Voting Power Techniques: What Do They Measure?

Abstract

Voting power science is a field of co-operative game theory concerned with calculating the influence a voter can exert on the outcome of a voting game. The techniques used to calculate voting power have names like the Shapley-Shubik index, and the Banzhaf measure. They are invaluable when used to design democratically fair voting games.

In this paper we examine these different techniques, with the specific aim of trying to understand what they are measuring. Many commentators have argued that the techniques are similar, albeit with different probability models. But by focusing upon the less well know differences that exist in the underlying measures themselves, it soon becomes apparent that the dissimilarities between the techniques extend far beyond their methods of counting voting coalitions.

The author would like to thank Maurice Salles and Sean Breslin for their help in the preparation of this manuscript.

Appendices

Appendix 1: Abstentions and More

The main body of the paper examined voting games in which the voters were only allowed to vote “yes” or “no”. The concept of abstention was completely ignored. In this section, not only will we incorporate abstentions into our block counting methodology, but we will also expand the number of voting choices available to the voter.

1.1 Abstention

So let’s start with the most basic change, instead of allowing a voter to vote “yes” or “no”, we will now allow “yes”, “no”, or “abstain”. The process of creating voting power measures is the same as we saw previously, first we will construct our indicator functions, and then integrate them using $\mathbb{P}$.

1.1.1 Indicator Functions

The indicator functions for the different criticalities are given below. We skip the tiresome listing of truth tables and simply state the functions instead.

$$\displaystyle\begin{array}{rcl} \mathbb{I}^{\mathit{DC}_{i}^{0} }(\omega )& =& \mathbb{I}^{\mathit{Win}}(\omega ) - \mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ no}}). {}\\ \mathbb{I}^{\mathit{IC}_{i}^{0} }(\omega )& =& \mathbb{I}^{i_{\mathit{no}} }(\omega )\left (\mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ yes}}) - \mathbb{I}^{\mathit{Win}}(\omega )\right ). {}\\ \mathbb{I}^{\mathit{DC}_{i}^{\delta } }(\omega )& =& \mathbb{I}^{\mathit{Win}}(\omega ) - \mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ no}}). {}\\ \mathbb{I}^{\mathit{IC}_{i}^{\delta } }(\omega )& =& \mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ yes}}) - \mathbb{I}^{\mathit{Win}}(\omega ). {}\\ \end{array}$$

Remarkably, these are exactly the same indicator functions we used in the simple “yes/no” voting games. The addition of abstention hasn’t changed the indicator functions.

1.1.2 $\mathbb{P}$ and $\Omega$

We can create the new set $\Omega$ relatively easily. We simply take every voter in the game and then “combine” them together to create every possible combination of voting actions. What do we mean by this? Imagine a game with two voters, Voter 1 can vote “yes” or “no”, and Voter 2 can vote ‘yes”, “no”, and “abstain”. If we were to “combine” them together we would end up with the following list of possible voter actions,

Voter 1	Voter 2
“yes”	“yes”
“yes”	“abstain”
“yes”	“no”
“no”	“yes”
“no”	“abstain”
“no”	“no”

(Each element of the set $\omega \in \Omega$ is represented as a separate line in this table.)

This example shows how to modify $\Omega$ (and by extension $\mathbb{P}$) to incorporate new voting choices. All we need do is “combine” every possible voter choice with every other possible voter choice, to create an enlarged set $\Omega$.

1.1.3 Integrating the Indicators

If there is one thing we’ve learnt from block counting, it’s that, even if $\Omega$ and $\mathbb{P}$ change, providing the indicator is unchanged, the statistic being calculated must be unchanged. When we added abstentions we didn’t need to change the indicators, so it follows that, even in a game with abstentions, the voting power measures are given by the expressions in Sect. 4.7.

1.2 And More…

OK, so if we added abstentions so easily into our methodology perhaps we could do more? What if we allow extra voting options like “25 % in favour”, or “maybe”? Why not take this idea to its logical conclusion and let the voters select from a possibly infinite range of options?

