The Swedish general election was held on September 14, 2014, to elect the *Riksdag*, all 21 county councils, and 290 municipal assemblies. The centre-right Alliance for Sweden coalition (comprising the Moderate Party, Liberal People's Party, Centre Party, and Christian Democrats) sought a third term in government. In contrast to the previous election, the three largest parties on the left (the Social Democrats, Green Party, and Left Party) ran independent campaigns, as did the far-right Sweden Democrats. The left-wing party, Feminist Initiative, did not secure the 4% threshold.

This article is going to investigate if there was an electoral fraud that gave an advantage to * Arbetarepartiet-Socialdemokraterna*, the party in power and which the the Prime Minister Stefan Löfven was elected from.

The Kingdom of Sweden has three levels of administrative divisions: County Council, Municipalities, and City Districts.

Sweden holds elections every four years to determine the makeup of the legislative bodies on those three levels of administrations. These elections determine the allocation of seats in the Riksdag. The Riksdag is the national legislative and the supreme decision-making body of Sweden. Elections to the 20 county councils and 290 municipal assemblies — all using roughly the same electoral system — are held concurrently with the legislative elections on the second Sunday in September.

The Riksdag is a unicameral legislature with 349 members, elected through a proportional representation electoral system. That is, the number of delegates awarded is proportional to the proportion of the vote that a party receives. For example, if Party A receives 40% of the votes and Party B receives 60% of the votes, then approximately 40% of the delegates are awarded to Party A and approximately 60% of the delegates are awarded to Party B. Opposingly, in winner-takes all electoral system, Party A will not be awarded any delegates while 100% of the delegates will be awarded to Party B in this case.

# Results and Analyses

According to Linda Berg and Henrik Osccarson,

On the basis of the election results, Social Democratic leader Stefan Löfven formed a weak minority coalition government together with the Green Party, with only 138 of the 349 parliamentary seats. Although Sweden is no stranger to minority governments (only 10 of the 30 governments in the post-war period have been majority governments), the new coalition is the fourth weakest in history in terms of parliamentary support.

The resulting composition of the Riksdag is provided in the chart below. Note that the two parties in the government, Social Democrats (S) and Greens (MP), do not hold a majority.

## The Benford Tests

The Benford Test is one of the tests to identify if there was a fraud in the election. Its use is controversial, since it makes the assumption that vote counts in free and fair elections follow a log-uniform distribution. This is highly suspect.

Using the Benford distribution and the chi-square goodness-of-fit test, the results suggested a "Type 1 error" in the second digit test.

For the votes that *Arbetarepartiet-Socialdemokraterna* received in 2014 election, the first digit test shows the p-value of 0.1091 and the second digit test shows the p-value of 0.03865. The p-value for the first digit test is slightly larger than the usual α value of 0.05. As such, we do not reject null hypothesis: there is no evidence of fraud for the first digit according to Benford's Law.

However, the p-value for the second digit is below the usual α value of 0.05. This suggests evidence of fraud in the Swedish general election. Does this make sense? Sweden has been politically stable and it is very unlikely for them to have an unfair election. Could this low p-value be due to the 5%-Rule? When we select α = 0.05, we are explicitly specifying that we are fine with claiming a fair election as fraudulent 5% of the time. This comes from the Type I Error rate.

We could relax the α level, or we could try to formulate a new digit test, one whose Type I Error rates are close to our claimed α. The Benford test has fallen into question, especially after work by Deckert et al. (2011). As such, we now begin to examine a replacement for the Benford test.

## The M²-1 Test

Since some well-known infinite integer sequences provably satisfy Benford's Law exactly and powers of almost any number is one of them, running the leading digit and second digit tests on the squared vote counts and then comparing them to the Benford distribution using the chi-square goodness-of-fit test might be an improvement over the Benford test. We shall call this test "M²-1 Test" for the leading digit test and "M²-2 Test" for the second digit test.

The procedure of these tests are just simply squaring the vote counts that we are looking at and then following the exact same steps as the leading digit and the second digit tests discussed in the previous section.

Using the Benford distribution and the chi-square goodness-of-fit test both for the leading digit and for the second digit of the squared vote counts reveal no evidence of fraud. For the votes that *Arbetarepartiet-Socialdemokraterna* received in 2014 election, the M²-1 Test shows the p-value of 0.5523 and the M²-2 Test shows the p-value of 0.2560. Both of them are rather large, especially compared to the usual α value of 0.05. As such, we cannot reject null hypothesis: there is no evidence that the leading digits or the second digits of the squared vote counts violate Benford's Law.

## Invalidation Rates Tests

The Invalidation Rates Tests are another tests to identify if there was a fraud in the election. This test examines if there is a statistically significant correlation between the number of the votes that a certain party received and the number of invalidated votes over multiple divisions or districts.

To perform this test, we use Binomial regression, with null hypothesis being π = constant, where π is the probability of a vote get invalidated in all the divisions/districts.

Also we look at the slope of the regression to determine if the invalidation of the votes either hurt or helped the candidate/party. Looking at the GLM for *Arbetarepartiet-Socialdemokraterna*, p-value is 0.613, and for Feminist Initaive, p-value is 0.0753, which are greater than alpha and we fail to reject the null hypothesis. Thus, we conclude that there is no evidence that there was differential invalidation regarding the *Arbetarepartiet-Socialdemokraterna* and Feminist Initiative parties.

# Conclusion

This article gave a brief overview of the 2014 Swedish election, including a look at how the Sweden actually elects its Riskdag. We also looked to see if there was evidence of fraud in the 2014 Swedish Election.

To detect the election fraud, the Benford tests were performed on the votes that the *Arbetarepartiet-Socialdemokraterna* party received in each of the districts.

Even though the Benford test for the second digit rejected the null hypothesis, it rather suggested a Type I error in Benford test than an unfairness in this election, since we are aware that the political background of Sweden very stable and the p-value was below 0.05. We then briefly introduced the new M-1 and M-2 tests which could be a replacement for the Benford test. Neither of these two tests indicate fraud in this election.

Also by looking for evidence of differential invalidations for *Arbetarepartiet-Socialdemokraterna* and Feminist Initiative parties, because the p-values are greater than alpha, we concluded that there is no evidence to support the assertion that there was fraud.

Official electoral body | Valmyndigheten | |
---|---|---|

Election | September 14, 2014 | |

Results | Official |