# It Is Easy For Multi-Issue Bundles To Advance Anti-Democratic Agendas

Matthew I. Jones, Matthew Chervenak, Nicholas A. Christakis

## Abstract

When confronted with a host of issues, groups often save time and energy by
compiling many issues into a single bundle when making decisions. This reduces
the time and cost of group decision-making, but it also leads to suboptimal
outcomes as individuals lose the ability to express their preferences on each
specific issue. We examine this trade-off by quantifying the value of bundled
voting compared to a more tedious issue-by-issue voting process. Our research
investigates multi-issue bundles and their division into multiple subbundles,
confirming that bundling generally yields positive outcomes for the group.
However, bundling and issue-by-issue voting can easily yield opposite results
regardless of the number of votes the bundle receives. Furthermore, we show
that most combinations of voters and issues are vulnerable to manipulation if
the subundling is controlled by a bad actor. By carefully crafting bundles,
such an antagonist can achieve the minority preference on almost every issue.
Thus, naturally occurring undemocratic outcomes may be rare, but they can be
easily manufactured. To thoroughly investigate this problem, we employ three
techniques throughout the paper: mathematical analysis, computer simulations,
and the analysis of American voter survey data. This study provides valuable
insights into the dynamics of bundled voting and its implications for group
decision-making. By highlighting the potential for manipulation and suboptimal
outcomes, our findings add another layer to our understanding of voting
paradoxes and offer insights for those designing group decision-making systems
that are safer and more fair.

##  1 Introduction

Groups are often required to make decisions about multiple issues at the same
time. In a world in which voters have unlimited time and energy, every issue
could be decided in isolation, thus ensuring that the will of the majority is
carried out on every issue and maximizing value for the group as a whole.
However, for a host of reasons ranging from the intrinsic relatedness of some
issues to time, convenience, and self-interested manipulation of the system,
groups often make decisions on a range of issues with one vote. This requires
placing multiple issues into a single _bundle_ and giving each group member
one vote for the entire set of issues.

Our recurring toy example will be a group of friends ordering a single pizza
with many available toppings. To decide which toppings to order, the group
could go down the list of toppings and do a vote on each. “Pepperoni, yes.
Mushrooms, no. Sausage, no…” and so on. This process would take a long time,
but in the end, the group should be satisfied with the pizza, as each topping
was added (or left off) by the will of the majority.

Now suppose everybody is hungry and rushed. Instead of taking the time to vote
on each topping separately, one person might propose a vote between ordering
the pizza with everything on it or nothing on it. If most members of the group
prefer the “supreme” pizza to a plain cheese, the group orders a pizza with
all the toppings, even if some are actually opposed by the majority. Is this
pizza a good choice for the group? How does this pizza compare to the best
possible pizza the group could have ordered? Was the sacrifice in quality
worth the speed and ease with which the group was able to make this choice?
Here, we provide a mathematical framework to quantify how group utility is
impacted by bundling multiple issues together and explore other more serious
applications.

Multi-issue bundling carries the risk of manipulation. A bad actor with
control of the voting agenda of the group can choose to place certain issues
in a bundle to prevent popular issues from passing. Therefore, throughout this
analysis, we will consider a theoretical non-voting antagonistic agenda-setter
(NVAAS) who has a preference for the unpopular position on every issue. Of
course, the agenda-setter could instead prefer the popular position on every
issue. This would end very well for the group, but this is unlikely to happen
and impossible to guarantee, so we focus on the worst-case scenario in which
the agenda-setter is opposed to the group on every issue. This NVAAS will
attempt to bundle issues together so that the final voting outcome is as far
from the issue-by-issue majority rule as possible. Our NVAAS could be a
disgruntled pizza shop owner who wants to sell generally unappetizing toppings
by bundling them together, or, in more realistic settings, a lawmaker trying
to pass unpopular legislation or a carmaker attempting to sell unwanted
amenities by bundling them with popular ones.

Early investigations into bundling demonstrated that voting on bundles can
benefit the minority, rather than majority, position on each issue [1], but
that if division of the question (dividing a bundle into its constituent
parts) is automatically applied to a bundle, the bundle made up of the median
position on each issue is the Condorcet winner [2]. Later, a series of related
“voting paradoxes” expanded upon the distinctions between bundle and single-
issue votes, including the Ostrogorski Paradox[3], Anscombe’s Paradox [4], and
the Paradox of Multiple Elections [5]. However, these paradoxes are sometimes
dismissed as being of little concern, either because they are trivial or
extremely unlikely [6]. Our work runs counter to this narrative, showing that
anti-democratic outcomes are quite flexible and can be easily manufactured and
maintained by an NVAAS.

Unsurprisingly, political scientists have studied the bundling of multiple
issues in the context of legislation, an area full of complex bills and high-
stakes decisions with the potential for manipulation [7]. There is substantial
empirical literature studying the omnibus bill in particular [8, 9], but
examining the multifaceted nature of bills is of general interest to the
political science community [10]. The compilation of multiple legislative
issues into a single bundle has also been proposed as a work-around to avoid
the detrimental effect of loss aversion [11]. Bundling can also (for better or
worse) save legislators from having to read and consider every minor detail of
a bill [12], and it can simplify decision-making for single issue voters [13].
Other recent work considers how multiple issues can be formed into bundles by
strategic actors seeking to maximize their own utility [14, 15].

Of course, there are other fields besides politics in which many issues are
grouped into a single choice. The bundling of amenities, like parking included
in rent, results in a non-optimal distribution of resources because renters
without cars are forced to pay for parking spaces they don’t need [16]. The
same holds for the packages of amenities, called trim levels, that are sold
with new cars [17]. While sometimes counter-productive, bundling products is a
good way to increase sales of individual products. Consumers may be willing to
buy a whole collection of products to get the one item they really desire
[18], but consumers purchasing a bundle of products expect them to be
discounted because they are in a bundle [19]. While companies claim these
bundles are for the benefit of consumers, it is worth considering them as
NVAASes, aiming to maximize profit at the consumer’s expense.

Here, we focus on the intrinsic value that bundling creates for the whole
group when individuals are voting sincerely, with or without an NVAAS. We show
that, in the average case, large bundles have a small but positive value,
suggesting there is some merit to placing large number of issues together.
However, when the bundling is controlled by a bad-faith actor, it is a
powerful tool to subvert the majority will on most issues. Using techniques
from the study of gerrymandering, we confirm that bundling can lead to anti-
democratic outcomes when antagonistic agents are given complete information
and control of the bundling agenda. Finally, we use American voter survey data
and demonstrate how this technique could be used to manipulate real groups of
real people making decisions on real political issues. Our results have
consequences for group decision-making, particularly in the area of politics.

##  2 Methods

In this section, we describe our model of multi-issue voting and highlight the
main features of the model. We begin with $n$ group members or voters and $m$
binary (we call the two choices “yes” and “no” for simplicity) issues. To
avoid some minor idiosyncrasies from ties and tiebreaking, we assume that $n$
and $m$ are both odd unless specifically mentioned.

We make two assumptions in our model: _separability_ and equal utility.
Separability means these issues are independent for each voter, so a voter’s
preferences on one issue do not depend on the outcome of another issue. This
may be true for some issues (a car buyer may only be interested in paying for
leather seats if they get other luxury amenities like heated seats and large
infotainment screens), but not for others (the quantity of cup holders is
unlikely to affect a buyer’s desire for advanced safety features). Equal
utility means that all issues have the same value to every voter which is
normalized to 1 for simplicity. These are standard assumptions when studying
multi-issue voting, and in the Discussion section, we consider these
assumptions at greater depth, why they are necessary, and what they imply
about the model.

