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Hamilton's Rule

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Inclusive Fitness Kin Selection and Hamilton's Rule→→Evolution of Altruism

Core Idea

Hamilton's rule (rb > c) predicts that altruistic behavior evolves when the benefit to recipients weighted by relatedness (rb) exceeds the cost to the altruist (c). This quantitative framework explains why sterile castes evolve in eusocial insects and how cooperation can be evolutionarily stable.

Explainer

From inclusive fitness, you already understand that an organism's evolutionary success is not measured solely by its own offspring but includes the reproduction of relatives who share its genes. Hamilton's rule translates this insight into a precise, testable inequality: rb > c. Here, r is the coefficient of relatedness between the altruist and the recipient (the probability that they share a given allele by common descent), b is the reproductive benefit the recipient gains from the altruistic act, and c is the reproductive cost the altruist pays. When the left side exceeds the right, the gene for altruism spreads — even though the altruist personally suffers a fitness cost.

The intuition behind the rule is bookkeeping. Imagine you carry an allele that makes you help your sibling at a cost to yourself. Your sibling shares half your genes (r = 0.5), so any reproductive benefit they gain has a 50% chance of propagating copies of that same allele. If helping your sibling doubles their reproductive output (b = 2 extra offspring) at a cost of one of your own offspring (c = 1), the math works out: 0.5 × 2 = 1 > 1... actually that is exactly equal. The allele breaks even. But if the benefit is slightly larger or the cost slightly lower, the allele spreads. The biologist J.B.S. Haldane reportedly quipped that he would lay down his life for two brothers or eight cousins — a colorful statement of Hamilton's rule, since r = 0.5 for siblings (2 × 0.5 = 1) and r = 0.125 for cousins (8 × 0.125 = 1).

The most dramatic application of Hamilton's rule is explaining eusociality in the Hymenoptera (ants, bees, wasps). Due to their haplodiploid sex-determination system, female hymenopterans share three-quarters of their genes with their sisters (r = 0.75) but only half with their own daughters (r = 0.5). Hamilton's rule predicts that a female in a haplodiploid species gains more inclusive fitness by helping her mother produce sisters than by reproducing herself — which is exactly the behavior of sterile worker castes. This elegant explanation of one of evolution's greatest puzzles was Hamilton's original motivation for formalizing the rule. However, eusociality has also evolved in diploid species (naked mole-rats, some shrimp), indicating that high relatedness facilitates but does not require haplodiploidy.

Hamilton's rule is powerful because it is general: it applies to any social behavior where costs and benefits can be quantified in fitness terms. It predicts not only when altruism should evolve but when it should *not* — organisms should be less altruistic toward distant relatives and more willing to help when costs are low. Empirical tests across species — from alarm calls in ground squirrels to food sharing in vampire bats — broadly support these predictions. The rule also clarifies the boundary between kin-selected altruism and selfishness: when rb < c, selection favors individuals who refuse to help, even at the expense of relatives. Understanding Hamilton's rule provides the quantitative foundation for all subsequent work on the evolution of cooperation, conflict, and social behavior.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → One-to-One Correspondence → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Making 10 as an Addition Strategy → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Iterated Integrals and Fubini's Theorem → Double Integrals in Cartesian Coordinates → Double Integrals in Polar Coordinates → Double Integrals in Polar Coordinates → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Postulates of Special Relativity → Time Dilation → Length Contraction → Lorentz Transformation → Relativistic Velocity Addition → Relativistic Momentum and Energy → Mass-Energy Equivalence and E=mc² → Photons as Particles with Energy and Momentum → Planck-Einstein Relation: Energy and Frequency → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → The Measurement Problem → Interpretations of Quantum Mechanics → Postulates of Quantum Mechanics → Observables and Quantum Operators → Commutators and Commutation Relations → Quantum Angular Momentum → Quantum Mechanical Treatment of Hydrogen → Solving the Schrödinger Equation for Hydrogen Atom → Quantum Numbers → Electron Configuration → Periodic Trends → Covalent Bonding → Electronegativity and Bond Polarity → Ionic Bonding → Lewis Structures → VSEPR Theory and Molecular Geometry → Molecular Geometry and Electron Pair Geometry → Molecular Polarity and Dipole Moments → Intermolecular Forces → States of Matter and Phase Changes: Melting, Boiling, and Sublimation → Gas Laws and the Ideal Gas Equation → Gas Stoichiometry and Volume-Volume Calculations → Thermochemistry and Enthalpy → Heat Capacity and Calorimetry → Entropy and Molecular Disorder → Spontaneity and ΔG → Entropy and Gibbs Free Energy → Chemical Equilibrium → Acid-Base Chemistry → Weak Acid Ionization → Weak Base Ionization → Acid and Base Strength: Ka, Kb, and Ionization → Leaving Groups and Nucleofugality → SN2 Substitution Reactions → SN1 Substitution Reactions → E1 Elimination Reactions → Alcohols and Ethers: Structure, Properties, and Nomenclature → Reactions of Alcohols → Aldehydes and Ketones: Structure and Reactivity → Oxidation Reactions in Organic Chemistry → Oxidation of Alcohols to Aldehydes and Ketones → Aldehyde and Ketone Structure and Nomenclature → Nucleophilic Addition to Aldehydes and Ketones → Carboxylic Acids and Their Derivatives → IUPAC Nomenclature of Carbonyls and Carboxylic Acids → IUPAC Nomenclature of Alkenes → Electrophilic Addition to Alkenes → Aromaticity and Benzene → Electrophilic Aromatic Substitution (EAS) → Nucleophilic Aromatic Substitution (SNAr) → Nucleophilic Acyl Substitution → Amines: Structure, Basicity, and Reactions → Amine Reactivity: Nucleophilicity and Basicity → Amino Acid Structure and Properties → Peptide Bonds and Polypeptide Formation → Protein Primary Structure → Protein Secondary Structure → Protein Tertiary Structure → Enzyme Structure and Function → Transcription: DNA to RNA → RNA Types and Structure → RNA Structure and Intramolecular Base Pairing → RNA Processing and Splicing → Translation: RNA to Protein → Ribosomes: Protein Synthesis Machines → Translation: Initiation and Elongation → Post-Translational Modifications → Proteasomal Degradation and Ubiquitin-Mediated Marking → Cell Cycle Regulation and Checkpoints → Mitosis → Cytokinesis → Meiosis → Chromosomal Theory of Inheritance → Mendelian Genetics → Dominance, Recessiveness, and Allelic Interactions → Sex-Linked Inheritance → Non-Mendelian Inheritance Patterns → Population Genetics and Hardy-Weinberg Equilibrium → Natural Selection → Genetic Drift → Evolutionary Genetics Foundations → Allele Frequency Change and Evolutionary Dynamics → Gene Flow and Population Structure → Gene Flow and Selection: Opposing Forces → Gene Flow → Hardy-Weinberg Equilibrium → Speciation → Phylogenetics and Evolutionary Trees → Cladistics and Biological Classification → Molecular Evolution and Molecular Clocks → The Neutral Theory of Molecular Evolution → Nearly Neutral Evolution and Drift-Selection Balance → Codon Usage Bias and Selection → Kin Selection Theory → Inclusive Fitness → Hamilton's Rule

Longest path: 241 steps · 1403 total prerequisite topics

Prerequisites (2)

Inclusive Fitnesshard Kin Selection and Hamilton's Rulesoft

Leads To (1)

Evolution of Altruismhard