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Introduction: 2 player games", "Title", TextAlignment->Left, FontFamily->"Arial", FontSize->40, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.500008], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell["Purpose :", "Section", FontSize->25], Cell[TextData[{ "In this first section we will use the simple two-player game as a basis to \ introduce Frank's methods. This allows us to demonstrate how his approach is \ related to game theory and kin selection. A relevant paper that you could \ read beforehand is Wenseleers & Ratnieks (submitted) ", StyleBox["\"Towards a general theory of conflict: the sociobiology of \ mendelian segregation\"", FontSlant->"Italic"], "." }], "Text", FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell["Biological examples :", "Section", FontSize->25], Cell["\<\ As biological examples of two-player games you can think of two males that \ may either fight (play \"hawk\") or cooperate (play \"dove\") when they \ compete for mates, of two ant queens that may either cooperate or fight \ during nest founding or of 2 parasitoid larvae that may either kill or \ tolerate each other within their host (Godfray 1987). Note that the players \ can be of any kind, e.g. it could also be two groups or nations, or two \ homologous chromosomes at a locus that compete for gamete production \ (Wenseleers & Ratnieks, submitted). That is, concepts of cooperation and \ conflict apply at any level. Below we will use two simple games to introduce the method; in the Questions \ & Answers section you can find applications to related problems (e.g. \ reciprocal altruism and siblicide in parasitoid wasps).\ \>", "Text", FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell["Assumptions :", "Section", FontSize->25], Cell[TextData[{ "We first assume that any individual (or group, or cell, or chromosome) can \ adaptively adjust its probability of cooperating. This model derives what is \ referred to as a ", StyleBox["mixed ESS", FontSlant->"Italic"], ". In genetic terms, it assumes that there are an infinite number of \ alleles each specifying a particular probability of cooperating, and that the \ allele that fixates near the ESS cannot be invaded by any other allele that \ specifies a slightly different probability of cooperating. The mixed ESS \ therefore corresponds to a situation with ", StyleBox["weak selection", FontSlant->"Italic"], " (population geneticists often use the term ", StyleBox["'low penetrance'", FontSlant->"Italic"], ", i.e. a situation in which any invading allele has very small selective \ effects).\nIn a second model we assume that any player is either of the \ cooperative or the defecting type and that the equilibrium will then consist \ of a stable mix between the two. This situation is referred to as a ", StyleBox["pure or discrete ESS ", FontSlant->"Italic"], "model. Population geneticists call this a ", StyleBox["'high penetrance'", FontSlant->"Italic"], " model, because alleles in this case have large effects, and selection can \ no longer be assumed to be weak. It is also for this case that Hamilton's \ rule has been shown to fail (Bulmer 1994, p. 193-194).\nIn both models we \ assume that both players are members of the same species and that payoffs are \ symmetrical. That is, that the payoffs to player 1 when it fights player 2 \ are the same as the payoffs to player 2 when it fights player 1. \nIn a third \ model it is shown how asymmetrical payoffs can be accomodated for and how the \ method can be extended to illuminate interactions between different species." }], "Text", FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["1.1 Mixed ESS model", "Section", FontSize->25], Cell["\<\ Parameters: w1 = direct fitness of player 1 w2 = direct fitness of player 2 y1 = phenotype of player 1 (probability of cooperating) y2 = phenotype of player 2 (probability of cooperating) g1 = genotype or breeding value (predictor of phenotype) of player 1 g2 = genotype/breeding value of player 2 B = benefit of having a cooperative partner C = cost of cooperating r = relatedness between the two players\ \>", "Text", FontSize->18], Cell["\<\ Let us consider the following payoff matrix: \t \t\t PLAYER 2 \t\t\t Defect Cooperate PLAYER 1 Defect\t0 B \t Cooperate -C B-C (payoffs are to player 1) As we will see, it is this payoff matrix that is implicit in Hamilton's rule. \ A specific property of this matrix is that the behaviour of each player is \ taken to have additive effects on the focal player's reproduction. This \ assumption, as we will see, is often violated.\ \>", "Text", FontSize->16], Cell[TextData[{ StyleBox["Frank's approach for calculating when a particular behaviour is \ favoured has 3 steps :", FontWeight->"Bold"], "\n1. Write an individual's direct fitness w as a function of its own \ behaviour and the behaviour of social inter-actants. Direct fitness is \ sometimes called 'neighbour modulated fitness'.\n2. Calculate when having the \ gene for the behaviour has positive effects on your own reproduction. \ Formally, this is done by checking when the regression (or the analytical \ equivalent - the total derivative) of individual fitness w on individual \ genotype g (or breeding value, as in quantitative genetics) is positive. If \ phenotypes of social interactants influence the focal player's fitness, then \ these will enter the equation as indirect effects. The rationale for this \ method stems from the Price equation (Frank 1997, 1998).\n3. The obtained \ equation says when a particular behaviour is favoured; setting the equation \ to zero allows the derivation of the equilibrium, i.e. the ESS." }], "Text", FontSize->16], Cell[TextData[{ StyleBox["A note on direct and inclusive fitness\n", FontWeight->"Bold"], "For now, we will assume that all individuals express the altruistic (or \ other) trait under study. The type of social selection that then occurs is \ sometimes referred to as \[OpenCurlyQuote]correlated selection\ \[CloseCurlyQuote] (Frank 1997, 1998) because the relatedness coefficient \ that arises in this context measures phenotypic correlation in behaviour \ between social interactants (also see Queller 1984). Problems of this type \ are most intuitively looked at from a neighbour modulated or direct fitness \ angle. In Section 3 we will look at a problem where the class of individuals \ that expresses the trait influences the direct fitness of a passive class of \ relatives. In that case, an inclusive fitness perspective is more \ appropriate, and the type of selection that occurs