gridvoting 0.10.2

Spatial Voting simulations on a grid with random challengers

gridvoting

This software is a library module for our research publication (open-access):

  Brewer, P., Juybari, J. & Moberly, R. 
  A comparison of zero- and minimal-intelligence agendas in majority-rule voting models. 
  J Econ Interact Coord (2023). https://doi.org/10.1007/s11403-023-00387-8

This software helps set up, calculate, and plot stationary probability distributions for sequential voting simulations (with ZI or MI random challengers) that take place on a 2D grid of feasible outcomes.

In our paper, we used the simulations to show that adding intelligence to the agenda of a collection of voting bots does not necessarily improve the fairness or reasonableness of outcomes. We're not claiming adding intelligence is always bad, since one cannot deduce such generalities from a few simulations. But in some well-known scenarios, the simulations demonstrate cases where adding intelligence to the voting agenda can increase the variance and decrease the equality of outcomes for equally situated agents.

License

The software is provided under the standard MIT License.

You are welcome to try the software, read it, copy it, adapt it to your needs, and redistribute your adaptations. If you change the software, be sure to change the module name somehow so that others know it is not the original. See the LICENSE file for more details.

Disclaimers

The software is provided in the hope that it may be useful to others, but it is not a full featured turnkey system for conducting arbitrary voting simulations. Additional coding is required to define a specific simulation.

Testing of the software is limited. While there are some runtime assertion tests in the software, we do not include a fully automated test suite because our free testing provider does not include a GPU in the test environment.

Some manual post-build testing has been done, but is not extensive.

As with any code where extensive testing is not done, it may contain bugs, unexpected behaviors, or have other issues.

The MIT License also includes this disclaimer: THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

code and data for specific simulations

Code specific to the spatial voting and budget voting portions of our research publication above -- as well as output data -- is deposited at: OSF Dataset for A comparison of zero and minimal Intelligence agendas in majority rule voting models and is freely available.

use of Google Colab

We used Google Colab, a cloud-based service that runs Python-based analyses on Google's servers and GPUs, for conducting most of the research reported in the publication above. When using Google Colab, the local computer does NOT need to have a GPU.

The software has also run (without Colab) on a local computer with a Nvidia gaming GPU, and remote computers with industrial Nvidia A100 GPUs.

requirements

• Nvidia GPU with minimum of 16GB GPU memory to duplicate simulations reported in the above paper
• Nvidia CUDA drivers (except on Google Colab, where CUDA is pre-installed)
• Python 3
• all of these Python-3 scientific computing modules (all except cupy are pre-installed on Google Colab, and cupy can be installed from these instructions):
  • numpy
  • pandas
  • matplotlib
  • scipy
  • cupy
• familiarity with Python language / scientific computing / gpu Nvidia-CUDA setup

Random sequential voting simulations

This follows section 2 of our research paper

A simulation consists of a sequence of times: t=0,1,2,3,... a finite feasible set of alternatives F, a set of voters who have preferences over the alternatives and vote truthfully, a rule for voting and selecting challengers, and a mapping of the set of alternativies F into a 2D grid.

The active or status quo alternative at time t is called f[t].

At each t, there is a majority-rule vote between alternative f[t] and a challenger alternative c[t]. The winner of that vote becomes the next status quo f[t+1].

Randomness enters the simulation through two possible rules for choosing the challenger c[t]. When zi=True, c[t] is chosen from the "Zero Intelligence" agenda which consists of a uniform random distribution over F. When zi=False, c[t] is chosen from the "Minimal Intelligence" agenda which is a uniform distribution over the status quo f[t] and the possible winning alternatives given the status quo f[t].

Principles of Operation

This section extends a higher, more mathematical description from section 3 of our research paper. with an overview of the code in the gridvoting module.

The gridvoting software module is designed to separate various concerns and manage the overlap of concepts.

