AuditVisualiser

We have a set of assertions which imply that the announced winner won. We could (but we won’t) verify each assertion by manually examining all the ballot papers. For example, Diego NEB Chuancould be verified by manually counting all the first preferences for Diego and checking that that tally was greater than the (manually counted) total number of times Chuan was preferenced without being preceded by Diego. Similarly, NEN assertions could be verified by counting the tallies ignoring candidates not specified in the assertion. For example, we could verify NEN: Bob > Chuan if only {Bob, Chuan, Diego} remainby sorting every ballot into a tally pile according to which of Bob, Chuan, and Diego was the highest preference, ignoring any preference for Alice, and then checking that Bob’s total was higher than Chuan’s. If we did this for every assertion in the set, it would be a logically sound way to verify the election result, but it would be very inefficient.

Instead, we test each of the assertions using an RLA at some risk limit α. If the audit accepts them all, we conclude the audit and accept the IRV election result.

If the announced winner is wrong, then at least one of the assertions must be false. Since we test each assertion with an RLA at risk limit α, the RLA for the wrong assertion will mistakenly accept it with probability at most α. Hence the overall process is a valid Risk Limiting Audit—it will mistakenly accept the wrong outcome with probability at most α. Both types of assertions—NEB and NEN—can be tested with standard RLA systems, but they need to be carefully transformed into the right form.

NEB Assertions

Scoring NEB assertions

An NEB assertion, for example Alice NEB Bob, says that Alice’s first preferences exceed the total number of mentions of Bob that are not preceded by a higher preference for Alice. We start with a set of CVRs and count them as follows:

This fits naturally into any existing Risk-limiting audit process, except that our two candidates are “first preferences for Alice” and “mentions of Bob not preceded by Alice.”

Auditing NEB assertions

Consider again the assertion Alice NEB Bob. To conduct a ballot-level comparison audit of this assertion, the process for randomly selecting ballots for audit is the same as any other RLA. When a ballot is selected, overstatements are errors that advantage the “first preferences for Alice” candidate, while understatements are errors that advantage the “mentions of Bob (not preceded by a higher preference for Alice)” candidate. An overstatement is an error that either mistakenly records a first preference for Alice, or mistakenly omits a mention of Bob not preceded by Alice.

For example, if a CVR says that a vote is a first-preference for Alice, but the ballot paper shows only a second preference for her (and a first preference for some other candidate, say Diego), then this is a one-vote overstatement. If the ballot paper actually shows a mention of Bob, not preceded by a higher preference for Alice, then the error is a two-vote overstatement.

NEN Assertions

Scoring NEN assertions

An NEN assertion, for example NEN: Alice > Bob if only {Alice, Bob, Chuan} remain, says that Alice beats Bob when only Alice, Bob, and Chuan are the only continuing candidates.

This is also easy to fit into any existing Risk-limiting audit process, except that our two candidates are “Alice’s tally when only Alice, Bob, Chuan remain” and “Bob’s tally when only Alice, Bob, Chuan remain.”

We start with a set of CVRs and count them as follows:

We simply allocate the vote as if Diego has been eliminated. The same works when sets of more than one candidate have been eliminated, or when there are more than 4 candidates: simply score the vote for the first-ranked candidate among those continuing.

Auditing NEN assertions

For an NEN assertion NEN: Alice > Bob if only {S} remain, an overstatement is an error that advantages Alice by mistakenly listing her as the highest preference in set S, or disadvantages Bob by mistakenly not listing him first among S. An understatement is the opposite.

For example, for the assertion NEN: Alice > Bob if only {Alice, Bob, Chuan} remain, if the CVR was (Diego, Alice, Bob, Chuan), but the ballot paper actually contained (Diego, Bob, Alice, Chuan), that would be a two-vote overstatement. (The first preference for Diego is ignored because we consider only {Alice, Bob, Chuan} as continuing.)

Assertion scoring summary

Table above shows how to score each ballot for each possible kind of assertion. Note that the score is a function of the assertion and the vote only - it does not depend on the apparent outcome.

The overstatement counts (discrepancies) are derived by simply subtracting the ballot paper score from the CVR score (and vice versa for understatements—understatements are derived from subtracting the CVR score from the ballot paper score). For example, if a CVR says that a vote contained a first preference for w, but the actual ballot contains a mention of l that precedes w, then it overstates the w NEB l assertion by 1 - -1 = 2. By convention, overstatements are written as a positive discrepancy, while understatements are negative.