Two-candidate plurality¶

$b_i$ is $i$ th ballot card, $N$ cards in all.

mark for Alice but not Bob, $A_{\mathrm{Alice, Bob}} (b_i) := 1$ .
mark for Bob but not Alice, $A_{\mathrm{Alice, Bob}} (b_i) := 0$ .
marks for both (overvote) or neither (undervote) or doesn't contain contest,

$A_{\mathrm{Alice, Bob}} (b_i) := 1/2$ .

$\bar{A}_{\mathrm{Alice, Bob}}^b := \frac{1}{N} \sum_{i=1}^N A_{\mathrm{Alice, Bob}} (b_i).$

Mean of a finite list of $N$ bounded, nonnegative numbers.

Alice won iff $\bar{A}_{\mathrm{Alice, Bob}}^b > 1/2$ .

General Plurality & Approval Voting¶

$K \ge 1$ winners, $C>K$ candidates in all.

Candidates $\{w_k\}_{k=1}^K$ are reported winners.

Candidates $\{\ell_j\}_{j=1}^{C-K}$ reported losers.

Outcome correct iff

$\bar{A}_{\mathrm{w_k, \ell_j}}^b > 1/2, \;\; \mbox{ for all } 1 \le k \le K, \;\; 1 \le j \le C-K$

Means of $K(C-K)$ lists of nonnegative, bounded numbers.

Same approach works for D'Hondt, Hamilton, & other proportional representation schemes. (Stark & Teague 2015;

Super-majority¶

$f \in (0, 1]$ .

Alice won iff

$\mbox{(votes for Alice)} > f \times \left ( \mbox{valid votes for anyone} \right )$

Set

$A(b_i) := \left \{ \begin{array}{ll} \frac{1}{2f}, & \mbox{$b_i$ has a mark for Alice and no one else} \\ 0, & \mbox{$b_i$ has a mark for exactly one candidate, not Alice} \\ \frac{1}{2}, & \mbox{otherwise}. \end{array} \right .$

Alice won iff

$\bar{A}^b > 1/2.$

Borda count, STAR-Voting, & other additive weighted schemes¶

Winner is the candidate who gets most "points" in total.

$s_{\mathrm{Alice}}(b_i)$ : Alice's score on ballot $i$ .

$s_{\mathrm{cand}}(b_i)$ : another candidate's score on ballot $i$ .

$s^+$ : upper bound on the score any candidate can get on a ballot.

Alice beat the other candidate iff Alice's total score is bigger than theirs:

$A_{\mathrm{Alice, c}}(b_i) := \frac{s_{\mathrm{Alice}}(b_i) - s_{\mathrm{c}}(b_i) + s^+}{2s^+}.$

Alice won iff $\bar{A}_{\mathrm{Alice, c}}^b > 1/2$ for every other candidate $c$ .

Ranked-Choice Voting, Instant-Runoff Voting (RCV/IRV)¶

2 types of assertions (Blom et al. 2018):

Candidate $i$ has more first-place ranks than candidate $j$ has total mentions.
After a set of candidates $E$ have been eliminated from consideration, candidate $i$ is ranked

higher than candidate $j$ on more ballots than vice versa.

Both can be written $\bar{A}^b > 1/2$ .

Finite set of such assertions implies reported outcome is right.

More than one set suffices; can optimize expected workload.

Auditing assertions¶

Test complementary null hypothesis $\bar{A}^b \le 1/2$ sequentially.

Audit until either all complementary null hypotheses about a contest are rejected at significance

level $\alpha$ or until all ballots have been tabulated by hand.

No multiplicity adjustment needed.

The SHANGRLA framework for Risk-Limiting Election Audits¶

Half-average nulls¶