1.2.1 The Indicator Functions

Giving a voter an infinite range of options to choose from means that we might no longer have an option that we can definitively call “yes”, or an option that we can definitively call “no”. So instead we define two new options called i _max and i _min . These are the generalised equivalents of voting “yes” and “no”, and change the likelihood of voter i’s desired outcome by the greatest amount. Using this new terminology we can give the new indicator functions as,

$$\displaystyle\begin{array}{rcl} \mathbb{I}^{\mathit{DC}_{i}^{0} }(\omega )& =& \mathbb{I}^{\mathit{Win}}(\omega ) - \mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}). {}\\ \mathbb{I}^{\mathit{IC}_{i}^{0} }(\omega )& =& \mathbb{I}^{i_{\mathit{min}} }(\omega )\left (\mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}}) - \mathbb{I}^{\mathit{Win}}(\omega )\right ). {}\\ \mathbb{I}^{\mathit{DC}_{i}^{\delta } }(\omega )& =& \mathbb{I}^{\mathit{Win}}(\omega ) - \mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}). {}\\ \mathbb{I}^{\mathit{IC}_{i}^{\delta } }(\omega )& =& \mathbb{I}^{\mathit{Win}}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}}) - \mathbb{I}^{\mathit{Win}}(\omega ). {}\\ \end{array}$$

1.2.2 $\mathbb{P}$ and $\Omega$

Adding potentially infinite options to each voter clearly changes $\Omega$ and $\mathbb{P}$. Just like before, all we do is “combine” the different voters to create the new set $\Omega$, and the new $\mathbb{P}$.

1.2.3 Integrating the New Indicators

Once again, the new $\mathbb{P}$ functions will not affect the statistic being calculated, however we are using slightly different indicators. When we integrate these new indicators we get the following,

$$\displaystyle\begin{array}{rcl} \Pr (\mathit{IC}_{i}^{0})& =& \Pr (((\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}})\mathrm{\;is\;Winning}) \cap i_{\mathit{min}}) -\Pr (\mathrm{Winning} \cap i_{\mathit{min}}). {}\\ \Pr (\mathit{DC}_{i}^{0})& =& \Pr (\mathrm{Winning}) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}})\mathrm{\;is\;Winning}). {}\\ \Pr (\mathit{TC}_{i}^{0})& =& \Pr (\mathrm{Winning}) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}})\mathrm{\;is\;Winning}) + {}\\ & & \Pr (((\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}})\mathrm{\;is\;Winning}) \cap i_{\mathit{min}}) -\Pr (\mathrm{Winning} \cap i_{\mathit{min}}). {}\\ \Pr (\mathit{IC}_{i}^{\delta })& =& \Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}})\mathrm{\;is\;Winning}) -\Pr (\mathrm{Winning}). {}\\ \Pr (\mathit{DC}_{i}^{\delta })& =& \Pr (\mathrm{Winning}) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}})\mathrm{\;is\;Winning}). {}\\ \Pr (\mathit{TC}_{i}^{\delta })& =& \Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}})\mathrm{\;is\;Winning}) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}})\mathrm{\;is\;Winning}). {}\\ \end{array}$$

And in those voting games where other voters are not affected by the way voter i votes,

