Table 1.
Descriptive statistics of polling stations in the 2017 Turkish constitutional referendum and the 2018 elections.
We show the mean value 〈xi〉(the average is taken over all polling stations i) and its standard deviation σ(xi) for five different variables xi, namely the number of voters Ni, turnout Ti, votes for winner Vi, relative turnout ti and the vote percentage vi.
Fig 1.
‘Yes’ and ‘Erdoğan’ votes as a function of turnout.
For a given level of turnout, the cumulative vote percentage of stations with this level or lower is shown. In 2017, a majority of more than 50% is achieved with the inclusion of high turnout stations (blue line). For 2018 we find similar results with slightly higher vote shares (red line).
Fig 2.
Election forensic fingerprints for recent Turkish elections.
The fingerprints for (A) 2017 and (B) 2018 show the joint vote–turnout distribution where the blue color intensity indicates the number of stations with a given vote and turnout. Both distributions are smeared out towards high vote and high turnout numbers, which is characteristic for ballot stuffing. A box plot (red horizontal boxes) shows the 25th, 50th, and 75th percentiles of the turnouts associated with a given level of votes, next to whiskers (red dashed lines) that indicate the 95% confidence interval. (C) Standardized fingerprints as defined in the text for 2017; they can be used to adjust for geographic heterogeneities in the data. (D) Traces of voter rigging can be identified by comparing the standardized fingerprints of small (red lines) and large (blue) polling stations. Small stations are particularly susceptible to voter coercion and intimidation, which results in their displacement toward inflated votes and turnout (shift of small stations shown as red lines toward the upper right corner). (E,F) The standardized fingerprints for 2018 are similar to results from 2017.
Fig 3.
Results for the statistical test for voter rigging.
(A) An accepted region for the displacements is constructed from the confidence interval of displacements observed in the reference set of trustworthy elections. There is a significant displacement δ(p) between small and large polling stations with values that lie outside this accepted region for Turkey 2017 (full dark magenta line) and 2018 (full light magenta line). The displacement sizes are substantially smaller than those observed in Russian or recent Venezuelan elections (shown as blue and red dashed lines). Reference elections are shown as dotted lines. (B) We rank all stations in the Turkish elections by their size and show the cumulative vote percentages cumi(v) which are computed over all stations with a size larger than the given rank. For higher ranks i, an increasing number of small stations is included, leading to a characteristic “hockey stick”. In 2017 it is the addition of small units with inflated votes and turnouts that pushes the results over the 50% line and leads to a majority of Yes’ votes (highlighted by a red circle). In the insets, we show the same relationship for other elections that (left) show significant displacements or (right) belong to the set of reference elections.