http://www.freeassociations.org/
Smith&Price (1973) “The Logic of Animal Conflict”
John Maynard Smith - Game theory: The war of attrition game (46/102)
Download Citation on ResearchGate | Game theory and evolution (1973) | In 1973 Maynard Smith and Price published ...
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Smith (1972) “Game Theory and the Evolution of Fighting” and Maynard. Smith and Price (1973) “The Logic of Animal Conflict”. The theory was formulated for understanding the ...
Evolution and Game Theory
The theory of evolution goes back to Darwin’s classic, The Origins of Species (and to Wallace).
Darwin focused mostly on evolution and adaptation of an organism to the environment in which it was situated. But in The Descent of Man, in the context of sexual selection, he anticipated many of the ideas of evolutionary game theory.
Evolutionary game theory was introduced by John Maynard Smith in Evolution and the Theory of Games, and in his seminal papers, Maynard Smith (1972) “Game Theory and the Evolution of Fighting” and Maynard Smith and Price (1973) “The Logic of Animal Conflict”.
The theory was formulated for understanding the behavior of animals in game-theoretic situations (to a game theorist, all situations). But it can equally well be applied to modeling “myopic behavior” for more complex organisms—such as humans.
Price (1973) “The Logic of Animal Conflict”
(Adobe PDF)
ー(スミス的プレイヤーと呼んでおく Maynard Smith 1973, 1983)は合理性を仮定しない。スミス的 ..... 表3 Ⅳ -c タカ・ハトゲーム.
(Adobe PDF)
ゲーム的状況ではある個体の適応度(利得)は、自分の戦略と相手の戦略の両方. によって決まる( ... Maynard-Smith は進化の結果、「進化的に安定な戦略( Evolutionarily Stable Strategy: ... 儀式的な闘争とタカハトゲーム. 儀式的 .... 合理的な選択の結果、.
http://www.est.ryukoku.ac.jp/est/kondoh/lcts/lcts_siryo/evoeco/evo_6.pdf
進化生態学(第6講) 今日の話題:進化的に安定な戦略と軍拡競争
1. 儀式的な闘争 同種動物間の闘争は、普通、儀式的であって、双方ともに相手にひどい怪我を負わせ ることはない。かつてこれは種の保存のためであると考えられてきたが、これは自然 選択の考えに反する。自然選択の理論によると、進化すべき形質(戦略)は種全体の生 存を高める戦略ではなく、種内のその個体の生存・繁殖を高める戦略だからだ。では、 どうして儀式的な闘争が適応戦略になり得たのだろうか?John Maynard-Smithはこの 問題をゲーム理論を用いて説明することを思いついた。
2. 進化とゲーム理論
