{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T03:47:48Z","timestamp":1778644068510,"version":"3.51.4"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T00:00:00Z","timestamp":1585699200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,4,4]],"date-time":"2020-04-04T00:00:00Z","timestamp":1585958400000},"content-version":"vor","delay-in-days":3,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100013296","name":"Max Planck Institute for Mathematics in the Sciences","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100013296","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull Math Biol"],"published-print":{"date-parts":[[2020,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Models of adaptive bet-hedging commonly adopt insights from Kelly\u2019s famous work on optimal gambling strategies and the financial value of information. In particular, such models seek evolutionary solutions that maximize long-term average growth rate of lineages, even in the face of highly stochastic growth trajectories. Here, we argue for extensive departures from the standard approach to better account for evolutionary contingencies. Crucially, we incorporate considerations of volatility minimization, motivated by interim extinction risk in finite populations, within a finite time horizon approach to growth maximization. We find that a game-theoretic competitive optimality approach best captures these additional constraints and derive the equilibria solutions under straightforward fitness payoff functions and extinction risks. We show that for both maximal growth and minimal time relative payoffs, the log-optimal strategy is a unique pure strategy symmetric equilibrium, invariant with evolutionary time horizon and robust to low extinction risks.<\/jats:p>","DOI":"10.1007\/s11538-020-00729-8","type":"journal-article","created":{"date-parts":[[2020,4,4]],"date-time":"2020-04-04T12:02:26Z","timestamp":1586001746000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Adaptive Bet-Hedging Revisited: Considerations of Risk and Time Horizon"],"prefix":"10.1007","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6370-6786","authenticated-orcid":false,"given":"Omri","family":"Tal","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5299-7023","authenticated-orcid":false,"given":"Tat Dat","family":"Tran","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,4,4]]},"reference":[{"key":"729_CR1","doi-asserted-by":"publisher","first-page":"1495","DOI":"10.1038\/s41559-017-0295-3","volume":"1","author":"B Ashby","year":"2017","unstructured":"Ashby B, Watkins E, Louren\u00e7o J, Gupta S, Foster KR (2017) Competing species leave many potential niches unfilled. Nat Ecol Evol 1:1495\u20131501","journal-title":"Nat Ecol Evol"},{"key":"729_CR2","doi-asserted-by":"crossref","unstructured":"Aucamp D (1977) An investment strategy with overshoot rebates which minimizes the time to attain a specified goal. Manag Sci 23(11)","DOI":"10.1287\/mnsc.23.11.1234"},{"issue":"6","key":"729_CR3","doi-asserted-by":"publisher","first-page":"724","DOI":"10.1287\/mnsc.34.6.724","volume":"34","author":"R Bell","year":"1988","unstructured":"Bell R, Cover TM (1988) Game-theoretic optimal portfolios. Manag Sci 34(6):724\u2013733","journal-title":"Manag Sci"},{"issue":"2","key":"729_CR4","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1287\/moor.5.2.161","volume":"5","author":"RM Bell","year":"1980","unstructured":"Bell RM, Cover TM (1980) Competitive optimality of logarithmic investment. Math Oper Res 5(2):161\u2013166","journal-title":"Math Oper Res"},{"issue":"Supplement 3","key":"729_CR5","doi-asserted-by":"publisher","first-page":"10860","DOI":"10.1073\/pnas.1400817111","volume":"111","author":"TC Bergstrom","year":"2014","unstructured":"Bergstrom TC (2014) On the evolution of hoarding, risk-taking, and wealth distribution in nonhuman and human populations. Proc Natl Acad Sci 111(Supplement 3):10860\u201310867","journal-title":"Proc Natl Acad Sci"},{"key":"729_CR6","unstructured":"Breiman L (1961) Optimal gambling systems for favorable games. In: Proceedings of the 4th Berkeley symposium on mathematical statistics and probability, vol 1, pp 63\u201368"},{"issue":"3","key":"729_CR7","doi-asserted-by":"publisher","first-page":"118","DOI":"10.3905\/joi.2016.25.3.118","volume":"25","author":"E