The Condorcet Jury Theorem Under Ambiguity

Bele Wollesen; Bele Wollesen

doi:10.3998/ergo.9868

1. Introduction

This paper evaluates the Condorcet Jury Theorem (CJT) under ambiguity. It demonstrates that, under conditions of ambiguity, one of the theorem’s key assumptions—namely, the competence assumption—can fail in non-trivial ways. The CJT is one of the most important theorems in social choice theory and is well-known beyond the field. Informally, the CJT in its classic form states that if there are two verdicts, one of which is correct, and voters are competent (i.e., better than a coin flip at voting for the correct verdict) and independent of each other, then the probability that the majority vote outcome tracks the correct verdict increases with the size of the group, tending towards one if the group is infinite. Let us call this the epistemic benefits described by the CJT.

The influence of the CJT and its epistemic benefits is hard to overstate. The CJT has been applied to juries (e.g., Penrod and Hastie 1979; Urken and Traflet 1983), organizations (Nitzan and Paroush 1982), crowd-sourced peer review (Arvan et al. 2025), methodological triangulation (Heesen et al. 2019), and generally serves as one of the arguments for democratic decision-making and the use of majority rule (Goodin and Spiekermann 2018). In the literature, it is common practice to rigorously justify the applicability of the independence assumption when using the CJT. However, such rigorous justification is often deemed less necessary for the competence assumption in the absence of specific factors (e.g., misleading propaganda).¹ I demonstrate that this is a mistake and that the competence assumption warrants more attention. It is important not to conflate an assumption of epistemic competence with a behavioural assumption of competence, which is required for the CJT. In this regard, the paper develops two important contributions to the literature: First, it shows that the conditions under which we can guarantee voter competence become obscure once we adopt a less idealised uncertainty framework. And second, there is a fruitful research project that identifies collective decision procedures better suited to less idealised uncertainty frameworks. For example, we will see that allowing abstention can help secure the epistemic benefits of voting.

There already exists a body of work checking the robustness of the CJT, leading to extensions of the original result and identifying the limits of its application. For example, it has been shown that not all individuals in the group need to be competent² (see, among others Boland 1989; Grofman et al. 1983). We also know that certain sorts of dependencies between judgements can retrieve the CJT (Dietrich and Spiekermann 2013; Ladha 1992; 1993). Moreover, List and Goodin (2001) show that the CJT can be generalized from the majority rule over two verdicts to the plurality rule over many verdicts; the plurality rule allows every voter to cast one vote, and the verdict with the most votes is elected.

However, the literature has yet to consider how the CJT holds up in an ambiguous world. In decision theory, ambiguity refers to a type of uncertainty where we cannot assign a specific probability to an event. If we can assign a precise probability to an event, this is called decision-making under risk. For example, in roulette, you can assign a precise probability to each outcome. Traditionally, this is contrasted with decision-making under uncertainty. In such cases, you have no information about an event’s probability. Typically, we are not completely ignorant of the truth but need more information to assign a precise probability. Arguably, we often find ourselves in this situation when we cast a vote. For instance, you might think that if a certain party is elected, it is more likely than not to support investment in green energy, even if you cannot assign a precise probability. In decision theory, there is an increasing acceptance of the descriptive as well as normative inadequacy of modeling agents’ epistemic states through precise probabilities (e.g., Kyburg 1983; Levi 1985; Seidenfeld 2004; Joyce 2010; Bradley 2017). This paper aims to take these concerns seriously and asks under which conditions we can still hope for competent voters and, thus, the epistemic benefits of voting. Prior work on voter competence modeled voters’ epistemic states through precise probabilities. It also assumed some form of maximizing expected utility (e.g., Austen-Smith and Banks 1996). Both assumptions are not directly applicable when voters need to make decisions based on ambiguous attitudes. Thus, this paper will deviate from previous work by modeling epistemic states as sets of probabilities.

