AgentBased Evolutionary Dynamics in 2 populations
Abed2pop is a modeling framework designed to simulate the evolution of two populations of agents who play a 2player 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 Abed2pop
To use Abed2pop, 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 (abed2pop.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 abed2pop.nlogo. The figure below shows Abed2pop's interface.
Overview
We use bold green fonts to denote
Abed2pop is a modeling framework designed to simulate the evolution of two populations of agents who recurrently play a 2player game defined by a
The simulation runs in discrete timesteps 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
probrevision , which is the probability that each individual agent revises his strategy in every tick.  By setting parameter
nofrevisionspertick , 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:
 the
noftrials agents make, and  whether these are conducted with or without replacement (
trialswithreplacement? ).
Strategy revision
Agents are occasionally given the opportunity to revise their strategy. The specific protocol used by revising agents is determined mainly by the following parameters:

candidateselection , which indicates whether the protocol isimitative ordirect (PGED, p. 9), 
popinofcandidates , which determine the number of candidates (agents or strategies) that revising agents in population i will consider to update their strategy, and 
decisionmethod , which determines how to single out one instance out of the set of candidates. Abed2pop implements the following methods:best ,logit ,positiveproportional ,pairwisedifference ,lineardissatisfaction andlinearattraction .
We briefly explain the meaning of each of these terms below.
Imitative Protocols
Under
Imitative protocols: Specifying how to compile the set of candidate agents to copy
There are parameters to specify:
 the number of candidate agents that revising agents will consider to copy (
pop1nofcandidates andpop2nofcandidates ),  whether the selection of other candidates (besides the revising agent) is conducted with or without replacement (
imitateeswithreplacement? ), and  whether the selection of other candidates (besides the revising agent) considers the revising agent again or not (
considerimitatingself? ).
Direct Protocols
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 (
pop1nofcandidates andpop2nofcandidates ),  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 (
singlesample? ).
Decision methods
A
 one particular agent out of a set of candidates, whose strategy will be adopted by the revising agent (in
imitative protocols), or  one particular strategy out of a set of candidate strategies, which will be adopted by the revising agent (in
direct protocols).
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 parametertiebreaker . 
logit : A random weighted choice is conducted among the candidates. The weight for each candidate is e^{(payoff / (10 ^ lognoiselevel))}. 
positiveproportional : A candidate is randomly selected with probabilities proportional to payoffs. Thus, payoffs should be nonnegative if thisdecisionmethod is used. 
pairwisedifference : In this method the set of candidates contains two elements only: the one corresponding to the revising agent's own strategy (or to himself inimitative protocols) and another one. The revising agent adopts the other candidate strategy only if the other candidate's payoff is higher than his own, and he does so with probability proportional to the payoff difference. 
lineardissatisfaction : In this method the set of candidates contains two elements only: the one corresponding to the revising agent's own strategy (or to himself inimitative protocols) and another one. The revising agent adopts the other candidate strategy with probability proportional to the difference between the maximum possible payoff he could get and his own payoff. (The other candidate's payoff is ignored.) 
linearattraction : In this method the set of candidates contains two elements only: the one corresponding to the revising agent's own strategy (or to himself inimitative protocols) and another one. The revising agent adopts the other candidate strategy with probability proportional to the difference between the other candidate's payoff and the minimum possible payoff he could get. (The revising agent's payoff is ignored.)
Parameters
Parameters to set up the population and its initial strategy distribution
 If
randominitialcondition? isOn , population 1 will consist ofpop1nofagents agents and population 2 will consist ofpop2nofagents agents, each of them with a random initial strategy.  If
randominitialcondition? isOff , population i will be created from the list given aspopinofagentsforeachstrategy . 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 ofpopinofagents is then set to the total number of agents in the population.
Parameters that determine when agents revise their strategy
 If
useprobrevision? isOn , each individual agent revises his strategy with probabilityprobrevision in every tick.  If
useprobrevision? isOff , thennofrevisionspertick agents 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 2player game. Entry A_{ij} in the matrix is a 2element 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.
If
If
Parameter
Parameter
Parameters that determine how agents revise their strategy
With probability
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.
Parameter
The multiset of candidate agents contains the revising agent (which is always included) plus a sample of (

imitateeswithreplacement? indicates whether the sample is taken with or without replacement. 
considerimitatingself? 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 (
Parameter
The user can set the value of this parameter only if
Parameter
Specifically, the chosen
The candidate with the highest payoff is returned. Ties are resolved using the selected

stickuniform : 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. 
stickmin : 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. 
randomwalk : This tiebreaker 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 rwagents, plus a set of n socalled committed rwagents, one for each of the n strategies in the population. The committed rwagents never change strategy. In each iteration of this random walk, one uncommitted rwagent is selected at random to imitate another (committed or uncommitted) rwagent, also chosen at random. We run one iteration 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 / (10 ^ lognoiselevel))}.
A candidate is randomly selected, with probabilities proportional to payoffs. Because of this, when using
In
The revising agent will consider adopting the other candidate strategy only if the other 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).
In
The revising agent will adopt the other candidate strategy with probability equal to the difference between the maximum possible payoff that someone could get in that population and his own payoff, divided by the maximum possible payoff minus the minimum possible payoff (which someone could get in that population). (The other candidate's payoff is ignored.)
In
The revising agent will adopt the other candidate strategy with probability equal to the difference between the other candidate's payoff and the minimum possible payoff that someone could get in that population, divided by the maximum possible payoff minus the minimum possible payoff (which someone could get in that population). (The revising agent's payoff is ignored.)
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 Abed2pop.
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 Abed2pop 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 Abed2pop 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
License
Abed2pop (AgentBased Evolutionary Dynamics in 2 populations) is a modeling framework designed to simulate the evolution of two populations of agents who play a 2player 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.
Copyright (C) 2017 Luis R. Izquierdo, Segismundo S. Izquierdo & Bill Sandholm
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 021101301, USA.
Contact information:
Luis R. Izquierdo
University of Burgos, Spain.
email: lrizquierdo@ubu.es
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.
Acknowledgements
Financial support from the U.S. National Science Foundation, the U.S. Army Research Office, the Spanish Ministry of Economy and Competitiveness, and the Spanish Ministry of Science and Innovation is gratefully acknowledged.
References
Sandholm, W. H. (2010). Population Games and Evolutionary Dynamics. MIT Press, Cambridge.