Busseti","year":"2016","unstructured":"Busseti E, Ryu E-K, Boyd S (2016) Risk constrained kelly gambling. J Invest 25(3):118\u2013134","journal-title":"J Invest"},{"issue":"1","key":"729_CR8","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/0022-5193(66)90188-3","volume":"12","author":"D Cohen","year":"1966","unstructured":"Cohen D (1966) Optimizing reproduction in a randomly varying environment. J Theor Biol 12(1):119\u201329","journal-title":"J Theor Biol"},{"key":"729_CR9","volume-title":"Elements of information theory","author":"TM Cover","year":"2006","unstructured":"Cover TM, Thomas JA (2006) Elements of information theory, 2nd edn. Wiley, Hoboken","edition":"2"},{"issue":"2","key":"729_CR10","first-page":"219","volume":"119","author":"MC Donaldson-Matasci","year":"2010","unstructured":"Donaldson-Matasci MC, Bergstrom CT, Lachmann M (2010) The fitness value of information. Oikos (Copenhagen, Denmark) 119(2):219\u2013230","journal-title":"Oikos (Copenhagen, Denmark)"},{"key":"729_CR11","unstructured":"Garivaltis A (2018) Game-theoretic optimal portfolios in continuous time. Econ Theory Bull 1\u20139"},{"issue":"3","key":"729_CR12","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1111\/ele.12241","volume":"17","author":"JR Gremer","year":"2014","unstructured":"Gremer JR, Venable DL (2014) Bet hedging in desert winter annual plants: optimal germination strategies in a variable environment. Ecol Lett 17(3):380\u2013387","journal-title":"Ecol Lett"},{"issue":"1","key":"729_CR13","doi-asserted-by":"publisher","first-page":"517","DOI":"10.2307\/2330126","volume":"6","author":"N Hakansson","year":"1971","unstructured":"Hakansson N (1971) Capital growth and the mean-variance approach to portfolio selection. J Financ Quant Anal 6(1):517\u2013557","journal-title":"J Financ Quant Anal"},{"key":"729_CR14","doi-asserted-by":"crossref","unstructured":"Hopper KR (2018) Bet hedging in evolutionary ecology with an emphasis on insects. In: Reference module in life sciences. Elsevier","DOI":"10.1016\/B978-0-12-809633-8.90409-6"},{"issue":"1","key":"729_CR15","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1137\/080741124","volume":"1","author":"C Kardaras","year":"2010","unstructured":"Kardaras C, Platen E (2010) Minimizing the expected market time to reach a certain wealth level. SIAM J Financ Math 1(1):16\u201329","journal-title":"SIAM J Financ Math"},{"key":"729_CR16","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1002\/j.1538-7305.1956.tb03809.x","volume":"35","author":"JL Kelly Jr","year":"1956","unstructured":"Kelly JL Jr (1956) A new interpretation of information rate. Bell Syst Tech J 35:917\u2013926","journal-title":"Bell Syst Tech J"},{"issue":"4","key":"729_CR17","doi-asserted-by":"publisher","first-page":"560","DOI":"10.1016\/j.tpb.2007.08.006","volume":"72","author":"O King","year":"2007","unstructured":"King O, Masel J (2007) The evolution of bet-hedging adaptations to rare scenarios. Theor Popul Biol 72(4):560\u201375","journal-title":"Theor Popul Biol"},{"issue":"4","key":"729_CR18","doi-asserted-by":"publisher","first-page":"1807","DOI":"10.1534\/genetics.104.035352","volume":"169","author":"E Kussell","year":"2005","unstructured":"Kussell E, Kishony R, Balaban N, Leibler S (2005) Bacterial persistence: a model of survival in changing environments. Genetics 169(4):1807\u201314","journal-title":"Genetics"},{"key":"729_CR19","doi-asserted-by":"publisher","first-page":"1835","DOI":"10.1111\/j.1558-5646.2007.00170.x","volume":"61","author":"R Lande","year":"2007","unstructured":"Lande R (2007) Expected relative fitness and the adaptive topography of fluctuating selection. Evolution 61:1835\u20131846","journal-title":"Evolution"},{"key":"729_CR20","first-page":"217","volume-title":"Sexual reproduction as bet-hedging","author":"X-Y Li","year":"2017","unstructured":"Li X-Y, Lehtonen J, Kokko H (2017) Sexual reproduction as bet-hedging. Springer, Cham, pp 217\u2013234"},{"issue":"3","key":"729_CR21","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1086\/701786","volume":"193","author":"E Libby","year":"2019","unstructured":"Libby E, Ratcliff WC (2019) Shortsighted evolution constrains the efficacy of long-term bet hedging. Am Nat 193(3):409\u2013423 PMID: 30794447","journal-title":"Am Nat"},{"issue":"1","key":"729_CR22","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s10818-017-9253-z","volume":"20","author":"A