This paper is structured as follows. I introduce the CJT with a simple model for agents who can be described using precise probabilities. For these agents, voter competence is guaranteed under assumptions of rationality, honesty, and epistemic competence. Next, I motivate and expand the model. I explicate how we model the epistemic states of rational agents that involve ambiguity and how to fill in our assumptions given those epistemic states. We explore why voter competence cannot be guaranteed under these conditions. Finally, we revisit assumptions, propose alternatives, and explore how to retrieve voter competence.

2. CJT and a Simple Model of Voter Competence

Imagine a researcher named Gyde who has to brief a policymaker about new insights in climate science. This briefing will include many agenda points, including whether the global temperature will rise over 1.5°C between 2030 and 2052 if we continue as we do now. As they are short on time, they ask their research assistants to read a paper each and email them back whether the statement that global warming will reach 1.5°C between 2030 and 2052 if it continues to increase at the current rate or the statement that global warming will not reach 1.5°C between 2030 and 2052 if it continues to increase at the current rate should be included in the briefing.³ Would it be a good idea for them to simply follow whatever the majority of their research assistants advise? To help answer this, let us take a closer look at the CJT and the assumptions needed to apply it.

2.1. The Condorcet Jury Theorem

While the core idea of the CJT is simple, we will need to introduce some notation at this point to rely upon later.⁴ Let us start by assuming that we have n voters, labelled $1, … n$ . These voters have to make a decision over $χ$ . In our example, $χ$ consists of two opposing statements that can be included in a briefing, and our voters are the research assistants. Generally, $χ$ is a set that contains multiple verdicts x, of which voters need to pick one. Yet, only one x is the correct verdict. The verdict may be correct according to an epistemic, moral, or any other standard. In this paper, we assume that to be correct is to match an (external) truth. Let us call the correct verdict in $χ$ , x. We say that ${T, F} = χ$ , and thus $X = T$ or $X = F$ . For each voter i, we write $V i$ to denote voter i’s vote, where $V i$ can take the values $T$ or $F$ . Here, $V i = T$ represents a vote for verdict $T$ from voter i, while $V i = F$ represents a vote for verdict $F$ . We measure voter competence for each voter i and each possible truth x as $P r (V i = x | X = x)$ . If, then, the two following assumptions hold, we can derive the CJT.

Voter competence is satisfied if for every voter i, $P r (V i = x | X = x)$ exceeds $0.5$ .
Voter independence is satisfied if the votes of all voters, $V 1, V 2, …, V n$ , are mutually independent, conditional on the truth.⁵

Additionally, for any voters $i ≠ j$ , we assume $P r (V i = x | X = x) = P r (V j = x | X = x)$ . Informally, this says that each voter is better than random at voting for the truth and equally proficient at doing so. While not needed to derive the CJT, we will also assume for any voter i, $P r (V i = T | X = T) = P r (V i = F | X = F)$ .⁶ Let us call the outcome of a majority vote of the n voters $O$ . The outcome $O$ is $O = T$ if there are more voters with $V i = T$ than with $V i = F; O = F$ if there are more voters with $V i = F$ than with $V i = T$ ; and $O = 0.5$ if there is a tie. The CJT then guarantees what we called the epistemic benefits of majority voting.

Epistemic benefits are satisfied when, for each possible truth x, $P r (O = x | X = x)$ increases and converges to 1 as the number n of voters increases.

2.2. A Simple Model of Voter Competence

Discussions of the CJT’s applicability usually focus on the independence assumption (e.g., Ladha 1992; 1993). Although crucial for deriving the CJT, the voter competence assumption is silent on how voters make decisions and is rarely questioned.⁷ In fact, voter competence is a behavioural assumption that conflates several assumptions, which is useful to tease apart, especially when considering the effect of ambiguity.⁸