これまでに扱ってきた最適戦略の問題では、それぞれの個体が他個体とは無関係に最適 戦略を決めている。しかし、餌や交尾相手の取り合いでは同種他個体がどのような行動 をとっているかに依存して最適な行動は変わるであろう。例えば、囲碁や将棋、ジャン ケンのように相手のあるゲームでは、最適な戦略は相手の戦略に依存する。このような 状況をゲームとよぶ。 ゲーム的状況ではある個体の適応度(利得)は、自分の戦略と相手の戦略の両方 によって決まる(じゃんけんのように)。自分が戦略 X をとり相手が戦略 Y をとって いるときの個体の適応度を E(X, Y)と表すことにしよう。
…
Maynard-Smithは進化の結果、「進化的に安定な戦略(Evolutionarily Stable Strategy: ESS)」に行き着くはずだと考えた。自然選択による進化とは、これまでとは異なる有 利な戦略を持つ個体が突然変異によって生じ、それが集団中に広まることである。ある 戦略 A をとっている個体の集まりがあって、別の戦略 X をとる少数(割合 p)の個体 が現れたとしよう。このとき戦略 A、X をとる個体の適応度の期待値は:
[A の適応度] = p×E(A, X) + (1 - p)×E(A, A) (1a) [X の適応度] = p×E(X, X) + (1 - p)×E(X, A) (1b)
ここで p が十分に小さいときに、[A の適応度]が[X の適応度]よりも大きければ A は集 団中でその数をのばすことができず、戦略 A は将来にわたって安泰であろう。このよ うな戦略を ESS とよぶのである。したがって、戦略 A が ESS になる条件は以下のいず れかを満たすことである:
(条件1)E(A, A) > E(X, A) (2a)
(条件2)E(A, A) = E(X, A) かつ E(A, X) > E(A, A) (2b)
p が十分に小さいとき X と出会うことはほとんどない。したがって、適応度のほとんど は多数派である A と出会ったときにどんな利得を得られるかで決まる。それを表した のが条件1である。A と出会ったときの利得が同じであるなら、少数派の X とであっ たときの利得が問題になってくる。それを表したのが条件2である。
3. 儀式的な闘争とタカハトゲーム
儀式的な闘争を説明するためにMaynard-Smithは次のようなゲームを考えた。ここに餌を巡ってあらそう2匹の生き物がいる。相手にどのような態度で臨むべきか?Maynard-Smithが考えた2つの戦略は「タカ戦略」と「ハト戦略」である。タカ戦略は相手がタカだろうとハトだろうとおかまいなしに攻撃する。その結果、相手がハトならば餌をまるごと手に入れることができる。相手がタカならば激しい闘争に発展し、その結果餌を手に入れる確率が1/2、怪我をして餌も手に入れられない確率が1/2とする。ハト戦略は平和的に振る舞う。相手がハトならば餌を半分ずつで分け合う。相手がタカなら餌をあきらめて、逃げるので怪我はしない。これら4通りの利得を前の例に習 って次のように表そう:
タカの利得:E(タカ, ハト)、E(タカ, タカ) (3a) ハトの利得:E(ハト, ハト)、E(ハト, タカ) (3b)
3-1. 餌の価値が闘争のコストよりも高い場合
餌の価値が F、闘争による怪我のコストが-I で表され、F > I が成り立つとしよう。それ
ぞれの利得は以下のように表される:
E(タカ, ハト) = F (4a) E(タカ, タカ)=(F/2)–(I/2) (4b) E(ハト, ハト)=F/2 (4c) E(ハト, タカ) = 0 (4d)
ハト戦略は ESS だろうか?E(ハト, ハト) > E(タカ, ハト)は成り立たないので、ESS で はないことがわかる。では、タカ戦略は ESS か?E(タカ, タカ) > E(ハト, タカ)が成り 立つので、この条件下ではタカ戦略は ESS であることがわかる。つまり一旦この集 団中にタカ戦略が広がったらハト戦略はこの集団に侵入することができない。
3-2. 餌の価値が闘争のコストよりも低い場合
餌の価値が F、闘争による怪我のコストが-I で表され、F < I が成り立つとしよう。この
とき、それぞれの利得は上と同じように(4a)~(4d)によって表される。E(タカ, タカ) >
E(ハト, タカ)は成り立たないので、タカ戦略は ESS ではないことがわかる。では、ハ
ト戦略が ESS なのだろうか?E(ハト, ハト) > E(タカ, ハト)が成り立たないのでタカ戦
略も ESS ではないことがわかる。すなわち、この条件ではタカ戦略もハト戦略も進
化的に安定ではないのである。では、いったい何がおこるのだろうか?Maynard-Smith