here is genuine \ \[OpenCurlyQuote]kin selection\[CloseCurlyQuote] (Frank 1997, 1998). In fact, \ most of the classical problems in social evolution are of this type, e.g. \ worker policing where workers eat nephews at the benefit of brothers; workers \ express the trait, and nephews and brothers are affected, but do not \ themselves have any phenotype. The passive class does not express the gene \ under study but may transmit copies of the actor\[CloseCurlyQuote]s genes." }], "Text", FontSize->16], Cell["\<\ Step 1: Write the individual reproductive success w1 and w2 of both players \ as a function of the behaviour of each\ \>", "Definition", FontSize->16], Cell["\<\ From the matrix above, the direct fitness of player 1 is given by \ \>", "Text", FontSize->16], Cell[BoxData[ \(\(\(w1 = \(-C\)*y1 + B*y2;\)\(\ \)\( (*\ \(+\ a\)\ constant, \ but\ constants\ can\ be\ left\ out\ *) \)\)\)], "Input"], Cell["\<\ Similarly, because we assume symmetrical payoffs, the direct fitness of \ player 2 is .\ \>", "Text", FontSize->16], Cell[BoxData[ \(\(w2 = \(-C\)*y2 + B*y1;\)\)], "Input"], Cell["\<\ Step 2: Analyse when the regression of player 1's fitness on its genotype is \ positive\ \>", "Definition", FontSize->16], Cell["\<\ The statistical regression of player 1's fitness (w1) on its genotype g1 is a \ bit awkward to calculate, but can more easily be approximated by its \ analytical equivalent - the total derivative (Frank 1998). This is exact when \ the genetic variants that occur in the population have almost identical \ phenotypes, and is therefore correct for the mixed and continuous strategy \ case (Frank 1998, Wenseleers & Ratnieks submitted). This works as follows : [note: Dt calculates total derivatives, D calculates partial derivatives, /. \ means Replace by]\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(dw1dg1[y_] = D[w1, y1]*1 + \(\(D[w1, y2]\)\(*\)\(r\)\(\ \ \ \ \)\)\)], "Input"], Cell[BoxData[ \(\(-C\) + B\ r\)], "Output"] }, Open ]], Cell[TextData[{ "where D[w1,y1] is the partial effect of your own behaviour (altruism) on \ your own reproduction, i.e. the personal cost of altruism, and D[w1,y2] is \ the benefit of the other player's altruism (because of the symmetrical \ payoffs, D[w1,y2] also equals D[w2,y1], the benefit of player 1 to player 2). \ The 1 and ", StyleBox["r ", FontSlant->"Italic"], "arise as total derivatives of own behaviour on own genotype (by definition \ 1), and as the slope of partner phenotype on individual genotype, a common \ measure of the relatedness coefficient (Orlove & Wood 1978). Note, however, \ that r = Dt[y2,g1] may be positive either due to common descent (Dt[g2,g1]>0) \ or due to purely phenotypic correlation (D[y2,y1]>0), arising e.g. from \ common environment, or prior knowledge of how player 2 is going to behave. ", "Relatedness in this case measures information about the behavior of social \ interactants. For this reason, this type of selection is sometimes thought of \ as being distinct from kin selection. Frank (1997, 1998) calls it 'correlated \ selection', and in concept it has close parallels to the idea of correlated \ equilibrium and Bayesian rationality in economic game theory (Aumann 1987; \ Skyrms 1994, 1996; their conclusion is that the Prisoner's dilemma may be \ overcome if social interactants have better than random information about \ what each will do, e.g. based on reputation, Nowak et al. 2000)." }], "Text", FontSize->14], Cell["\<\ Or in a single step, using a total derivative and replacement rules :\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(dw1dg1[ y_] = \(\(Dt[w1, g1]\)\(/.\)\({Dt[y2, g1] -> r, Dt[y1, g1] -> 1, Dt[B, g1] \[Rule] 0, Dt[C, g1] \[Rule] 0, y1 \[Rule] y, y2 \[Rule] y}\)\(\ \ \ \ \)\)\)], "Input"], Cell[BoxData[ \(\(-C\) + B\ r\)], "Output"] }, Open ]], Cell["Step 3: Solve to yield the ESS", "Definition", FontSize->16], Cell["\<\ The equation above means that when B.r > C, altruism is selected for - a \ simple rederivation of Hamilton's rule (Hamilton 1964, 1975). In this case, \ once this rule is fullfilled, it will favour ever greater degrees of \ altruism, until all players in the population will cooperate with probability \ 1. In many other cases, however, the population will tend towards an ESS with \ each individual being favoured to be cooperative with a particular \ probability.\ \>", "Text", FontSize->14], Cell["\<\ A good example is the hawk-dove game, characterised by the following payoff \ matrix : \t \t\t PLAYER 2 \t\t\t Dove Hawk PLAYER 1 Dove\tB/2 0 \t Hawk \tB \t(B-C)/2 which can be renormalised (x 2 -B) to\ \>", "Text", FontSize->14], Cell[TextData[{ "\t \t\t PLAYER 2\n\t\t\t Dove Hawk\nPLAYER 1 \ Dove\t0 \t-B\n\t Hawk \tB \t-C\n", StyleBox["(payoffs are to player 1)", FontSize->15] }], "Text"], Cell[TextData[{ StyleBox["The idea of this game is that when two individuals compete for a \ limited resource (e.g. food, a nesting site, etc\[Ellipsis]), they may either \ fight over it (play 'hawk') to gain a greater than average share, or behave \ peacefully (play 'dove'), and take no more than their fair share (Maynard \ Smith 1982). Hawk has an advantage in a population of all-doves (benefit=", FontSize->14], StyleBox["B", FontSize->14, FontSlant->"Italic"], StyleBox[") but in a population of all-hawks, it will cause resources to be \ wasted on fighting (fighting cost=", FontSize->14], StyleBox["C", FontSize->14, FontSlant->"Italic"], StyleBox[").\nThe fitness of player 1 is now given as ", FontSize->14] }], "Text"], Cell[BoxData[ \(\(w1 = \(-C\)*y1*y2 + B*y1*\((1 - y2)\) - B*\((1 - y1)\)*y2;\)\)], "Input"], Cell["\<\ if y1 and y2 are the probabilities with which player 1 and player 2 play \ hawk.