To reason about agents, we typically make three assumptions: what the voter values, the voter’s epistemic states, and how the voter acts based on their values and epistemic states. Ambiguity affects voters’ epistemic states rather than directly influencing their behaviour. An agent’s epistemic states involve two assumptions that are often conflated. For our purposes, it is useful to separate them. The first assumption concerns the internal structure of epistemic states. Common ways to model the internal structure of the epistemic states of epistemically competent agents include, for example, all-out beliefs or credence functions. In principle, we can assume that the epistemic states of our agents are well-ordered without saying anything about how their epistemic states relate to the world. Ideally, however, we also want to address how epistemic states relate to the world or, at the very least, to the evidence available to the agent. An agent whose epistemic states are well-ordered but unresponsive to the evidence around them is only competent in a very thin sense. Thus, when I refer to the epistemic competence of agents, I am referring to the combination of two assumptions: the internal structure of epistemic states and the relation of epistemic states to the truth. In the following, I will outline assumptions about honesty and practical rationality that underlie voter competence in cases of risk, before addressing more complex scenarios involving ambiguity.

2.2.1. Internal Structure of Epistemic States: Credence Functions

The first assumption concerns an agent’s internal epistemic representation. We assume that the epistemic states of a voter can be represented by credence functions. These are rational credences in the sense that they are between 0 and 1 and follow the probability axioms.⁹ We assume that voters have a certain credence over $X = T$ and $X = F$ (i.e., $p (T)$ and $p (F)$ , respectively), and because $X = F$ and $X = T$ are exhaustive and mutually exclusive, it holds that $p (T) = 1 – p (F)$ .¹⁰ For illustrative purposes, let us assume that agents receive one probabilistic private signal that determines their epistemic state over the truth of $T$ and $F$ (e.g., Ladha 1992).¹¹ Let us trace this assumption back to our previous example.

2.2.2. Relation of Epistemic States to the Truth: Minimal Fidelity of Credences

We assumed that individuals have private information about the true state of the world. Yet, to avoid trivial cases of voter competence failure, we need minimal safeguards in place to ensure the quality of these signals.¹² In other words, we need to state assumptions about how credences relate to the truth. Formally, this means imposing restrictions on the probability distributions of credence functions. A voter who can be modelled by credence functions but is exposed to highly misleading propaganda will, trivially, not be guaranteed to be competent at voting. For example, imagine a research environment where critical research is suppressed by state action. In the most extreme case, this would mean that no research indicating global warming is available in this country ( $P r (p i (T) > 0.5 | X = T) = 0$ ). Now consider a less extreme example, where research on global warming is not directly suppressed but there is a severe lack of funding for public research. Additionally, private actors (e.g., oil companies) produce research aimed at supporting the verdict that global warming is less severe than it truly is. It is not the case then that some research is by chance misleading but research is systematically misleading; we have a case of biased science ( $P r (p i (T) > 0.5 | X = T) < 0.5$ ). To exclude such trivial cases of voter competence failure, let us formulate some assumptions about how credence functions relate to the truth.

Minimal Fidelity of Credences: Formally, we say that individual i holds a credence, $p i (T) ∈ [0, 1]$ , about the true state of the world. Individual credences are independent draws from a state-dependent distribution satisfying $P r (p i (T) > 0.5 | X = T) > 0.5$ and $P r (p i (T) < 0.5 | X = F) > 0.5$ , where $p i (T)$ is the credence for individual i about the truth of statement $T$ .

Minimal Fidelity of Credences (MFC) is a restriction on the probability distribution of credences—just as voter competence is a restriction on the probability distribution of voting behaviour. If the true state of the world is $T$ , then it is more likely that an arbitrary voter assigns a probability between $0.5$ and $1$ , whereas if the true state is $F$ , it is more likely that such a voter assigns a probability between $0$ and $0.5$ to verdict $T$ . Let us introduce the concept of a voter favouring a certain verdict to mean that the voter ascribes a probability higher than $0.5$ to that verdict. To summarize, MFC shields voters from systematically misleading epistemic states, such as being in an epistemic environment where certain research is censored or unlikely to be published.¹³ For now, this means we assume that the assistants reading the articles can sometimes favour the incorrect verdict. That is, Gyde’s assistants deal with uncertainty and may tell us that the incorrect verdict is more probable. However, they are better than a coin flip at favouring the correct verdict.