###  2.1 Bundled Vote Scores

Table 1 shows an example of a utility profile with five voters and three
issues. Voters $v_{1}$, $v_{2}$, and $v_{3}$ have net positive utility from
the bundle and therefore all three issues pass, even though issues $t_{2}$ and
$t_{3}$ are opposed by a majority of voters. Was this a good outcome for the
group, or would it have been better to vote down the whole bundle? We need a
measure to determine how much value is generated for the group by a decision
to accept or reject a bundle of issues, so we introduce two metrics: utility
scores and issue scores.

| $t_{1}$ | $t_{2}$ | $t_{3}$
---|---|---|---
$v_{1}$ | 1 | 1 | 1
$v_{2}$ | 1 | $\pagecolor{red!25}-1$ | 1
$v_{3}$ | 1 | 1 | $\pagecolor{red!25}-1$
$v_{4}$ | 1 | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$
$v_{5}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$
Table 1: A utility profile for five voters and three issues. Three voters
($v_{1}$, $v_{2}$, and $v_{3}$) support the entire bundle so it will pass, but
only one of the issues, $t_{1}$, is supported by a majority of voters.
Throughout this paper, the cells with negative utility are shaded red to
increase readability.

The utility score begins with a single basic idea: the utility for a group is
just the sum of utilities of each of the group’s members. Therefore, if a
bundle is passed, the utility for the group is the sum of all of the entries
in the utility profile. If the bundle fails, it is the negative of this sum.
Finally, we normalize this value by dividing by the total number of voter-
issue pairs, $nm$, so that this measure takes values between $-1$ and $1$. The
utility profile in Table 1 has a utility score of $\frac{8-7}{3\cdot
5}=\frac{1}{15}$.

On the other hand, proponents of democracy could argue that the goal of group
decision-making is not to maximize social utility but to carry out the will of
the majority on every issue. Instead of looking to maximize utility, a good
group decision process aims to satisfy the majority on as many _issues_ as
possible. The issue score is the number of issues where the bundled choice
agrees with issue-by-issue voting, subtracted by the number of issues where
the two disagree, and normalized by dividing by the number of issues. In Table
1, the bundle passes, but only one of the three issues would pass by itself.
Therefore, the bundle’s issue score is $\frac{1-2}{3}=-\frac{1}{3}$.

Though these scores are clearly similar, they are not always in agreement. The
profile in Table 1 has a positive utility score, suggesting that approving the
bundle was the correct decision. But it also has a negative issue score,
suggesting the bundle should have been rejected. This is one of many profiles
where these two measures lead to opposite conclusions, so we will consider
both.

###  2.2 Voter Utility Models

These scores allow us to study bundled voting, but we still need a method of
forming utility profiles like Table 1 to compute the scores. We focus on two
models of voter utility formation.

The first is the independent, identically distributed (IID) model [20, 21]. In
this model, the value of each cell in the utility profile is $+1$ with
probability $p$ and $-1$ with probability $1-p$. This model is analytically
tractable, and all the closed-form equations that follow use the IID model. In
the special case of $p=\frac{1}{2}$, each utility is a coin flip, and the
entire bundle will be very competitive. Approximately half of the issues will
be supported and half opposed by the majority even in the limit of large $n$
and large $m$. Later, we show that this highly competitive domain produces the
lowest-value bundles.

For a higher-fidelity (and less tractable) model, we use the well-known
spatial model of voting [22, 23]. For each issue, we randomly choose two
points in a $k$-dimensional hypercube to be the “yes” and “no” positions. Each
voter is also assigned a random ideal point, and her utility for a given issue
is determined by the closeness of the voter to the “yes” and “no” points for
that issue. Pizza toppings could be placed in a three-dimensional space by
rating them on how salty, sweet, and savory they are. A diner who likes salty
and savory things and dislikes foods that are too sweet may prefer toppings
like bacon and pepperoni, while another diner who prefers sweet over salty may
vote down bundles with things like olives, instead supporting toppings like
pineapple or peppers. In this model, utility profiles across voters are highly
correlated since two voters with similar points in space will agree on almost
all issues. The results shown use $k=2$ dimensions, but the results do not
significantly change when using higher dimensions (see Supplementary
Information for more details).

To model a growing majority in the IID model, we simply adjusted the parameter
$p$. As $p$ grew larger, more utilities in the profile are positive,
suggesting the voters are becoming more similar and the issues more popular.
For the spatial model, we introduce a parameter $q$ that controls the
distribution of ideal points in space. When $q=\frac{1}{2}$, all points are
uniformly distributed across the entire space, and when $q$ grows close to
$1$, almost all voters will have positive utility on almost all issues. More
details on the spatial model and the parameter $q$ are in the Supplementary
Information.

##  3 Results

We divide our results into three sections. First, we examine the implications
of placing all issues in a single bundle. Then, we consider splitting a bundle
into two _subbundles_ of popular and unpopular issues, and finally, we
generalize this by considering any number of subbundles. In each case, we
study the average effect of the bundling process and determine the potential
for manipulation by well-informed antagonistic agents. At the end, we test
these methods on real data and confirm that these results hold in realistic
settings. Our results incorporate large ensembles of computer-generated
profiles for simulations alongside mathematical analyses.

###  3.1 Single Bundle

Placing a larger number of issues into a single bundle provides two
significant advantages for the decision-maker: time and processing
requirements. Returning to our pizza example, no group wants to spend precious
minutes voting on each individual topping when that time could be spent eating
instead. Furthermore, bundling can simplify the decision process: a voter who
loves veggies and sees a proposed pizza with broccoli, spinach, and peppers
may not even need to think about the rest of the toppings because she already
knows that she will enjoy the pizza. These are tangible benefits, but bundling
also has a significant downside. With a bundle, voters are no longer able to
provide any information about which issues they support and which they oppose.
As a result, voters are forced to support issues with negative utility and
vice versa. Depending on the situation, this can be a serious impediment to
social value.

In our hypothetical group trying to order a pizza, if the group is evenly
divided on each topping, then about half the group will be unhappy with every
decision. However, suppose the group is entirely made up of vegetarians. In
this case, the group will unanimously reject bacon as a topping, pleasing
everyone. They will do the same with all of the meat toppings, and accept many
or all of the veggie toppings. As the issues become less divisive and most
issue-by-issue votes turn out the same way, a bundled vote will align with a
higher fraction of the issue-by-issue votes.

Similarly, in the IID model, when $n$ and $m$ are large and
$p\neq\frac{1}{2}$, each issue vote will go the same way. Each issue has
roughly $pn$ supporters and $(1-p)n$ detractors. The expected utility of
passage is

$E(\text{Utility Score})=m|\frac{pn-(1-p)n}{nm}|=|2p-1|$ (1)

With both models and both measures of bundle value, we see an improvement in
score as the majority becomes stronger. We conclude that a bundled vote has
positive expected value and that the value increases as groups become less
heterogeneous. As one faction gains influence and agenda-setting control,
bundling erases less value. For the remainder of the paper, we will consider
only the highly competitive (and more interesting) case where $p=\frac{1}{2}$
or $q=\frac{1}{2}$.