\ \>", "Text", FontSize->14], Cell["Or after rearrangement", "Text", FontSize->14], Cell[BoxData[ \(\(w1 = B*y1 - B*y2 - C*y1*y2;\)\)], "Input"], Cell["\<\ In general, simplified equations can be obtained using the Simplify[] or \ FullSimplify[] commands\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[w1]\)], "Input"], Cell[BoxData[ \(B\ y1 - \((B + C\ y1)\)\ y2\)], "Output"] }, Open ]], Cell["\<\ However, unlike in the previous game, the consequences of playing hawk or \ dove now depend on what the opponent does - the central idea in all of game \ theory. Formally, the personal benefit of playing hawk, D[w1,y1] is now \ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(D[w1, y1]\)], "Input"], Cell[BoxData[ \(B - C\ y2\)], "Output"] }, Open ]], Cell["\<\ That is, unlike in Hamilton's rule it is no longer a constant. Similarly, the \ effect of the opponent's behaviour on player 1 is\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(D[w1, y2]\)], "Input"], Cell[BoxData[ \(\(-B\) - C\ y1\)], "Output"] }, Open ]], Cell["\<\ The non-additivity or non-constancy of costs and benefits will in this case \ cause the population to evolve towards a mixed ESS. This ESS can be \ calculated as follows : Playing hawk with a higher probability is favoured when \ Dt[w1,g1]=D[w1,y1].1+D[w1,y2].r > 0. Thus the direction of selection for hawk \ behaviour is given by \ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(selectionforhawk = \((D[w1, y1] + D[w1, y2]*r)\) /. {y1 \[Rule] y, y2 \[Rule] y}\)], "Input"], Cell[BoxData[ \(B - C\ y + r\ \((\(-B\) - C\ y)\)\)], "Output"] }, Open ]], Cell["\<\ where y1 and y2 have been set to the population average phenotype y, because \ we allow phenotypes to deviate by only a small amount from the population \ average. The mixed ESS is reached when this equation is zero, which is for\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(mESS[B_, C_, r_] = \(\(\(FullSimplify[ Solve[selectionforhawk == 0, y]]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(\(B - B\ r\)\/\(C + C\ r\)\)], "Output"] }, Open ]], Cell[TextData[{ "Note: the [[1]][[1]][[2]] just serve to get rid of some unneeded brackets. \ The normal output of Solve[...] would be ", Cell[BoxData[ \({{y \[Rule] \(B - B\ r\)\/\(C + C\ r\)}}\)]], ". The [[1]] means take the first element of the list {}, the second [[1]] \ does this one more time, and the [[2]] then takes the second part of the \ remaining ", Cell[BoxData[ \(y \[Rule] \(B - B\ r\)\/\(C + C\ r\)\)]], ", i.e. ", Cell[BoxData[ \(\(B - B\ r\)\/\(C + C\ r\)\)]], "." }], "Text", FontSize->14], Cell["\<\ Setting y to zero (all-dove=ancestral situation), it can be checked that in \ an all-dove population, hawk is always positively selected for\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(selectionforhawk /. {y \[Rule] 0}\)], "Input"], Cell[BoxData[ \(B - B\ r\)], "Output"] }, Open ]], Cell["\<\ (since B-B.r is always >0). Setting y to 1 (all-hawk=ancestral situation), it \ can be checked that dove is favoured in an all-hawk population when \ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\((selectionforhawk /. {y \[Rule] 1})\) < 0\)], "Input"], Cell[BoxData[ \(B - C + \((\(-B\) - C)\)\ r < 0\)], "Output"] }, Open ]], Cell["\<\ i.e. when C(1+r)>B.(1-r); when relatedness is absent this is when C > B, i.e. \ when the fighting cost exceeds the benefit of sole use of a resource.\ \>", "Text", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["1.2 Pure ESS Model", "Section", FontSize->25], Cell["\<\ The mixed strategy assumes that a player can adopt a probabilistic phenotype, \ randomly expressing one strategy or another. Alternatively, the genotype may \ fix a player's strategy, but different genotypes may express different \ strategies. In the mixed strategy case, individuals are mixed, but the \ population is pure (near the ESS); in the pure strategy case, individuals are \ pure, but the population is mixed. Pure strategy games do not generally \ predict the same invasion criteria and equilibrium conditions as their mixed \ strategy counterparts (Grafen 1979, Queller 1984, Wenseleers & Ratnieks \ submitted). This is a major source of confusion, because most population \ genetic models analyse the pure strategy case, whereas most inclusive \ fitness/optimisation models analyse the mixed strategy case. In any case, one \ should always specify which of the two that is modelled.\ \>", "Text", FontSize->14], Cell["\<\ Analysis of the pure strategy case is more difficult than analysis of the \ mixed strategy case because the assumption of little genetic variance no \ longer holds. The consequence is that the regression of fitness on genotype \ can no longer be approximated using a total derivative. One way around this \ is to calculate regressions directly - below it is shown how to do this using \ the equilibration method of Frank (1998, p. 91-92). Wenseleers & Ratnieks \ (submitted) show how this regression can be partitioned into social \ components to yield a Hamilton's rule that is correct for both the mixed and \ pure strategy case (Wenseleers & Ratnieks submitted). \ \>", "Text", FontSize->14], Cell["\<\ The method again consists of three steps; the hawk-dove game is used as an \ example. \ \>", "Text", FontSize->14], Cell["\<\ Step 1: Write individual reproductive success of both players as a function \ of the behaviour of each\ \>", "Definition", FontSize->16], Cell["\<\ From the hawk-dove matrix above, the direct fitness of player 1 is given by \ \ \>", "Text", FontSize->14], Cell[BoxData[ \(\(\(w1 = B*y1 - B*y2 - C*y1*y2;\)\(\ \)\)\)], "Input"], Cell["And similarly, the direct fitness of player 2 is", "Text", FontSize->14], Cell[BoxData[ \(\(\(w2 = B*y2 - B*y1 - C*y1*y2;\)\(\ \)\)\)], "Input"], Cell["\<\ Step 2: Analyse when the regression of player 1's fitness on its genotype is \ positive\ \>", "Definition", 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Relatedness, as before, is the regression of partner phenotype on actor \ genotype. 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0`00002T600H00000:@H004000030000D?@B00410@4HmA80fh3kMhR^n7OoooooI7H02000000U0000 300000<0000X0000300000@0000W0000600000@000000000oooo0000000U0000300000@0000V0000 700000D00000000000000000000000009@0000`0000500008P0000`0003ooooo8P0000`0003ooooo 9@0000`0000100009@0000`0000700209@0000`000000020<00000`0000?00209@0000`0000=0020 B`000100000000001@0002P0000<00000@0002P0000<00000P0002P0000<00000`0002P0000<0000 100002P0000<00001@0000h0000D000000000100000D0000 \>"], "Graphics", ImageSize->{213, 27.0625}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "By consequence, the expected behaviour of player 2 y2 is ", StyleBox["p", FontSlant->"Italic"], "-", StyleBox["r.p", FontSlant->"Italic"], " when player 1 is of the dove type (g1=y1=0), but ", StyleBox["p+r.