Lo","year":"2017","unstructured":"Lo A, Orr H, Zhang R (2017) The growth of relative wealth and the kelly criterion. J Bioecon 20(1):49\u201367","journal-title":"J Bioecon"},{"key":"729_CR23","doi-asserted-by":"crossref","unstructured":"MacLean LC, Thorp EO, Ziemba WT (2011) Good and bad properties of the kelly criterion, chapter 39. In: World scientific handbook in financial economics series, pp 563\u2013572","DOI":"10.1142\/9789814293501_0039"},{"key":"729_CR24","doi-asserted-by":"crossref","unstructured":"Markowitz H (2006) Samuelson and investment for the long run. In: Samuelsonian economics and the twenty-first century. Oxford University Press, pp 252\u2013261","DOI":"10.1093\/acprof:oso\/9780199298839.003.0018"},{"issue":"4","key":"729_CR25","first-page":"6","volume":"4","author":"D Morgan","year":"2015","unstructured":"Morgan D (2015) An alternative mathematical interpretation and generalization of the capital growth criterion. J Financ Invest Anal 4(4):6","journal-title":"J Financ Invest Anal"},{"issue":"2","key":"729_CR26","doi-asserted-by":"publisher","first-page":"286","DOI":"10.2307\/1969529","volume":"54","author":"J Nash","year":"1951","unstructured":"Nash J (1951) Non-cooperative games. Ann Math 54(2):286\u2013295","journal-title":"Ann Math"},{"key":"729_CR27","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198815082.001.0001","volume-title":"Agents and goals in evolution","author":"S Okasha","year":"2018","unstructured":"Okasha S (2018) Agents and goals in evolution. Oxford University Press, Oxford"},{"issue":"1669","key":"729_CR28","doi-asserted-by":"publisher","first-page":"2963","DOI":"10.1098\/rspb.2009.0500","volume":"276","author":"H Olofsson","year":"2009","unstructured":"Olofsson H, Ripa J, Jonz\u00e9n N (2009) Bet-hedging as an evolutionary game: the trade-off between egg size and number. Proc Biol Sci 276(1669):2963\u20132969","journal-title":"Proc Biol Sci"},{"issue":"1","key":"729_CR29","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s10818-017-9254-y","volume":"20","author":"H Orr","year":"2017","unstructured":"Orr H (2017) Evolution, finance, and the population genetics of relative wealth. J Bioecon 20(1):29\u201348","journal-title":"J Bioecon"},{"issue":"1\u20132","key":"729_CR30","first-page":"1","volume":"2","author":"SR Proulx","year":"2001","unstructured":"Proulx SR, Day T (2001) What can invasion analyses tell us about evolution under stochasticity? Selection 2(1\u20132):1\u201315","journal-title":"Selection"},{"issue":"6","key":"729_CR31","doi-asserted-by":"publisher","first-page":"E1174","DOI":"10.1073\/pnas.1719171115","volume":"115","author":"Y Ram","year":"2018","unstructured":"Ram Y, Liberman U, Feldman MW (2018) Evolution of vertical and oblique transmission under fluctuating selection. Proc Natl Acad Sci 115(6):E1174\u2013E1183","journal-title":"Proc Natl Acad Sci"},{"issue":"6","key":"729_CR32","doi-asserted-by":"publisher","first-page":"1509","DOI":"10.1890\/0012-9658(2002)083[1509:RAKSRT]2.0.CO;2","volume":"83","author":"D Reznick","year":"2002","unstructured":"Reznick D, Bryant MJ, Bashey F (2002) r- and k-selection revisited: the role of population regulation in life-history evolution. Ecology 83(6):1509\u20131520","journal-title":"Ecology"},{"issue":"6","key":"729_CR33","doi-asserted-by":"publisher","first-page":"1124","DOI":"10.1007\/s10955-011-0166-2","volume":"142","author":"O Rivoire","year":"2011","unstructured":"Rivoire O, Leibler S (2011) The value of information for populations in varying environments. J Stat Phys 142(6):1124\u20131166","journal-title":"J Stat Phys"},{"key":"729_CR34","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1016\/j.jtbi.2017.09.020","volume":"435","author":"IN Rubin","year":"2017","unstructured":"Rubin IN, Doebeli M (2017) Rethinking the evolution of specialization: a model for the evolution of phenotypic heterogeneity. J Theor Biol 435:248\u2013264","journal-title":"J Theor Biol"},{"issue":"7","key":"729_CR35","doi-asserted-by":"publisher","first-page":"2090","DOI":"10.1287\/mnsc.2015.2228","volume":"62","author":"N Rujeerapaiboon","year":"2015","unstructured":"Rujeerapaiboon N, Kuhn D, Wiesemann W (2015) Robust growth-optimal portfolios. Manag Sci 62(7):2090\u20132109","journal-title":"Manag Sci"},{"key":"729_CR36","doi-asserted-by":"crossref","unstructured":"Rujeerapaiboon