Positional Scoring Rules¶

Suppose that $\mathbf{P}$ is an anonymous profile of linear orders (i.e., a Profile object). A scoring vector for $m$ candidates is a tuple of numbers $\langle s_1, s_2, \ldots, s_m\rangle$ where for each $l=1,\ldots, m-1$, $s_l \ge s_{l+1}$.

Suppose that $a\in X(\mathbf{P})$ and $R\in \mathcal{L}(X(\mathbf{P}))$ is linear order on the set of candidates. The rank of $a$ in $R$ is one more than the number of candidates ranked above $a$ (i.e., $|\{b\mid b\in X(\mathbf{P})\mbox{ and } b\mathrel{R}a\}| + 1$). The score of $a$ given $R$ is $score(R,a)=s_r$ where $r$ is the rank of $a$ in $R$.

For each anonymous profile $\mathbf{P}$, $a\in X(\mathbf{P})$ and scoring vector $\vec{s}$ for the number of candidates in $\mathbf{P}$, let

\[ Score_\vec{s}(\mathbf{P},x)= \sum_{R\in\mathcal{L}(X(\mathbf{P}))} \mathbf{P}(R) * score(R, x) \]

A positional scoring rule is defined by specifying a scoring vector for each number of candidates. The winners in $\mathbf{P}$ according to the positional scoring rule is the set of candidates that maximize their scoring according to the scoring vector for the number of candidates in $\mathbf{P}$. That is, if $F$ is a positional scoring rule, then for each profile $\mathbf{P}$,

\[ F(\mathbf{P}) = \mathrm{argmax}_{a\in X(\mathbf{P})} Score_\vec{s}(\mathbf{P}, a). \]

where $\vec{s}$ is the scoring vector for the number of candidates in $\mathbf{P}$.

The most well-known positional scoring rules are:

Plurality: the positional scoring rule for $\langle 1, 0, \ldots, 0\rangle$.
Borda: the positional scoring rule for ${\langle m-1, m-2, \ldots, 1, 0\rangle}$, where $m$ is the number of candidates.
Anti-Plurality: the positional scoring rule for ${\langle 0, 0, \ldots, -1\rangle}$.

from pref_voting.profiles import Profile
from pref_voting.scoring_methods import plurality, borda, anti_plurality

prof = Profile([[0, 2, 1], [0, 1, 2], [2, 1, 0], [1, 2, 0]], [3, 2, 1, 4])

prof.display()
plurality.display(prof)
borda.display(prof)
anti_plurality.display(prof)

+---+---+---+---+
| 3 | 2 | 1 | 4 |
+---+---+---+---+
| 0 | 0 | 2 | 1 |
| 2 | 1 | 1 | 2 |
| 1 | 2 | 0 | 0 |
+---+---+---+---+
Plurality winner is {0}
Borda winner is {1}
Anti-Plurality winner is {2}

An arbitrary scoring rule can be defined using the scoring_rule function. For instance, the Two-Approval voting method is a positional scoring rule in which scores are assigned as follows: candidates ranked either in first- or second-place are given a score of 1, otherwise the candidates are given a score of 0.

from pref_voting.profiles import Profile
from pref_voting.scoring_methods import plurality, borda, anti_plurality, scoring_rule

prof = Profile([[0, 2, 1], [0, 1, 2], [2, 1, 0], [1, 2, 0]], [3, 2, 1, 4])

two_approval_score = lambda num_cands, rank: 1 if rank == 1 or rank == 2 else 0

prof.display()
scoring_rule.display(prof, score = two_approval_score)

# for comparison, display the Plurality winner
plurality.display(prof)

+---+---+---+---+
| 3 | 2 | 1 | 4 |
+---+---+---+---+
| 0 | 0 | 2 | 1 |
| 2 | 1 | 1 | 2 |
| 1 | 2 | 0 | 0 |
+---+---+---+---+
Scoring Rule winner is {2}
Plurality winner is {0}

One problem with the above code is that the name of the scoring rule is “Scoring Rule” rather than “Two-Approval”. To define a voting method using the scoring_rule function, use the @vm decorator.

from pref_voting.profiles import Profile
from pref_voting.scoring_methods import plurality, borda, anti_plurality, scoring_rule
from pref_voting.voting_method import vm

@vm(name="Two-Approval")
def two_approval(profile, curr_cands = None):
    """Returns the list of candidates with the largest two-approval score in the profile restricted to curr_cands.
    """

    two_approval_score = lambda num_cands, rank: 1 if rank == 1 or rank == 2 else 0

    return scoring_rule(profile, curr_cands = curr_cands, score=two_approval_score)

prof = Profile([[0, 2, 1], [0, 1, 2], [2, 1, 0], [1, 2, 0]], [3, 2, 1, 4])

prof.display()
two_approval.display(prof)

+---+---+---+---+
| 3 | 2 | 1 | 4 |
+---+---+---+---+
| 0 | 0 | 2 | 1 |
| 2 | 1 | 1 | 2 |
| 1 | 2 | 0 | 0 |
+---+---+---+---+
Two-Approval winner is {2}

Plurality¶

pref_voting.scoring_methods.plurality(profile, curr_cands=None)[source]¶

The Plurality score of a candidate $c$ is the number of voters that rank $c$ in first place. The Plurality winners are the candidates with the largest Plurality score in the profile restricted to curr_cands.