(1-p)", FontSlant->"Italic"], " when it is of the hawk type (g1=y1=1). This means that with positive \ relatednes (r > 0), player 1 will not just play against the population at \ random, but will more likely play against its own type (i.e. it will play \ with a probability less than ", StyleBox["p", FontSlant->"Italic"], " against hawks if it is itself a dove, and with a probability higher than \ ", StyleBox["p", FontSlant->"Italic"], " against hawks if it is itself a hawk). \nWith these identities we can now \ calculate the expected fitness of player 1 given that it is of the hawk or \ the dove type :" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(wH = w1 /. {y1 \[Rule] 1, y2 \[Rule] p + r*\((1 - p)\)}\)], "Input"], Cell[BoxData[ \(B - B\ \((p + \((1 - p)\)\ r)\) - C\ \((p + \((1 - p)\)\ r)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(wD = w1 /. {y1 \[Rule] 0, y2 \[Rule] p - r*p}\)], "Input"], Cell[BoxData[ \(\(-B\)\ \((p - p\ r)\)\)], "Output"] }, Open ]], Cell["For hawk to be favoured over dove, wH>wD. ", "Text", FontSize->14], Cell[BoxData[ \(\(selectionforhawk = wH - wD;\)\)], "Input"], Cell["Step 3: Solve to yield the ESS", "Definition", FontSize->16], Cell["A pure ESS is reached when wH=wD, which is for p=", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(pESS[B_, C_, r_] = \(\(\(FullSimplify[ Solve[wH \[Equal] wD, p]]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(1 + B\/C + 1\/\(\(-1\) + r\)\)], "Output"] }, Open ]], Cell["\<\ As one can see, for nonzero relatedness, the predicted equilibrium is \ different from, and tends to be less hawkish, than the corresponding mixed \ ESS (y*=(B/C).(1-r)/(1+r), see secton 1.1). The reasons for this are \ discussed more fully in Grafen (1979), Queller (1984) and Wenseleers & \ Ratnieks (submitted). Setting p to zero (ancestral all-dove population), it can also be checked \ that hawk will spread in an all-dove population when \ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\((selectionforhawk /. {p \[Rule] 0})\) > 0\)], "Input"], Cell[BoxData[ \(B - B\ r - C\ r > 0\)], "Output"] }, Open ]], Cell["\<\ (this is more stringent than for the mixed strategy case, where hawk would \ always spread). Setting p to 1 (ancestral all-hawk population), it can be checked that dove \ is favoured in an all-hawk population when \ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\((selectionforhawk /. {p \[Rule] 1})\) > 0\)], "Input"], Cell[BoxData[ \(\(-C\) + B\ \((1 - r)\) > 0\)], "Output"] }, Open ]], Cell["\<\ Note, however, that at the pure ESS, individual fitnesses are not at a local \ maximum with respect to variations in the probability that a particular \ individual with play hawk or dove. This suggests that if we release the \ constraint that individuals express only pure strategies, and allow \ individuals to express mixed strategies, the population will evolve from the \ pure to the mixed equilibrium (Frank 1998, p. 42 and 92). Therefore, if only \ enough alleles are played out against each other over the course of \ evolution, the final endpoint will correspond to the mixed ESS of \ optimisation models (Eshel 1996). Thus the short-term outcome of selection \ may be a pure ESS, but in the long term it seems likely that the population \ will tend towards the mixed strategy equilibrium, which is indepdentent of \ the detailed genetics (e.g. dominance assumptions etc\[Ellipsis]). One could \ also consider the mixed strategy analysis a first order approximation of the \ pure strategy case.\ \>", "Text", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 1.3 Mixed ESS Model with Asymmetrical Payoffs (battle of the sexes game)\ \>", "Section", FontSize->25], Cell[TextData[{ "Above we assumed that payoffs are symmetrical. In many cases, however, the \ two interactants do not have the same behavioural options. Think e.g. about \ hosts and parasites where the parasite can have one of two virulence genes \ and the host one of two resistance genes. In that case, payoffs will be \ asymmetrical. \nOr in case of ", StyleBox["Polistes", FontSlant->"Italic"], " wasps, a resident wasp might either accept or reject a joining wasp, and \ the joining wasp might either cooperate in nest building or fight to obtain a \ greater share of the reproduction. Here, owner/intruder asymmetries will also \ cause asymmetrical payoffs.\nBelow it is shown how to analyse such problems. \ We will use the", StyleBox[" 'battle of the sexes'", FontSlant->"Italic"], " game introduced by Dawkins (1976) to illustrate the method (also see \ Maynard Smith 1982, p. 130)." }], "Text", PageWidth->WindowWidth, FontSize->16], Cell[TextData[{ StyleBox["The \"battle of the sexes\" game. ", FontSize->17, FontWeight->"Bold", FontSlant->"Italic"], "The idea is as follows. Suppose that the succesful raising of an offspring \ is worth +15 to each parent. The cost of raising an offspring is -20, which \ can be borne by one parent only, or shared equally between the two. The cost \ of a long courtship is -3 to both participants. Females can be 'coy' or \ 'fast'; males can be 'faithful' or 'philanderer'. Coy females insist on a \ long courtship, whereas fast females do not; all females care for the \ offspring they produce. Faithful males are willing, if necessary, to engage \ in a long courtship, and also care for the offspring. Philanderers are not \ prepared to engage in a long courtship, and do not care for their offspring. \ With these assumptions, we get the following payoff matrix :\n\t \t\t \ FEMALE\n\t\t\t Coy\t Fast\n\ MALE\t Faithful\t(+15-20/2-3=2,+15-20/2-3=2)\t (+15-20/2=5,+15-20/2=5)\ \n\t Philanderer\t(0,0)\t \ (+15,+15-20=-5)\n(the first number is the payoff to the male, the second is \ the payoff to the female)\nThis game illustrates quite neatly that the best \ strategy from either player's perspective will depend on what the other \ player does. In this case, we see that this will likely lead to cyclical \ dynamics, because if\nfemales are coy, it pays males to be faithful\nmales \ are faithful, it pays females to be fast\nfemales are fast, it pays males to \ philander\nmales philander, it pays females to be coy" }], "Text", PageWidth->WindowWidth, FontSize->16], Cell["\<\ Parameters: w1 = direct fitness of player 1 (the male) w2 = direct fitness of player 2 (the female) y1 = phenotype of player 1 (probability of philandering) y2= phenotype of player 2 (probability that female is fast) r = relatedness between male and female\ \>", "Text", FontSize->18], Cell["\<\ Step 1: write individual reproductive success of both players as a function \ of the behaviour of each :\ \>", "Definition", FontSize->16], Cell["Player 1's (the male's) direct fitness :", "Text", FontSize->14], Cell[BoxData[ \(\(w1 = 2*\((1 - y1)\)*\((1 - y2)\) + 5*\((1 - y1)\)*y2 + 0*y1*\((1 - y1)\) + 15*y1*y2;\)\)], "Input"], Cell["Player 2's (the female's) direct fitness :", "Text", FontSize->14], Cell[BoxData[ \(\(w2 = 2*\((1 - y1)\)*\((1 - y2)\) + 5*\((1 - y1)\)*y2 + 0*y1*\((1 - y1)\) - 5*y1*y2;\)\)], "Input"], Cell["\<\ Step 2: analyse when the regression of each player's fitness on their \ genotype is positive\ \>", "Definition", FontSize->16], Cell["\<\ We now have 2 equations: one specifying when an increase in male philandery \ is favoured, the other specifying when females are selected to be less coy. \ But since each of these is dependent on what the opossite sex does, they will \ need to be simultaneously maximised to provide the joint ESS.