https://github.com/giakoumoglou/game_theory_spatial_pd
[Nature 1992] Evolutionary Games and Spatial Chaos
https://github.com/giakoumoglou/game_theory_spatial_pd
evolutionary-games game-theory martin matlab nowak simulation
Last synced: about 1 year ago
JSON representation
[Nature 1992] Evolutionary Games and Spatial Chaos
- Host: GitHub
- URL: https://github.com/giakoumoglou/game_theory_spatial_pd
- Owner: giakoumoglou
- Created: 2020-06-10T11:38:47.000Z (about 6 years ago)
- Default Branch: master
- Last Pushed: 2024-07-15T14:42:30.000Z (almost 2 years ago)
- Last Synced: 2025-03-28T06:04:18.339Z (over 1 year ago)
- Topics: evolutionary-games, game-theory, martin, matlab, nowak, simulation
- Language: MATLAB
- Homepage: https://www.nature.com/articles/359826a0
- Size: 3.52 MB
- Stars: 8
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
Awesome Lists containing this project
README
# 1. Game Theory: Spatial Prisoner's Dilemna
[](https://www.mathworks.com/products/matlab.html)
[](https://github.com/giakou4/game_theory_spatial_PD/LICENSE)
[](https://www.mathworks.com/matlabcentral/fileexchange/76781-spatial-prisoners-dilemna)
[](https://img.shields.io/github/forks/giakou4/game_theory_spatial_PD.svg)
[](https://img.shields.io/github/stars/giakou4/game_theory_spatial_PD.svg)
[](https://img.shields.io/github/issues/giakou4/game_theory_spatial_PD.svg)
[](https://img.shields.io/github/issues-closed/giakou4/game_theory_spatial_PD.svg)
[](https://img.shields.io/github/languages/code-size/giakou4/game_theory_spatial_PD)
This script is a simulation of Martin A. Nowak and Robert M. May paper about "Evolutionary Games and Spatial Chaos" 1992. It is created by N. Giakoumoglou, M. Demetriou and P. Manouselis for a presentation in Game Theory course in May 2020.
## 1.1 The Spatial Prisoner's Dilemna
In spatial prisoner's dileman there are two players those who always cooperate, C, and those who always defect, D. We place those players on a two dimensional lattice (grid), each lattice site is occupied either by a C or a D. In each round of the game (each generation), the players play the PD game with nearest neighboring sites and with one's own site (thus we define these sites as a territory – a 3x3 grid). The score for each player is the sum of the payoffs in these encounters with neighbors. At the start of the next generation, each lattice-site is occupied by the player with the highest score among the previous owner and the immediate neighbors. Boundaries are fixed but we can also define the lattice as a torus. Conclusions we will deduct remain true if players interact only with the four orthogonal neighbors in square lattices or self-interactions are included.
## 1.2 The Prisoner's Dilemna Game
The PD can be formulated in tabular form as follows, where T > R > P ≥ S
| | C | D |
|----------|:-------------:|------:|
| C | R=1 | S=0|
| D | T=b>1 | P=0 |
## 1.3 Chaos in the Spatial PD game
The dynamical behavior of the system depends on the parameter b.
* (b > 1.8) 2x2 or larger cluster of D will continue to grow at the corners.
* (b < 1.8) big D cluster will shrink
* (b < 2) 2x2 or larger cluster of C will continue to grow
* (b > 2) C clusters do not grow
* (2 > b > 1) C clusters can grow in regions of D and vice versa
Chaos persists in shifting patterns C → D, D → C, D → D, C → C
## 1.4 Some Examples
Color assignments:
* C → C blue
* D → D red
* D → C yellow
* C → D green
fc = frequency of cooperators
limT→inf fc = 12log2 – 8 = 0.318 can be proven
Although this approximation always works when we have 10% random D and 1.8
### 1.4.2 10% D randomly at 99x99 lattice, T=200, b=1.9
### 1.4.3 10% D randomly at 99x99 lattice, T=200, b=2.5
### 1.4.4 1 D at the center of the 99x99 lattice, T=2000, b=1.9
### 1.4.5 8 Neighbors (thus self interaction exluded)
* “Interesting Region" is 5/3>b>8/5 (here b=1.62 with 10% random D)
* Similar symmetric patterns
* fc → 0.299
### 1.4.6 5 Neighbors (including self)
* “Interesting Region” is 2>b>5/3 (here b=1.8 with 10% random D)
* Similar symmetric patterns
* fc → 0.374
### 1.4.7 4 Neighbors (thus self interaction exluded)
* “Interesting Region” is 3/2>b>4/3 (here b=1.4 with 10% random D)
* Similar symmetric patterns
* fc → 0.374
## 1.5 Conclusions and Applications
Although the details of the patterns depend on the value of b, a wide range of values leads to chaotic patterns whose nature is almost always independent of the initial proportions of C and D.
Such deterministically generated spatial structures may model and describe pre-biotic evolution of cooperation (among molecules, cells or organisms) as well as Turing models and 2-state Ising models.
## 2. Code
* ```Spatial_PD_4_NN.m```: In this implementation, the spatial PD game is played among 4 orthogonal neighbours. One can define the following parameters for the PD game:
* ```b```: Defection payoff
* ```torus```: If ~0, define the lattice as a torus, 0 else
* ```p```: Proportion of defectors in [0, 1]
* ```self_interaction```: If 0, self interaction is excluded, if ~0, included
* ```rounds```: Number of rounds/generation/time step
* ```n``` : Squared lattice side size
* ```printLattice``` : If 1, prints the lattice over rounds
* ```printFc```: If 1, prints the frequency of cooperators over rounds
* ```limit```: If >0, prints limit in fc as a horizontal line
* ```Spatial_PD_8_NN.m```: In this implementation, the spatial PD game is played among 4 orthogonal neighbours. One can define the following parameters for the PD game:
* ```b```: Defection payoff
* ```torus```: If ~0, define the lattice as a torus, 0 else
* ```p```: Proportion of defectors in [0, 1]
* ```self_interaction```: If 0, self interaction is excluded, if ~0, included
* ```rounds```: Number of rounds/generation/time step
* ```n```: Squared lattice side size
* ```flag```: If 1, places a single D in the center of the nxn lattice (n must be odd to work correctly)
* ```printLattice```: If 1, prints the lattice over rounds
* ```printFc```: If 1, prints the frequency of cooperators over rounds
* ```limit```: If >0, prints limit in fc as a horizontal line
* ```main.m```: Includes the reproduction of the figures of the paper and much more!
## 3. Support
Reach out to me:
- [Giakoumoglou's email](mailto:nikolaos.giakoumoglou@gmail.com "nikolaos.giakoumoglou@gmail.com")
## 4. Citation
* Nowak, M., May, R. Evolutionary games and spatial chaos. Nature 359, 826–829 (1992). https://doi.org/10.1038/359826a0