$$\displaystyle\begin{array}{rcl} \Pr (\mathit{IC}_{i}^{0})& =& \Pr (i_{\mathit{ min}}) \times (\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{max}}) -\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{min}})). {}\\ \Pr (\mathit{DC}_{i}^{0})& =& \Pr (\mathrm{Winning}) -\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{ min}}). {}\\ \Pr (\mathit{TC}_{i}^{0})& =& \Pr (\mathrm{Winning}) -\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{ min}}) + {}\\ & & \Pr (i_{\mathit{min}}) \times \left (\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{max}}) -\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{min}})\right ). {}\\ \Pr (\mathit{IC}_{i}^{\delta })& =& \Pr (\mathrm{Winning}\;\vert \;i_{\mathit{ max}}) -\Pr (\mathrm{Winning}). {}\\ \Pr (\mathit{DC}_{i}^{\delta })& =& \Pr (\mathrm{Winning}) -\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{ min}}). {}\\ \Pr (\mathit{TC}_{i}^{\delta })& =& \Pr (\mathrm{Winning}\;\vert \;i_{\mathit{ max}}) -\Pr (\mathrm{Winning}\;\vert \;i_{\mathit{min}}). {}\\ \end{array}$$

The keen eyed reader will no doubt have spotted that these expressions are the same as the ones we generated for the simple “yes/no” voting games with the terms i _no and i _yes replaced by i _min and i _max .

Appendix 2: Multiple Outcomes and Complex Non-monotonic Decision Rules

Up to now we have been dealing with games that can be either “Winning” or “Losing”. But can we generalise our ideas to encompass more complex games? Games with more than two outcomes? Perhaps games that give some kind of ranking of alternatives? Once again, we find that we can do this, and more, with a minimum of fuss. But before we look at expanding the number of possible outcomes, let’s discuss the voting decision rule. Even though it was never explicitly stated before, there is no restriction on the decision rule. There is no requirement for it to be weighted, monotonic, or in any way sensible. It could be the most complex, non-monotonic rule you can think of. It will not affect our block counting methodology.

Now, back to expanding the number of possible outcomes. If we have a game with more than two mutually exclusive outcomes, then it becomes necessary to stipulate with respect to which particular outcome power is being measured. The reason for this is simple, in the most general types of games, with the potential for arbitrarily complex decision rules, the power of a voter might change from outcome to outcome.

Actually, this doesn’t complicate things very much. All we need to do is change our indicator functions slightly. We now have to specify with respect to which particular outcome we are measuring criticality. We will use the symbol O to represent the specified outcome. And we also need to change the definitions of voter actions i _min and i _max so that they are given with respect to outcome O. We will use the symbols i _min ^O and i _max ^O to do this.

$$\displaystyle\begin{array}{rcl} \mathbb{I}^{O{ \_}\mathit{DC}_{i}^{0} }(\omega )& =& \mathbb{I}^{O}(\omega ) - \mathbb{I}^{O}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}^{O}). {}\\ \mathbb{I}^{O{ \_ }\mathit{IC}_{i}^{0} }(\omega )& =& \mathbb{I}^{i_{\mathit{min}}^{O} }(\omega )\left (\mathbb{I}^{O}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}}^{O}) - \mathbb{I}^{O}(\omega )\right ). {}\\ \mathbb{I}^{O{ \_}\mathit{DC}_{i}^{\delta } }(\omega )& =& \mathbb{I}^{O}(\omega ) - \mathbb{I}^{O}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}^{O}). {}\\ \mathbb{I}^{O{ \_}\mathit{IC}_{i}^{\delta } }(\omega )& =& \mathbb{I}^{O}(\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}}^{O}) - \mathbb{I}^{O}(\omega ). {}\\ \end{array}$$

