Spatial cooperation games

Basic problem

Altruism in biological systems is defined as behaviour by an individual that increases the fitness of another individual while decreasing the fitness of the actor. There are many well known examples, including food sharing in vampire bats. One proposed explanation is "reciprocal altruism", where individuals provide benefit to each other in expectation of future reciprocation. However, the evolution of reciprocal altruism is synonymous with the evolution of cooperation, and mechanisms must exists that would exclude "cheaters" that reap the benefits of cooperation without actually contributing themselves.

General approach

We will use game theory in general and iterated cooperation games (prisoner's dilemma, snow drift) in particular to study conditions under which cooperation evolves. By developing models in R, we will simulate the competition of organisms with basic, pure strategies (cooperate or defect) in unstructured and spatially structured populations. We will record and analyze the population dynamics and the prevalence of cooperation for different parameter values and settings.

What can be learned?

Concepts:

• Effect of spatial structure on the evolution of cooperation.
• Effects of particular rules and pay-offs of the game on the evolution of cooperation.

Methods:

• Spatially explicit simulation of population interactions on a lattice.
• Synchronous and asynchronous updating.

Starting point

Download the reader and the starting script. Develop R scripts for the spatial prisoner's dilemma and snowdrift games according to instructions in the reader.

Interesting questions that you can investigate

• How does spatial structure affect the evolution of cooperation in the iterated prisoner's dilemma and snow drift games?
• What is the effect of the payoff parameters (cost, benefit)?

Advanced questions:
Investigate the effects of

• neighbourhood size
  
• updating scheme (synchronous vs. asynchronous; pair-wise vs. multiple competitions)
• population size
  
• heterogeneous environment

… on the evolution of cooperation and the significance of spatial structure.

Glossary

• Game theory. A branch of applied mathematics, often used in economics and biology. It provides a formal approach to modelling conflicts in which multiple players choose different actions in the attempt to maximize their returns. It was developed in early and mid 20th century by some of the most prestigious mathematicians of the time, such as John von Neumann and John Nash.
• Payoff matrix.  In the general case, the payoff matrix is an m x n matrix, which defines the outcomes of a two-person game, where player A has m and player B has n possible moves (strategies). The element at position [x,y] in the matrix represents the score obtained by A when playing with B, given they have strategies x and y.   
• Zero-sum game.  A game in which the loss of one participant is exactly balanced by the gain(s) of the other participant(s).
• Prisoner's dilemma (PD). One of the most famous examples of games from the domain of game theory. A two-player game with two possible strategies for each player, either C, 'cooperate' or D, 'defect'. If we note the payoff of A when playing with B as (A,B), then PD is a subset of all zero-sum two-player games for which: (D,C) > (C,C) > (D,D) > (C,D)
• Snow drift game (SD).  A variant of the PD, differing just in the relationship between two entries of the payoff matrix. SD can be defined as a two-player, zero-sum game, with C and D strategies, and the following payoff relationships: (D,C) > (C,C) > (C,D) > (D,D). Note that the difference from PD is the reversal of the (D,D) and (C,D) values.
• Iterated game.  When players are playing the same game (such as PD or SD) multiple times, the whole process is referred to as an iterated game. While not explored in this module, this also allows for strategies that are contingent on the past moves and game outcomes. 
• Pure strategy. Strategy in a game that does not change over time and is independent of any previous interactions.
• Neighbourhood.  In a spatial PD or SD, the location of adjacent individuals with which a particular individual can interact. The most commonly used neighbourhood on a square lattice is a 3 x 3 square, where the individual in the centre can interact with all the organisms with adjoining vertices and edges. 
• Spatial game. A game in which players interact preferentially with other individuals in their vicinity (their neighbourhood).

Literature & Weblinks

• Axelrod, R. (1984) The evolution of cooperation. Basic Books, New York.
• Nowak, M.A & May, R.M. (1992) Evolutionary games and spatial chaos. Nature 359, 826-829. See also News and Views by Karl Sigmund.
  
• Nowak, M.A., Bonhoeffer, S. & May, R.M. (1994) Spatial games and the maintenance of cooperation. Proc Natl Acad Sci U S A. 91, 4877-81.
  
• Hauert, C & Doebeli, M. (2004) Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643-646
• Doebeli, M. & Hauert, C (2005) Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game. Ecology Letters 8, 748-766.
• Hauert, C. (2005) Interactive tutorials on evolutionary game theory.