Agent-Based Evolutionary Dynamics in 2 populations
Abed-2pop is a modeling framework designed to simulate the evolution of two populations of agents who play a 2-player game. Agents in one population play the game with agents in the other population. Every agent is occasionally given the opportunity to revise his strategy.
How to install Abed-2pop
To use Abed-2pop, you will have to install NetLogo (free and open source) 6.0.1 or higher and download this zip file. The zip file contains the model itself (abed-2pop.nlogo) and also this very webpage (which contains a description of the model). To run the model, you just have to click on the file abed-2pop.nlogo. The figure below shows Abed-2pop's interface.
We use bold green fonts to denote
Abed-2pop is a modeling framework designed to simulate the evolution of two populations of agents who recurrently play a 2-player game defined by a
The simulation runs in discrete time-steps called ticks. Within each tick:
- The set of revising agents is chosen
- The game is played by the revising agents and all the agents involved in their revision
- The revising agents simultaneously update their strategies
This sequence of events, which is explained below in detail, is repetead iteratively.
Selection of revising agents
All agents are equally likely to revise their strategy in every tick. There are two ways in which the set of revising agents can be chosen:
- By setting parameter
prob-revision, which is the probability that each individual agent revises his strategy in every tick.
- By setting parameter
n-of-revisions-per-tick, which is the number of randomly selected agents that will revise their strategy in every tick. These agents are selected without taking into account the population they belong to.
Computation of payoffs
The payoff that each agent obtains in each tick is computed as the average payoff over
Specifying how payoffs are computed
There are parameters to specify:
n-of-trialsagents make, and
- whether these are conducted with or without replacement (
Agents are occasionally given the opportunity to revise their strategy. The specific protocol used by revising agents is determined by two parameters:
candidate-selection, which indicates whether the protocol is imitativeor direct(PGED, p. 9), and
decision-method, which can be best, logit, positive-proportionalor pairwise-difference.
We briefly explain the meaning of each of these terms below.
Imitative protocols: Specifying how to compile the set of potential imitatees
There are parameters to specify:
- the number of agents that revising agents will consider imitating (
- whether these are selected with or without replacement (
- whether agents may imitate themselves or not (
In contrast, under
Direct protocols: Specifying how to test strategies
There are parameters to specify:
- the number of strategies that revising agents consider, including their current strategy (
- whether the strategies in the test-set are compared according to a payoff computed using one single sample of counterparts from the other population (i.e. the same set of counterparts for every strategy), or whether the payoff of each strategy in the test-set is computed using a freshly drawn random sample of counterparts (
- one particular agent out of a set of candidates, whose strategy will be adopted by the revising agent (in
- one particular strategy out of a set of candidate strategies, which will be adopted by the revising agent (in
The selection is based on the (average) payoff obtained by each of the candidates (agents or strategies) in the input set. In
best: The candidate with the highest payoff is selected. Ties are resolved according to the procedure indicated with parameter tie-breaker.
logit: A random weighted choice is conducted among the candidates. The weight for each candidate is e(payoff / eta).
positive-proportional: A candidate is randomly selected with probabilities proportional to payoffs. Thus, payoffs should be non-negative if this decision-methodis used.
pairwise-difference: In this method the set of candidates contains two elements only: the one corresponding to the revising agent's own strategy (or to himself in imitativeprotocols) and another one. Then, the revising agent adopts the other candidate's strategy only if the other candidate's payoff is higher than his own, and he does so with probability proportional to the payoff difference.
Parameters to set up the population and its initial strategy distribution
random-initial-condition?is On, population 1 will consist of pop-1-n-of-agentsagents and population 2 will consist of pop-2-n-of-agentsagents, each of them with a random initial strategy.
random-initial-condition?is Off, population i will be created from the list given as pop-i-n-of-agents-for-each-strategy. Let this list be [x1 x2 ... xn]; then the initial population will consist of x1 agents playing strategy 1, x2 agents playing strategy 2, ... , and xn agents playing strategy n. The value of pop-i-n-of-agentsis then set to the total number of agents in the population.
Parameters that determine when agents revise their strategy
use-prob-revision?is On, each individual agent revises his strategy with probability prob-revisionin every tick.
use-prob-revision?is Off, then n-of-revisions-per-tickagents are randomly selected to revise their strategy in every tick. These agents are selected without taking into account the population they belong to.
The value of these three parameters can be changed at runtime with immediate effect on the model.
Parameters that determine how payoffs are calculated
This flowchart is a summary of all the parameters that determine how payoffs are calculated.
The idea is to start at the top of the flowchart, give a value to every parameter you encounter on your way and, at the decision nodes (i.e. the red diamonds), select the exit arrow whose [label] indicates the value of the parameter you have just chosen. Thus, note that, depending on your choices, it may not be necessary to set the value of all parameters.
Every parameter is explained below.