はこの問題を解決するために混合戦略というものを考えた。
混合戦略とはある確率である一方の戦略をとるような戦略のことである。上の状況では、 確率 [F / I] でタカ戦略をとり、確率 [1 – (F / I)] でハト戦略をとる混合戦略が ESSとなっている(レジュメ最後の添付資料を参照)。このことは、餌(あるいは別の資源) の価値が闘争のコストよりも低い場合には、戦いを挑まれたら戦わずに逃げるハト戦略 が混合戦略の一環として ESS になりうることを示している。言い換えれば、闘争がオ オゴトになるような「強い生物」でこそ儀式的闘争が進化しうるのである。
ゲーム理論で解明されたユダヤの知恵
http://nam-students.blogspot.jp/2012/11/blog-post_28.html
ゲーム理論 新版 岡田章 2011
https://nam-students.blogspot.com/2019/01/2011.html進化ゲーム(しんかげーむ、英: evolutionary game)とは、非協力ゲーム理論を動学化した理論体系である。
非協力ゲーム理論はナッシュ均衡という均衡概念を用いて分析を行う静学的な理論体系であるが、進化ゲーム理論は戦略分布や戦略プロファイルで表現される社会状態の変化を分析の対象とする動学的な体系である。プレーヤの出生死滅で集団中の戦略分布が変化することを想定するレプリケータダイナミクスモデルや、プレーヤの学習によって戦略分布や戦略プロファイルが変化する学習ダイナミクスモデルが考えられている。
たとえば、次のチキンゲームを考えてみよう。
このゲームのナッシュ均衡は、<1がAで2がB>、<1がBで2がA>、<1と2がそれぞれ確率0.6でA、確率0.4でBをとる>の3通りである。いずれかの状態からプレーヤ1もプレーヤ2も逸脱する誘因を持たない。その意味でそれぞれ安定な状態である。
では、実際にこのゲームをプレーする状況を考えてみよう。よーいドンで手を出し合ったときいずれかのナッシュ均衡が実現するだろうか。実際に実験をしてみると<両者B>や<両者A>もかなりの確率で発生し、必ずしもナッシュ均衡が実現するとは限らない。しかし、その場合も何度かプレーを繰り返しているうちに<1がAで2がB>の状態か<1がBで2がA>のいずれかの状態に落ち着くケースが増えてくる。このとき、プレーヤの手の組み合わせ(戦略プロファイル)がどのように変化していくのかを記述する動学モデルがあれば、ある初期状態から出発したときにどのようなナッシュ均衡に収束しやすいか、あるナッシュ均衡に収束するためにはどのような初期状態が必要であるのかといった問題を考察することができる。また状態変化の収束点になりうる安定なナッシュ均衡や、収束点にはなりえない不安定なナッシュ均衡を識別することも可能になる。
ナッシュ自身、このような動学モデルの必要性を認識していた形跡があるが、プレーヤの出生死滅やプレーヤによる学習を仮定することで集団中の戦略頻度の変化や戦略プロファイルの変化を明示的に分析できるように工夫された動学モデルの体系が進化ゲーム理論である。
対象となる系の様相の進化的安定性を進化ゲームではレプリケータダイナミクスと呼ぶ。最も一般的な連続時間系のレプリケータダイナミクスは以下のような微分方程式で書かれる。
ここで、xは集団中のiの比率、fはiの適応度を表している。または集団の平均適応度である。
時間間隔の離散時間系のレプリケータダイナミクスとして以下のような差分方程式がよく用いられる。
https://en.wikipedia.org/wiki/Evolutionary_game_theory
Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.[1]
Evolutionary game theory differs from classical game theory in focusing more on the dynamics of strategy change.[2] This is influenced by the frequency of the competing strategies in the population.[3]
Evolutionary game theory has helped to explain the basis of altruistic behaviours in Darwinian evolution. It has in turn become of interest to economists, sociologists, anthropologists, and philosophers.
