A Three Strategy Game:
The
Bourgeois vs. Hawks vs. Doves Simulation
Contents
Introduction
This page contains a description of how to use the simulation of the
Bourgeois vs. Hawks and Doves game. Do not attempt this game until you
thoroughly understand the Hawk
and Doves game and you are familiar with the basic operation of the Hawks and Doves simulation -- this
simulation has many similarities. Also be sure that you understand the "new"
strategy Bourgeois. Take the
time to review the materials first or you will not get much out of this
simulation and you will probably have trouble answering the questions.
One major difference between this simulation and the Hawks vs. Doves
simulation is that there is no plot of fitness vs. freq. With three
strategies, such a plot is difficult to make (requiring either three axes
or fixation of the frequency of one strategy) but more importantly, unlike
a two strategy game, there may be no pure or mixed ESS outcome. Depending
on the initial conditions (payoffs, frequencies) a number of outcomes are
possible -- pure, mixed or no ESS! So, we will only look at the result in
terms of evolution.
| A Note from the Programmer: Maynard Smith and many other game
theorists usually plot three strategy game evolution results as barycentric
plots. While very elegant, these take some getting used to and so I
have decided to use the more intuitive plots of frequency vs. time. I will
probably add this type of a plot in the future, depending on the feedback I get on this site. |
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About the Simulation
Once you have loaded the simulation and have moved from the introductory
window, a new window like the one below will appear:
This window is divided into three sections:
- The Left Panel contains a blue button "Instructions"
which explains how to use the window.
- The Central Panel, "Info on Calc. of Payoffs", has
nine buttons each labeled with the symbolic notation for a particular payoff.
Pressing one of these buttons will tell you how a particular payoff is
calculated.
| ! Important Note: As with the Hawks
and Doves simulation, you will not be able to alter the actual definition
of any strategy nor can you modify the formulae used to calculate the payoffs
for a particular contest. As in the Hawk and Dove simulation, the only way
that you will be able to modify the payoffs is by changing the Benefits
and Costs. |
- The Right Panel contains controls for changing benefits and
costs:
As usual, simply enter the values you wish for the resource value (GAIN)
and two types of costs. You should use the same conventions for assigning values to W, I, and
D as with Hawks and Doves since both of these strategies are found in
this game and B is combination
of the two strategies.
The button labeled "Reset to Default" will set the gains and
costs back to their initial values which are the same default values that
we used in the Hawks and Doves game.
Finally, pressing the red "Calc. Matrix" will send you on to
the next window and will calculate the payoff matrix.
Use the same conventions as before to assign Benefits
and Costs:
Benefits > or = 0
Costs < or = 0 |
Once you are satisfied with the Benefits and Costs, press the red button
and you will see the next window which reviews the payoff matrix:
- The Left Panel gives three buttons
- Information about using the page
- A button that will take you back to the previous window to revise the
payoff matrix by changing the benefits and costs
- A red button labeled "Continue -- Set Freqs. and Run" --
which takes you to the next window.
- The Right Panel contains the payoff matrix (gold) calculated
using the Benefits and Costs you set on the last page. The blue button
above each payoff can be pressed to give you the formula used to calculate
each payoff (see note above).
When you are satisfied, press the red button which takes you to the next
window:
Once again this window is a "tryptic" (apologies to all of
those great painters for appropriating the term):
- The Left Panel has two buttons
- Information on using the page and
- A button that will allow you to go back to review the last window in
case you want to see the payoff matrix before you set strategy frequencies
(from the previous window, you will be able to reset the benefits and costs
and thereby change the payoffs as was noted above).
- The Center Panel contains three text fields for entering the
frequencies of each of the strategies:
- Be certain that the frequencies add to 1.0. If they do not add to 1.0
or if you enter non-numerical data, you will see the following warning
window which you should close and then re-enter your data:
- If you wish to run a two strategy game (e.g., H vs. B) enter a value
of 0.0 for the strategy you wish to exclude but be sure that the other
two add to 1.0.