2.2.3. Motivation: Honesty

Next, we address what motivates voters. Without this, little can be said about voting behaviour. Voter competence would fail trivially if voters enjoyed voting against their beliefs. To prevent these failures, we introduce honesty. We model voters’ motivation with utility assignments, ensuring symmetric payoffs for the truth of $X = T$ and $X = F$ . Any voter i gets a payoff of $1$ if they vote for the correct outcome and $– 1$ if they vote for the incorrect outcome. This means that if verdict $T$ is correct, individuals receive a payoff of $1$ if they voted for $T$ and a payoff of $– 1$ otherwise, and vice versa for verdict $F$ . Formally, any voter i has utilities: $U i (T, T) = U i (F, F) = 1$ and $U i (T, F) = U i (F, T) = – 1$ . The first argument of $U i$ represents their vote, and the second represents the correct verdict.¹⁴ Hence, all individuals would like to vote for verdict $T$ if $X = T$ and vote for verdict $F$ if $X = F$ . Thus, Gyde may assume that their research assistants care only about sending the correct answer to them, and not about whether the research assistants, as a collective, got it correct.

2.2.4. Practical Rationality: Expected Utility Maximiser

However, note that even these assumptions about our voters’ epistemic states and motivation still do not guarantee voter competence. Crucially, we need to make assumptions about how voters act on their epistemic states and motivation. In line with the standard rationality assumption in the literature, we assume that an individual determines which of the two verdicts provides the higher expected utility. Thus the decision of voter i is simply the one that maximises the expected utility given by $E U i (V i = T) = p i (T) ⋅ U i (T, T) + (1 – p i (T)) ⋅ U i (T, F), E U i (V i = F) = p i (T) ⋅ U i (F, T) + (1 – p i (T)) ⋅ U i (F, F)$ . Any voter i will maximize their expected utility and hence choose to vote for $T$ if $E U i (V i = T) > E U i (V i = F)$ and vote for $F$ if $E U i (V i = T) < E U i (V i = F)$ . This is illustrated in Figure 1 below.

Figure 1: X-axis: Probability of $X = T$ , left extreme $p (T) = 0$ and right extreme $p (T) = 1$ . Y-axis: Expected utility of the respective actions. The blue dotted line marks the probability that voter i assigns to $X = T$ (i.e., $p i (T) = 0.7$ ).

Of course, Gyde will never get the correct result with certainty, as they do not have infinite research assistants (or academic papers to draw upon). Yet, if epistemic competency, practical rationality, and honesty are satisfied, voter competence is satisfied (see appendix for proof). In that case, the accuracy of the voting result will improve with every research assistant (see Bradley and Thompson 2012).¹⁵ Next, we will motivate why this model of voter competence may be too simple—descriptively or normatively—and propose a more general model.

3. Ambiguity and Voter Competence

Imagine you consider playing the lottery and ask your friend how likely it is to win. They reply: “I am not sure, but I am certain that it is something between 1 in a million and 1 in 10 million. Anyway, something incredibly small—definitely not worth your money!” Later, if someone asked you how likely it is to win the lottery, what would you answer?

In everyday life, the information we receive is often of the sort you received from your friend and is not necessarily suited to being reduced to a precise probability. Ellsberg (1961) used this type of uncertainty over probabilities in a series of experiments, demonstrating that participants’ choices do not align with subjective expected utility theory. This type of problem has been discussed in decision theory and economics¹⁶ under the term ambiguity (see, e.g., Barberis and Thaler 2003; Frisch and Baron 1988; Gilboa and Schmeidler 1989; Heath and Tversky 1991). The Ellsberg paradox¹⁷ has inspired an extensive empirical literature. This literature highlights a concern in decision theory: Precise probabilities fail to represent severe uncertainties correctly. They are inadequate on either descriptive or normative grounds (see, e.g., Kyburg 1983; Levi 1985; Bradley 2009; Seidenfeld 2004; Joyce 2010; Gilboa et al. 2009). Among several concerns, one is that precise credences are not the rationally correct attitude to adopt given some types of evidence. If precise credences are not rationally required, voters may not base their decisions on them. Consequently, prior models that use precise credences to flesh out the conditions under which we can assume voter competence are not always applicable. Therefore, there is a lacuna in the literature regarding whether and when we can rely on the epistemic benefits of majority voting in the face of ambiguity. To close this lacuna, we should move on to models that can account for imprecision regarding the grounds on which voters presumably make a range of voting decisions. In the next section, we will expand the simple model introduced earlier. By incorporating the possibility of ambiguity, we will demonstrate why voter competence is no longer guaranteed under these assumptions.