Classical non-cooperative game theory was conceived by John von Neumann to determine optimal strategies in competitions between adversaries. A contest involves players, all of whom have a choice of moves. Games can be a single round or repetitive. The approach a player takes in making his moves constitutes his strategy. Rules govern the outcome for the moves taken by the players, and outcomes produce payoffs for the players; rules and resulting payoffs can be expressed as decision trees or in a payoff matrix. Classical theory requires the players to make rational choices. Each player must consider the strategic analysis that his opponents are making to make his own choice of moves.[4][5]
Evolutionary game theory started with the problem of how to explain ritualized animal behaviour in a conflict situation; "why are animals so 'gentlemanly or ladylike' in contests for resources?" The leading ethologists Niko Tinbergen and Konrad Lorenz proposed that such behaviour exists for the benefit of the species. John Maynard Smithconsidered that incompatible with Darwinian thought,[6] where selection occurs at an individual level, so self-interest is rewarded while seeking the common good is not. Maynard Smith, a mathematical biologist, turned to game theory as suggested by George Price, though Richard Lewontin's attempts to use the theory had failed.[7]
Maynard Smith realised that an evolutionary version of game theory does not require players to act rationally – only that they have a strategy. The results of a game shows how good that strategy was, just as evolution tests alternative strategies for the ability to survive and reproduce. In biology, strategies are genetically inherited traits that control an individual's action, analogous with computer programs. The success of a strategy is determined by how good the strategy is in the presence of competing strategies (including itself), and of the frequency with which those strategies are used.[8] Maynard Smith described his work in his book Evolution and the Theory of Games.[9]
Participants aim to produce as many replicas of themselves as they can, and the payoff is in units of fitness (relative worth in being able to reproduce). It is always a multi-player game with many competitors. Rules include replicator dynamics, in other words how the fitter players will spawn more replicas of themselves into the population and how the less fit will be culled, in a replicator equation. The replicator dynamics models heredity but not mutation, and assumes asexual reproduction for the sake of simplicity. Games are run repetitively with no terminating conditions. Results include the dynamics of changes in the population, the success of strategies, and any equilibrium states reached. Unlike in classical game theory, players do not choose their strategy and cannot change it: they are born with a strategy and their offspring inherit that same strategy.[10]
EGT encompasses Darwinian evolution, including competition (the game), natural selection (replicator dynamics), and heredity. EGT has contributed to the understanding of group selection, sexual selection, altruism, parental care, co-evolution, and ecological dynamics. Many counter-intuitive situations in these areas have been put on a firm mathematical footing by the use of these models.[11]
The common way to study the evolutionary dynamics in games is through replicator equations. These show the growth rate of the proportion of organisms using a certain strategy and that rate is equal to the difference between the average payoff of that strategy and the average payoff of the population as a whole.[12] Continuous replicator equations assume infinitepopulations, continuous time, complete mixing and that strategies breed true. The attractors (stable fixed points) of the equations are equivalent with evolutionarily stable states. A strategy which can survive all "mutant" strategies is considered evolutionarily stable. In the context of animal behavior, this usually means such strategies are programmed and heavily influenced by genetics, thus making any player or organism's strategy determined by these biological factors.[13][14]
Evolutionary games are mathematical objects with different rules, payoffs, and mathematical behaviours. Each "game" represents different problems that organisms have to deal with, and the strategies they might adopt to survive and reproduce. Evolutionary games are often given colourful names and cover stories which describe the general situation of a particular game. Representative games include hawk-dove,[1] war of attrition,[15] stag hunt, producer-scrounger, tragedy of the commons, and prisoner's dilemma. Strategies for these games include Hawk, Dove, Bourgeois, Prober, Defector, Assessor, and Retaliator. The various strategies compete under the particular game's rules, and the mathematics are used to determine the results and behaviours.