Parameters:

profile (Profile) – An anonymous profile of linear orders on a set of candidates
curr_cands (List[int], optional) – If set, then find the winners for the profile restricted to the candidates in curr_cands

Returns:

A sorted list of candidates

Borda¶

pref_voting.scoring_methods.borda(edata, curr_cands=None, algorithm='positional')[source]¶

The Borda score of a candidate is calculated as follows: If there are $m$ candidates, then the Borda score of candidate $c$ is $\sum_{r=1}^{m} (m - r) * Rank(c,r)$ where $Rank(c,r)$ is the number of voters that rank candidate $c$ in position $r$. The Borda winners are the candidates with the largest Borda score in the profile restricted to curr_cands. :param edata: An anonymous profile of linear orders or a MarginGraph. :type edata: Profile, MarginGraph :param curr_cands: If set, then find the winners for the profile restricted to the candidates in curr_cands :type curr_cands: List[int], optional :param algorithm: if “positional”, then the Borda score of a candidate is calculated from each voter’s ranking as described above. If “marginal”, then the Borda score of a candidate is calculated as the sum of the margins of the candidate vs. all other candidates. The positional scores and marginal scores are affinely equivalent. :type algorithm: String

Returns:: A sorted list of candidates

Borda for ProfilesWithTies¶

pref_voting.scoring_methods.borda_for_profile_with_ties(profile, curr_cands=None, borda_scores=<function symmetric_borda_scores>)[source]¶: Borda score for truncated linear orders using different ways of defining the Borda score for truncated linear orders.

Anti-Plurality¶

pref_voting.scoring_methods.anti_plurality(profile, curr_cands=None)[source]¶

The Anti-Plurality score of a candidate $c$ is the number of voters that rank $c$ in last place. The Anti-Plurality winners are the candidates with the smallest Anti-Plurality score in the profile restricted to curr_cands.

Parameters:

profile (Profile) – An anonymous profile of linear orders on a set of candidates
curr_cands (List[int], optional) – If set, then find the winners for the profile restricted to the candidates in curr_cands

Returns:

A sorted list of candidates

Example:

from pref_voting.profiles import Profile
from pref_voting.scoring_methods import anti_plurality

prof1 = Profile([[2, 1, 0], [2, 0, 1], [0, 1, 2]], [3, 1, 2])
prof1.display()
print(anti_plurality(prof1)) # [1]
anti_plurality.display(prof1)

prof2 = Profile([[2, 1, 0], [2, 0, 1], [0, 2, 1]], [3, 1, 2])
prof2.display()
print(anti_plurality(prof2)) # [2]
anti_plurality.display(prof2)

+---+---+---+
| 3 | 1 | 2 |
+---+---+---+
| 2 | 2 | 0 |
| 1 | 0 | 1 |
| 0 | 1 | 2 |
+---+---+---+
[1]
Anti-Plurality winner is {1}
+---+---+---+
| 3 | 1 | 2 |
+---+---+---+
| 2 | 2 | 0 |
| 1 | 0 | 2 |
| 0 | 1 | 1 |
+---+---+---+
[2]
Anti-Plurality winner is {2}

Dowdall¶

pref_voting.scoring_methods.dowdall(profile, curr_cands=None)[source]¶

The first-ranked candidate gets 1 point, the second-ranked candidate gets 1/2 point, the third-ranked candidate gets 1/3 point, and so on. The Dowdall winners are the candidates with the greatest overall score in the profile restricted to curr_cands.

Parameters:

profile (Profile) – An anonymous profile of linear orders on a set of candidates
curr_cands (List[int], optional) – If set, then find the winners for the profile restricted to the candidates in curr_cands.

Returns:

A sorted list of candidates

Note

This system is used in Nauru. See, e.g., Jon Fraenkel & Bernard Grofman (2014), “The Borda Count and its real-world alternatives: Comparing scoring rules in Nauru and Slovenia,” Australian Journal of Political Science, 49:2, 186-205, DOI: 10.1080/10361146.2014.900530.

Scoring Rule¶

pref_voting.scoring_methods.scoring_rule(profile, curr_cands=None, score=lambda num_cands, rank: ...)[source]¶

A general scoring rule. Each voter assign a score to each candidate using the score function based on their submitted ranking (restricted to candidates in curr_cands). Returns that candidates with the greatest overall score in the profile restricted to curr_cands.

Parameters:

profile (Profile) – An anonymous profile of linear orders on a set of candidates
curr_cands (List[int], optional) – If set, then find the winners for the profile restricted to the candidates in curr_cands
score (function) – A function that accepts two parameters num_cands (the number of candidates) and rank (a rank of a candidate) used to calculate the score of a candidate. The default score function assigns 1 to a candidate ranked in first place, otherwise it assigns 0 to the candidate.

Returns:

A sorted list of candidates

Important

The signature of the score function is:

def score(num_cands, rank):
    # return an int or float

Example:

from pref_voting.profiles import Profile
from pref_voting.scoring_methods import scoring_rule, plurality, borda, anti_plurality

prof = Profile([[0, 1, 2], [1, 0, 2], [2, 1, 0]], [3, 1, 2])
prof.display()
scoring_rule.display(prof) # Uses default scoring function, same a Plurality
plurality.display(prof)

scoring_rule.display(prof, score=lambda num_cands, rank: num_cands - rank) # same a Borda
borda.display(prof)

scoring_rule.display(prof, score=lambda num_cands, rank: -1 if rank == num_cands else 0) # same as Anti-Plurality
anti_plurality.display(prof)

+---+---+---+
| 3 | 1 | 2 |
+---+---+---+
| 0 | 1 | 2 |
| 1 | 0 | 1 |
| 2 | 2 | 0 |
+---+---+---+
Scoring Rule winner is {0}
Plurality winner is {0}
Scoring Rule winners are {0, 1}
Borda winners are {0, 1}
Scoring Rule winner is {1}
Anti-Plurality winner is {1}