水曜日, 1月 09, 2019

Smith&Price (1973) “The Logic of Animal Conflict”進化ゲーム


進化ゲーム

Smith&Price (1973) “The Logic of Animal Conflict”



John Maynard Smith - Game theory: The war of attrition game (46/102)




Game theory and evolution (1973) - ResearchGate

www.researchgate.net/.../251172760_Game_theory_an...

Download Citation on ResearchGate | Game theory and evolution (1973) | In 1973 Maynard Smith and Price published ...

Evolution and learning in games - MIT OpenCourseWare

 

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ocw.mit.edu/...game-theory.../MIT6_254S10_lec10.pdf

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 ...

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-254-game-theory-with-engineering-applications-spring-2010/lecture-notes/MIT6_254S10_lec10.pdf

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”


  1.  Maynard-Smith, J.; Price, G. R. (1973). "The Logic of Animal Conflict". Nature246 (5427): 15–18. Bibcode:1973Natur.246...15Sdoi:10.1038/246015a0.

進化ゲームと社会制度に関するノート

 

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hannan-u.repo.nii.ac.jp/index.php?...

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.

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 ...

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www.est.ryukoku.ac.jp/est/kondoh/lcts/lcts_siryo/.../evo_6.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 AX 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。

  



もっとも美しい数学 ゲーム理論 (文春文庫)

もっとも美しい数学 ゲーム理論 (文春文庫)

ゲーム理論が進化生物学で成功したのは,効用を戦略の頻度増加という形できちんと定義できたことによるところが大きい.その利得が次世代の戦略頻度分布を変え,さらに戦略の平均利得を変えるという構造になっているから厳密に解析可能になり,そしてそれが,頻度依存する戦略の進化の分析を可能にしたのだ.その最初の適応例であるタカハトゲームは個体が自分の利益を優先するときにタカハトの混合戦略というESS解がある(つまり常に抗争するという状況より平和的になり得る)ことを示している.
しかし囚人ジレンマの話はまったく文脈が別だ.このゲームのESSはランダム対戦の場合「常に裏切り戦略」であることがはっきりしている.繰り返しゲーム,あるいは空間構造があるゲームの問題は,対戦者同士の戦略に何らかの相関が現れたときに協力が進化しうるという問題で,基本的には包括適応度理論の問題なのだ.ジーグフリードはここが理解できていない.*2 そのためにこの部分は大変浅い記述にとどまっている.



             (リンク:::::::::数学経済学

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
http://www.yuhikaku.co.jp/books/detail/9784641163829

岡田 章 (一橋大学教授)/著 2011年12月発売
A5判並製カバー付 , 496ページ 
定価 4,104円(本体 3,800円)
ISBN 978-4-641-16382-9
Game Theory

第1章 ゲーム理論とは何か
第2章 戦略形ゲーム ☆
第3章 展開形ゲーム
第4章 完全均衡点
第5章 情報不完備ゲーム
第6章 繰り返しゲーム
第7章 期待効用理論
第8章 交渉ゲーム
第9章 コアの理論
第10章 他の協力ゲーム解
第11章 進化ゲーム
第12章 ゲーム理論のフロンティア 


#11-405スミス&プライス
下のスミスとは別人

#11-440
鹿狩りゲーム

https://nam-students.blogspot.com/2019/01/blog-post_58.html
瀧澤弘和『現代経済学』より[上下逆にして改変]: 
図2-4 ゲーム理論の影響を受けた諸分野とノーベル賞

        1988 モーリス・アレ 市場と資源の効率的利用に関する理論 

*       1991 ロナルド・コース 取引費用経済学
ゲーム理論___________________________________
1994                   人間行動・制度     I    I
ハーサニ/ナッシュ
/ゼルテン                  行動ゲーム理論 I
      \                         実験ゲーム理論 I 
       \インセンティブ・制度設計               I    I
        \                          I    I
        /\___情報の非対称性
               I    I
       /  \  1996マーリーズ/ヴィックリー      I    I
      /    \ 2001アカロフ/スペンス/スティグリッツ I    I
     /      \              I       I    I
    /        \             I 行動経済学実験経済学
 対立と協力        \            I 2002       I
 2005         /\           I カーネマン/スミス  I
 オーマン/シェリング  /  \          I I          I
            /    \         I I          I
  マーケット・デザイン  メカニズム・デザイン   I I          I
       / \      2007       I I 経済ガバナンスの理論
      /   \   ハーヴィッツ/マスキン/ I I 2009
サーチ理論 2010 \  マイヤーソン  \    I I オストロム
 ダイアモンド/    \          \   I I ウィリアムソン
 モーテンセン/ピサリデス\          \  I I
マッチング理論 2012  オークション理論  契約理論 I 
 ロス/シャープレイ              2014 I
                       ティロール I
                        2016 行動経済学 ナッジ理論
                ハートホルムストローム 2017
                             リチャード・セイラー