\ \>", "Text", FontSize->14], Cell["\<\ Selection for philander males (derived using Hamilton's rule) :\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(dw1dg1[r_, y1_, y2_] = \[IndentingNewLine]\ \ \ D[w1, y1]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ effect\ of\ own\ behaviour\ on\ own\ fitness\ *) \[IndentingNewLine] + \ \[IndentingNewLine]\(\(D[w1, y2]\)\(*\)\(r\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\( (*\ effect\ of\ partner\ behaviour\ on\ own\ fitness\ \ \[IndentingNewLine]\t\t\t\t\tx\ the\ slope\ of\ partner\ behaviour\ on\ own\ \ genotype, \ \[IndentingNewLine]\t\t\t\t\t\ the\ direct\ fitness\ definition\ \ of\ relatedness\ *) \)\)\)], "Input"], Cell[BoxData[ \(r\ \((3\ \((1 - y1)\) + 15\ y1)\) - 2\ \((1 - y2)\) + 10\ y2\)], "Output"] }, Open ]], Cell["\<\ Selection for fast females (derived using automatic total derivative approach \ and replacement rules) :\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(dw2dg2[r_, y1_, y2_] = FullSimplify[ Dt[w2, g2] /. {Dt[y2, g2] -> 1, Dt[y1, g2] -> r}]\)], "Input"], Cell[BoxData[ \(3 - 8\ y1 - 2\ r\ \((1 + 4\ y2)\)\)], "Output"] }, Open ]], Cell["Step 3: Joint maximisation to provide the joint equilibrium", \ "Definition", FontSize->16], Cell[TextData[{ "Setting the eqns. above to zero, and solving the system of equations \ yields the equilibrium behaviour of both players (male and female).This can \ be done using the Solve[] function of ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(equil[r_] = Solve[{dw1dg1[r, y1, y2] \[Equal] 0, dw2dg2[r, y1, y2] \[Equal] 0}, {y1, y2}]\)], "Input"], Cell[BoxData[ \({{y1 \[Rule] \(-\(\(9 - 10\ r + 6\ r\^2\)\/\(24\ \((\(-1\) + r\^2)\)\)\)\), y2 \[Rule] \(-\(\(4 - 15\ r + 6\ r\^2\)\/\(24\ \((\(-1\) + r\^2)\)\)\)\)}}\)], "Output"] }, Open ]], Cell["\<\ Evaluating this equilibrium for r=0 gives y1=3/8=37.5% and y2=1/6=16.7%, i.e. \ males should philander with a probability of 3/8 and females should be fast \ with a probability of 1/6 (Dawkins 1976). But is this equilibrium globally \ attracting, i.e. is it a true ESS? This is a question that can most easily be \ investigated graphically (see below).\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(equil[0]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \({{y1 \[Rule] 3\/8, y2 \[Rule] 1\/6}}\)], "Output"] }, Open ]], Cell["\<\ To investigate the dynamics of the system one can draw a field plot, since \ for each pair of y1 and y2 we get a direction of selection. \ \>", "Text", FontSize->14], Cell[BoxData[{ \( (*\ load\ the\ PlotField\ package\ to\ enable\ you\ to\ draw\ field\ plots\ \ and\ the\ ImplicitPlot\ package\ to\ allow\ you\ to\ make\ plots\ of\ the\ \ equilibrium\ lines\ *) \[IndentingNewLine]<< Graphics`PlotField`\), "\[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\)}], "Input"], Cell[BoxData[ \(\(\(\(rex = 0\) \)\(;\)\(\ \ \)\( (*\ let' s\ set\ r\ to\ zero\ *) \)\)\)], "Input"], Cell[BoxData[ \(\(fieldplot = PlotVectorField[{dw1dg1[rex, y1, y2], \ dw2dg2[rex, y1, y2]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ {y1, \ 0, \ 1}, \ {y2, \ 0, \ 1}, PlotPoints -> 17];\)\)], "Input"], Cell[BoxData[ \(\(equillines = ImplicitPlot[{dw1dg1[rex, y1, y2] \[Equal] 0, \ dw2dg2[rex, y1, y2] \[Equal] 0}, {y1, 0, 1}, {y2, 0, 1}, PlotStyle \[Rule] {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(\(eq = Graphics[{\[IndentingNewLine]{RGBColor[0.5, 0, 0.5], Disk[{3/8, 1/6}, 0.02]}, \[IndentingNewLine]{RGBColor[0.5, 0, 0.5], Text["\", {0.32, 0.2}]}, \[IndentingNewLine]}];\)\(\ \ \ \ \ \)\)\)], "Input"], Cell["\<\ Now display the whole lot together (field plot, equilibrium lines and \ equilibrium). 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Less clearly visible is whether \ the equilibrium is stable (globally attracting), or whether the dynamics of \ this system are characterised by limit cycles (ever lasting cyclical \ dynamics). In fact, as it turns out, the system is characterised by limit \ cycles, i.e. an ever lasting arms race. In sum - as Dawkins (1989, p. 303) \ put it - 'The behaviour of lovers is oscillating like the moon, and \ unpredictable as the weather.'. Or in other words, although there is an \ equilibrium, it is not strictly speaking an ESS (note that Dawkins (1976) got \ this wrong in the first edition of ", StyleBox["'The Selfish Gene'", FontSlant->"Italic"], "). For other payoffs one can sometimes get a globally attracting \ equilibrium - a true ESS. Other arms-races (e.g. host-parasite dynamics) are \ often characterised by similar dynamics.\nThe trajectory that the population \ will follow can also be illustrated by representing the two selection \ equations as differential equations, which can then be solved numerically. 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This is done using \ the function DSolve[]. No need to go into this at this point, but may be \ useful for other problems (see talk D. Sumpter).\ \>", "Subsubsection", CellDingbat->None, FontSize->16, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(Y1'\)[t] \[Equal] dw1dg1[rex, Y1[t], Y2[t]], \ \(Y2'\)[t] \[Equal] dw2dg2[rex, Y1[t], Y2[t]]}, {Y1[t], Y2[t]}, t]\)], "Input"], Cell[BoxData[ \({{Y1[t] \[Rule] 1\/8\ \((8\ C[1]\ Cos[4\ \@6\ t] + 3\ Cos[4\ \@6\ t]\^2 + 4\ \@6\ C[2]\ Sin[4\ \@6\ t] + 3\ Sin[4\ \@6\ t]\^2)\), Y2[t] \[Rule] 1\/6\ \((6\ C[2]\ Cos[4\ \@6\ t] + Cos[4\ \@6\ t]\^2 - 2\ \@6\ C[1]\ Sin[4\ \@6\ t] + Sin[4\ \@6\ t]\^2)\)}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "In this case, you can even get a solution for any value of r, an \ analytical result which I think would be challenging to derive by hand. Not \ that this matters here, it just shows the power of ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Subsubsection", CellDingbat->None, FontSize->16, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(Y1'\)[t] \[Equal] dw1dg1[r, Y1[t], Y2[t]], \ \(Y2'\)[t] \[Equal] dw2dg2[r, Y1[t], Y2[t]]}, {Y1[t], Y2[t]}, t]\)], "Input"], Cell[BoxData[ \({{Y1[ t] \[Rule] \((\(\[ExponentialE]\^\(\(-2\)\ \((\(-r\) + \ \@\(\(-24\) + 25\ r\^2\))\)\ t\)\ \((\(-5\)\ r + \@\(\(-24\) + 25\ \ r\^2\))\)\)\/\(2\ \@\(\(-24\) + 25\ r\^2\)\) + \(\[ExponentialE]\^\(2\ \((r + \ \@\(\(-24\) + 25\ r\^2\))\)\ t\)\ \((5\ r + \@\(\(-24\) + 25\ \ r\^2\))\)\)\/\(2\ \@\(\(-24\) + 25\ r\^2\)\))\)\ \ \((\((\[ExponentialE]\^\(\(-2\)\ \((r + \@\(\(-24\) + 25\ r\^2\))\)\ t\)\ \((\ \(-48\) + 