A primary concern is data transformation between different tasks. The VotingModel and related Markov chain require a probabilistic vector space of dim number_of_alternatives. Plotting occurs in a rectangular area. The alternatives then also need to have coordinates within a rectangular area. Utility functions for the alternatives might also depend on their coordinates. If the alternatives' coordinates are not equivalent to a complete rectangular grid, then there needs to be an embedding function from some smaller space into the rectangular area in order to communicate the results from the Markov chain calculation to the plotting routines to the viewers' screen. The Python3 numpy module creates arrays that are easy to reshape and manipulate, and is used intenally for many of the data transfomations. Knowledge of how numpy can be used to manipulate data will be useful in writing additional code to further manipulate or analyze simulation data. The cupy module is a drop-in replacement for numpy for accelerating data manipulation by using a hardware GPU.

The various concepts are coordinate grids, shapes within a grid, defining a voting simulation, and calculating the stationary distribution of the simulation by a GPU-based MarkovChain calculation. For example, one can have a voting simulation without a grid, or one can have a simulation where the feasible alternatives are points within a triangle (from a budget constraint) itself embedded within a square grid for plotting purposes.

The VotingModel class manages simulations. Each simulation is an instance of VotingModel. The constructor requires various properties of the simulation, such as the number of voters, number of alternatives, and voters' utility functions. Utility functions are defined not as Python3 functions but as numeric arrays of dim (number_of_voters, number_of_alternatives) defining the utility of each voter for each outcome as a number where more is better. It is the ordering and not the values that are important.
Besides plotting and answering simple questions about the alternatives, the class also provides code for calculating the transition matrix and providing it as an input to the MarkovChainGPU class below.

The Grid class manages rectangular grids. An instance of VotingModel will usually specify a Grid instance for defining utility functions and plotting/visualization purposes. It is also possible to use VotingModel without specifying any kind of grid or coordinate mapping, for an example see class CondorcetCycle.

The MarkovChainGPU class manages a Markov Chain calulation on a GPU. This class is called internally from VotingModel. The class contains two methods for calculating the stationary distribution of a Markov Chain: the power method (default), and an algebraic method (optional).

Classes

Grid

constructor

gridvoting.Grid(x0, x1, xstep=1, y0, y1, ystep=1)

Constructs a 2D grid in x and y dimenstions and provides helpful methods for accessing the grid.

• x0 the leftmost grid x-coordinate
• x1 the rightmost grid x-coordinate
• xstep=1 optional, default value 1. grid spacing in x dimenstion
• y0 the lowest grid y-coordinate
• y1 the highest grid y-coordinate
• ystep=1 optional, default value 1. grid spacing in y dimension

Example:

import gridvoting
grid = gridVoting.Grid(x0=-5,x1=5,y0=-7,y=7)

Defines a grid where -5<=x<=5 and -7<=y<=7

This grid will have 11*15 = 165 alternatives corresponding to integer points in the grid.

Instance Properties:

• parameters from the constructor call are available as instance properties
  • grid.x0
  • grid.x1
  • grid.xstep
  • grid.y0
  • grid.y1
  • grid.ystep
• calculated properties
  • grid.x -- 1D numpy array containing the x-coordinate of each grid point in typewriter order
  • grid.y -- 1D numpy array containing the y-coordinate of each grid point in typewriter order
  • grid.gshape -- natural shape (number_of_rows,number_of_cols) where rows represent y-axis and cols represent x-axis
  • grid.extent -- equal to the tuple (x0,x1,y0,y1) for use with matplotlib.pyplot.plt
  • grid.len -- equal to the number of points on the grid

methods

grid.as_xy_vectors()

Returns a 2-D array of coordinates for each grid point in typewriter order

Example:

import gridvoting
grid = gridVoting.Grid(x0=-5,x1=5,y0=-7,y=7)
vectors = grid.as_xy_vectors()
print(vectors.shape)
print(vectors)

vectors.shape will be (165,2) and vectors will be an array of 165 2D coordinates [[-5,7],[-4,7],[-3,7],...,[5,7],[-5,6],...,[5,6],...,[5,-7]]

---

grid.index(x,y)

Locates the index in the grid's coordinate array for the point (x,y)

Example:

import gridvoting
grid = gridVoting.Grid(x0=-5,x1=5,y0=-7,y=7)
idx = grid.index(x=-4,y=6)
print(idx)

idx will be 12, because [-4,6] is entry [12] of grid.as_xy_vectors()

---

grid.spatial_utilities(voter_ideal_points, metric='sqeuclidean', scale=-1)

VotingModel

MarkovChainGPU

Other Functions

TO BE DOCUMENTED