Integrating these new indicators gives us the following,

$$\displaystyle\begin{array}{rcl} \Pr (O{ \_}\mathit{IC}_{i}^{0})& =& \Pr (((\omega ^{N\setminus \{i\}} \times i_{ max}^{O})\mathrm{\;is\;}O) \cap i_{\mathit{ min}}^{O}) -\Pr (O \cap i_{\mathit{ min}}^{O}). {}\\ \Pr (O{ \_}\mathit{DC}_{i}^{0})& =& \Pr (O) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}^{O})\mathrm{\;is\;}O). {}\\ \Pr (O{ \_}\mathit{TC}_{i}^{0})& =& \Pr (O) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}^{O})\mathrm{\;is\;}O) + {}\\ & & \Pr (((\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}}^{O})\mathrm{\;is\;}O) \cap i_{\mathit{ min}}^{O}) -\Pr (O \cap i_{\mathit{ min}}^{O}). {}\\ \Pr (O{ \_}\mathit{IC}_{i}^{\delta })& =& \Pr ((\omega ^{N\setminus \{i\}} \times i_{ max}^{O})\mathrm{\;is\;}O) -\Pr (O). {}\\ \Pr (O{ \_}\mathit{DC}_{i}^{\delta })& =& \Pr (O) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}^{O})\mathrm{\;is\;}O). {}\\ \Pr (O{ \_}\mathit{TC}_{i}^{\delta })& =& \Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ max}}^{O})\mathrm{\;is\;}O) -\Pr ((\omega ^{N\setminus \{i\}} \times i_{\mathit{ min}}^{O})\mathrm{\;is\;}O). {}\\ \end{array}$$

And in those voting games where other voters are not affected by the way voter i votes,

$$\displaystyle\begin{array}{rcl} \Pr (O{ \_}\mathit{IC}_{i}^{0})& =& \Pr (i_{\mathit{ min}}^{O}) \times (\Pr (O\;\vert \;i_{\mathit{ max}}^{O}) -\Pr (O\;\vert \;i_{\mathit{ min}}^{O})). {}\\ \Pr (O{ \_}\mathit{DC}_{i}^{0})& =& \Pr (O) -\Pr (O\;\vert \;i_{\mathit{ min}}^{O}). {}\\ \Pr (O{ \_}\mathit{TC}_{i}^{0})& =& \Pr (O) -\Pr (O\;\vert \;i_{\mathit{ min}}^{O}) + {}\\ & & \Pr (i_{\mathit{min}}^{O}) \times \left (\Pr (O\;\vert \;i_{\mathit{ max}}^{O}) -\Pr (O\;\vert \;i_{\mathit{ min}}^{O})\right ). {}\\ \Pr (O{\_}\mathit{IC}_{i}^{\delta })& =& \Pr (O\;\vert \;i_{\mathit{ max}}^{O}) -\Pr (O). {}\\ \Pr (O{ \_}\mathit{DC}_{i}^{\delta })& =& \Pr (O) -\Pr (O\;\vert \;i_{\mathit{ min}}^{O}). {}\\ \Pr (O{ \_}\mathit{TC}_{i}^{\delta })& =& \Pr (O\;\vert \;i_{\mathit{ max}}^{O}) -\Pr (O\;\vert \;i_{\mathit{ min}}^{O}). {}\\ \end{array}$$

Once again we see that these are almost the same expressions we generated for the simple “yes/no” voting games. The most obvious difference being that power is now specified with respect to a given outcome O, and the idea of voting “yes” or “no” has been replaced with the action that most favours outcome O, and the action that least favours outcome O.

With these expressions we can now calculate voting power statistics in practically any voting game we desire. The decision rule of the game can be arbitrarily complex, the game can have many different possible outcomes, and every voter can have an infinite range of different voting actions to choose from.

Appendix 3: Definitions

This paper has deliberately simplified some of the more rigorous mathematical terms in order to ease comprehension of the material. In this section we give the required formal definitions.

Rather than restrict our analysis to a specific voting system, we will introduce here the concept of a generalised voting game. This generalised voting game encompasses all possible voting games of interest, in that it allows for any voting rule, any number of possible voting outcomes, and any probability distribution of the voters. In keeping with the spirit of generalisation, we will henceforth refer to the voters as players. The definitions of probability and product spaces are taken from Pollard (2003).

Definition 1.

