# Space of possible elections

For a given type of ballot, number of voters and number of candidates, there is a finite and computable number of potential elections that can possibly occur. Due to the combinatorial nature of the problem this number is typically extremely large, even for modest electorate sizes.

Nevertheless, understanding the construction of this **space of possible elections** may be helpful to others looking for analyzing elections computationally, and may also be instructive for generating alternative parameterizations and analyses of election models.

This article details the space of possible elections for certain ballot types, as well as presents expressions for the total number of possible elections.

# Notation and parameters

Throughout this article, the following conventions will be used:

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}**: the number of voters**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}**: the number of candidates**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}**: the number of possible scores in a rated ballot**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}}**: the set of possible ballots**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}(\mathcal{B}, n, c)}**: the set of possible elections using ballots from**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}}**,**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}**candidates and**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}**voters**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\cdot|}**: the cardinality (or size) of a set, e.g.**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathcal{B}|}**is the number of possible ballots.**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(k)}**: the set of partitions for the integer**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}**, with**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |p(k)|}**the partition number of**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}**.**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(k)}**: the set of compositions (ordered partitions) for the integer**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}**, with**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |C(k)|}**the number of compositions of**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in p(k)}**: a specific partition of**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(k)}****Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa \in C(k)}**: a specific composition of**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(k)}****Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{N}{k} = \frac{N!}{(N-k)!k!}}**: the binomial coefficient, the number of non-ordered ways to pick**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N}**, without replacement.

# Number of possible ballots

Since each voter may cast any of a finite set of valid ballots, we must first determine how many valid ballots are available.

## Single-mark ballots

Under single-mark ballots (1), like in plurality voting, the number of ballots is trivially **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^1(c)| = c}**
.

## Cardinal (rated) systems

Let **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}^C(c,r)}**
be the set of cardinal (*C*) ballots with **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}**
possible evaluations for candidates. In this case, the total number of ballots is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathcal{B}^C(c,r)| = r^c}**

For approval voting, **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = 2}**
.

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}**2 3 4 5 6 7 8 9 10 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^C(c,2)|}**2 4 8 16 32 64 128 512 1,024 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^C(c,5)|}**25 125 625 3,125 15,625 78,125 390,625 1,953,125 9,765,625 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^C(c,6)|}**36 216 1,296 7,776 46,656 279,936 1,679,616 1,0077,696 60,466,176 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^C(c,10)|}**100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 10,000,000,000 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^C(c,11)|}**121 1,331 14,641 161,051 1,771,561 19,487,171 214,358,881 2,357,947,691 25,937,424,601

If blanks are allowed, they can be considered an extra degree of rating (so 0-5 scores + blank makes 7 total evaluations).

## Ordinal (ranked) systems

For ranked (>) systems, we must consider the possibility of equal rankings vs. strict rankings. Partial rankings may also be possible, which leads to a more complex computation.

### Full strict rankings

Let **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}^{>}(c)}**
be the set of ranked ballots with strict, full rankings. In this case, the total number of ballots is simply

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathcal{B}^{>}(c)| = c!}**

These are the factorials or permutation numbers (OEIS sequence A000142).

2 3 4 5 6 7 8 9 10 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^{>}(c)|}**2 6 24 120 720 5,040 40,320 362,880 3,628,800

### Partial strict rankings

If limited to top

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathcal{B}^{>\emptyset}_{k}(c)| = \frac{c!}{(c-k)!}}**

But if voters are allowed to cast arbitrary partial rankings, the number becomes

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathcal{B}^{>\emptyset}(c)| = \sum_{k=1}^{c} |\mathcal{B}^{>\emptyset}_{k}(c)| = \sum_{k=1}^{c} \frac{c!}{(c-k)!} = \mathrm{e} \Gamma(c+1,1) - 1}**

Here, **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(x, z) = \int_{z}^{\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{d}t}**
is the incomplete gamma function, and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}}**
is Euler's number.

These are OEIS sequence A007526.

2 3 4 5 6 7 8 9 10 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\mathcal{B}^{>\emptyset}(c)|}**4 15 64 325 1,956 13,699 109,600 986,409 9,864,100

### Full rankings, equality allowed

Let **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}^{\ge}(c)}**
be the set of ranked ballots with full rankings, possibly with equalities. Every equality creates an equivalence class between candidates, within which the order does not matter. There can be at least 1 equivalence class (all candidates are ranked equal), and at most

The equivalence classes are determined by the compositions **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa \in C(c)}**
. For example, for we may have the compositions (3), (2,1), (1,2) and (1,1,1), giving the ballots of the form (_ = _ = _), (_ = _ > _), (_ > _ = _) and (_ > _ > _). Thus, there are equivalence classes in a given composition, the size of each is given by .

Each of the "slots" in a ballot may be filled with a candidate, but the order should not matter within an equivalence. Therefore, for the 1st equivalence with slots, there are possible choices of candidates. For the second, , for the third and so on, exhausting the possibilities at every level.

The total number of ballots is then given by

These are known as Fubini numbers or ordered Bell numbers (OEIS sequence A000670).

2 3 4 5 6 7 8 9 10 3 13 75 541 4,683 47,293 545,835 7,087,261 102,247,563

## Remarks

In addition to what was outlined above, specific voting systems may induce further structure to the ballots. For example, a cardinal system which normalizes ballots would reduce the number of possible ballots significantly. A cardinal system based entirely on margins would have a different number of *effective* ballots, despite the ballots themselves being different. Similar things may happen under ranked systems under additional constraints.

One may also want to exclude some ballots for having no effect. For example, the ballot with all equal ratings or rankings will have no effect in most voting systems. However, these ballots are only a vanishingly small number from the total possible ballots.

# The space of possible elections

Each of the voters may cast any one of the ballots, but since voters casting identical ballots are exchangeable the number of elections is smaller than .

The number of total unique elections depends on the possible partitions of the voters into different groups, each casting identical ballots. For each partition of the voters we have groups of voters, each casting a unique ballot. Therefore, we have to pick a different possible ballot for each of these groups. For the first group there are possibilities, for the second , for the third and so on. If the number of ballots is smaller than the number of voters, this sum must be truncated. Therefore, the total number of unique elections is then given by

As the partition number may be approximated by , we can estimate for cardinal elections

and for full strict ranking elections

These numbers are exceptionally large for even modest number of candidates and voters.

# Final remarks

Note that impartial culture models of voters do not sample this space uniformly, as there are many ballot combinations which produce identical elections due to the exchange symmetry between voters casting the same ballot.

If there are more ballots than voters, the impartial culture model would on average split the voters into as many groups as there are voters, each voter casting a unique ballot. Due to strong correlations between voters and candidates in real-life elections, most possible ballots are not expected to occur. The structure outlined here can be used to develop alternative sampling models which are agnostic to specific models of utility or voter distributions.

Furthermore, if candidates themselves are also exchangeable we obtain the number of unique election 'scenarios'. Due to potentially distinct number of candidates in each ballot this computation is usually more convoluted. As a rule of thumb for ballots which specify all candidates, divide the number by .

The impartial anonymous culture model (also known as Dirichlet model) also fails to capture this structure properly, as sampling this space uniformly would in general produce too many unique ballots. The ideal approach would be to generate a random integer partition of voters and sample it through the Dirichlet model.

# References & external links

- T. Nicolaus Tideman, Florenz Plassmann.
*The Structure of the Election-Generating Universe*

- Ya. V. Uspensky.
*Asymptotic expressions of numerical functions occurring in problems concerning the partition of numbers into summands*. Bull. Acad. Sci. de Russie, 14(6):199–218, 1920