Figure 1: The effect of a strong majority in the IID model (a,b) and the 2D
spatial model (c,d) on utility scores (a,c) and issue scores (b,d). The IID
model was adjusted by varying $p$ between $\frac{1}{2}$ and $\frac{13}{20}$.
The spatial model moved $q$ from $\frac{1}{2}$ to $1$. In (a), simulations
confirm our analytic results in Equation (1), except at $p=\frac{1}{2}$, where
Equation (2) holds. Parameter values: $n=101,m=51$.

In the Supplementary Information, we derive the expected utility score for a
bundle with a large number of voters and issues under the IID model with
$p=\frac{1}{2}$ to be

$E(\text{utility score})=\frac{2}{\pi\sqrt{nm}}.$ (2)

Figure 2: Distribution of utility scores (a,c) and issues scores (b,d) for the
$p=\frac{1}{2}$ IID (a,b) and $p=\frac{1}{2}$ 2D spatial (c,d) models of voter
utility. Each was generated with 10,000 simulated bundles where $n=m=101$.
Notice the excellent agreement on mean utility score for the IID model in (a)
between the analytic approximation from Equation (2) and the simulations.

This slight positive value of bundled votes occurs in perfectly balanced
competition, where the expected utility of a bundle is exactly zero. It may
seem that these two facts are in contradiction, but they are not. If all the
pizza toppings are bad, the group gets a cheese pizza (positive value for not
having unpopular toppings) and if the toppings are all good, the group gets
the supreme pizza with all the delicious toppings (also positive value).
Either way, the utility score will usually be positive. See the Supplementary
Information for more details about the value of this marginal utility.

With simulations, we see the distribution of both models and both scores in
Figure 2. In all panels, the average is small but positive, suggesting that
bundles are at least slightly effective, and that this conclusion is robust to
model parameters.

All the results so far use random bundles. Instead, suppose an NVAAS, perhaps
a pizza maker trying to sell unpopular ingredients, gets to custom design a
bundle similar to specialty pizzas like meat lovers, chicken-bacon-ranch, etc.
If these ingredients are all opposed by a majority of the group, how much can
the malevolent agent accomplish with the limited apparatus of bundling? As an
example, consider the utility profile in Table 2; the entire bundle is voted
down unanimously, but only two of the issues are actually opposed by a
majority. Using a single bundle, our NVAAS successfully prevented seven
popular issues from passing! This example is similar to the Ostrogorski
paradox, in which the winning party has minority views on every issue [3].

| $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $t_{5}$ | $t_{6}$ | $t_{7}$ | $t_{8}$ | $t_{9}$
---|---|---|---|---|---|---|---|---|---
$v_{1}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$ | $1$ | $1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$
$v_{2}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$ | $1$ | $1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$
$v_{3}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$ | $1$ | $1$ | $\pagecolor{red!25}-1$
$v_{4}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$ | $1$ | $1$
$v_{5}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$ | $1$
$v_{6}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$
$v_{7}$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$ | $1$ | $1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $\pagecolor{red!25}-1$ | $1$
Table 2: A heavily manipulated utility profile of seven voters and nine
issues. Negative utilities are highlighted to increase readability. Despite
unanimous opposition to the bundle, only two of the nine issues are not
majority-supported.

The utility score for Table 2 is a respectable $\frac{1}{9}$, while the issue
score is $-\frac{5}{9}$. The reason for this discrepancy in score is clear
from the pattern of positive and negative utilities in the utility profile.
There is enough positive utility for seven of the issues to have a slim
majority support, but they are spread out across all seven voters evenly, so
each voter narrowly votes down the bundle. Table 2 was created using the same
“packing” and “cracking” techniques that are used to create gerrymandered
political redistricting plans [24].

Because these utility profiles can be strongly gerrymandered in this way, the
fraction of voters supporting a bundle provides very limited information about
how few or how many of the issues have majority support. Suppose we have a
bundle with $n$ voters and $m$ issues, and $\overline{n}$ of the voters
support the bill. When $\overline{n}$ is small,

$\text{Max number of issues supported:
}\left\lfloor\frac{(n-\overline{n})(m-1)}{n+1-2\overline{n}}\right\rfloor.$
(3)

On the other hand, when $\overline{n}$ is close to $n$,

$\text{Min number of issues supported:
}\left\lceil\frac{\overline{n}(m+1)-m(n-1)}{n+1}\right\rceil.$ (4)

See the Supplementary Information for the derivation. Critically, these bounds
are extremely weak. For most values of $\overline{n}$, these bounds become
zero and $n$, meaning we can say nothing about the number of majority-
supported issues for certain. For example, if a bundle of 15 issues is
supported by only 3 of 21 voters, it is possible that all issues could be
majority-supported. Even with unanimous support or dissent, we can only be
sure that two or three of the issues are actually decided correctly by the
bundled vote. Collecting multiple issues into a single bundle could save a
group time and energy, but it also exposes them to potentially severe
manipulation by a well-informed bad actor, and not even unanimous agreement
can provide a guarantee in the value of the decision. These bounds are related
to, but distinct from, previous work categorizing the conditions under which
voting paradoxes can occur [25, 26].

###  3.2 Two Subbundles

Next, we consider the implications of breaking a bundle into two or more
subbundles. It is tempting to think that bundle division could be a magic
bullet. If we can somehow quickly detect which issues should be passed and
which should be rejected, we can put them into two separate subbundles, pass
the majority-supported subbundle, and reject the minority-supported subbundle
to achieve a perfect issue score of 1. Unfortunately, even if we ignore the
question of how to determine which issues have majority support and which do
not, an NVAAS could manipulate both in a similar manner to Table 2.

|  $t_{1}$ |  $t_{2}$ |  $t_{3}$ |  $t_{4}$ |  $t_{5}$ |  $t_{6}$ |  $t_{7}$ |  $t_{8}$ |  $t_{9}$ |  $t_{10}$ |  $t_{11}$ |  $t_{12}$ |  $t_{13}$ |  $t_{14}$ |  $t_{15}$ |  $t_{16}$ |  $t_{17}$ |  $t_{18}$ |  $t_{19}$ |  $t_{20}$ |  $t_{21}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{2}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{3}$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{4}$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{5}$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{6}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{7}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$
$v_{8}$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$
$v_{9}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$
$v_{10}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$
$v_{11}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$
$v_{12}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$
$v_{13}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$
Table 3: A large bundle of 21 issues for 13 voters. The first 11 issues form
the majority-supported subbundle and the remaining have only minority support.
However, the first subbundle has only two supporters while the second has
support from all but three voters.

Table 3 shows how dividing the bundle into majority- and minority-supported
subbundles can achieve the opposite of the intended result. This set of voters
has a total of 273 voter-issue pairs, and almost exactly half (137) are
positive, while the other half (136) are negative. This suggests there should
be a number of popular and unpopular issues in this bundle. Indeed, the first
11 of the 21 issues have majority support and the last 10 do not. However,
when we divide the bundle into the majority and minority supported issues, we
make things worse, not better! The majority support subbundle fails with only
two supporters, and the minority support subbundle passes with only three
dissenters. The resulting subbundle vote has an issue score of -1, the worst
possible outcome! In pizza terms, the group agreed to all the bad toppings and
rejected all the good ones.

Applying a similar method to the bounds in Equations (3) and (4) (see
Supplementary Information for derivations), we can determine how many
supporters the majority subbundle _must_ have and the maximum number of
supporters the minority subbundle _can_ have.

