# Hypothesis

Test faster, fix more

Hypothesis is, of course, a library for writing tests.

But from an implementation point of view this is hardly noticeable. Really it’s a library for constructing and exploring data and using it to prove or disprove hypotheses about it. It then has a small testing library built on top of it.

It’s far more widely used as a testing library, and that’s really where the focus of its development lies, but with the find function you can use it just as well to explore your data interactively.

In this article we’ll go through an example of doing this, by using it to take a brief look at one of my other favourite subjects: Voting systems.

We’re going to focus entirely on single winner preferential voting systems: You have a set of candidates, and every voter gives a complete ordering of the candidates from their favourite to their least favourite. The voting system then tries to select a single candidate and declare them the winner.

The general Python interface for a voting system we’ll use is things that look like the following:

``````def plurality_winner(election):
counts = Counter(vote[0] for vote in election)
alternatives = candidates_for_election(election)
winning_score = max(counts.values())
winners = [c for c, v in counts.items() if v == winning_score]
if len(winners) > 1:
return None
else:
return winners[0]
``````

That is, they take a list of individual votes, each expressed as a list putting the candidates in order, and return a candidate that is an unambiguous winner or None in the event of a tie.

The above implements plurality voting, what most people might think of as “normal voting”: The candidate with the most first preference votes wins.

The other main voting system we’ll consider is Instant Runoff Voting ( which you might know under the name “Alternative Vote” if you follow British politics):

``````def irv_winner(election):
candidates = candidates_for_election(election)
while len(candidates) > 1:
scores = Counter()
for vote in election:
for c in vote:
if c in candidates:
scores[c] += 1
break
losing_score = min(scores[c] for c in candidates)
candidates = [c for c in candidates if scores[c] > losing_score]
if not candidates:
return None
else:
return candidates[0]
``````

In IRV, we run the vote in multiple rounds until we’ve eliminated all but one candidate. In each round, we give each candidate a score which is the number of voters who have ranked that candidate highest amongst all the ones remaining. The candidates with the joint lowest score drop out.

At the end, we’ll either have either zero or one candidates remaining ( we can have zero if all candidates are tied for joint lowest score at some point). If we have zero, that’s a draw. If we have one, that’s a victory.

It seems pretty plausible that these would not produce the same answer all the time (it would be surpising if they did!), but it’s maybe not obvious how you would go about constructing an example that shows it.

Fortunately, we don’t have to because Hypothesis can do it for us!

We first create a strategy which generates elections, using Hypothesis’s composite decorator:

``````import hypothesis.strategies as st

@st.composite
def election(draw):
candidates = list(range(draw(st.integers(2, 10))))
return draw(
st.lists(st.permutations(candidates), min_size=1)
)
``````

This first draws the set of candidates as a list of integers of size between 2 and 10 (it doesn’t really matter what our candidates are as long as they’re distinct, so we use integers for simplicity). It then draws an election as lists of permutations of those candidates, as we defined it above.

We now write a condition to look for:

``````def differing_without_ties(election):
irv = irv_winner(election)
if irv is None:
return False
plurality = plurality_winner(election)
if plurality is None:
return False
return irv != plurality
``````

That is, we’re interested in elections where neither plurality nor IRV resulted in a tie, but they resulted in distinct candidates winning.

We can now run this in the console:

``````>>> from hypothesis import find
>>> import voting as v
>>> distinct = find(v.election(), v.differing_without_ties)
>>> distinct
[[0, 1, 2],
[0, 1, 2],
[1, 0, 2],
[2, 1, 0],
[0, 1, 2],
[0, 1, 2],
[1, 0, 2],
[1, 0, 2],
[2, 1, 0]]
``````

The example is a bit large, mostly because we insisted on there being no ties: If we’d broken ties arbitrarily (e.g. preferring the lower numbered candidates) we could have found a smaller one. Also, in some runs Hypothesis ends up finding a slightly smaller election but with four candidates instead of three.

We can check to make sure that these really do give different results:

``````>>> v.irv_winner(distinct)
1

>>> v.plurality_winner(distinct)
0
``````

There are a lot of other interesting properties of voting systems to explore, but this is an article about Hypothesis rather than one about voting, so I’ll stop here. However the interested reader might want to try to build on this to:

1. Find an election which has a Condorcet Cycle
2. Find elections in which the majority prefers the plurality winner to the IRV winner and vice versa.
3. Use @given rather than find and write some tests verifying some of the classic properties of election systems.

And the reader who isn’t that interested in voting systems might still want to think about how this could be useful in other areas: Development is often a constant series of small experiments and, while testing is often a good way to perform them, sometimes you just have a more exploratory “I wonder if…?” question to answer, and it can be extremely helpful to be able to bring Hypothesis to bear there too.