TheoreticalPopulation Biology 59, 3-6 ( 2001 ) doi:10.1006/tpbi.2000.1501,available online at http://www.idealibrary.comon lllE.b-l@ TPB William D. Hamilton's Work in Evolutionary Game Theory K. Sigmund Institutefor Mathematics,Uniaersity of Vienna,Vienna A-1090, Austria ReceivedAugust 23, 2000 William D. Hamilton's impact on contemporaryviews At that time, there had been severalattempts to use of evolution has beennothing short of revolutionary.His game theory in evolutionary biology, most notably a contributions to our reasoningon kin selection,genomic paper by Lewontin (1960)which interpretedlife-history conflicts, parasitism, and costs of sex dominate these traits as strategiesand reproductivesuccess as a payoff fields. He also had an important, if somewhat less function and which showedthat the optimal solution of obvious, influence on evolutionary game theory. This a certain game "against nature" was a mixed strategy. field, which is now well-establishedboth in biology and Another forerunnerwas Verner (1965),who treated sex in economics,is usually (and rightly) attributed to John ratios as strategiesand aimed to show that, in the Maynard Smith: the canonicalreferences are a briefjoint presenceof fluctuations in the overall sex ratio, an paper with George Price (Maynard Smith and Price, individual's 1:1sex ratio wasoptimal. Theseearly papers 1973), and the book "Evolution and the Theory of did not yet apply game theory to model competition Games" (Maynard Smith, 1982).But W. D. Hamilton within a population and did not evenimplicitly consider played an important, and indeedpioneering, role in the population dynamics. developmentof this subject. Decadesearlier, both oftheseaspects had beenused by In the early sixties,game theory was looking back on R. A. Fisher, Bill Hamilton's "hero of twentieth century more than a decadeof headyprogress. The seminalbook evolution" (Hamilton, 1996).Fisher's argument on sex- on "The Theory of Games and Economic Behaviour" ratio selection(Fisher, 1930)was later couchedin terms (von Neumann and Morgenstern, 1944) had inspired which made in appearas a paragon of evolutionary game someof the best mathematiciansof the time. It had also theory (see,e.g., Maynard Smith 1982).But when Fisher fired public imagination and been subject to some of the first conceivedhis argument, he knew nothing about hype later lavishedon catastrophetheory and chaos.The gametheory: in fact, in (Fisher, 1934),he proved from responseof economists,to whom gametheory was offered scratcha result about the optimal play in a defunct card as a tool, turned out to be somewhatmore guarded.Bill game,obviously unaware of John von Neumann'smini- Hamilton had encountered the von Neumann and max theoremfrom 1930.Robert MacArthur (1965)used Morgensternbook "in idle reading" (as he later wrote, an approachsimilar to Fisher'sin discussingsex ratios, Hamilton 1996)while he was a student of geneticsand again without using conceptsfrom game theory. returned to it "almost as a leisure distraction" when he Bill Hamilton applied game theory to reveal the had landed at the London School of Economicsfor his essenceand the limitations of Fisher'sargument and to doctoral thesis (officially in human demography).He explain the widespreadoccurrence of extraordinary sex later wrote that "the'idea of fitness being treated as ratios. This article, which many (including himself) con- equivalent to "payofl" in a biological version of von sider his best paper, appearedin 1967in Science,butit Neumann'sgame theory had crossedmy mind as soonas was basedon ideashe had already developedby 1963. 'lrnexpectedly I read his earliestaccount." Hamilton had also read the Hamilton stressedthe close similarity to standard textbook of the time (Luce and Raiffa 1957), certain types of situations consideredin the theory of which included the concept of the Nash equilibrium and games"(Hamilton, 1967),frrst "in the senseof a play by a discussionof the Prisoner'sDilemma. the individual againstthe population" (what nowadaysis 0040-5809/01s35.00 CopyrighrO 2001 by Acäde.nicP|s All rightsof roproductionin any fm reserred. 