Truel: what happens when three people duel?

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ACKNOWLEDGEMENT

I would take this opportunity to thank my research supervisor, family and friends for their support and guidance without which this research would not have been possible.

DECLARATION

I, [type your full first names and surname here], declare that the contents of this dissertation/thesis represent my own unaided work, and that the dissertation/thesis has not previously been submitted for academic examination towards any qualification. Furthermore, it represents my own opinions and not necessarily those of the University.

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ABSTRACT

The persistence of extensive variation in nature seems to stand against the most general principle of evolution by natural selection: in antagonistic interactions, the stronger type is expected to replace the weaker. Game theory shows that, however, in contrast to this intuitive expectation for interactions between two players, strategic considerations on fitness maximization in repeated pairwise interactions between three players (truels) or more (N-person duels) lead to what can be dubbed “survival of the weakest”: the weakest individual can have the highest fitness. A paradox arises: competitive skills cannot be improved by natural selection, unless we assume mutations with strong effects or unless we assume that interactions are exclusively between two individuals. The paradox disappears, however, with more realistic assumptions (a mixture of duels and truels; the attacked individual backfires; the contest can end without a winner; defensive and offensive skills are correlated; players not directly involved in the contest suffer collateral damage). An unexpected new result emerges: the weaker types can persist in a population in the absence of recurrent mutations, migration, and fluctuating selection. Game theory and the analysis of N-person duels, therefore, help understand one of the most enduring puzzles in evolutionary biology: the maintenance of variation under constant selection.

TABLE OF CONTENT

ACKNOWLEDGEMENTii

DECLARATIONiii

ABSTRACTiv

CHAPTER 1: INTRODUCTION6

The Truel: Survival Of The Weakest7

CHAPTER 2: LITERATURE REVIEW11

All about Truels12

The Duels14

Truel Strategies15

Truels as a Model of Opinion Spreading19

Distribution of Winners20

Generalization to N Players20

Truels with Spatial Dependence21

CHAPTER 3: METHODOLOGY24

Truels In Evolving Populations24

Limited Truels: Contests Can End Without A Winner25

Interference Truels: Collateral Damage26

REFERENCES28

CHAPTER 1: INTRODUCTION

The most general principle of evolution by natural selection is what Darwin (1869) dubbed “survival of the fittest” (Spencer 1864): the stronger types in the struggle for survival and reproduction increase in frequency, thereby leading to the design we see in nature. Although in certain cases selection can improve robustness (e.g., Wilke et al. 2001) or antirobustness (e.g., Archetti 2009) of genotypes at the expenses of fitness, it is understood that if a stronger type has an advantage over a weaker type in an antagonistic interaction it also has the highest probability of surviving.

Consider the duel as a model for antagonistic interactions. Two individuals, A and B, shoot at each other, with accuracies (probabilities to hit the opponent) a and b, respectively. If they shoot at the same time and a > b, clearly A has a higher probability of winning the contest. Consider then a sequential, repeated duel: at the beginning, and after each shot, who shoots next is chosen at random: again clearly A has a higher ...