$\text{Number of voters supporting majority
subbundle}\geq\left\lceil\frac{m_{1}+nc}{m_{1}+c}\right\rceil.$ (5)

$\text{Number of voters supporting minority
subbundle}\leq\left\lfloor\frac{m_{2}(n-1)}{m_{2}+c}\right\rfloor.$ (6)

Here, $c=1$ for the subbundle with an odd number of issues and $c=2$ for the
subbundle with an even number. Like before, these bounds are not very
restrictive. In the voting profile in Table 3, we have $n=13$, $m_{1}=11$, and
$m_{2}=10$ and these bounds are equalities, so we know that these subbundles
are maximally manipulated.

There are still strong strategic considerations when deciding to subdivide a
bundle. In extreme cases, a bundle that is going to pass is divided into two
subbundles, both of which fail (or vice versa). A pizza metaphor would be the
following example: A group consists of three herbivores, three carnivores, and
one omnivore. The herbivores are turned off by the meats on the supreme pizza
and the carnivores are turned off by the veggies, so the group chooses to get
plain cheese. The pizza seller, disappointed by the lack of topping sales,
first asks if the group would like to add just the meat toppings. The
carnivore and omnivore coalition of the group agrees to this and the
vegetarians are outvoted. Then the group is asked if they would like to add
just the veggie toppings and they agree again, this time the herbivores
allying with the omnivore. By making it two choices instead of one, the groups
decision on every topping has been reversed. An example of this is shown in
Table 4. Bundles like this example are easy to come by; the probabilities of
this occurring to a random bundle division can be seen for various $n$ and $m$
can be seen in Table 5, ranging from 2.5% to 8.5%.

|  $t_{1}$ |  $t_{2}$ |  $t_{3}$ |  $t_{4}$ |  $t_{5}$ |  $t_{6}$ |  $t_{7}$ |  $t_{8}$ |  $t_{9}$ |  $t_{10}$ |  $t_{11}$ |  $t_{12}$ |  $t_{13}$ |  $t_{14}$ |  $t_{15}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$
$v_{2}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{3}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$
$v_{4}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{5}$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$
$v_{6}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$ |  $\pagecolor{red!25}-1$
$v_{7}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$
Table 4: A bundle that is rejected by the group 1-6, but when divided into two subbundles, both pass 4-3. | $m=11$ | $m=101$ | $m=1001$
---|---|---|---
$n=11$ | 3.3 (1.5) | 5.7 (2.3) | 5.9 (2.5)
$n=101$ | 5.6 (2.1) | 7.0 (2.1) | 7.7 (2.6)
$n=1001$ | 5.8 (2.3) | 6.9 (2.8) | 8.6 (2.6)
Table 5: The percent of IID utility profiles where the bundled vote is
completely reversed when randomly divided into two subbundles. A bootstrapping
method with 100 iterations of 100 samples was used for each pair of $n$ and
$m$, with standard deviation shown in parentheses.

###  3.3 Subbundling Schemes of Many Subbundles

The reason bundling is less effective than issue-by-issue voting is that
voters are typically forced to vote for issues they do not like to gain a vote
for the marginally larger number of issues they do support (or vice versa).
The larger a bundle is, the more issues a voter will be forced to vote for (or
against), in spite of their preferences on that particular issue. Therefore,
it stands to reason that as we divide a bundle into smaller bundles, the
expected value from the sum of the subbundle votes should be greater than the
value of the bundle. After all, the issue-by-issue vote, which has maximal
value in both utility and issue scores, can be considered a vote on $m$
single-issue bundles.

Equation (2) shows that if we divide a bundle into $k$ subbundles (randomly,
not according to an agenda-setter), the expected utility score increases by
$\frac{k}{\sqrt{k}}=\sqrt{k}$, so there is a law of diminishing returns on the
number of subbundles. This approximation breaks down as the subbundles quickly
get small until $k=m$, when this becomes simple issue-by-issue voting. We see
the results of simulations and the corresponding analytic approximations in
Figure 3. The main takeaway is that _random_ division of a bundle into
subbundles results in immediate added value for the group. We have shown (and
will continue to show) that an NVAAS can find anti-democratic bundles and
subbundles, but these are rare, and the average bundle division has positive
value. Therefore, when the bundle division is done randomly, and not by a
potentially malevolent actor, there is value to be gained that only needs to
be weighed against the cost of additional votes.

Figure 3: The utility scores for subbundle votes as a function of how many
bundles were used. In (a), issues are randomly assigned to one of $k$
subbundles. The utility score of the split bundle vote does not reach the
value of issue-by-issue voting, even when $k=m$, because many subbundles have
no issues while other have many. In (b), subbundles are created ensuring that
each subbundle is roughly the same size (while also making sure each subbundle
has an odd number of issues for tie-breakers). For example, if we have 21
issues into 5 subbundles, instead of creating subbundles of size 5, 4, 4, 4,
and 4, where tie-breaking effects would be substantial, we create them with
sizes 5, 5, 5, 3, and 3. When subbundles are created this way, we get much
more value when the number of subbundles is large compared to the purely
random method, since there are no wasted subbundles with zero issues.

We have shown that when utilities are random, bundles tend to be well-behaved
with positive value on average, but can be counterproductive when manipulated
by an NVAAS. Our previous examples of manipulation in Tables 2, 3, and 4 have
been carefully constructed by placing $1$s and $-1$s. However, this last
result suggests that, with surprising frequency, even uniformly generated
bundles in the average case can be manipulated to achieve anti-majoritarian
goals by dividing them up into subbundles. This is a significant shift,
showing manipulation is not just _possible_ but (relatively) _easy_.

To study the potential for ordinary bundles to be abused by bad actors, we
create utility profiles using the IID utility model and then examine each
possible subbundle. Using the efficiency gap score from gerrymandering [27],
we evaluate the reasonableness of that subbundle. We also compute the issue
score, our bespoke measure designed specifically for multi-issue group
decision-making.

Figure 4: Prevalence of potential manipulation in an average utility profile.
In (a), we see that most subbundles are well-behaved according to the
efficiency gap metric. However, this is not the whole story, as we can see in
(b) that many subbundles have very low issue scores, suggesting anti-
democratic outcomes. In (a) and (b) results are sorted by the size of the
subbundle. Then we add the additional restraint that each issue must appear in
one of the subbundles in a scheme. (c) shows us the distribution of issue
scores for subbundle schemes. The vast majority have positive scores, and it
is difficult to see how low the issue scores can go. (d) solves this problem
by showing the distribution of the lowest possible issue scores, and we see
that most utility profiles can be significantly manipulated, with around
$\frac{3}{4}$ of issues having anti-democratic outcomes, and potentially
almost all issues.

Again, the average case is promising. For reference, the subbundles in Table 3
have efficiency gap scores of 0.45 and 0.4 and issue scores of -1. A typical
subbundle has a a small efficiency gap (Figure 4a), and a positive issue score
(Figure 4b). Just like large bundles, random subbundles still generate
positive value. However, inside every large bundle, there are specific
subbundles that NVAASes could call up for a vote in order to pass a
significant minority agenda. Many subbundles have large negative issue scores.