The first game that Maynard Smith analysed is the classic Hawk Dove[a] game. It was conceived to analyse Lorenz and Tinbergen's problem, a contest over a shareable resource. The contestants can be either Hawk or Dove. These are two subtypes or morphs of one species with different strategies. The Hawk first displays aggression, then escalates into a fight until it either wins or is injured (loses). The Dove first displays aggression, but if faced with major escalation runs for safety. If not faced with such escalation, the Dove attempts to share the resource.[1]
Given that the resource is given the value V, the damage from losing a fight is given cost C:[1]
The actual payoff however depends on the probability of meeting a Hawk or Dove, which in turn is a representation of the percentage of Hawks and Doves in the population when a particular contest takes place. That in turn is determined by the results of all of the previous contests. If the cost of losing C is greater than the value of winning V (the normal situation in the natural world) the mathematics ends in an ESS, a mix of the two strategies where the population of Hawks is V/C. The population regresses to this equilibrium point if any new Hawks or Doves make a temporary perturbation in the population. The solution of the Hawk Dove Game explains why most animal contests involve only ritual fighting behaviours in contests rather than outright battles. The result does not at all depend on good of the species behaviours as suggested by Lorenz, but solely on the implication of actions of so-called selfish genes.[1]
In the Hawk Dove game the resource is shareable, which gives payoffs to both Doves meeting in a pairwise contest. Where the resource is not shareable, but an alternative resource might be available by backing off and trying elsewhere, pure Hawk or Dove strategies are less effective. If an unshareable resource is combined with a high cost of losing a contest (injury or possible death) both Hawk and Dove payoffs are further diminished. A safer strategy of lower cost display, bluffing and waiting to win, is then viable – a Bluffer strategy. The game then becomes one of accumulating costs, either the costs of displaying or the costs of prolonged unresolved engagement. It is effectively an auction; the winner is the contestant who will swallow the greater cost while the loser gets the same cost as the winner but no resource.[15] The resulting evolutionary game theory mathematics leads to an optimal strategy of timed bluffing.[16]
This is because in the war of attrition any strategy that is unwavering and predictable is unstable, because it will ultimately be displaced by a mutant strategy which relies on the fact that it can best the existing predictable strategy by investing an extra small delta of waiting resource to ensure that it wins. Therefore, only a random unpredictable strategy can maintain itself in a population of Bluffers. The contestants in effect choose an acceptable cost to be incurred related to the value of the resource being sought, effectively making a random bid as part of a mixed strategy (a strategy where a contestant has several, or even many, possible actions in his strategy). This implements a distribution of bids for a resource of specific value V, where the bid for any specific contest is chosen at random from that distribution. The distribution (an ESS) can be computed using the Bishop-Cannings theorem, which holds true for any mixed strategy ESS.[17] The distribution function in these contests was determined by Parker and Thompson to be:
The result is that the cumulative population of quitters for any particular cost m in this "mixed strategy" solution is:
as shown in the adjacent graph. The intuitive sense that greater values of resource sought leads to greater waiting times is borne out. This is observed in nature, as in male dung flies contesting for mating sites, where the timing of disengagement in contests is as predicted by evolutionary theory mathematics.[18]
In the War of Attrition there must be nothing that signals the size of a bid to an opponent, otherwise the opponent can use the cue in an effective counter-strategy. There is however a mutant strategy which can better a Bluffer in the War of Attrition Game if a suitable asymmetry exists, the Bourgeois strategy. Bourgeois uses an asymmetry of some sort to break the deadlock. In nature one such asymmetry is possession of a resource. The strategy is to play a Hawk if in possession of the resource, but to display then retreat if not in possession. This requires greater cognitive capability than Hawk, but Bourgeois is common in many animal contests, such as in contests among mantis shrimps and among speckled wood butterflies.
Games like Hawk Dove and War of Attrition represent pure competition between individuals and have no attendant social elements. Where social influences apply, competitors have four possible alternatives for strategic interaction. This is shown on the adjacent figure, where a plus sign represents a benefit and a minus sign represents a cost.
