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CharityWorld-JASSS is a model of wealth redistribution in which a number of spatially embedded agents begin with a highly unequal distribution of wealth. This unequal distribution is then redistributed by the agents using a simple local rule determining when and to whom money should be given. Each of the four quadrants in the grid represents a different implementation of the same model.

CharityWorld-JASSS is an unofficial and reduced version of CharityWorld. CharityWorld-JASSS should be used for illustration purposes only. You can find the official version of CharityWorld (written in Objective-C) in http://www.macaulay.ac.uk/fearlus/floating-point/charity-world/.

In this model there is a grid of default size 13x13 (each of the quadrants) which is filled with stationary agents, one per cell. Agents are endowed with an initial wealth of 2 coins.

At the beginning of the simulation agents participate in a lottery for which they have to pay 2 coins as the ticket price. The agent at the centre of the grid happens to be the winner of the lottery so, after the lottery has taken place (i.e. after clicking on

At any point in time an agent is locally rich (if and only if the agent is richer than its neighbourhood average), locally average (if and only if the agent's wealth is equal to its neighbourhood average), or locally poor (if and only if the agent is poorer than its neighbourhood average). An agent's neighbours are defined using the Moore neighbourhood, which consists of those agents with whom the agent shares an edge or a corner. Thus, an agent which is not at the edge of the grid has 8 neighbours. The grid does not wrap around.

Our agents have a strong craving for equality, so if they are locally rich, they are prepared to donate a coin to their neighbours who are not locally rich. More specifically, an agent considers its neighbours in ascending order of wealth, and gives one coin to each of them (one neighbour at a time) as long as the donating agent is locally rich and the considered neighbour is not locally rich. This process is called the

Scheduling in the model works as follows: at every time-step a locally rich agent is selected at random and told to perform one cycle of redistribution.

So, to clarify things, the first cycle of redistribution in a simulation run should consist of the lottery winner (who is the only locally rich agent right after the lottery has taken place) giving one coin to each of its neighbours (who are not locally rich). Subsequent cycles of redistribution will also be conducted by the lottery winner until one of its neighbours becomes locally rich and is selected (at random) to perform the cycle of redistribution. At that point, wealth will spread even further, eventually making other neighbouring agents locally rich, and therefore turning them into potential donors. The question is: will wealth be completely redistributed using only this simple local rule?

my-wealth ** <---IS COMPARED WITH---> ** (sum map [wealth-of ?] my-neighbours) / (length my-neighbours)

my-wealth * (length my-neighbours) ** <---IS COMPARED WITH---> ** (sum map [wealth-of ?] my-neighbours)

my-wealth ** <---IS COMPARED WITH---> ** ((sum map [wealth-of ?] my-neighbours) + my-wealth) / ((length my-neighbours) + 1)

my-wealth * ((length my-neighbours) + 1) ** <---IS COMPARED WITH---> ** ((sum map [wealth-of ?] my-neighbours) + my-wealth)

*Tolerance Windows*: The use of tolerance windows consists in replacing the comparison operators as shown below:- x > y ---> (x > y +
*epsilon*) - x < y ---> (x < y –
*epsilon*) - x = y ---> (x >= y -
*epsilon*) && (x <= y +*epsilon*)

*epsilon*is a non-negative floating point number.- x > y ---> (x > y +
*Round to*: This technique consists in, just before any comparison takes place, rounding each of the two numbers involved in the comparison to the (floating-point representation of the) closest number with*n*significant digits*n**significant decimal digits*. If the number is whithin the range [-

This parameter determines the value of the coin used to calculate every agent's wealth.

Please realise that if the model ran without any rounding errors, then the value of the coin would not make any difference whatsoever. Using binary numbers like 0.5, 1.0, 1.5, 2.0, or any other multiple of 0.5, prevents any errors from happening. Thus, the behaviour of the model using any of those numbers is exactly the same (using the same random seed). On the contrary, using numbers like 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, ... causes the appearance of rounding errors and of its undesirable consequences in the emergent properties of the model.

2. Click on

At this point every agent will be endowed with 2 coins.

3. Optionally, input your own random seed by clicking on

To replicate runs, use the same random seed.

4. Click on

Every agent gives the 2 coins to the agent in the centre.

5. Click on

A locally rich agent is selected at random and asked to perform the cycle of redistribution.

6. Explore the two

*Absolute wealth*: Each agent is coloured according to the following legend:- wealth > 2 * coin-value: green shades. The more wealth, the darker the shade of green.
- wealth = 2 * coin-value: yellow.
- wealth < 2 * coin-value: red shades. The less wealth, the darker the shade of red.
- Blue overrides when: (wealth > 1.5 * coin-value AND wealth < 2.5 * coin-value AND wealth != 2 * coin-value). Blue will not appear in runs that do not suffer floating-point errors.

*Relative (local) wealth*: Each agent is coloured according to the following legend:- locally rich: turquoise
- locally average: orange
- locally poor: magenta

7. Explore the effects of using different

*Tolerance Windows*, with different values of*epsilon*.*Round to*, with different values of*n*significant digits*significant-digits*and*epsilon*.

Under floating-point arithmetic, however, the story is very different. There are many parameter values (in fact most of them) for which the coin value makes a difference, the specific implementation makes a difference, and wealth is never redistributed. Some coin values that show this type of behaviour are: 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, and 0.9.

You can also check that when the

CharityWorld is written in a manner that follows a fairly standard simple Swarm model implementation, with the exception that all floating point numbers are represented using objects rather than the standard double C data type. These objects all belong to the class DoubleSimple, which contains a double instance variable, and methods to replace the arithmetic operators {+, -, *, /} and the comparison operators. Subclasses of DoubleSimple implement various techniques to deal with floating-point problems (e.g. tolerance windows, offsets, interval arithmetic, strings...), and DoubleSimple features creation methods that cause all new floating point objects to belong to one of these subclasses rather than DoubleSimple itself. The user can therefore specify which subclass of Double-Simple they wish to use throughout a particular simulation, and hence which technique will be used to manage floating point issues.

For a detailed description of CharityWorld, see:

Polhill, J.G., Izquierdo, L.R. and Gotts, N.M. (2006)

What every agent-based modeller should know about floating point arithmetic.

Environmental Modelling & Software, 21 (3), March 2006. pp. 283-309.

http://www.sciencedirect.com/science/journal/13648152

Copyright (C) 2005 Luis R. Izquierdo & J. Gary Polhill

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 2 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.

Contact information:

Luis R. Izquierdo

University of Burgos, Spain.

e-mail: luis@izquierdo.name

If you are interested in the effects of floating-point errors on agent-based models you might also want to read the following papers:

Polhill, J.G., Izquierdo, L.R. and Gotts, N.M. (2006)

What every agent-based modeller should know about floating point arithmetic.

Environmental Modelling & Software, 21 (3), March 2006. pp. 283-309.

http://www.sciencedirect.com/science/journal/13648152

Polhill, J.G., Izquierdo, L.R. and Gotts, N.M. (2005)

The ghost in the model (and other effects of floating point arithmetic).

Journal of Artificial Societies and Social Simulation 8(1).

http://jasss.soc.surrey.ac.uk/8/1/5.html

Polhill, J.G. and Izquierdo, L.R. (2005)

Lessons learned from converting the Artificial Stock Market to interval arithmetic.

Journal of Artificial Societies and Social Simulation 8(2).

http://jasss.soc.surrey.ac.uk/8/2/2.html