72\ r + 50\ r\^2 - 75\ r\^3 - 18\ \@\(\(-24\) + 25\ r\^2\) + 22\ r\ \@\(\(-24\) + 25\ r\^2\) - 15\ r\^2\ \@\(\(-24\) + 25\ r\^2\))\))\)/\((4\ \ \((\(-24\) + 25\ r\^2)\)\ \((r + \@\(\(-24\) + 25\ \ r\^2\))\))\) + \((\[ExponentialE]\^\(2\ \((\(-r\) + \@\(\(-24\) + 25\ r\^2\))\ \)\ t\)\ \((48 - 72\ r - 50\ r\^2 + 75\ r\^3 - 18\ \@\(\(-24\) + 25\ r\^2\) + 22\ r\ \@\(\(-24\) + 25\ r\^2\) - 15\ r\^2\ \@\(\(-24\) + 25\ r\^2\))\))\)/\((4\ \ \((\(-24\) + 25\ r\^2)\)\ \((\(-r\) + \@\(\(-24\) + 25\ r\^2\))\))\) + C[1])\) + \((\(-\(\(3\ \[ExponentialE]\^\(\(-2\)\ \((\(-r\) \ + \@\(\(-24\) + 25\ r\^2\))\)\ t\)\)\/\@\(\(-24\) + 25\ r\^2\)\)\) + \(3\ \ \[ExponentialE]\^\(2\ \((r + \@\(\(-24\) + 25\ r\^2\))\)\ t\)\)\/\@\(\(-24\) \ + 25\ r\^2\))\)\ \((\(\[ExponentialE]\^\(2\ \((\(-r\) + \@\(\(-24\) + 25\ \ r\^2\))\)\ t\)\ \((\(-72\) + 48\ r + 75\ r\^2 - 50\ r\^3 - 8\ \@\(\(-24\) + \ 25\ r\^2\) + 27\ r\ \@\(\(-24\) + 25\ r\^2\) - 10\ r\^2\ \@\(\(-24\) + 25\ \ r\^2\))\)\)\/\(4\ \((\(-24\) + 25\ r\^2)\)\ \((\(-r\) + \@\(\(-24\) + 25\ \ r\^2\))\)\) + \(\[ExponentialE]\^\(\(-2\)\ \((r + \@\(\(-24\) + 25\ r\^2\))\)\ \ t\)\ \((72 - 48\ r - 75\ r\^2 + 50\ r\^3 - 8\ \@\(\(-24\) + 25\ r\^2\) + 27\ \ r\ \@\(\(-24\) + 25\ r\^2\) - 10\ r\^2\ \@\(\(-24\) + 25\ r\^2\))\)\)\/\(4\ \ \((\(-24\) + 25\ r\^2)\)\ \((r + \@\(\(-24\) + 25\ r\^2\))\)\) + C[2])\), Y2[t] \[Rule] \((\(2\ \[ExponentialE]\^\(\(-2\)\ \((\(-r\) + \ \@\(\(-24\) + 25\ r\^2\))\)\ t\)\)\/\@\(\(-24\) + 25\ r\^2\) - \(2\ \ \[ExponentialE]\^\(2\ \((r + \@\(\(-24\) + 25\ r\^2\))\)\ t\)\)\/\@\(\(-24\) \ + 25\ r\^2\))\)\ \((\((\[ExponentialE]\^\(\(-2\)\ \((r + \@\(\(-24\) + 25\ \ r\^2\))\)\ t\)\ \((\(-48\) + 72\ r + 50\ r\^2 - 75\ r\^3 - 18\ \@\(\(-24\) + 25\ r\^2\) + 22\ r\ \@\(\(-24\) + 25\ r\^2\) - 15\ r\^2\ \@\(\(-24\) + 25\ r\^2\))\))\)/\((4\ \ \((\(-24\) + 25\ r\^2)\)\ \((r + \@\(\(-24\) + 25\ \ r\^2\))\))\) + \((\[ExponentialE]\^\(2\ \((\(-r\) + \@\(\(-24\) + 25\ r\^2\))\ \)\ t\)\ \((48 - 72\ r - 50\ r\^2 + 75\ r\^3 - 18\ \@\(\(-24\) + 25\ r\^2\) + 22\ r\ \@\(\(-24\) + 25\ r\^2\) - 15\ r\^2\ \@\(\(-24\) + 25\ r\^2\))\))\)/\((4\ \ \((\(-24\) + 25\ r\^2)\)\ \((\(-r\) + \@\(\(-24\) + 25\ r\^2\))\))\) + C[1])\) + \((\(\[ExponentialE]\^\(2\ \((r + \@\(\(-24\) + \ 25\ r\^2\))\)\ t\)\ \((\(-5\)\ r + \@\(\(-24\) + 25\ r\^2\))\)\)\/\(2\ \ \@\(\(-24\) + 25\ r\^2\)\) + \(\[ExponentialE]\^\(\(-2\)\ \((\(-r\) + \ \@\(\(-24\) + 25\ r\^2\))\)\ t\)\ \((5\ r + \@\(\(-24\) + 25\ \ r\^2\))\)\)\/\(2\ \@\(\(-24\) + 25\ r\^2\)\))\)\ \((\(\[ExponentialE]\^\(2\ \ \((\(-r\) + \@\(\(-24\) + 25\ r\^2\))\)\ t\)\ \((\(-72\) + 48\ r + 75\ r\^2 - \ 50\ r\^3 - 8\ \@\(\(-24\) + 25\ r\^2\) + 27\ r\ \@\(\(-24\) + 25\ r\^2\) - 10\ \ r\^2\ \@\(\(-24\) + 25\ r\^2\))\)\)\/\(4\ \((\(-24\) + 25\ r\^2)\)\ \((\(-r\ \) + \@\(\(-24\) + 25\ r\^2\))\)\) + \(\[ExponentialE]\^\(\(-2\)\ \((r + \@\(\ \(-24\) + 25\ r\^2\))\)\ t\)\ \((72 - 48\ r - 75\ r\^2 + 50\ r\^3 - 8\ \ \@\(\(-24\) + 25\ r\^2\) + 27\ r\ \@\(\(-24\) + 25\ r\^2\) - 10\ r\^2\ \ \@\(\(-24\) + 25\ r\^2\))\)\)\/\(4\ \((\(-24\) + 25\ r\^2)\)\ \((r + \ \@\(\(-24\) + 25\ r\^2\))\)\) + C[2])\)}}\)], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Plotting the trajectory on the field plot, we can see that our concern was \ warranted: the system does not tend towards the equilibrium in the middle but \ is characterised by limit cycles. 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Investigate the evolution of reciprocal altruism. For this, replace the \ 'Cooperate' strategy in the 2-player mixed strategy game by 'TIT-FOR-TAT' \ (TFT, cooperate on the first move and then do what your opponent did on the \ previous move); let defect stand for unconditional defection ('always \ defect', AD). Assume that the game is played an unknown number of times, and \ that after each play of Prisoner's dilemma, the next play will occur with \ probability p, with 0 < p < 1 (this game is known as the iterated Prisoner's \ dilemma). Calculate the payoffs E(AD,AD), E(AD,TFT), E(TFT, AD) and E(TFT, \ TFT), and check (1) when TFT can invade an all-defecting population and (2) \ when AD can invade an all-TFT population. Assume for simplicity that the \ players are unrelated to each other.\ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ StyleBox["The payoff of TFT playing against itself (E(TFT,TFT)) is\n\ (b-c)+p(b-c)+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["(b-c)+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^3\)]], StyleBox["(b-c)+\[Ellipsis]=(b-c)/(1-p) (see proof below)\nThe other \ payoffs remain unaffected: E(AD,AD)=0, E(TFT,AD)=-c and E(AD,TFT)=b. \nWith \ these payoffs it is easy to check that (1) TFT can never invade an \ all-defecting population (because you need to be two to gain any advantage) \ and (2) that AD invades an all-TFT population when b.p < c. Note the \ similarity of the latter condition to Hamilton's rule, with p taking over the \ role of relatedness ", FontSlant->"Italic"], StyleBox["[in fact p is a nongenetic type of relatedness (Skyrms 1996); as \ Hamilton (1971, p. 65) put it: \[OpenCurlyDoubleQuote]Rather than continue in \ the jangling partnership, the disillusioned cooperator can part quietly from \ the selfish companion at the first clear sign of unfairness and try his luck \ in another union. The result would be some degree of assortative pairing.\ \[CloseCurlyDoubleQuote]]. ", FontSlant->"Italic"], StyleBox["Putting (genetic) relatedness into the equation one would find \ (1) that TFT can only invade an all-defecting population when b.r > c and (2) \ that AD invades an all-TFT population when b.(p+r)/(1+p.r) < c. All this \ means that if non-cooperation is the primitive condition, it is difficult to \ see how reciprocal altruism could evolve from it. Axelrod & Hamilton (1981) \ suggest that cooperative behaviour might originate as altruism between \ relatives selected by kin selection and then spread to encompass \ nonrelatives, or it might spread from a small cluster of cooperative \ individuals. But clearly, the transition from nonnoncooperation to reciprocal \ altruism is a difficult one.