Kin Selection

Status

Python-backed. This page describes the moran_models/nowak_mechanisms/kin_selection/ package in the sibling EvolvedCooperation repository. It is not yet a browser replay case study.

Kin selection is the first of Nowak's five mechanisms for the evolution of cooperation. Cooperation spreads when the benefit delivered to a recipient, weighted by genetic relatedness, exceeds the private cost paid by the actor — Hamilton's rule: $rB > C$.

How It Is Implemented Here

Relatedness is operationalised through lineage labels. Every site carries an inherited lineage identifier. The positive routing kernel assigns higher weight to same-lineage neighbors than to other-lineage neighbors:

$$K^+_{j \to i} \propto \begin{cases} w_{\text{same}} & \text{if lineage}_j = \text{lineage}_i \\ w_{\text{other}} & \text{otherwise} \end{cases}$$

with row normalization applied afterward. Because offspring inherit the parent's lineage label, cooperator clusters accumulate same-lineage neighbors over time, which progressively recirculates more of the cooperative benefit back toward cooperators — the positive feedback that makes cooperation viable.

One Step

One full synchronous kin-selection update runs as follows. All sites update simultaneously.

Kin Selection Simulation Step

One synchronous grid update from step t to step t + 1 under kin-selection routing.

1

Start from the current grid state

Each site stores one cooperation trait $h$ and one inherited lineage label.

↓

2

Compute cooperative output and private cost

$B_i^+ = B_{\text{plus}} \cdot h_i, \qquad C_i = C_{\text{scale}} \cdot h_i$

↓

3

Build the kin-weighted routing kernel

Assign raw weight $w_{\text{same}}$ to same-lineage neighbors and $w_{\text{other}}$ to other-lineage neighbors, then row-normalize:

$w_{ij} = \begin{cases} w_{\text{same}} & \text{if } \text{lineage}_i = \text{lineage}_j \\ w_{\text{other}} & \text{otherwise} \end{cases} \qquad K_{ij}^+ = \frac{w_{ij}}{\sum_k w_{ik}}$

↓

4

Accumulate lineage-weighted receipts

Each site sums the kin-weighted share of cooperative benefit received from its neighbors:

$R_i^+ = \sum_j K_{ji}^+ \cdot B_j^+$

↓

5

Compute site fitness

$W_i = w_0 + R_i^+ - C_i$

Baseline fitness $w_0$ dampens selection intensity.

↓

6

Sample a local parent by softmax over fitness, copy trait and lineage

Parent selection is restricted to the site's local neighborhood. The offspring inherits trait $h$ (with small Gaussian mutation) and lineage label — same-lineage clusters grow when cooperators reproduce locally.

Display 1: One synchronous kin-selection update from step t to step t + 1.

Production

Each site produces cooperative output and pays a private cost proportional to its trait:

$B_i^+ = B_{\text{plus}} \cdot h_i$

$C_i = C_{\text{scale}} \cdot h_i$

Kernel construction

For each producer $i$, a raw routing weight $w_{ij}$ is assigned to each neighbor $j$ based on lineage match, then row-normalized so weights sum to 1:

$w_{ij} = \begin{cases} w_{\text{same}} & \text{if } \text{lineage}_i = \text{lineage}_j \\ w_{\text{other}} & \text{otherwise} \end{cases}$

$K_{ij}^+ = \frac{w_{ij}}{\sum_k w_{ik}}$

Routing

Each site receives the lineage-weighted share of every neighbor's production:

$R_i^+ = \sum_j K_{ji}^+ \cdot B_j^+$

Fitness score

$W_i = w_0 + R_i^+ - C_i$

Local replacement

Each site samples a parent from its local neighborhood via softmax over $W$. The offspring inherits the parent's trait $h$ (with small Gaussian mutation) and lineage label. Because the lineage label is inherited, same-lineage clusters grow when local cooperators outcompete their neighbors — the feedback that sustains cooperation.

Variable definitions:

• $h_i$ is site $i$'s cooperation trait in [0, 1]
• $B_i^+$ is the cooperative benefit produced by site $i$
• $C_i$ is the private cost paid by site $i$
• $K_{ij}^+$ is the normalized routing weight from producer $i$ to recipient $j$
• $R_i^+$ is the total routed benefit received by site $i$
• $W_i$ is the fitness score used for local replacement
• $w_0$ is the fixed baseline fitness shared by all sites, which dampens selection intensity

Key Parameters

Parameter	Default	Role
kin_weight_same_lineage	0.8	Routing weight toward same-lineage neighbors
kin_weight_other_lineage	0.2	Routing weight toward other-lineage neighbors
B_plus_scale	1.0	Scales cooperative benefit produced per unit trait
C_scale	0.2	Private cost per unit trait

Hamilton's rule maps onto these parameters as $r \approx w_{\text{same}} / (w_{\text{same}} + w_{\text{other}})$, $B =$ B_plus_scale, $C =$ C_scale.

Python Module Layout

moran_models/nowak_mechanisms/kin_selection/
  __init__.py
  kin_selection_model.py
  kin_selection_pygame_ui.py
  config/
    kin_selection_config.py
  utils/
    sweep_kin_selection_phase.py

Usage

./.conda/bin/python -m moran_models.nowak_mechanisms.kin_selection.kin_selection_model

Live viewer:

./.conda/bin/python -m moran_models.nowak_mechanisms.kin_selection.kin_selection_pygame_ui

References

• Hamilton, W. D. (1964). The genetical evolution of social behaviour. I. Journal of Theoretical Biology, 7(1), 1–16. https://doi.org/10.1016/0022-5193(64)90038-4
• Nowak, M. A. (2006). Five rules for the evolution of cooperation. Science, 314(5805), 1560–1563. https://doi.org/10.1126/science.1133755