A player is a probability space $(\mathcal{X}_{i},\mathcal{A}_{i}, \mathbb{P}_{i})$, where $\mathcal{X}_{i}$ is a set, $\mathcal{A}_{i}$ is a sigma-field of subsets of $\mathcal{X}_{i}$, and $\mathbb{P}_{i}$ is a countably additive, nonnegative measure with $\mathbb{P}_{i}(\mathcal{X}_{i}) = 1$. Given a set of N players, where | N |  = n, the set of all ordered n-tuples (x ₁, …, x _n ), with $x_{j} \in \mathcal{X}_{j}$ for each j ∈ 1, … n is denoted as $\mathcal{X}_{1} \times \cdots \times \mathcal{X}_{n}$ and abbreviated to $\Omega ^{N}$. Given a player i, the set of all ordered (n − 1)-tuples $(x_{1},\ldots,x_{i-1},x_{i+1},x_{n})$, with $x_{j} \in \mathcal{X}_{j}$ for each $j \in 1,\ldots,i - 1,i + 1,\ldots n$ is denoted as $\mathcal{X}_{1} \times \cdots \times \mathcal{X}_{i-1} \times \mathcal{X}_{i+1} \times \cdots \times \mathcal{X}_{n}$ and abbreviated to $\Omega ^{N\setminus \{i\}}$. The action of creating a single (n − 1)-tuple, denoted as ω ^N∖{i}, from a single n-tuple ω ^N by removing the element x _i is represented as ω ^N ∖ x _i . The action of creating a single n-tuple, denoted as ω ^N, from a single (n − 1)-tuple ω ^N∖{i} by adding an element $x_{i} \in \mathcal{X}_{i}$ is represented as ω ^N∖{i} × x _i .

Definition 2.

Given a set of N players, where | N |  = n, a set of the form $A_{1} \times \cdots \times A_{n} =\{ (x_{1},\ldots,x_{n}) \in \mathcal{X}_{1} \times \cdots \times \mathcal{X}_{n}: x_{i} \in A_{i}$ for each i}, with $A_{i} \in \mathcal{A}_{i}$ for each i, is called a measurable rectangle. The product sigma field $\mathcal{A}_{1} \times \cdots \times \mathcal{A}_{n}$ on $\mathcal{X}_{1} \times \cdots \times \mathcal{X}_{n}$ is defined to be the sigma field generated by all measurable rectangles. Let the product space $(\mathcal{X}_{1} \times \cdots \times \mathcal{X}_{n},\mathcal{A}_{1} \times \cdots \times \mathcal{A}_{n})$ be denoted as $(\Omega,\mathcal{F})$.

Definition 3.

A generalised voting game is a quadruple $(\Omega,\mathcal{F}, \mathbb{P},\mathcal{W})$ such that $(\Omega,\mathcal{F}, \mathbb{P})$ is the product space generated by a set of N players, $\mathbb{P}$ is the product measure, and $\mathcal{W}$ is a $\mathcal{F}\setminus \mathcal{O}$ measurable function, where the elements $O \in \mathcal{O}$ are called outcomes. Such a game is denoted as a $\mathbf{GVG}(\Omega,\mathcal{F}, \mathbb{P},\mathcal{W})$.

Definition 4.

For a $\mathit{GVG}(\Omega,\mathcal{F}, \mathbb{P},\mathcal{W})$, a player i is increasingly critical with respect to an outcome $O \in \mathcal{O}$ in an event $\omega ^{N} \in \Omega ^{N}$ if, and only if, $\mathcal{W}(\omega ^{N})\neq O$ and there exists an $\{x_{i}^{{\prime}}\}\in \mathcal{X}_{i}$ such that $\mathcal{W}((\omega ^{N}\setminus \{x_{i}\}) \times \{ x_{i}^{{\prime}}\}) = O$. Let O_IC _i denote the set of increasingly critical events for a player i with respect to an outcome O.

Definition 5.