\nproof: \nfor any p\[NotEqual]1, 1+p+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["+\[Ellipsis]+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^\(m - 1\)\)]], StyleBox["=(1-", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^m\)]], StyleBox[")/(1-p) since (1-p)(1+p+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["+\[Ellipsis]+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^\(m - 1\)\)]], StyleBox[")=1-", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^m\)]], "\n", StyleBox["now, as m goes to infinity, ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^m\)]], StyleBox[" goes to zero so that 1+p+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^3\)]], StyleBox["+\[Ellipsis]=1/(1-p)", FontSlant->"Italic"] }], "Text"], Cell[TextData[StyleBox["Or you can have Mathematica do all the work :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((b - c)\)*\(\[Sum]\+\(i = 0\)\%\[Infinity] p\^i\)\)], "Input"], Cell[BoxData[ \(\(b - c\)\/\(1 - p\)\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 2a. Godfray (1987) uses an explicit genetic model to derive when parasitoid \ wasp larvae should either tolerate or eat each other within their host. Show \ that Godfray's results can be derived in a much easier way using Steve \ Frank's methods. For simplicity, model only the case with a clutch size c of \ 2. To parallel Godfray's notation, assume the following payoffs to larva 1: \ E(fighter,fighter)=1/2 (it survives half of the time), E(fighter,tolerant)=1 \ (the fighter larva gets the host for itself), E(tolerant,fighter)=0 (the \ tolerant larva gets killed) and E(tolerant, tolerant)=f where f is the ratio \ of the reproduction as a pair over the reproduction alone. Furthermore, \ assume that competing larvae are of a random sex and that their mother is \ singly mated so that the average relatedness among them is \.bd. Calculate \ (1) when tolerance can invade in an all-fighter population and (2) when \ fighting will invade in an all-tolerant population assuming either discrete \ or mixed strategies. Also calculate the mixed and pure strategy ESSs, not \ calculated in Godfray's paper. 2b. Assume the parasitoid wasp mother mates twice. How would this affect the \ likely evolution of tolerance? If the level of siblicide among larvae were \ conditional on relatedness structure, how would this affect the op-timal \ mating strategy of the mother? 2c. In parasitoid wasps, after egg deposition, larvae are left on their own, \ i.e. there is no parental care. How could parental care affect the possible \ outcome? \ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["2a.\nBulmer (1994, p. 193) mentions that this problem \ represents \"an extreme example of a situation in which inclusive-fitness \ arguments cannot be used, because the costs and benefits of altruism are not \ constant\". But let's see how we can tackle it using the methods above.", FontSlant->"Italic"]], "Text"], Cell[TextData[StyleBox["First, write the fitness of larva 1 as a function of \ its own (y1) and its partner (y2) phenotype (whether or not to behave \ tolerant), as in ", FontSlant->"Italic"]], "Text"], Cell[BoxData[ \(\(w1 = \((1 - y1)\)*\((1 - y2)\)*\((1/2)\) + \((1 - y1)\)*y2*1 + y1*y2*f;\)\)], "Input"], Cell[TextData[StyleBox["MIXED STRATEGY CASE :\nUnder 'relaxing assumptions', \ point 7 'Incomplete penetrance' (p. 229) Godfray mentions this case.\nAn \ increase in the probability of playing 'tolerant' is favoured when (eqn. 1.1) \ :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(selectiontol = \((D[w1, y1]*1 + D[w1, y2]*r)\) /. {y1 \[Rule] y, y2 \[Rule] y}\)], "Input"], Cell[BoxData[ \(1\/2\ \((\(-1\) + y)\) - y + f\ y + r\ \((1 + 1\/2\ \((\(-1\) + y)\) - y + f\ y)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Where the cost of playing tolerant = D[w1,y1]= \ -(1/2)(1-y2)-(1-f).y2 and the benefit of having a tolerant partner = \ D[w1,y2]=(1/2)(1-y1)+f.y1. Again, as in the hawk-dove game, one can see that \ the costs and benefit depend on what the opponent does so that Hamilton's \ assumption of constant costs & benefits is violated. But a Hamilton's rule \ defined as in eqn. 1.1 is always valid for the mixed or continuous strategy \ case (for a derivation of a Hamilton's rule that is also correct for the \ discontinuous strategy case, see Wenseleers & Ratnieks submitted). \nNear the \ ESS, y1=y2=y, so that the equation above can be solved to yield the ESS \ probability with which any larva should behave tolerantly : \ y*=(1-r)/[(2f-1)(1+r)]", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mESS[r_, f_] = \(\(\(Solve[selectiontol \[Equal] 0, y]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(\(-\(\(\(-1\) + r\)\/\(\(-1\) + 2\ f - r + 2\ f\ r\)\)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Tolerance will invade in an all-fighting population \ when", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((selectiontol /. {y \[Rule] 0})\) > 0\)], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\) + r\/2 > 0\)], "Output"] }, Open ]], Cell[TextData[StyleBox["which can never be the case, even not when two larvae \ would be identical twins ! (cf. Bulmer 1994, p. 194 and Godfray 1987 p. 229 \ 1st paragraph with penetrance z approaching zero)\nFighting will invade in an \ all-tolerant population when", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((selectiontol /. {y \[Rule] 1})\) < 0\)], "Input"], Cell[BoxData[ \(\(-1\) + f + f\ r < 0\)], "Output"] }, Open ]], Cell[TextData[StyleBox["i.e. when f < 2/3 for r = \.bd. (cf. Godfray 1987 p. \ 229 eqn. 12 with clutch size c=2)", FontSlant->"Italic"]], "Text"], Cell[TextData[{ StyleBox["DISCRETE (PURE) STRATEGY CASE :\nTolerance is favoured over \ fighting when the expected reproduction of larva 1 is higher as a tolerant \ than as a fighter (eqn. 1.2). ", FontSlant->"Italic"], "\n", StyleBox["The fitness of larva 1 when it is of the tolerant type is ", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Wtolerant = w1 /. {y1 \[Rule] 1, y2 \[Rule] p + r*\((1 - p)\)}\)], "Input"], Cell[BoxData[ \(f\ \((p + \((1 - p)\)\ r)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["The fitness of larva 1 when it is of the fighter type \ is ", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Wfighter = w1 /. {y1 \[Rule] 0, y2 \[Rule] p + r*\((0 - p)\)}\)], "Input"], Cell[BoxData[ \(p - p\ r + 1\/2\ \((1 - p + p\ r)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Tolerance will invade in an all-fighting population \ (p=0) when", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\((Wtolerant - Wfighter)\) /. {p \[Rule] 0})\) > 0\)], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\) + f\ r > 0\)], "Output"] }, Open ]], Cell[TextData[StyleBox["i.e. when f > 1 for r =\.bd. This requires that an \ individual should have a higher fitness as one of a pair than when alone \ (implying some unlikely Allee effect). This result is identical to the \ population genetic result of Godfray 1987 (eqn. 2 p. 223), and shows how \ difficult it is to evolve tolerance once fighting has evolved. Bull & Charnov \ (1985) call this irreversible evolution.