Payoff matrix for the 2-player game. Entry Aij in the matrix is a 2-element list [p1 p2] which refers to the outcome where player row chooses strategy i and player column uses strategy j. Within the list [p1 p2], p1 denotes the payoff obtained by player row (in population 1) and p2 denotes the payoff obtained by player column (in population 2). The number of strategies for agents in population 1 (population 2) is assumed to be the number of rows (columns) in this payoff matrix.
Parameters that determine how agents revise their strategy
The idea is to start at the top of the flowchart, give a value to every parameter you encounter on your way and, at the decision nodes (i.e. the red diamonds), select the exit arrow whose [label] indicates the value of the parameter you have just chosen. Every parameter is explained below.
The multiset of candidate agents contains the revising agent (which is always included) plus a sample of
imitatees-with-replacement?indicates whether the sample is taken with or without replacement.
consider-imitating-self?indicates whether the sample is taken from the revising agent's whole population (i.e. including the reviser) or from the revising agent's whole population excluding the reviser. Note that, in any case, the reviser is always part of the multiset of candidate agents at least once.
The set of candidate strategies contains the revising agent's own strategy plus (
The user can set the value of this parameter only if
Specifically, the chosen
The candidate with the highest payoff is returned. Ties are resolved using the selected
stick-uniform: If the current strategy of the revising agent is in the tie, the revising agent sticks to it. Otherwise, a random choice is made using the uniform distribution.
stick-min: If the current strategy of the revising agent is in the tie, the revising agent sticks to it. Otherwise, the strategy with the minimum number is selected.
uniform: A random choice is made using the uniform distribution.
min: The strategy with the minimum number is selected.
random-walk: This tie-breaker makes use of an auxiliary random walk (one for each population) that is running in the background. Let N be the number of agents in the population and let n be the number of strategies. In the auxiliary random walk, there are N rw-agents, plus a set of n so-called committed rw-agents, one for each of the n strategies in the population. The committed rw-agents never change strategy. In each iteration of this random walk, one uncommitted rw-agent is selected at random to imitate another (committed or uncommitted) rw-agent, also chosen at random. We run floor(N × random-walk-speed) iterations of this process per tick. When there is a tie, the relative frequency of each strategy in the corresponding auxiliary random walk is used as a weight to make a random weighted selection among the tied candidates.
A random weighted choice is conducted among the candidates. The weight for each candidate is e(payoff /
A candidate is randomly selected, with probabilities proportional to payoffs. Because of this, when using
candidate-selection= imitative, the value of pop-i-n-to-consider-imitatingis automatically set to 1.
candidate-selection= direct, the value of pop-i-n-in-test-setis automatically set to 2.
Let candidate denote the other agent or strategy. Then, the revising agent will consider adopting the candidate's strategy only if the candidate's payoff is higher than his own, and he will actually adopt it with probability proportional to the payoff difference. In order to turn the difference in payoffs into a meaningful probability, we divide the payoff difference by the maximum possible payoff difference that could occur in that population (i.e. the maximum payoff an agent in that role could ever get minus the minimum payoff he could ever obtain).
How to save and load parameter files
Clicking on the button labeled , a window pops up to allow the user to save all parameter values in a .csv file.
Clicking on the button labeled , a window pops up to allow the user to load a parameter file in the same format as it is saved by Abed-2pop.
How to use it
Running the model
Once the model is parameterized (see Parameters section), click on the button labeled to initialize everything. Then, click on to run the model indefinitely (until you click on again). A click on runs the model one tick only.
The value of all parameters can be changed at runtime with immediate effect on the model
Plots and monitors
All plots in Abed-2pop show units of time in the horizontal axis. These are called seconds and milliseconds for simplicity, but they could be called "time units". One second is defined as the number of ticks over which the expected number of revisions equals the total number of agents (in both populations). This number of ticks is called
and is shown in a monitor. The other monitor in Abed-2pop shows the
that have been executed. Plots are updated every
For each population, there are two plots to show the strategy distribution of agents as ticks go by, like the ones below:
and two other plots to show the expected payoff of each strategy as ticks go by, like the ones below:
As you can see in the figures above, plots comes in two different flavors: one that shows the data from the beginning of the simulation (labeled complete history and active if
Abed-2pop (Agent-Based Evolutionary Dynamics in 2 populations) is a modeling framework designed to simulate the evolution of two populations of agents who play a 2-player game. Agents in one population play the game with agents in the other population. Every agent is occasionally given the opportunity to revise his strategy.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You can download a copy of the GNU General Public License by clicking here; you can also get a printed copy writing to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
Luis R. Izquierdo
University of Burgos, Spain.
Extending the model
You are most welcome to extend the model as you please, either privately or by creating a branch in our Github repository. If you’d like to see any specific feature implemented but you don’t know how to go about it, do feel free to email us and we’ll do our best to make it happen.
Sandholm, W. H. (2010). Population Games and Evolutionary Dynamics. MIT Press, Cambridge.