上とは別のスミス


進化ゲーム(しんかげーむ、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。

概要編集

たとえば、次のチキンゲームを考えてみよう。

1・2AB
A1,1-2,5
B5,-2-8,-8

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。

レプリケータダイナミクス編集

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。

{\dot {x_{i}}}(t)=x_{i}(t)\{f_{i}(x(t))-\phi (x(t))\},\quad \phi (x(t))=\sum _{i=1}^{n}{x_{i}(t)f_{i}(x(t))}

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\phi は集団の平均適応度である。

時間間隔\tau 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。

x_{i}(t+\tau )=x_{i}(t){\frac {f_{i}(x(t))}{\phi (x(t))}}


関連項目編集


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 economistssociologistsanthropologists, and philosophers.

Contents

HistoryEdit

Classical game theoryEdit

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]

The problem of ritualized behaviourEdit

The mathematical biologist John Maynard Smith modelled evolutionary games.

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 speciesJohn 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]

Adapting game theory to evolutionary gamesEdit

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]

Evolutionary gamesEdit

ModelsEdit

Evolutionary game theory analyses Darwinian mechanisms with a system modelwith three main components – Population, Game, and Replicator Dynamics. The system process has four phases:

1) The model (as evolution itself) deals with a Population (Pn). The population will exhibit Variation among Competing individuals. In the model this competition is represented by the Game.

2) The Game tests the strategies of the individuals under the “rules of the game”. These rules produce different payoffs – in units of Fitness (the production rate of offspring). The contesting individuals meet in pairwise contests with others, normally in a highly mixed distribution of the population. The mix of strategies in the population affects the payoff results by altering the odds that any individual may meet up in contests with various strategies. The individuals leave the game pairwise contest with a resulting fitness determined by the contest outcome, represented in a Payoff Matrix.

3) Based on this resulting fitness each member of the population then undergoes replication or culling determined by the exact mathematics of the Replicator Dynamics Process. This overall process then produces a New Generation P(n+1). Each surviving individual now has a new fitness level determined by the game result.

4) The new generation then takes the place of the previous one and the cycle repeats. The population mix may converge to an Evolutionarily Stable State that cannot be invaded by any mutant strategy.

EGT encompasses Darwinian evolution, including competition (the game), natural selection (replicator dynamics), and heredity. EGT has contributed to the understanding of group selectionsexual selectionaltruismparental careco-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 infinitepopulationscontinuous timecomplete 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 huntproducer-scroungertragedy 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.

Hawk DoveEdit

Solution of the Hawk Dove game for V=2, C=10 and fitness starting base B=4. The fitness of a Hawk for different population mixes is plotted as a black line, that of Dove in red. An ESS (a stationary point) will exist when Hawk and Dove fitness are equal: Hawks are 20% of population and Doves are 80% of the population.

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]

Payoff Matrix for Hawk Dove Game
meets Hawk meets Dove 
if HawkV/2 − C/2V
if Dove0V/2

Given that the resource is given the value V, the damage from losing a fight is given cost C:[1]

  • If a Hawk meets a Dove he gets the full resource V to himself
  • If a Hawk meets a Hawk – half the time he wins, half the time he loses...so his average outcome is then V/2 minus C/2
  • If a Dove meets a Hawk he will back off and get nothing – 0
  • If a Dove meets a Dove both share the resource and get V/2

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]

War of attritionEdit

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]

War of attrition for different values of resource. Note the time it takes for an accumulation of 50% of the contestants to quit vs. the Value(V) of resource contested for.

