Many election jurisdictions in the United States need to randomly select precincts, or batches of ballots, to audit after an election. This article provides step-by-step instructions for how to do this properly using dice. It also explains some of the advantages of using dice over other methods.
This article is intended for election officials, as well as members of the public interested in improving election auditing procedures in their area.
There are a number of correct ways of using dice to randomly select precincts (as well as many incorrect ways, if one isn't careful). The method described in this article is the one described in a 2006 paper by Arel Cordero, David Wagner, and David Dill (also see [CWD] referenced at the bottom). We chose to highlight this method both for its simplicity and efficiency over other methods. Also, we simplified the description so that, for example, familiarity with math notation isn't needed. (We do use letters to stand for numbers, though.) Lastly, the method we describe here is not suitable for risk-limiting audits. For those a different procedure is needed.
Advantages
Rolling dice is a good method for doing a random selection because it is simple and straightforward, easy to observe, and is observably random. Confidence can be increased by letting members of the public do some of the dice rolls.
Other processes like drawing slips of paper from a hat or barrel don't have these same advantages. For example, if drawing slips of paper, it is hard to know if the slips of paper are evenly mixed, and there is no clear way for the public to confirm this. Also, if you are choosing from among hundreds or dozens of precincts, there is no easy way for the public to know for sure that no slips of paper have been left out. Finally, it is also hard for members of the public to be sure that sleight-of-hand techniques aren't being used when choosing the pieces of paper.
Instructions
Even though the steps below can be used without change to select not just precincts but also batches of ballots, for simplicity we usually only say "precincts."
Preparation
We start by describing the steps you can do in advance.
Step 1: Purchase enough dice
You will be using ten-sided dice rather than the more commonly used six-sided dice. You can see a picture of what ten-sided dice look like on the right. They are widely available for purchase online, or you should be able to find them at a local games shop.
Since it can take time to purchase dice (e.g. if you're ordering them online), you should acquire them far enough in advance and make sure you will have enough. The number of dice you need depends on the number of precincts you will have. If you don't know in advance how many precincts you will have (especially likely in the case of batches), you should come up with an estimate of the absolute maximum number you might wind up having. It's okay to overestimate here.
If you will have no more than 100 precincts or batches, you will need two dice. If you will have no more than 1,000, you will need three. For 10,000, you will need four. And so on. (This is the same as the number \(R\) we will be discussing below.)
When purchasing dice, it's a good idea if each die is a different color. This way they can be rolled together all at once and you can still tell them apart. For example, if you need three dice, you can purchase three ten-sided dice in the colors red, white, and blue. (Ten-sided dice are sold in a variety of different colors.)
Step 2: Determine the helper numbers
Before doing the selection, you need to determine two key "helper" numbers. You can do this step once you know the exact number of precincts or batches.
The two numbers you need to determine are— (1) the number of dice to roll each time, which we call \(R\) (for "roll"), and (2) a "dividing factor" (also a whole number), which we call \(D\) (for "dividing"). These numbers depend on the total number of precincts you have. You will need a calculator.
To determine \(R\) (aka the number of dice to roll each time), take the total number of precincts you have, subtract 1, and then note the number of digits that number has. This number of digits is \(R\). For example—
- For 11 to 100 precincts, \(R\) is 2.
- For 101 to 1,000 precincts, \(R\) is 3.
- For 1,001 to 10,000 precincts, \(R\) is 4, and so on.
To determine \(D\) (aka the dividing factor), take the number 1 with \(R\) zeros after it (e.g. 1,000 if \(R\) is 3). Then, use a calculator to divide that number by the total number of precincts you have. Finally, round the answer down (i.e. by removing the decimal part, if any). For example, if the total number of precincts is 170, then \(R\) would be 3. Divide 1,000 by 170 to get 5.8823.... Then round that down to get 5 for \(D\). As another example, if the total number of precincts is 200, then \(R\) would again be 3. Dividing 1,000 by 200 gives 5 exactly. Since 5 has no decimal part, rounding down keeps it at 5. So \(D\) would still be 5.
Step 3: Make precinct lookup sheet
Next, you need to make a document we call the "precinct lookup sheet" (or "batch lookup sheet," in the case of batches). This is a document or spreadsheet that lets you look up the right precinct each time you roll the dice. Note that this document won't just be for the election administrators to use. Election observers and the public generally should also get a copy.
In its simplest form, the lookup sheet is simply a numbered list of the names of all precincts. The numbers should start at 0, go up by one with each precinct, and not skip any numbers. We call each number the "index" of that precinct. The figure at the right shows an example of what this might look like. Because this lookup sheet is so simple, we call it the "simple" form.
There is another style of lookup sheet you can make we call the "spelled out" form. It is slightly more work to make, but it makes the actual selection process a little bit easier to do.
To make this style of lookup sheet, add two more columns to the grid called "Min Roll" and "Max Roll." To fill the "Min Roll" column for a certain row, multiply the index number for that precinct by the number \(D\) you determined above. To fill the "Max Roll" column, add \(D - 1\) to the "Min Roll" value. In other words, "Max Roll" is one less than "Min Roll" of the next row. The Min Roll and Max Roll numbers will let you and any election observers see immediately what precinct corresponds to a roll of the dice, without having to do any additional arithmetic. Finally, for convenience, you can add a row at the end that says "ROLL AGAIN" for any higher numbers.
The figure above shows an example of what the spelled out form might look like. After reading the rest of this document, you can decide for yourself whether you'd like to use the simple form or the spelled out form.
TODO: finish the rest.
References
- [CWD] Arel Cordero, David Wagner, and David Dill, "The Role of Dice in Election Audits - Extended Abstract," June 16, 2006, 8 pages, preprint.
