This Page implements the tools for finding out the **Risk Limits** for auditing of multi-level
elections as is the case in India. It features a one-time contest for electing a single Party or Coalition
to power in the Central Government.

To hide or show everything but the tools

Theory

A **Risk-Limiting Audit**, also sometimes referred to as the Result-Conforming Audit
is a method of post-election audit designed to confirm the election results by examining a small sample of
the votes. The **Risk Limit** is the maximum probability that the audit will not detect
any error in election result when it is present. The error here refers to the case when the
candidate/party declared as the winner in a given constituency is not the actual winner who
would be reported by a full-hand manual count of the votes.

There are quite a many tools for performing risk-limiting audit, the prominent of them being the
Ballot Polling Audit. Tools for performing that is present in Vanessa's webpage titled
**Simple Ballot Polling Audit**. You could alternatively use Stark's Ballot Polling tools at
**Stark's Ballot Polling Audit**.

Multi-level elections can be efficiently audited. The desired confidence level in the overall elections can be shared by individual constituencies. The product of the risk limits in the individual constituencies results in the overall risk limit. When a coalition wins the elections, auditing is done in those constituencies where the parties belonging to that particular coalition has won. The sharing of the risk limits is done assuming that the election outcome is correct in the audited constituencies. If during the audit the result in any constituency does change, then the risk limits to which the other constituencies have been audited must be changed. This will require auditing more number of votes in those constituencies. The new risk-limits will be reflected in the tools developed by us by simply feeding in new election results for that particular constituency.

The tools on this page work as follows:

- Choose the number of constituencies in the overall elections (In India, for eg. 543)
- Select the number of parties contesting in the elections.
- Enter the contest name in the specified field.
- Enter the candidate names in the specified fields and the number of seats won by that party.
- Select the checkboxes beside the parties to group them into the winning coalition.
- In the Election Results table fill in the results for the respective constituencies.
- Set the desired risk limit.
- Click on compute when you have successfully filled in all the fields.
- Audit the constituencies simultaneously using one of two Ballot Polling tools whose links are present on the top of this webpage. Update the table in case any constituency outcome changes and hence recompute the risk limits and continue the audit.

Risk Computing Tool

Risk-limits for various constituencies

Technical Notes

Say the desired overall confidence level is $$ 1 - \epsilon $$

We can audit the constituency *i* in which the winner has won to confidence level
$$1-\epsilon^{\alpha_i} $$

where \(\alpha_i\) is some constant between 0 and 1. A *flipset* is a set of constituencies
whose result when flipped will alter the election results. The \(\alpha_i\) must be such that
in any *flipset*, the sum of the \(\alpha_i\) of the constituencies must sum up to greater than
1.

The cost of auditing constituency i to risk limit \(\alpha_i\) where the diluted margin is \(\mu_i\) is proportional to \(\frac{\alpha_i}{\mu_i}\). Our tool solves the Linear Program of minimizing the total cost of auditing the constituencies in which the winner won subject to the constraints that \(0<=\alpha_i<=1\) and for every minimal flipset, \(\sum \alpha_i >=1\).