965) Differential Games. Wiley, New York"},{"key":"427_CR21","volume-title":"Introduction to Finite Mathematics","author":"J Kemeny","year":"1974","unstructured":"Kemeny J, Snell L, Thompson G (1974) Introduction to Finite Mathematics, 3rd edn. Prentice-Hall, New Jersey","edition":"3"},{"issue":"3","key":"427_CR22","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1287\/opre.45.3.357","volume":"45","author":"WS Lim","year":"1997","unstructured":"Lim WS, Alpern S, Beck A (1997) Rendezvous search on the line with more than two players. Oper Res 45(3):357\u2013364","journal-title":"Oper Res"},{"issue":"3","key":"427_CR23","doi-asserted-by":"publisher","first-page":"696","DOI":"10.2307\/3214008","volume":"23","author":"T Nakai","year":"1986","unstructured":"Nakai T (1986) A search game with one object and two searchers. J Appl Probab 23(3):696\u2013707","journal-title":"J Appl Probab"},{"key":"427_CR24","volume-title":"Geometric Games and Their Applications","author":"W Ruckle","year":"1983","unstructured":"Ruckle W (1983) Geometric Games and Their Applications. Pitman, Boston"},{"key":"427_CR25","unstructured":"Simanjuntak M (2014) A Network Dispersion Problem for Non-communicating Agents . In: Doctoral Consortium - DCAART, (ICAART 2014) 66-72"},{"key":"427_CR26","doi-asserted-by":"publisher","first-page":"m1379","DOI":"10.1136\/bmj.m1379","volume":"369","author":"A Venkatesh","year":"2020","unstructured":"Venkatesh A, Edirappuli S (2020) Social distancing in covid-19: what are the mental health implications? BMJ 369:m1379. https:\/\/doi.org\/10.1136\/bmj.m1379","journal-title":"BMJ"},{"issue":"1","key":"427_CR27","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1287\/moor.1110.0528","volume":"37","author":"R Weber","year":"2012","unstructured":"Weber R (2012) Optimal symmetric rendezvous search on three locations. Math Oper Res 37(1):111\u2013122","journal-title":"Math Oper Res"},{"key":"427_CR28","first-page":"228","volume":"13","author":"MI Zelikin","year":"1972","unstructured":"Zelikin MI (1972) On a differential game with incomplete information. Soviet Math. Dokl. 13:228\u2013231","journal-title":"Soviet Math. Dokl."}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-022-00427-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13235-022-00427-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-022-00427-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,18]],"date-time":"2022-03-18T17:31:14Z","timestamp":1647624674000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13235-022-00427-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,2]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["427"],"URL":"https:\/\/doi.org\/10.1007\/s13235-022-00427-1","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"type":"print","value":"2153-0785"},{"type":"electronic","value":"2153-0793"}],"subject":[],"published":{"date-parts":[[2022,2,2]]},"assertion":[{"value":"3 January 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 February 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}