Finally, the Right Panel contains two controls:
- a pull-down menu that allows you to set the number of generations
in the evolution simulation. The default is 50, but experience will
show that in some cases you may want to use fewer generations (to
get a better view of the changes) or more generations (when equilibrium
has not yet been reached)
- and a red button that when pressed will take you to the evolution
simulation
| If you want to review the concepts of fitness and frequency
and especially if you want see an example of how the "evolve"
graph is calculated, press
here. |
As with the evolution simulation in the Hawks and Doves Game, there are
two plots -- the left is a plot of the relative fitnesses and the right
is a plot of the strategy frequencies:
A key at the bottom gives the color and symbol labels for each strategy
and a message will tell you how many generations were required to reach
equilibrium (if at all).
A few things to remember when viewing the
plot: As with the Hawk vs. Dove game, there are a couple of things
to notice about the plot.
- First, fitness is expressed
relatively. Thus, it always has a value between zero and 1.0.
- Secondly, be aware that there are some rounding errors and so
the graph and hash marks on the axes have some slight errors. For instance,
the fitness of the Hawk line does not start exactly at 1.0 on the y-axis
nor does it end exactly on the x axis at a frequency of 1.0 as it should.
Nonetheless, the errors are not large.
- Thirdly, below the graph, a text print out will tell you whether
or not there is a pure or mixed ESS and if mixed, what the equilibrial
freq. will be. Unlike the Hawks and Doves game, this program determines
equilibrium has occurred when there is no change in two successive generations
in the frequencies of all three strategies. Please note that biologically-speaking,
equilibrium would probably arrive at a different, earlier number of generations.
The program does not assign equilibrium until the frequency of H
remains constant to 38 places for two successive generations!
- Finally, please note that when the game is only played with
two strategies (e.g., H and B), the relative fitnesses of all three
strategies are still displayed. This is simply to help you envision what
would happen if the third strategy was added.
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Questions to Address and Things to
Try
The speed of this simulation will allow you to answer all of these questions
rapidly -- take the time to consider each in detail and record your answers
or thoughts and questions in your course notes for discussion in class.
| ! Try to answer all of the questions below. Discussion material
is provided for some of the questions (follow the link after the question,
but please try to answer the question first). If you have trouble answering
other questions, ask about them in class. |
See how Bourgeois does against just Hawk, just Dove and finally against
both.
In a systematic manner, start with initially different frequencies
of H, D and B.
- For example, try H at 0.9, D at 0.09 and (therefore) B at 0.01.
- Reverse the frequencies of H and D.
- Try nearly equal frequencies of H and D and low B.
- Satisfy yourself that in each case B is still an ESS -- can you alter
the values of winning the resource, injury and display costs in any meaningful
way to prevent B from being a pure ESS? Use the same sorts of modifications
that you made in the H and D game to make one or the other (in one case,
unrealistically) a pure ESS.
- Discussion
of these experiments.
More about frequencies: Review the situation with the default
payoff matrix and with H at 0.9, D at 0.09 and (therefore) B at 0.01.
- Set the number of generations to 10. Describe what happens to H, D
and B over this time.
- Set the number of generations to 50. Describe what happens to H, D
and B between gen 10 and 50.
- Set the number of generations to 150. Describe what happens to H, D
and B between gen 50 and 150.
- Repeat this experiment with the initial frequencies of H and D reversed
(and therefore the same initial freq. of B).
- Between generations 10 and 50, what were the approximate frequencies
of H and D?
- Were they the same regardless of whether or not you started with H
or D at 0.9?
- Have you observed these frequencies before?
- What is going on here?
- If B is a pure ESS why does it take so long for it to fix?
- What would you need to do to make B fix faster, (given a starting frequency)?
- Discussion
of these results.
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Differences Between the Application
and Simulation
There are a few differences between the stand-alone application and the
web-based applet. Here they are:
- Launching:
- Applet:, simply press the appropriate link.
- Application: you must first download the application (see navigation window)
and be sure that it is unpacked (your web browser should do this automatically,
but follow the instructions that can be found on the download window).
Once it's unpacked, double click on it and it'll launch (provided you have
a java interpreter installed in your OS -- if you use some version of Windows
32 this may be a bit more complicated; see notes on the download page).
- Quitting:
- Applet: you simply close all windows, this will exit you from
the simulation.
- Application: you must go to the File menu and select "Quit"
(Mac) or "Exit" (windows).
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Copyright © 1999 by Kenneth N. Prestwich
College of the Holy Cross, Worcester, MA USA 01610
email: kprestwi@holycross.edu
About Fair
Use of these materials
Last modified 2 - 22 - 99 |