For a $\mathit{GVG}(\Omega,\mathcal{F}, \mathbb{P},\mathcal{W})$, a player i is decreasingly critical with respect to an outcome $O \in \mathcal{O}$ in an event $\omega ^{N} \in \Omega ^{N}$ if, and only if, $\mathcal{W}(\omega ^{N}) = O$ and there exists an $\{x_{i}^{{\prime}}\}\in \mathcal{X}_{i}$ such that $\mathcal{W}((\omega ^{N}\setminus \{x_{i}\}) \times \{ x_{i}^{{\prime}}\})\neq O$. Let O_DC _i denote the set of decreasingly critical events for a player i with respect to an outcome O.

Definition 6.

For a $\mathit{GVG}(\Omega,\mathcal{F}, \mathbb{P},\mathcal{W})$, a player i is totally critical with respect to an outcome $O \in \mathcal{O}$ in an event $\omega ^{N} \in \Omega ^{N}$ if it is either increasingly critical or decreasingly critical, with respect to the aforementioned outcome and event. Let O_TC _i denote the set of totally critical events for a player i with respect to an outcome O. For any given event ω ^N, it is not possible to be simultaneously both increasingly and decreasingly critical with respect to a given outcome O, therefore (O_IC _i ∩ O_DC _i ) = ∅.

Definition 7.

Criticality $\boldsymbol{\delta }$—With this assumption there is no restriction on how player i can vote between the two different events that define it as critical. The set of criticality δ increasingly critical events for player i, with respect to an outcome O, is denoted by O_IC _i ^δ, and the set of criticality δ decreasingly critical events for player i, with respect to an outcome O, is denoted by O_DC _i ^δ.

Definition 8.

Criticality 0—With this assumption one of the two events that define player i as being critical must have player i voting with its lowest possible support for outcome O. The set of criticality 0 increasingly critical events for player i, with respect to an outcome O, is denoted by O_IC _i ⁰, and the set of criticality 0 decreasingly critical events for player i, with respect to an outcome O, is denoted by O_DC _i ⁰.

In simple “yes/no” voting games Criticality 0 and Criticality δ are equivalent. However, if any of the players are allowed to abstain this equivalence will be lost, and it will be necessary to understand which criticality assumption you wish to measure.

Definition 9.

For a $\mathit{GVG}(\Omega,\mathcal{F}, \mathbb{P},\mathcal{W})$, a player i, and an outcome $O \in \mathcal{O}$, let $\mathbb{I}^{O}: \Omega ^{N} \rightarrow \{\{ 0\},\{1\}\}$ be the indicator function that an event ω ^N is classified as outcome O, i.e. when $\mathcal{W}(\omega ^{N}) = O$. Then, given an $\omega ^{N\setminus \{i\}} \in \Omega ^{N\setminus \{i\}}$, define $\{x_{i}^{O_{\mathrm{max}}}\}$ such that for all $x_{i} \in \mathcal{X}_{i}$,

$$\displaystyle{\mathbb{I}^{O}\left (\omega ^{N\setminus \{i\}} \times \;\{ x_{ i}^{O_{\mathrm{max}} }\}\right ) \geq \mathbb{I}^{O}\left (\omega ^{N\setminus \{i\}} \times \; x_{ i}\right ).}$$

Likewise, define $\{x_{i}^{O_{\mathrm{min}}}\}$ such that for all $x_{i} \in \mathcal{X}_{i}$,

$$\displaystyle{\mathbb{I}^{O}\left (\omega ^{N\setminus \{i\}} \times \;\{ x_{ i}^{O_{\mathrm{min}} }\}\right ) \leq \mathbb{I}^{O}\left (\omega ^{N\setminus \{i\}} \times \; x_{ i}\right ).}$$

$x_{i}^{O_{\mathrm{min}}}$ and $x_{i}^{O_{\mathrm{max}}}$ are generalised equivalents of voting “yes” and “no”. They need not be unique elements within $\mathcal{X}_{i}$, and could instead be subsets. Should this turn out to be the case, the elements $\{x_{i}^{O_{\mathrm{min}}}\}$ and $\{x_{i}^{O_{\mathrm{max}}}\}$ can be taken as any appropriate element within said subsets.