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:

{\displaystyle p(x)={\frac {e^{-x/V}}{V}}.}

The result is that the cumulative population of quitters for any particular cost m in this "mixed strategy" solution is:

{\displaystyle p(m)=1-e^{-m/V},}

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]

Asymmetries that allow new strategiesEdit

Dung Fly (Scatophaga stercoraria) – a War of Attrition player
The mantis shrimp guarding its home with the Bourgeois Strategy
Animal Strategy Examples: by examining the behaviours, then determining both the Costs and the Value of resources attained in a contest the strategy of an organism can be verified

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.

Social behaviourEdit

Alternatives for game theoretic social interaction

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.

  • In a Cooperative or Mutualistic relationship both "donor" and "recipient" are almost indistinguishable as both gain a benefit in the game by co-operating, i.e. the pair are in a game-wise situation where both can gain by executing a certain strategy, or alternatively both must act in concert because of some encompassing constraints that effectively puts them "in the same boat".
  • In an Altruistic relationship the donor, at a cost to himself provides a benefit to the recipient. In the general case the recipient will have a kin relationship to the donor and the donation is one-way. Behaviours where benefits are donated alternatively (in both directions) at a cost, are often called altruistic, but on analysis such "altruism" can be seen to arise from optimised "selfish" strategies
  • Spite is essentially a "reversed" form of altruism where an ally is aided by damaging the ally's competitor(s). The general case is that the ally is kin related and the benefit is an easier competitive environment for the ally.  Note: George Price, one of the early mathematical modellers of both altruism and spite, found this equivalence particularly disturbing at an emotional level.[19]
  • Selfishness is the base criteria of all strategic choice from a game theory perspective – strategies not aimed at self-survival and self-replication are not long for any game. Critically however, this situation is impacted by the fact that competition is taking place on multiple levels – i.e. at a genetic, an individual and a group level.

Contests of selfish genesEdit

Female Belding's ground squirrelsrisk their lives giving loud alarm calls, protecting closely related female colony members; males are less closely related and do not call.[20]

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.

The Mathematics of Kin Selection
The concept of Kin Selection is that:
inclusive fitness=own contribution to fitness + contribution of all relatives.

Fitness is measured relative to the average population; for example, fitness=1 means growth at the average rate for the population, fitness < 1 means having a decreasing share in the population (dying out), fitness > 1 means an increasing share in the population (taking over).

The inclusive fitness of an individual wi is the sum of its specific fitness of itself ai plus the specific fitness of each and every relative weighted by the degree of relatedness which equates to the summation of all rj*bj....... where rj is relatedness of a specific relative and bj is that specific relative's fitness – producing:

w_{i}=a_{i}+\sum _{{j}}r_{j}b_{j}.

Now if individual ai sacrifices his "own average equivalent fitness of 1" by accepting a fitness cost C, and then to "get that loss back", wi must still be 1 (or greater than 1)...and if we use R*B to represent the summation we get:

1< (1-C)+RB ....or rearranging.....  R>C/B.[23]

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]

Eusociality and kin selectionEdit

Meat ant workers (always female) are related To mother or father=0.5 To sister+=0.75 To own daughter or son=0.5 To brother=0.25....... Therefore, it is more advantageous to help produce a sister than to have a child oneself.

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]

Prisoner's dilemmaEdit

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.

Payoffs in two varieties of prisoner's dilemma game. 
Prisoner's dilemma: Co-operate or Defect? 
Payoff (Temptation in Defecting vs. Co-operation) > Payoff (Mutual Co-operation) > Payoff(Joint Defection) > Payoff(Sucker co-operates but opponent defects)

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.

Routes to altruismEdit

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:

TypeApplies to:SituationMathematical effect
Kin Selection – (inclusive fitness of related contestants)Kin – genetically related individualsEvolutionary Game participants are genes of strategy. The best payoff for an individual is not necessarily the best payoff for the gene. In any generation the player gene is NOT only in one individual, it is in a Kin-Group. The highest fitness payoff for the Kin Group is selected by natural selection. Therefore, strategies that include self-sacrifice on the part of individuals are often game winners – the evolutionarily stable strategy. Animals must live in kin-group during part of the game for the opportunity for this altruistic sacrifice ever to take place.Games must take into account Inclusive Fitness. Fitness function is the combined fitness of a group of related contestants – each weighted by the degree of relatedness – relative to the total genetic population. The mathematical analysis of this gene centric view of the game leads to Hamilton's rule, that the relatedness of the altruistic donor must exceed the cost-benefit ratio of the altruistic act itself:[32]
R>c/b R is relatedness, c the cost, b the benefit
Direct reciprocityContestants that trade favours in paired relationshipsA game theoretic embodiment of "I'll scratch your back if you scratch mine". A pair of individuals exchange favours in a multi-round game. The individuals are recognisable to one another as partnered. The term "direct" applies because the return favour is specifically given back to the pair partner only.The characteristics of the multi-round game produce a danger of defection and the potentially lesser payoffs of cooperation in each round, but any such defection can lead to punishment in a following round – establishing the game as repeated prisoner's dilemma. Therefore, the family of tit-for-tat strategies come to the fore.[33]
Indirect ReciprocityRelated or non related contestants trade favours but without partnering. A return favour is "implied" but with no specific identified source who is to give it.This behaviour is akin to "I'll scratch your back, you scratch someone else's back, another someone else will scratch mine (probably)". The return favour is not derived from any particular established partner. The potential for indirect reciprocity exists for a specific organism if it lives in a cluster of individuals who can interact over an extended period of time.