This susceptibility to manipulation persists even if we require that all
issues must be voted on and included in a subbundle. In Figure 4c-d, we
analyze the ability of an individual with minority views on all issues to
create a _subbundle scheme_ by dividing a bundle into any number of subbundles
to maximize the number of issues passed or failed against the majority.
Beginning with a random utility profile, we consider every possible subbundle
scheme that uses only odd sized subbundles (to avoid tiebreaker effects). The
most extreme case seen in Figure 4d is a subbundling scheme in which 12 of the
13 issues are passed/failed against the majority, for a total issue score of
$\frac{-11}{13}$. Almost all utility profiles have a subbundle scheme in which
more than $\frac{2}{3}$ of all issues are passed or failed against the
majority. This is on par with the extreme case bounds that we derived earlier,
suggesting that the majority of bundles have the potential for strategic
manipulation on the same level as the most extreme bundles.

Returning one last time to the pizza example, suppose the owner has 13
toppings, none of which are liked by the majority of the customers. By
carefully bundling ingredients, the group will agree to almost all of the
ingredients. And this can be done with almost any group (although a new
subbundling scheme will be needed for each group), not just very particular
groups like the one in Table 4.

###  3.4 A Realistic Scenario

To demonstrate the effectiveness of this method, we create a hypothetical
committee of lawmakers using real American voters and test how well an NVAAS
could manipulate the group into an undemocratic result. Our data for
individual preferences comes from the ANES’s 2020 time series data,
specifically the post-election survey [28]. There were several phases of
processing the data to get to our final result, which we describe here.

This survey included hundreds of questions, including 15 questions that
revealed an respondent’s utility for an legislative issue. For computational
reasons (an exhaustive search of all possible subbundle schemes gets
computationally expensive quickly as $m$ increases), we reduced this set to 11
issues by removing the 4 issues that had the most consensus. The details of
the issues we used, including their ANES variable codes, can be found in the
SI. We collected the responses and created a large utility profile of all the
respondents. However, each question allowed a neutral response (neither
support or opposition). Since our model does not allow utility 0, we discarded
all respondents who were indifferent to one of the 11 issues. The initial
sample of voters was fairly evenly distributed across party identification
(2318 Democrats, 2250 Republicans, and 3712 other). After filtering out the
indifferent respondents, however, our population was more imbalanced, with 436
Democrats, 130 Republicans, and 278 other. To restore balance to our group,
have a more general statement about group utilities, and also reduce the
population for computational reasons, we sampled from this larger group to
create small committees of 10 Democrats, 10 Republicans, and 11 others. For
each of these 31 voter groups, we examine all possible subbundle schemes to
find the one most beneficial to an NVAAS.

All 1,000 simulated committees had a subbundle scheme with a negative issue
score. Most had scores of $\frac{-5}{11}$ or $\frac{-3}{11}$, meaning the
NVAAS was successful on 8 or 7 issues, respectively. One group, shown in
Figure 5, had an issue score of $\frac{-9}{11}$ in which only one of the 11
issues had a majority-preferred outcome! The full distribution of minimum
issue scores, as well as a more representative subbundling scheme, can be
found in the Supplementary Information.

Figure 5: A subbundling scheme for 11 issues and 31 voters. Color represents
if the subbundle/issue is preferred (blue) or opposed (red) by a majority, and
the vote result on each subbundle/issue is shown in parentheses. This is the
most heavily manipulated subbundle scheme found from the ANES data, with the
majority only happy about the outcome on one issue (free trade agreements). On
the rest, the NVAAS was able to achieve the minority preference by rejecting
popular issues.

This example showcases the power of antagonistic subbundling to subvert the
will of the majority, even when the group’s utility profile has an underlying
structure as individual preferences are driven by political orientation.

##  4 Discussion and Conclusion

The benefits and costs of bundling a large number of decisions into a single
vote to simplify the group decision process can be hard to assess. The group
is saved from having to query its members $m$ times and instead gets to hold a
single vote. The group members only have to make one determination of value
instead of $m$ separate choices. However, this economical decision-making
comes at a cost. By reducing to a single yes/no vote, members lose the ability
to draw distinctions between issues, which typically results in an outcome
with a lower utility for the group than issue-by-issue voting. This method
also leaves the group vulnerable to manipulation by an NVAAS, as these issues
can usually be bundled in very specific ways to achieve anti-democratic ends
in which most issues are accepted or rejected against the will of the
majority.

If the preferences of each individual on each issue (a complete utility
profile) is collected, that information should be used to automatically carry
out issue-by-issue voting. Random bundling is a benign process that can
genuinely save time and energy for the entire group, but if an agent is given
that information and then allowed to set the agenda by creating subbundles,
the result will largely be aligned with the utility of that agent, not the
group as a whole. This information is at least partially known in many
situations: congressional committee chairs likely know a great deal about the
preferences of the members of their committee and for-profit companies spend a
great deal of money analyzing consumer preferences and trends. Thus, most
groups are potentially vulnerable to this manipulation, regardless of any
underlying structure in the utility profile.

###  4.1 Assumptions and Consequences

As mentioned earlier, this model makes two assumptions about human
preferences: separability and equal utility. These are strong assumptions, so
we take some extra time to define and justify these assumptions, as well as
discuss the implications of these conditions.

The first assumption is separability, which means voter preference on one
issue is not dependent on the success or failure of any other issue [29, 30].
This condition is essential for studying issue-by-issue voting; if a voter
only prefers ham on a pizza if there is already pineapple on the pizza, there
is no clean, principled way to determine how she should vote on ham when the
outcome of the pineapple vote is unknown. Voter preferences are separable more
often than not, even on related issues like abortion access and Aid to
Families with Dependent Children programs [30]. Removing this assumption would
overburden our model, require further assumptions about voter behavior, and
obscure our results even further. We would need to take into account voters
who play it safe (voting no on ham just in case pineapple is not approved) or
who risk it all (vote yes for ham in the hopes that pineapple will also be
approved) to every type of behavior in the middle. This is a valuable area for
future research.

The second assumption, that all issues are equally important to each voter, is
necessary to remove the temptation for voters to trade votes. Variation in
issue utility leads to self-interested voters giving away votes on low-utility
issues in exchange for votes on high-utility issues which opens up all sorts
of interesting dynamics [20, 31, 32]. The precise way in which bundling
interacts with vote trading is an open question and is also a natural
direction for future work.

These assumptions also simplify other aspects of the multi-issue decision-
making process.

  * •

There are no single-issue voters [13] because all issues have equal weight.

  * •

We avoid the issue of exernalities by incorporating any “unintended
consequences” in the utilities of the issues. For example, if one member of
the group really likes sausage but knows that their vegan friend will not be
able to eat it, they may have a $-1$ for that utility despite personally
liking it.

  * •

Brinksmanship on large issues like the U.S. debt ceiling [33] appears when
groups cannot decide which issues should be bundled together. Since we
implicitly assume that a bundle has already been created, we do not have any
brinksmanship concerns. See also the following point.

  * •

Because issues are separable, voting behavior is independent of the sequence
in which issue-by-issue votes are cast. Therefore, we can assume they are
simultaneous.

  * •

With rational voters making binary decisions, simple majority rule satisfies
all the conditions for sincere voting [34], and there is no reason for a voter
to vote for a single issue/bundle that they oppose or vice versa.

  * •

We assume that the bundle in question is competing against the status quo. If
in fact it is two separate bundles that voters are deciding between, because
issues are separable, we can look only at the issues in which the two bundles
differ. One bundle can be labelled as the status quo and let voters decide to
pass or fail the other bundle, so our analysis here holds even for competition
between bundles with no status quo.