At first glance it may appear that the contestants of evolutionary games are the individuals present in each generation who directly participate in the game. But individuals live only through one game cycle, and instead it is the strategies that really contest with one another over the duration of these many-generation games. So it is ultimately genes that play out a full contest – selfish genes of strategy. The contesting genes are present in an individual and to a degree in all of the individual's kin. This can sometimes profoundly affect which strategies survive, especially with issues of cooperation and defection. William Hamilton,[21] known for his theory of kin selection, explored many of these cases using game theoretic models. Kin related treatment of game contests[22] helps to explain many aspects of the behaviour of social insects, the altruistic behaviour in parent/offspring interactions, mutual protection behaviours, and co-operative care of offspring. For such games Hamilton defined an extended form of fitness – inclusive fitness, which includes an individual's offspring as well as any offspring equivalents found in kin.
Hamilton went beyond kin relatedness to work with Robert Axelrod, analysing games of co-operation under conditions not involving kin where reciprocal altruism comes into play.[24]
Eusocial insect workers forfeit reproductive rights to their queen. It has been suggested that Kin Selection, based on the genetic makeup of these workers, may predispose them to altruistic behaviour.[25] Most eusocial insect societies have haplodiploidsexual determination, which means that workers are unusually closely related.[26]
This explanation of insect eusociality has however been challenged by a few highly noted evolutionary game theorists (Nowak and Wilson)[27] who have published a controversial alternative game theoretic explanation based on a sequential development and group selection effects proposed for these insect species.[28]
A difficulty of the theory of evolution, recognised by Darwin himself, was the problem of altruism. If the basis for selection is at individual level, altruism makes no sense at all. But universal selection at the group level (for the good of the species, not the individual) fails to pass the test of the mathematics of game theory and is certainly not the general case in nature.[29] Yet in many social animals, altruistic behaviour exists. The solution to this paradox can be found in the application of evolutionary game theory to the prisoner's dilemma game – a game which tests the payoffs of cooperating or in defecting from cooperation. It is certainly the most studied game in all of game theory.[30]
The analysis of prisoner's dilemma is as a repetitive game. This affords competitors the possibility of retaliating for defection in previous rounds of the game. Many strategies have been tested; the best competitive strategies are general cooperation with a reserved retaliatory response if necessary.[31] The most famous and one of the most successful of these is tit-for-tat with a simple algorithm.
procedure tit-for-tat EventBit:=Trust; do while Contest=ON; if Eventbit=Trust then Cooperate else Defect; if Opponent_Move=Cooperate then EventBit:=Trust else Eventbit:=NOT(Trust); end;
The pay-off for any single round of the game is defined by the pay-off matrix for a single round game (shown in bar chart 1 below). In multi-round games the different choices – Co-operate or Defect – can be made in any particular round, resulting in a certain round payoff. It is, however, the possible accumulated pay-offs over the multiple rounds that count in shaping the overall pay-offs for differing multi-round strategies such as Tit-for-Tat.
Example 1: The straightforward single round prisoner's dilemma game. The classic prisoner's dilemma game payoffs gives a player a maximum payoff if he defect and his partner co-operates (this choice is known as temptation). If however the player co-operates and his partner defects, he gets the worst possible result (the suckers payoff). In these payoff conditions the best choice (a Nash equilibrium) is to defect.
Example 2: Prisoner's dilemma played repeatedly. The strategy employed is Tit-for-Tat which alters behaviors based on the action taken by a partner in the previous round – i.e. reward co-operation and punish defection. The effect of this strategy in accumulated payoff over many rounds is to produce a higher payoff for both players co-operation and a lower payoff for defection. This removes the Temptation to defect. The suckers payoff also becomes less, although "invasion" by a pure defection strategy is not entirely eliminated.