It has been argued that human behaviours in establishing moral system as well as the expending of significant energies in human society for tracking individual reputation is a direct effect of societies reliance on strategies of indirect reciprocation.[34]

The game is highly susceptible to defection, as direct retaliation is impossible. Therefore, indirect reciprocity will not work without keeping a social score, a measure of past co-operative behaviour. The mathematics leads to a modified version of Hamilton's Rule where:
q>c/b where q (the probability of knowing the social score) must be greater than the cost benefit ratio[35][36]

Organisms that use social score are termed Discriminators, and require a higher level of cognition than strategies of simple direct reciprocity. As evolutionary biologist David Haig put it – "For direct reciprocity you need a face; for indirect reciprocity you need a name".

The evolutionarily stable strategyEdit

The Payoff Matrix for the Hawk Dove Game with the addition of the Assessor Strategy. This "studies its opponent", behaving as a Hawk when matched with an opponent it judges "weaker", like a Dove when the opponent seems bigger and stronger. Assessor is an ESS, since it can invade both Hawk and Dove populations, and can withstand invasion by either Hawk or Dove mutants.

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:

  • An optimal strategy: that would maximize Fitness, and many ESS states are far below the maximum fitness achievable in a fitness landscape. (see Hawk Dove graph above as an example of this)
  • A singular solution: often several ESS conditions can exist in a competitive situation. A particular contest might stabilize into any one of these possibilities, but later a major perturbation in conditions can move the solution into one of the alternative ESS states.
  • Always present: it is possible for there to be no ESS. An evolutionary game with no ESS is Rock-Scissors-Paper, as found in species such as the side-blotched lizard (Uta stansburiana).
  • An unbeatable strategy: the ESS is only an uninvadeable strategy.
Female funnel web spiders (Agelenopsis aperta) contest with one another for the possession of their desert spider webs using the Assessor strategy.[40]

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:

WHawkV·(1-p)+(V/2-C/2)·p

Similarly for a Dove:

WDoveV/2·(1-p)+0·(p)

so....

WDoveV/2·(1-p) 

Equating the two fitnesses, Hawk and Dove

V·(1-p)+(V/2-C/2)·pV/2·(1-p) 

... and solving for p

pV/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.

Unstable games, cyclic patternsEdit

Rock-paper-scissorsEdit

Rock Paper Scissors
Mutant Invasion for Rock Paper Scissors payoff matrix – an endless cycle

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.

A computer simulation of the Rock Scissors Paper game. The associated RPS Game Payoff Matrix is shown. Starting with an arbitrary population the percentage of the three morphs builds up into a continuously cycling 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 lizardEdit

The side-blotched lizard (Uta stansburiana) is polymorphic with three morphs[45] that each pursues a different mating strategy

The side-blotched lizard effectively uses a rock-paper-scissors mating strategy.
1) The orange throat is very aggressive and operates over a large territory – attempting to mate with numerous females within this larger area
2) The unaggressive yellow throat mimics the markings and behavior of female lizards, and "sneakily" slips into the orange throat's territory to mate with the females there (thereby taking over the population), and
3) The blue throat mates with and carefully guards one female – making it impossible for the sneakers to succeed and therefore overtakes their place in a population…

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]

Signalling, sexual selection and the handicap principleEdit

The peacock's tail may be an instance of the handicap principle in action.