  * •

In this work, we completely ignore questions of information and information
asymmetry between voters. This is because voting is sincere, and therefore a
voter’s actions do not depend on what she perceives other voters’ intentions
to be. However, an NVAAS must have total information in order to successfully
manipulate the group.

Many of these things result in new and interesting dynamics, and this section
provides many future directions of exploration by loosening specific
assumptions to see how rational behavior changes.

###  4.2 Legislative Implications

This paper describes mathematically how majority rule decision-making
processes for bundled decisions may result in drastically different outcomes
than majority rule decision-making processes without bundling. In a
legislative context, where bundled legislation is the norm and dividing
legislation is difficult, our findings imply that when legislation passes by
majority vote, thereby achieving a theoretically positive aggregate utility,
it is mathematically possible that nearly all of the individual issues within
those legislative bundles are unsupported by a majority of legislators.
Majority rule by bundling and majority rule on individual issues could lead to
radically different legislative outcomes. This result implies that a high
proportion of existing law, were it divided into component measures, would not
be supported by any majority. The study further shows that partially dividing
legislation cannot solve this problem.

These findings raise fundamental questions about the design of legislative
processes. Should the goal of a legislature be to maximize the aggregate
utility of its legislators, or to pass legislation that garners majority
support? What is the role of bundling in the theoretical and practical pursuit
of those goals? By optimizing for aggregate utility, lawmakers get more of
what they want. However, this comes at the expense of passing a large body of
law that a majority of members do not want. Optimizing instead for majority
rule by issue, notwithstanding practicality concerns, would likely reduce the
amount of law created, but would ensure that resulting law was backed by true
legislative majorities. This method would preserve the majority rule principle
that many, perhaps falsely, believe is reflected in current practice but might
alienate minorities that receive fewer benefits from the legislative process.
Additionally, legislatures deal with highly nonseparable issues. Conflict can
also arise if, for example, a legislature votes to increase the number of
police or teachers but refuses to increase taxes to pay for them in a separate
vote.

Beyond the disconnect in majority rule between bundled and unbundled
legislation, the mathematics presented here show that NVAASes with asymmetric
information advantages could design legislation that maximizes or minimizes
either the aggregate utility or the number of underlying issues supported by a
majority without any apparent difference in the final vote tally. Lower cost
legislation, that is, legislation composed of issues with fewer supporters,
could be favored by bundlers seeking to maximize their own utility. A question
to be explored in future work addresses what information sharing or
legislative processes might mitigate this hazard. One option is voting
reforms. There is evidence to suggest that other voting methods like approval
voting [35] or yes-no voting [36] are less susceptible to bundling-related
manipulation than simple plurality voting.

## References

  * [1] Claude Hillinger.  Voting on issues and on platforms.  Behavioral Science, 16(6):564–566, 1971.
  * [2] Joseph B. Kadane.  On division of the question.  Public Choice, 13(1):47–54, 1972.
  * [3] Douglas W. Rae and Hans Daudt.  The ostrogorski paradox: A peculiarity of compound majority decision *.  European Journal of Political Research, 4(4):391–398, 1976.
  * [4] Carl Wagner.  Anscombe’s paradox and the rule of three-fourths.  Theory and Decision, 15(3):303–308, 1983.
  * [5] Steven J. Brams, D. Marc Kilgour, and William S. Zwicker.  The paradox of multiple elections.  Social Choice and Welfare, 15(2):211–236, 1998.
  * [6] William V. Gehrlein and Vincent Merlin.  On the probability of the ostrogorski paradox.  Studies in Choice and Welfare, page 119–135, 2020.
  * [7] William H. Riker.  The art of political manipulation.  Yale Univ. Pr., 1986.
  * [8] Glen S. Krutz.  Hitching a ride: Omnibus legislating in the US congress.  Ohio State University Press, 2001.
  * [9] Glen S. Krutz.  Omnibus legislating in the u.s. congress.  Legisprudence Library, page 35–51, 2021.
  * [10] Barry S. Rundquist and Gerald S. Strom.  Bill construction in legislative committees: A study of the u. s. house.  Legislative Studies Quarterly, 12(1):97, 1987.
  * [11] Katherine L. Milkman, Mary Carol Mazza, Lisa L. Shu, Chia-Jung Tsay, and Max H. Bazerman.  Policy bundling to overcome loss aversion: A method for improving legislative outcomes.  Organizational Behavior and Human Decision Processes, 117(1):158–167, 2012.
  * [12] Hanah Metchis Volokh.  Read-the-bill rule for congress, a.  Missouri Law Review, 76(1):135–184, 2011.
  * [13] Roger D. Congleton.  Information, special interests, and single-issue voting.  Public Choice, 69(1):39–49, 1991.
  * [14] Ricardo Alonso and Odilon Câmara.  Persuading voters.  American Economic Review, 106(11):3590–3605, 2016.
  * [15] Odilon Câmara and Jon X. Eguia.  Slicing and bundling.  The Journal of Politics, 79(4):1460–1464, 2017.
  * [16] C. J. Gabbe and Gregory Pierce.  Hidden costs and deadweight losses: Bundled parking and residential rents in the metropolitan united states.  Housing Policy Debate, 27(2):217–229, 2016.
  * [17] Kevin Doyle.  Sticker shock: Navigating car trim levels, Aug 2018.
  * [18] Alfons Priessner and Nina Hampl.  Can product bundling increase the joint adoption of electric vehicles, solar panels and battery storage? explorative evidence from a choice-based conjoint study in austria.  Ecological Economics, 167:106381, 2020.
  * [19] Roger M. Heeler, Adam Nguyen, and Cheryl Buff.  Bundles = discount? revisiting complex theories of bundle effects.  Journal of Product & Brand Management, 16(7):492–500, 2007.
  * [20] Dimitrios Xefteris and Nicholas Ziros.  Strategic vote trading in power sharing systems.  American Economic Journal: Microeconomics, 9(2):76–94, 2017.
  * [21] Helios Herrera, Massimo Morelli, and Thomas Palfrey.  Turnout and power sharing.  The Economic Journal, 124(574), 2014.
  * [22] Matthew I. Jones, Antonio D. Sirianni, and Feng Fu.  Polarization, abstention, and the median voter theorem.  Humanities and Social Sciences Communications, 9(1), 2022.
  * [23] Keith T. Poole and Howard Rosenthal.  Congress: A political-economic history of roll call voting.  Oxford University Press, 1997.
  * [24] Daryl R. DeFord, Nicholas Eubank, and Jonathan Rodden.  Partisan dislocation: A precinct-level measure of representation and gerrymandering.  Political Analysis, 30(3):403–425, 2021.
  * [25] Rajat Deb and David Kelsey.  On constructing a generalized ostrogorski paradox: Necessary and sufficient conditions.  Mathematical Social Sciences, 14(2):161–174, 1987.
  * [26] Carl Wagner.  Avoiding anscombe’s paradox.  Theory and Decision, 16(3):233–238, 1984.
  * [27] Moon Duchin.  Gerrymandering metrics: How to measure? what’s the baseline?  arXiv:1801.02064, 2018.
  * [28] American National Election Studies.  Anes 2020 time series study full release [dataset and documentation].  Technical report, ANES, 2021.
  * [29] Jérôme Lang and Lirong Xia.  Voting in combinatorial domains.  Handbook of Computational Social Choice, page 197–222, 2016.
  * [30] Dean Lacy.  A theory of nonseparable preferences in survey responses.  American Journal of Political Science, 45(2):239, 2001.
  * [31] William H. Riker and Steven J. Brams.  The paradox of vote trading.  American Political Science Review, 67(4):1235–1247, 1973.
  * [32] Alessandra Casella and Thomas Palfrey.  Trading votes for votes. a dynamic theory.  Econometrica, 87(2):631–652, Mar 2019.
  * [33] Helios Herrera, Antonin Macé, and Matías Núñez.  Political brinkmanship: Us debt ceiling.  preprint https://www.heliosherrera.com/Brexit.pdf, 2022.
  * [34] Kenneth J. Arrow.  Social Choice and individual values.  Yale University Press, 1963.
  * [35] Steven J. Brams.  Approval voting on bills and propositions.  The Good Society, 5(2):37–39, Spring 1995.
  * [36] Steven J. Brams and Peter C. Fishburn.  Yes-no voting.  Social Choice and Welfare, 10(1), 1993.