N, Ross Barmish B, Kuhn D (2018) On risk reduction in kelly betting using the conservative expected value. In: 2018 IEEE conference on decision and control (CDC), pp 5801\u20135806","DOI":"10.1109\/CDC.2018.8619186"},{"key":"729_CR37","unstructured":"Seger J, Brockmann HJ (1987) What is bet-hedging? In: Oxford surveys in evolutionary biology. Oxford University Press, Oxford, pp 182\u2013211"},{"issue":"1","key":"729_CR38","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1109\/TIT.1956.1056774","volume":"2","author":"C Shannon","year":"1956","unstructured":"Shannon C (1956) The bandwagon (edtl.). IRE Trans Inf Theory 2(1):3\u20133","journal-title":"IRE Trans Inf Theory"},{"key":"729_CR39","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1046\/j.1420-9101.2003.00530.x","volume":"16","author":"A Simons","year":"2003","unstructured":"Simons A, Johnston M (2003) Suboptimal timing of reproduction in lobelia inflata may be a conservative bet-hedging strategy. J Evol Biol 16:233\u2013243","journal-title":"J Evol Biol"},{"key":"729_CR40","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1038\/246015a0","volume":"246","author":"J Smith","year":"1973","unstructured":"Smith J, Price G (1973) The logic of animal conflict. Nature 246:15\u201318","journal-title":"Nature"},{"key":"729_CR41","doi-asserted-by":"publisher","first-page":"058101","DOI":"10.1103\/PhysRevLett.120.058101","volume":"120","author":"F Stollmeier","year":"2018","unstructured":"Stollmeier F, Nagler J (2018) Unfair and anomalous evolutionary dynamics from fluctuating payoffs. Phys Rev Lett 120:058101","journal-title":"Phys Rev Lett"},{"issue":"4","key":"729_CR42","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pcbi.1006529","volume":"15","author":"P Villa Martin","year":"2019","unstructured":"Villa Martin P, Munoz MA, Pigolotti S (2019) Bet-hedging strategies in expanding populations. PLOS Comput Biol 15(4):1\u201317","journal-title":"PLOS Comput Biol"},{"key":"729_CR43","doi-asserted-by":"crossref","unstructured":"Vince R, Zhu Q (2013) Inflection point significance for the investment size. Available at SSRN https:\/\/ssrn.com\/abstract=2230874","DOI":"10.2139\/ssrn.2230874"},{"issue":"2","key":"729_CR44","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1016\/j.jtbi.2004.11.020","volume":"234","author":"DM Wolf","year":"2005","unstructured":"Wolf DM, Vazirani VV, Arkin AP (2005) Diversity in times of adversity: probabilistic strategies in microbial survival games. J Theor Biol 234(2):227\u2013253","journal-title":"J Theor Biol"},{"issue":"2","key":"729_CR45","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/BF02515724","volume":"38","author":"J Yoshimura","year":"1996","unstructured":"Yoshimura J, Jansen VAA (1996) Evolution and population dynamics in stochastic environments. Res Popul Ecol 38(2):165\u2013182","journal-title":"Res Popul Ecol"},{"issue":"20","key":"729_CR46","doi-asserted-by":"publisher","first-page":"2611","DOI":"10.1016\/j.ecolmodel.2009.06.046","volume":"220","author":"J Yoshimura","year":"2009","unstructured":"Yoshimura J, Tanaka Y, Togashi T, Iwata S, ichi Tainaka K (2009) Mathematical equivalence of geometric mean fitness with probabilistic optimization under environmental uncertainty. Ecol Model 220(20):2611\u20132617","journal-title":"Ecol Model"}],"container-title":["Bulletin of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-020-00729-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11538-020-00729-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-020-00729-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,3]],"date-time":"2021-04-03T19:35:37Z","timestamp":1617478537000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11538-020-00729-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4]]},"references-count":46,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,4]]}},"alternative-id":["729"],"URL":"https:\/\/doi.org\/10.1007\/s11538-020-00729-8","relation":{"has-preprint":[{"id-type":"doi","id":"10.1101\/617084","asserted-by":"object"}]},"ISSN":["0092-8240","1522-9602"],"issn-type":[{"value":"0092-8240","type":"print"},{"value":"1522-9602","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4]]},"assertion":[{"value":"12 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 April 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"50"}}