Altruism takes place when one individual, at a cost C to itself, exercises a strategy that provides a benefit B to another individual. The cost may consist of a loss of capability or resource which helps in the battle for survival and reproduction, or an added risk to its own survival. Altruism strategies can arise through:
The evolutionarily stable strategy (ESS) is akin to Nash equilibrium in classical game theory, but with mathematically extended criteria. Nash Equilibrium is a game equilibrium where it is not rational for any player to deviate from his present strategy, provided that the others adhere to their strategies. An ESS is a state of game dynamics where, in a very large population of competitors, another mutant strategy cannot successfully enter the population to disturb the existing dynamic (which itself depends on the population mix). Therefore, a successful strategy (with an ESS) must be both effective against competitors when it is rare – to enter the previous competing population, and successful when later in high proportion in the population – to defend itself. This in turn means that the strategy must be successful when it contends with others exactly like itself.[37][38][39]
An ESS is not:
The ESS state can be solved for by exploring either the dynamics of population change to determine an ESS, or by solving equations for the stable stationary point conditions which define an ESS.[41] For example, in the Hawk Dove Game we can look for whether there is a static population mix condition where the fitness of Doves will be exactly the same as fitness of Hawks (therefore both having equivalent growth rates – a static point).
Let the chance of meeting a Hawk=p so therefore the chance of meeting a dove is (1-p)
Let WHawk equal the Payoff for Hawk.....
WHawk=Payoff in the chance of meeting a Dove + Payoff in the chance of meeting a Hawk
Taking the PAYOFF MATRIX results and plugging them into the above equation:
WHawk= V·(1-p)+(V/2-C/2)·p
Similarly for a Dove:
WDove= V/2·(1-p)+0·(p)
so....
WDove= V/2·(1-p)
Equating the two fitnesses, Hawk and Dove
V·(1-p)+(V/2-C/2)·p= V/2·(1-p)
... and solving for p
p= V/C
so for this "static point" where the Population Percent is an ESS solves to be ESS(percent Hawk)=V/C
Similarly, using inequalities, it can be shown that an additional Hawk or Dove mutant entering this ESS state eventually results in less fitness for their kind – both a true Nash and an ESS equilibrium. This example shows that when the risks of contest injury or death (the Cost C) is significantly greater than the potential reward (the benefit value V), the stable population will be mixed between aggressors and doves, and the proportion of doves will exceed that of the aggressors. This explains behaviours observed in nature.
An evolutionary game that turns out to be a children's game is rock-paper-scissors. The game is simple – rock beats scissors (blunts it), scissors beats paper (cuts it), and paper beats rock (wraps it up). Anyone who has ever played this simple game knows that it is not sensible to have any favoured play – the opponent will soon notice this and switch to the winning counter-play. The best strategy (a Nash equilibrium) is to play a mixed random game with any of the three plays taken a third of the time. This, in EGT terms, is a mixed strategy. But many lifeforms are incapable of mixed behavior – they only exhibit one strategy (known as a pure strategy). If the game is played only with the pure Rock, Paper and Scissors strategies the evolutionary game is dynamically unstable: Rock mutants can enter an all scissor population, but then – Paper mutants can take over an all Rock population, but then – Scissor mutants can take over an all Paper population – and on and on.... This is easily seen on the game payoff matrix, where if the paths of mutant invasion are noted, it can be seen that the mutant "invasion paths" form into a loop. This in triggers a cyclic invasion pattern.
Rock-paper-scissors incorporated into an evolutionary game has been used for modelling natural processes in the study of ecology.[42] Using experimental economics methods, scientists have used RPS game to test human social evolutionary dynamical behaviors in laboratory. The social cyclic behaviors, predicted by evolutionary game theory, have been observed in various laboratory experiments.[43][44]
The side-blotched lizard (Uta stansburiana) is polymorphic with three morphs[45] that each pursues a different mating strategy
However the blue throats cannot overcome the more aggressive orange throats. The overall situation corresponds to the Rock, Scissors, Paper game, creating a six-year population cycle. When he read that these lizards were essentially engaged in a game with rock-paper-scissors structure, John Maynard Smith is said to have exclaimed "They have read my book!"[46]
Aside from the difficulty of explaining how altruism exists in many evolved organisms, Darwin was also bothered by a second conundrum – why do a significant number of species have phenotypical attributes that are patently disadvantageous to them with respect to their survival – and should by the process of natural section be selected against – e.g. the massive inconvenient feather structure found in a peacock's tail? Regarding this issue Darwin wrote to a colleague "The sight of a feather in a peacock's tail, whenever I gaze at it, makes me sick."[47] It is the mathematics of evolutionary game theory, which has not only explained the existence of altruism but also explains the totally counterintuitive existence of the peacock's tail and other such biological encumbrances.