## Supplementary Information

##  Appendix A Details of the spatial model

There have been many variations on the spatial model of voting throughout the
years. Here, we use one of the simplest models, in which voters are assigned
ideal points in the $d$-dimensional hypercube, $[0,1]^{d}$. Issues are
assigned two points, a “yes” and a “no”, and each voter’s utility on the issue
is determined by the relative distance between her ideal point and the
“yes”/“no” points.

In most of the paper, these points are uniformly distributed throughout
$[0,1]^{n}$, as in Figure 6a. When studying the effect of a shifting majority,
however, we allow our population to drift to one corner of the hypercube by
adjusting a parameter $q$. Each coordinate of a voter’s ideal point is uniform
between 0 and $\frac{1}{2}$ with probability $q$ and uniform between
$\frac{1}{2}$ and 1 with probability $1-q$. Each issue’s “yes” position is
sampled the same way, while the “no” positions are reversed, being larger than
$\frac{1}{2}$ with probability $q$ and less than $\frac{1}{2}$ with
probability $1-q$. Therefore, when $q$ is much greater than $\frac{1}{2}$, the
population tends to be in one corner of the hypercube with all the “yes”
points, while all the “no” points are in the opposite corner, and most voters
support most issues, as in Figure 6b.

Figure 6: Examples of ideal point distributions for $n=21$, $m=11$. In (a), we
see the $q=\frac{1}{2}$ case, where voters and issues are all uniformly spread
out in the unit square. In (b), $q=0.85$, so voters and “yes” positions are
condensed in the bottom left quadrant while “no” positions are in the top
right. This set of voters and issues will have a large number of positive
utilities and only a few negative utilities.

The use of a spatial model requires choosing the dimension of the underlying
space. The natural choice for $d$ is application-specific. A bill deciding
whether or not to give extra funding to schools may be fairly one-dimensional
if each voter prefers to only fund schools who meet some standardized testing
threshold. On the other hand, a group deciding which snacks to get for a party
may have many different dimensions, such as sweet vs salty, healthy vs
unhealthy, etc. The results in the main paper used $d=2$, and fortunately,
this choice has very little impact on the value of the bundle, as can be seen
in Figure 7.

Figure 7: Robustness of bundle value with respect to dimension. As the number
of dimensions varies from 1 to 10, the utility (a) and issue (b) scores are
both unchanging, suggesting that the choice of dimension does not impact
bundle value.

##  Appendix B Derivation of bounds

Here we derive the upper and lower bounds for the number of number of issues
supported in a single bill with $\overline{n}$ supporters, as well as the
limits on the number of supporters of the majority and minority supported
subbundles.

###  B.1 Single Bundle

Let $n$ be the number of voters considering a bundle of $m$ issues. To avoid
ties, assume that $n$ and $m$ are both odd. Suppose the number of voters who
vote for the bundle (and therefore support over half the issues) is
$\overline{n}$. We would like bounds on the number of issues that are
supported, denoted $\overline{m}$.

####  B.1.1 Lower Bound

In the worst-case scenario, none of the $n-\overline{n}$ voters who did not
support the bundle support any of the issues and each of the $\overline{n}$
supporters support the fewest issues possible while still supporting the
bundle, $\frac{m+1}{2}$ issues. Therefore, the number of positive utilities in
the utility profile can be as low as $\overline{n}\frac{m+1}{2}$.

As far as issues are concerned, each issue can receive $\frac{n-1}{2}$ votes
and still be opposed by a majority. Therefore, if the number of positive
utilities is less than $m\frac{n-1}{2}$, all issues could have only minority
support. In the worst case scenario (for the lower bound), any excess utility
past this point is all given to the same issue until that issue has unanimous
support to avoid passing more issues than necessary.

$\text{Fewest possible total positive utility: }\overline{n}\frac{m+1}{2}$ (7)

$\text{Maximum positive utility before any issues pass: }m\frac{n-1}{2}$ (8)

$\text{Minimum excess positive utility:
}\overline{n}\frac{m+1}{2}-m\frac{n-1}{2}$ (9)

$\text{Min num issues passed:
}\left\lceil\frac{\overline{n}\frac{m+1}{2}-m\frac{n-1}{2}}{\frac{n+1}{2}}\right\rceil=\left\lceil\frac{\overline{n}(m+1)-m(n-1)}{n+1}\right\rceil$
(10)

Strictly speaking, these final two quantities could be negative (and in fact
are for most values of $\overline{n}$), in which case no issues need to be
passed.

####  B.1.2 Upper Bound

The computation for the upper bound is similar, except there is no longer any
concern about excess positive utility. Instead, “yes” votes are all cast for
the same issue until it reaches the necessary $\frac{n+1}{2}$ vote threshold
to be passed.

For the upper bound, suppose that the $\overline{n}$ supporters of the bundle
support every single issue, and the $n-\overline{n}$ voters that did not
support the bundle support $\frac{m-1}{2}$ issues, the maximum possible.
Therefore, we have $(n-\overline{n})\frac{m-1}{2}$ votes from non-supporters.
Each issue needs $\frac{n+1}{2}-\overline{n}$ votes from non-bundle-supporters
to gain a majority.

$\text{Maximum positive utility: }\overline{n}m+(n-\overline{n})\frac{m-1}{2}$
(11)

$\text{Maximum non-supporter positive utility: }(n-\overline{n})\frac{m-1}{2}$
(12)

$\text{Minimum non-supporter votes needed to pass an issue:
}\frac{n+1}{2}-\overline{n}$ (13)

$\text{Max num issues passed:
}\left\lfloor\frac{(n-\overline{n})\frac{m-1}{2}}{\frac{n+1}{2}-\overline{n}}\right\rfloor=\left\lfloor\frac{(n-\overline{n})(m-1)}{n+1-2\overline{n}}\right\rfloor$
(14)

####  B.1.3 Upper-Lower Bound Duality

As an interesting note, the placement of positive and negative utility to
minimize the number of issues passed with $\overline{n}$ bundle supporters is
exactly the opposite of the placement when trying to maximize the number of
issues passed with $n-\overline{n}$ supporters. This gives us the clean
relationship

$\text{lower}(\overline{n})=\text{upper}(n-\overline{n})$ (15)

###  B.2 Majority/Minority Supported Subbundles

Suppose we have $n$ voters and $m$ issues, $m_{1}$ of which have majority
support and $m_{2}=m-m_{1}$ of which do not. We assume $n$ and $m$ are odd to
avoid ties, but it is unavoidable that either $m_{1}$ or $m_{2}$ will be even.

