The German federal election of 2025 represents an interesting case for game-theoretic analysis because the parties agreed to exclude the second-largest party (AfD) from any coalition.
This had a few notable side effects:
In what follows, we briefly review the theory behind the analysis, present the data, and then dive into the outcomes.
This analysis is purely mechanical: it does not take into account political positions.
Power distribution is not only about the number of votes you get, but also about how the votes are distributed. For example, if one party gets 49% and another gets 51%, the smaller one has zero power in a simple majority system. However, if there are 10 other parties with 5% each, the 49% party holds almost all the power. Thus, depending on the distribution of votes among the rest, a party’s power can range anywhere between 0% and 100%.
How do we measure it? The common approach is to count the situations in which a party is decisive for forming a winning coalition.
There are different ways of doing this (e.g. Banzhaf index, Shapley-Shubik index etc). While they can differ in extreme cases, they usually produce similar results in practice (we use Banzhaf index for our analysis).
First, here are the election results:
| Party | 2021 | 2025 |
|---|---|---|
| Union | 24.1% | 28.5% |
| SPD | 25.7% | 16.4% |
| AfD | 10.3% | 20.8% |
| Green | 14.7% | 11.6% |
| BSW | - | 4.98% |
| Left | 4.9% | 8.8% |
| FDP | 11.4% | 4.3% |
Which led to the following distribution of seats (the last column shows the hypothetical distribution if BSW had passed the 5% threshold):
| Party | 2021 | 2025 | *2025+BSW |
|---|---|---|---|
| Union | 197 | 208 | 221 |
| SPD | 206 | 120 | 127 |
| AfD | 83 | 152 | 162 |
| FDP | 91 | 0 | 0 |
| Green | 118 | 85 | 90 |
| Left | 39 | 64 | 68 |
| SSW | 1 | 1 | 1 |
| BSW | 0 | 0 | 39 |
| sum | 735 | 630 | 708 |
| maj | 368 | 316 | 355 |
Side note: German election rules are relatively complicated. The number of seats depends on the vote distribution, Bavaria has a special system, the SSW always gets one seat, and there is a mixed system of candidate votes and party votes. We abstract from these details here.
With theory and data in hand, we can calculate the power distribution. We use the Python package powerindex.
# 2021
> px -w Union:197 SPD:206 AfD:83 FDP:91 Left:39 SSW:1 -q 368 --csv -r 3
Union,0.333
SPD,0.333
AfD,0.167
FDP,0.167
Left,0.0
SSW,0.0
Note the 0% power for the Left, despite its 39 seats, since it does not contribute to any winning coalition.
# 2025
> px -w Union:208 SPD:120 AfD:152 FDP:0 Green:85 Left:64 SSW:1 -q 316 --csv -r 3
Union,0.4
SPD,0.233
AfD,0.233
FDP,0.0
Green,0.067
Left,0.067
SSW,0.0
# 2025+BSW
> px -w Union:221 SPD:127 AfD:162 Green:90 Left:68 SSW:1 BSW:39 -q 355 --csv -r 3
Union,0.333
SPD,0.167
AfD,0.267
Green,0.1
Left,0.1
SSW,0.0
BSW,0.033
However, once AfD is excluded, the picture changes.
Technically, we keep the same threshold and set the excluded party’s seats to zero.
# 2025 excl AfD
> px -w Union:208 SPD:120 AfD:0 FDP:0 Green:85 Left:64 SSW:1 -q 316 --csv -r 3
Union,0.583
SPD,0.25
AfD,0.0
FDP,0.0
Green,0.083
Left,0.083
SSW,0.0
Putting it all together:
| Party | 2021 | 2025 | *2025+BSW | *2025-AfD |
|---|---|---|---|---|
| Union | 33.3% | 40% | 33.3% | 58.3% |
| SPD | 33.3% | 23.3% | 16.7% | 25% |
| AfD | 16.7% | 23.3% | 26.7% | 0% |
| FDP | 16.7% | 0% | 0% | 0% |
| Green | 0% | 6.7% | 10% | 8.3% |
| Left | 0% | 6.7% | 10% | 8.3% |
| SSW | 0% | 0% | 0% | 0% |
| BSW | 0% | 0% | 3.3% | 0% |
From a game-theoretic perspective, we can observe two effects on power distribution:
A few directions that might be interesting for further exploration:
(This comment doesn’t sound political to me, but it might be perceived that way, so I’ve moved it here.)
One might ask: what’s the point of calculating the “+BSW” scenarios? Yes, they missed the threshold by only 0.02%, but rules are rules — there’s no room to rig the election, so nothing can be done about it. Two counterarguments: