{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:08:52Z","timestamp":1760238532876,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,8,20]],"date-time":"2020-08-20T00:00:00Z","timestamp":1597881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>In the much-studied Centipede Game, which resembles the Iterated Prisoners\u2019 Dilemma, two players successively choose between (1) cooperating, by continuing play, or (2) defecting and terminating play. The subgame-perfect Nash equilibrium implies that play terminates on the first move, even though continuing play can benefit both players\u2014but not if the rival defects immediately, which it has an incentive to do. We show that, without changing the structure of the game, interchanging the payoffs of the two players provides each with an incentive to cooperate whenever its turn comes up. The Nash equilibrium in the transformed Centipede Game, called the Reciprocity Game, is unique\u2014unlike the Centipede Game, wherein there are several Nash equilibria. The Reciprocity Game can be implemented noncooperatively by adding, at the start of the Centipede Game, a move to exchange payoffs, which it is rational for the players to choose. What this interchange signifies, and its application to transforming an arms race into an arms-control treaty, are discussed.<\/jats:p>","DOI":"10.3390\/g11030035","type":"journal-article","created":{"date-parts":[[2020,8,20]],"date-time":"2020-08-20T21:13:42Z","timestamp":1597958022000},"page":"35","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Note on Stabilizing Cooperation in the Centipede Game"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3650-0392","authenticated-orcid":false,"given":"Steven J.","family":"Brams","sequence":"first","affiliation":[{"name":"Department of Politics, New York University, 19 W. 4th Street, New York, NY 10012, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D. Marc","family":"Kilgour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/0022-0531(81)90018-1","article-title":"Games of Perfect Information; Predatory Pricing and the Chain-Store Paradox","volume":"23","author":"Rosenthal","year":"1981","journal-title":"J. Econ. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/j.geb.2020.01.007","article-title":"Non-Equilibrium Play in Centipede Games","volume":"120","author":"Iriberri","year":"2020","journal-title":"Games Econ. Behav."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"803","DOI":"10.2307\/2951567","article-title":"An Experimental Study of the Centipede Game","volume":"60","author":"McKelvey","year":"1992","journal-title":"Econometrica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1006\/jmps.1998.1225","article-title":"Experimental Results on the Centipede Game in Normal Form: An Investigation on Learning","volume":"42","author":"agel","year":"1998","journal-title":"J. Math. Psychol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1111\/j.1540-4560.1987.tb00252.x","article-title":"Beyond Deterrence","volume":"43","author":"Lebow","year":"1987","journal-title":"J. Soc. Issues"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1619","DOI":"10.1257\/aer.99.4.1619","article-title":"Field Centipedes","volume":"99","author":"Volij","year":"2009","journal-title":"Am. Econ. Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1257\/aer.101.2.975","article-title":"Checkmate: Exploring Backward Induction among Chess Players","volume":"101","author":"Levitt","year":"2011","journal-title":"Am. Econ. Rev."},{"key":"ref_8","unstructured":"Axelrod, R. A. (1984). The Evolution of Cooperation, Basic Books."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1093\/qje\/qjx033","article-title":"Cooperation in the Finitely Repeated Prisoner\u2019s Dilemma","volume":"133","author":"Embrey","year":"2018","journal-title":"Q. J. Econ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1257\/jel.20160980","article-title":"On the Determinants of Cooperation in Infinitely Repeated Games: A Survey","volume":"56","year":"2018","journal-title":"J. Econ. Lit."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"10409","DOI":"10.1073\/pnas.1206569109","article-title":"Iterated Prisoner\u2019s Dilemma Contains Strategies that Dominate an Evolutionary Opponent","volume":"109","author":"Press","year":"2012","journal-title":"Proc. Acad. Natl. Sci. USA"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/11\/3\/35\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:03:54Z","timestamp":1760177034000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/11\/3\/35"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,20]]},"references-count":11,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["g11030035"],"URL":"https:\/\/doi.org\/10.3390\/g11030035","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2020,8,20]]}}}