Majority supported issue subbundle

The majority support subbundle has $m_{1}$ issues, each with majority support.
We want a lower bound on the number of supporters. To begin, we count the
number of positive utilities in the subbundle:

$\text{Amount of positive utility}\geq m_{1}\left(\frac{n+1}{2}\right).$ (16)

Depending on if $m_{1}$ is even or odd, each voter can support just under half
the issues before any voters support the subbundle.

$\text{positive utility before any supporters of subbundle}\leq
n\left(\frac{m-c}{2}\right)$ (17)

where $c=\begin{cases}1&m\text{ is odd}\\\ 2&m\text{ is even}\end{cases}.$

Therefore, the number of excess votes is bounded by

$\text{Amount of excess positive utility}\geq
m_{1}\left(\frac{n+1}{2}\right)-n\left(\frac{m-c}{2}\right).$ (18)

By dividing by the amount of positive utility each voter can take before
supporting every issue, we get our final bound:

$\text{Number of
supporters}\geq\left\lceil\frac{m_{1}\left(\frac{n+1}{2}\right)-n\left(\frac{m-c}{2}\right)}{\frac{m+c}{2}}\right\rceil=\left\lceil\frac{m_{1}(n+1)-n(m-c)}{m+c}\right\rceil$
(19)

Minority supported issue subbundle

This subbundle is similar. Because each issue is supported by a minority, we
have the following bound on the number of positive utilities:

$\text{Amount of positive utility}\leq m_{2}\frac{n-1}{2}.$ (20)

We also have a lower bound on how much positive utility a supporter needs:

$\text{Amount of positive utility per supporter}\geq\frac{m_{2}+c}{2}$ (21)

where $c$ is as above. Dividing these two, we get

$\text{Number of
supporters}\leq\left\lfloor\frac{m_{2}\frac{n-1}{2}}{\frac{m_{2}+c}{2}}\right\rfloor=\left\lfloor\frac{m_{2}(n-1)}{m_{2}+c}\right\rfloor$
(22)

##  Appendix C Derivation of bundle value

In the basic IID model, utility is $+1$ with probability $\frac{1}{2}$ and
$-1$ with probability $\frac{1}{2}$, for an expected value of zero. In fact,
the expected sum of all the utilities in a utility profile is zero. Despite
this, the expected utility score is not zero. To see this, imagine that
instead of voting for a bundle, we accept it if it has positive net utility
and reject it if it has negative net utility. Both decisions result in
positive value, so the expected value from this decision process is positive.
While bundled voting may not be quite as effective as this, it still functions
the same way, and manages to get a positive expected score by rejecting
negative-utility bundles and accepting those with positive utility.

Approximately half of voters will support the bundle and half will reject it.
However, by random chance and because $n$ is odd, the number of supporters
will be slightly different than the number of detractors. Consider the
difference between the number of supports and detractors. If these marginal
voters are supporters, the bill passes and they provide positive value. If
they are detractors, the bill fails and they also provide positive value (by
rejecting a bundle with negative utility). Either way, the non-marginal voters
cancel out in expectation (Figure 8 blocks A and B) and the marginal voters
provide positive value to the bundle.

Figure 8: Visual representation of marginal voters and marginal utility. Color
represents net positive or negative utility. The voters in A and B have equal
size and opposite utility, so they cancel out. The marginal voters are those
in blocks C-E. Each marginal voter has some positive and negative utility. The
utilities in blocks C and D are equal size and opposite for net zero utility.
What remains is block E, the marginal utility of the marginal voters, and this
is what provides the positive utility score to the bundled vote.

We can estimate the number of marginal voters by first observing that the
number of supporters is binomially distributed, where there are $n$ trials and
the probability of success is $\frac{1}{2}$. For large $n$, the binomial
distribution $B(n,\frac{1}{2})$ can be approximated by a normal distribution,
so the number of supporters is proportional to
$\mathcal{N}(n\frac{1}{2},n\frac{1}{2}\frac{1}{2})=\mathcal{N}(\frac{n}{2},\frac{n}{4})$.
The normal distribution has expected absolute deviation from the mean of
$\sqrt{\frac{2}{\pi}}\sigma$, so the number of expected supporters varies from
the mean by $\sqrt{\frac{2}{\pi}}\sqrt{\frac{n}{4}}$. Doubling this gives the
expected difference between supporters and detractors, aka the number of
marginal voters:

$E(\text{Number of marginal voters})=\sqrt{\frac{2n}{\pi}}.$ (23)

We know that each marginal voter votes the same way, and we can use the exact
same trick to determine how much value each of these voters adds to the
bundle. The number of positive utilities is binomially distributed
($B(m,\frac{1}{2})$) which can be approximated by a normal distribution
($\mathcal{N}(\frac{m}{2},\frac{m}{4})$) which has an expected absolute
deviation from the mean of $\sqrt{\frac{2}{\pi}}\sqrt{\frac{m}{4}}$ which we
double to get the marginal utility of voter:

$E(\text{Marginal utility of each voter})=\sqrt{\frac{2m}{\pi}}.$ (24)

The rest of the utility for each voter cancels out (Figure 8 blocks C and D),
so in the end the final utility of the bundle is just the number of marginal
voters times the marginal utility per voter:

$E(\text{Expected utility of bundle})=\frac{2}{\pi}\sqrt{mn}.$ (25)

The utility score of a bundle is normalized by the maximum possible utility,
so the expected utility score is the expected utility divided by $nm$:

$E(\text{Expected utility score of bundle})=\frac{2}{\pi\sqrt{mn}}.$ (26)

##  Appendix D ANES Data details

The 11 final issues from the survey data (with their ANES variable name in
parentheses) were:

  * •

Limits on foreign imports (V202231x)

  * •

Change in immigration levels (V202232)

  * •

Preferential hiring for blacks (V202252x)

  * •

Level of government regulation (V202256)

  * •

Government efforts to reduce income inequality (V202259x)

  * •

Higher taxes on millionaires (V202325)

  * •

Regulation of greenhouse gases (V202336x)

  * •

Ban on assault-style weapons (V202344x)

  * •

Free trade agreements (V202361x)

  * •

Universal basic income of 12k/year (V202376x)

  * •

Government spending on healthcare (V202380x)

For completeness, the four issues that we considered but discarded were:

  * •

Limits on campaign spending (V202225)

  * •

Vaccine requirements in schools (V202331x)

  * •

Background checks for guns (V202341x)

  * •

Government action on opioid addiction (V202350x)

The subbundling scheme shown in Figure 5 of the main paper was the most
extreme case found. All other subbundle schemes had a lesser degree of
manipulation. Figure 9 shows the distribution of minimum issue score for each
sample of 31 voters. A more representative example of a manipulated subbundle
scheme is shown in Figure 10, in which each subbundle has an issue with
majority approval.

Figure 9: For each sampled group of 31 voters, the distribution of minimum
possible subbundle scores across all schemes. The vast majority have issue
scores of $\frac{-5}{11}$ or $\frac{-3}{11}$, including the example shown in
Figure 5 of the main paper. Figure 10: A representative subbundling scheme
with an issue score of $\frac{-5}{11}$. Each bundle has one majority-preferred
outcome with the rest being beneficial for the NVAAS.