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 honestlyAmotz 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]

Co-evolutionEdit

Two types of dynamics have been discussed so far in this article:

  • Evolutionary games which lead to a stable situation or point of stasis for contending strategies which result in an evolutionarily stable strategy
  • Evolutionary games which exhibit a cyclic behaviour (as with RPS game) where the proportions of contending strategies continuously cycle over time within the overall population
Competitive Co-evolution - The rough-skinned newt (Tarricha granulosa) is highly toxic, due to an evolutionary arms race with a predator, the common garter snake(Thamnophis sirtalis), which in turn is highly tolerant of the poison. The two are locked in a Red Queen arms race.[50]
Mutualistic Coevolution - Darwin's orchid (Angraecum sesquipedale) and the moth Morgan's sphinx(Xanthopan morgani) have a mutual relationship where the moth gains pollen and the flower is pollinated.

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]

Extending the modelEdit

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:

A Spatial Game
In a spatial evolutionary game contestants meet in contests at fixed grid positions and only interact with immediate neighbors. Shown here are the dynamics of a Hawk Dove contest, showing Hawk and Dove contestants as well as the changes of strategy taking place in the various cells
Spatial GamesGeographic factors in evolution include gene flow and horizontal gene transfer. Spatial game models represent geometry by putting contestants in a lattice of cells: contests take place only with immediate neighbours. Winning strategies take over these immediate neighbourhoods and then interact with adjacent neighbourhoods. This model is useful in showing how pockets of co-operators can invade and introduce altruism in the Prisoners Dilemma game,[55] where Tit for Tat (TFT) is a Nash Equilibrium but NOT also an ESS. Spatial structure is sometimes abstracted into a general network of interactions.[56][57] This is the foundation of evolutionary graph theory.
Effects of having informationIn EGT as in conventional Game Theory the effect of Signalling (the acquisition of information) is of critical importance, as in Indirect Reciprocity in Prisoners Dilemma (where contests between the SAME paired individuals are NOT repetitive). This models the reality of most normal social interactions which are non-kin related. Unless a probability measure of reputation is available in Prisoners Dilemma only direct reciprocity can be achieved.[32] With this information indirect reciprocity is also supported.

Alternatively, agents might have access to an arbitrary signal initially uncorrelated to strategy but becomes correlated due to evolutionary dynamics. This is the green-beard effect or evolution of ethnocentrism in humans.[58] Depending on the game, it can allow the evolution of either cooperation or irrational hostility.[59]

From molecular to multicellular level, a signaling game model with information asymmetry between sender and receiver might be appropriate, such as in mate attraction[49] or evolution of translation machinery from RNA strings.[60]

Finite populationsMany evolutionary games have been modelled in finite populations to see the effect this may have, for example in the success of mixed strategies.

See alsoEdit

NotesEdit

  1. ^ Maynard Smith chose the name Hawk Dove from descriptions of political views current during the Vietnam war.

ReferencesEdit

  1. a b c d e Maynard-Smith, J.; Price, G. R. (1973). "The Logic of Animal Conflict". Nature246 (5427): 15–18. Bibcode:1973Natur.246...15Sdoi:10.1038/246015a0.
  2. ^ Newton, Jonathan (2018). "Evolutionary Game Theory: A Renaissance". Games9 (2). doi:10.3390/g9020031.
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Further readingEdit

  • Sigmund, Karl; "Games of Life", Penguin Books, also Oxford University Press, 1993, ISBN 0198547838
  • Dawkins, Richard; "The Selfish Gene", Oxford University Press, ISBN 0-19-929114-4
  • Davis, Morton,; "Game Theory – A Nontechnical Introduction", Dover Books, ISBN 0-486-29672-5
  • Kohn, Marek; "A Reason for Everything", Faber and Faber, ISBN 0-571-22393-1
  • Segerstrale, Ullica; "Nature's Oracle - The life and work of W.D. Hamilton", Oxford University Press, 2013, ISBN 978-0-19-860727-4
  • Dugatkin and Reeve; "Game Theory and Animal Behavior", Oxford University Press, ISBN 0-19-513790-6
  • Vincent and Brown; "Evolutionary Game Theory, Natural Selection and Darwinian Dynamics", Cambridge University Press, ISBN 0-521-84170-4
  • Hofbauer and Sigmund; "Evolutionary Games and Population Dynamics", Cambridge University Press, ISBN 0-521-62570-X

External linksEdit