\nFighting will invade in an \ all-tolerant population (p=1) when ", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\((Wfighter - Wtolerant)\) /. {p \[Rule] 1})\) > 0\)], "Input"], Cell[BoxData[ \(1 - f - r\/2 > 0\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["For r =\.bd this is when f < 3/4. By setting c=2 in eqn. 5 of \ Godfray 1987, it can again be checked that this is identical to the \ population genetic result. ", FontSlant->"Italic"], "\n", StyleBox["A pure strategy ESS is reached when the fitness of a fighting \ larva equals the fitness of a tolerant larva, which is when", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[Wtolerant \[Equal] Wfighter, p]\)], "Input"], Cell[BoxData[ \({{p \[Rule] \(-\(\(1 - 2\ f\ r\)\/\(\((\(-1\) + 2\ f)\)\ \((\(-1\) + r)\)\)\)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Overall it can be seen that this method is much easier, and \ vastly more economical and general than the corresponding population genetic \ model, e.g. you don't need to compile massive mating tables, it does not \ require any complex matrix algebra, is valid under any relatedness structure, \ etc\[Ellipsis] Nevertheless, if you want to get your papers into Am. Nat. you \ may be forced to give your models more of an air of mathematical sophistry \ (and pointing out that siblicide in parasitoid wasps is just like the \ hawk-dove game won't help).", FontSlant->"Italic"], "\n\n", StyleBox["2b. The average relatedness among the mother's offspring of \ random sex is given by r= \.bc.rMM+\.bc.rMF+\.bc.rFM +\.bc.rFF where rMM, \ rMF, rFM and rFF stand for relatedness between 2 male offspring, between a \ male and a female offspring, between a female and a male offspring and \ between 2 female offspring (we assume a 1:1 sex ratio in the brood). Under \ single mating rMM =\.bd , rMF =\.bd , rFM =\.bc and rFF =\.be , giving r= \ \.bd. But under double mating, rFF drops from \.be to \.bd so that r= 7/16, \ which is lower than \.bd. This lower relatedness will make it harder for \ tolerance to evolve (substitute in the above equations). \nIf larvae adjust \ their probability of being siblicidal based on the average relatedness within \ the brood, it might pay for mothers not to mate twice (or not to \ superparasitise a host, which may be another cause of low relatedness).", FontSlant->"Italic"], "\n\n", StyleBox["2c. If the parent rears her offspring to adulthood he/she might \ prevent offspring from behaving aggressively, or punish them if they behave \ aggressively. Punishment, in turn, may make it unprofitable for any offspring \ to even try to behave aggressively. This may be why so many parasitoid wasps \ have larvae with huge mandibles, adapted for fighting, something that is \ rarely seen in species with parental care. Hamilton (1964) suggested that \ bees and wasps might have evolved combs to prevent larvae from eating each \ other. ", FontSlant->"Italic"], "\n" }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["1.5 References", "Section", FontSize->25], Cell[TextData[{ "Aumann, R. J. 1987. Correlated equilibrium as an expression of Bayesian \ rationality. ", StyleBox["Econometrica", FontSlant->"Italic"], " 55: 1-18.\nAxelrod, R. & Hamilton, W. D. 1981. The evolution of \ cooperation. ", StyleBox["Science", FontSlant->"Italic"], " 211: 1390-1396.\nBull, J. J. & Charnov, E. L. 1985. Irreversible \ evolution. ", StyleBox["Evolution", FontSlant->"Italic"], " 39: 1149-1155.\nBulmer, M. 1994. ", StyleBox["Theoretical Evolutionary Ecology. ", FontSlant->"Italic"], "Sinauer Associates Publishers, Sunderland, Massachusetts.\nDawkins, R. \ 1976. ", StyleBox["The Selfish Gene. ", FontSlant->"Italic"], "Oxford University Press.\nFrank, S. A. 1994. Genetics of mutualism - the \ evolution of altruism between species. ", StyleBox["Journal of Theoretical Biology", FontSlant->"Italic"], " 170: 393-400.\nFrank, S. A. 1997. Multivariate analysis of correlated \ selection and kin selection, with an ESS maximization method. ", StyleBox["Journal of Theoretical Biology", FontSlant->"Italic"], " 189: 307-316.\nFrank, S.A. 1998. ", StyleBox["Foundations of Social Evolution. ", FontSlant->"Italic"], "Princeton University Press, Princeton, New Jersey.\nGodfray, H.C.J. 1987. \ The evolution of clutch size in parasitic wasps. ", StyleBox["American Naturalist ", FontSlant->"Italic"], "129: 221-233.\nGrafen, A. 1979. The hawk-dove game played between \ relatives. ", StyleBox["Animal Behaviour", FontSlant->"Italic"], " 27: 905-907.\nGrafen, A. 1985. A geometric view of relatedness. ", StyleBox["Oxford Surveys in Evolutionary Biology", FontSlant->"Italic"], " 2: 28-89.\nHamilton, W. D. 1964. The genetical evolution of social \ behaviour. I & II. ", StyleBox["Journal of Theoretical Biology", FontSlant->"Italic"], " 7: 1-52.\n", "Hamilton, W. D. 1971. Selection of selfish and altuistic behaviour in some \ extreme models. In: ", StyleBox["Man and Beast: Comparative Social Behavior", FontSlant->"Italic"], " (J. F. Eisenberg & W. S. Dillon, eds.), pp. 57-91. Smithsonian Press, \ Washington DC.", "\nHamilton, W. D. 1975. Innate social aptitudes in man: an approach from \ evolutionary genetics. ", StyleBox["Biosocial anthropology", FontSlant->"Italic"], " (R. Fox, eds), pp. 133-155. Wiley, New York.\nMaynard Smith, J. 1982. \ Evolution and the Theory of Games. Cambridge University Press, New York.\n\ Nowak, M. A., Page, K. M. & Sigmund, K. 2000. Fairness versus reason in the \ ultimatum game. ", StyleBox["Science", FontSlant->"Italic"], " 289: 1773-5.\nOrlove, M. J. & Wood, C. L. 1978. Coefficients of \ relationship and coefficients of relatedness in kin selection: a covariance \ form for the rho formula. ", StyleBox["Journal of Theoretical Biology", FontSlant->"Italic"], " 73: 679-86.\nSkyrms, B. 1994. Darwin meets the logic of decision - \ Correlation in evolutionary game theory. ", StyleBox["Philosophy of Science", FontSlant->"Italic"], " 61: 503-528.\nSkyrms, B. 1996. ", StyleBox["Evolution of the Social Contract.", FontSlant->"Italic"], " Cambridge University Press, Cambridge.\nWenseleers, T. & Ratnieks, F.L.W. \ Towards a general theory of conflict: the sociobiology of mendelian \ segregation. Submitted." }], "Text", FontSize->16] }, Closed]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 800}, {0, 555}}, WindowSize->{791, 400}, WindowMargins->{{Automatic, -3}, {58, Automatic}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, StyleDefinitions -> "Classroom.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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