On analysis, problems of biological life are not at all unlike the problems that define economics – eating (akin to resource acquisition and management), survival (competitive strategy) and reproduction (investment, risk and return). Game theory was originally conceived as a mathematical analysis of economic processes and indeed this is why it has proven so useful in explaining so many biological behaviours. One important further refinement of the EGT model that has economic overtones rests on the analysis of COSTS. A simple model of cost assumes that all competitors suffer the same penalty imposed by the Game costs, but this is not the case. More successful players will be endowed with or will have accumulated a higher "wealth reserve" or "affordability" than less successful players. This wealth effect in evolutionary game theory is represented mathematically by "resource holding potential (RHP)" and shows that the effective cost to a competitor with higher RHP are not as great as for a competitor with a lower RHP. As a higher RHP individual is more desirable mate in producing potentially successful offspring, it is only logical that with sexual selection RHP should have evolved to be signalled in some way by the competing rivals, and for this to work this signalling must be done honestly. Amotz Zahavi has developed this thinking in what is known as the handicap principle,[48] where superior competitors signal their superiority by a costly display. As higher RHP individuals can properly afford such a costly display this signalling is inherently honest, and can be taken as such by the signal receiver. Nowhere in nature is this better illustrated than in the magnificent and costly plumage of the peacock. The mathematical proof of the handicap principle was developed by Alan Grafen using evolutionary game-theoretic modelling.[49]
Two types of dynamics have been discussed so far in this article:
A third, co-evolutionary, dynamic combines intra-specific and inter-specific competition. Examples include predator-prey competition and host-parasite co-evolution, as well as mutualism. Evolutionary game models have been created for pairwise and multi-species coevolutionary systems.[51] The general dynamic differs between competitive systems and mutualistic systems.
In competitive (non-mutualistic) inter-species coevolutionary system the species are involved in an arms race – where adaptations that are better at competing against the other species tend to be preserved. Both game payoffs and replicator dynamics reflect this. This leads to a Red Queen dynamic where the protagonists must "run as fast as they can to just stay in one place".[52]
A number of EGT models have been produced to encompass coevolutionary situations. A key factor applicable in these coevolutionary systems is the continuous adaptation of strategy in such arms races. Coevolutionary modelling therefore often includes genetic algorithms to reflect mutational effects, while computers simulate the dynamics of the overall coevolutionary game. The resulting dynamics are studied as various parameters are modified. Because several variables are simultaneously at play, solutions become the province of multi-variable optimisation. The mathematical criteria of determining stable points are Pareto efficiency and Pareto dominance, a measure of solution optimality peaks in multivariable systems.[53]
Carl Bergstrom and Michael Lachmann apply evolutionary game theory to the division of benefits in mutualistic interactions between organisms. Darwinian assumptions about fitness are modeled using replicator dynamics to show that the organism evolving at a slower rate in a mutualistic relationship gains a disproportionately high share of the benefits or payoffs.[54]
A mathematical model analysing the behaviour of a system needs initially to be as simple as possible to aid in developing a base understanding the fundamentals, or “first order effects”, pertaining to what is being studied. With this understanding in place it is then appropriate to see if other, more subtle, parameters (second order effects) further impact the primary behaviours or shape additional behaviours in the system. Following Maynard Smith’s seminal work in EGT, the subject has had a number of very significant extensions which have shed more light on understanding evolutionary dynamics, particularly in the area of altruistic behaviors. Some of these key extensions to EGC are:
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