4 K. Sigmund called "playing the field") and then in the pairwisecom- much more explicit fashion (Maynard Smith, 1974) and petition of two parasitoidfemales laying their eggsin the in a context(pairwise conflicts) which wascloser to usual same host ("a refined version fof the model] which is examplesin gametheory. In Hamilton's 1967paper, the very realisticallygameJike"). Technically, the main tool unbeatability concept was aknost hidden behind too is that of an "unbeatable" strategy.Hamilton did not many other breakthrough ideas on intragenomic dehneit explicitly-in fact, he only spokeof sexratios as conflicts,levels-of-selection, and the cost of sex,all piling being unbeatable and always in the context of a specific on eachother. geneticor phenotypic model. As Hamilton (1996) later In later papers (Hamilton and May, 1977; Comins stated the notion required that no alternative strategy, et al., 1980),Hamilton used the concept of an ESS and "no matter what starting frequency in a mixture, would praisedits "combination of simplicity and generality''in be able to increase." helping to define o'final" strategies.But it should be A few years later, John Maynard Smith introduced the stressedthat there exists, by now, a confusing variety definition of an evolutionarily stablestrategy (or ESS), of versions of population stability which had all too stressingthat it had "beenderived in part from the theory often been illegitimately labeled ESS (for discussions, of games, and in part from the work of MacArthur and seeLessard, 1990; Eshel, 1996).Furthennore, the corre- Hamilton on the evolution of the sex ratio" (Maynard sponding equilibria need not always be the ultimate out- Smith and Price, 1973).The concept met immediately come of evolution (Eshel and Motro, 1981; Nowak, with resounding success.In contrast to the unbeatable 1990). strategy, it only required that an alternative strategy Hamilton was clearly less interestedin disentangling would not be ableto increasewhen its frequencywaslow. strategic and genetic viewpoints of great generality than In fact, it would be selectedagainst: a rare mutant in studying concrete examplesof applications of game invading a resident monomorphic population of ESS theory to socialbehavior. It was, as he wrote (Hamilton players would be eliminated under the influence of 1996),"a completesurprise to find an exactevolutionary natural selection.This notion was strongerthan that of a analogueto one of game theory's most famous puzzles Nash equilibrium, but weaker than that of an unbeatable coming to light right in the midst of my own work on strategy (and of a strict Nash equilibrium). Roughly population genetical dynamics": this is the Prisoner's speaking, two strategies exposed to selection can Dilemma game.He first discussedit in the proceedings (a) either coexist in equilibrium in the population, or volumeof a symposiumwhich took placein Washington, (b) one can dominate the other in the senseof always DC, in 1969(Hamilton, 1971). eliminating it from any mixture, or (c) they can mutually In this paper, Hamilton still had to struggle with the resistinvasion by the other.In case(c), that ofbistability, fact that game theory "presupposesbeings to think, and both strategieswould be ESSs,but not unbeatable;in potentially, to communicate," which implies that "in case(b), the dominating strategy is unbeatable.In a non-human biology, at least as regards preconsidered further case(d) of selectiveneutrality, both strategies strategies,its relevancemust be limited." The rationality have equal successagainst each other and against axiom of classical game theory was looming as an themselves.In many contexts,this caseis unlikely, but obstacle.But the Prisoner's Dilemma problem becomes overlooking it causes confusion. For instance, both actually easierwithin the evolutionary context-although Hamilton (Axelrod and Hamilton, 1981)and Maynard Hamilton adds:"I am doubtful whetherthe findingsfrom Smith ( 1982)published proofs of the purported "evolu- natural selectionthrow any light on the problem ofhow tionary stability" of the Tit for Tat strategywhich, in fact, it is rational to act." only showedthat it was a Nash equilibrium. Hamilton later referred to this proceedingspaper as The notion of ESSis particularly appropriatefor large, "my first intellectual wild oat" and warned the reader of well-mixed populations, for in that case any invading its shortcomings-including, in his own words, "lack of mutant is obviously rare. In highly structured popula- focusand lack of anything new to say" (Hamilton, 1996). tions, this is not necessarilythe eäse,and it could well be Lack of focus there may be; indeed, the paper reads, that Hamilton conceived the stronger condition of from its opening Nietzsche quotation onward, like an unbeatability becausehe was usedto thinking of viscous unbridled steeplechasethrough Bill Hamilton's brain. populations. But the main reason why the concept of But it darts over a vast amount of new things to say, with evolutionary stability spread faster than that of utterly original ideas taken up at full gallop and tossed unbeatability was probably causedless by its greater offat the next turn, including, in particular, anticipations generality than by the fact that it was introduced in a of seminalpapersbyJohn Maynard Smith, E. O. Wilson, William Hamilton's Evolutionary Game Theory 5 and Robert Trivers that weresoon to burst