3.1. A Less Simple Model of Voter Competence

3.1.1. Internal Structure of Epistemic States: Set of Credence Functions

Many economists and philosophers (Gilboa and Schmeidler 1989; Joyce 2010; Levi 1985) argue that the most natural way to understand the agent’s epistemic state in situations like Ellsberg’s Paradox is their inability to determine which among a set of possible probability distributions is the true one. That is, while we know that there is one true probability distribution describing the lottery, we don’t know which one it is. We now turn to how this translates into a formal model of epistemic states. When the probability distributions in the set of conceivable distributions can themselves be assigned probabilities, ambiguity in this sense can be expressed as second-order probability. However, when the distributions cannot be assigned probabilities, ambiguity is often expressed by a set of probabilities (Camerer and Weber 1992).¹⁸ While not everyone in the literature agrees on what exactly is rationally required, there is some minimal agreement on what is rationally permissible. That is, it is always rationally permissible to adopt an epistemic state that covers the full range of probabilities compatible with the evidence that you have at your disposal (Mahtani 2019). This translates into a model where we have a set of probability distributions that cannot be reduced to second-order probabilities about the true probability distribution of the lottery. We will adopt this model of ambiguity to examine its impact on voter competence.

In our new model, voters may adopt a set of probability functions $P$ . Sets of probabilities are also sometimes called imprecise probabilities (e.g., Jeffrey 1983; Joyce 2005) or a credal set (Levi 1974). $P (x)$ is a set of numbers, e.g., any number between 0.6 and 0.9, without putting any weight on any particular number. In this case, we call their credal set convex.¹⁹ A closed set includes its limit points (e.g., 0.6 and 0.9). A closed, convex set can formally represent what we intuitively mean by an agent adopting a full range. We will use this interval version of imprecise probabilities throughout this paper. Convex sets can be represented by their extreme points. Thus, when we refer to $P (x)$ , we may write $[P ¯ (x), P ¯ (x)]$ . Here, $P ¯ (x)$ denotes the lowest probability in verdict x, while $P ¯ (x)$ represents the highest probability in the set $P (x)$ .²⁰ Since every element of a credal set is a probability function, the highest probability of x equals 1 minus the lowest probability of $¬ x$ (i.e., $P ¯ (T) = 1 – P ¯ (F)$ ).²¹ For example, suppose you have the epistemic state $[P ¯ (T) = 0.6, P ¯ (T) = 1]$ . Then, your credal set contains the credence functions assigning any value between 0.6 and 1 to $X = T$ (e.g., $p (T) = 0.7, p (T) = 0.61, p (T) = 0.89$ ).²² See Figure 2 for an illustration of this.

Figure 2: X-axis: Probability that $X = T$ , with left extreme denoting probability 0 and right extreme 1. The dotted blue lines indicate $P i ¯ (T)$ and $P i ¯ (T)$ . Y-axis: Expected utility of the respective actions.

Let us now explicate how to spell out the conditions for how a set of credences relates to the truth.

3.1.2. Relation of Epistemic States to the Truth: Minimal Fidelity of Sets of Credence Functions

In the case of a single credence function, the intuition was quite straightforward. Voters have a better chance than a coin flip to favour (i.e., $p (T) > 0.5$ ) the true statement than to favour the incorrect statement (i.e., $p (T) < 0.5$ ). One can easily imagine constraints on credal set distributions that guarantee voter competence (e.g., $P r (P ¯ (T) = 1) = 1$ ). But what might be reasonable minimal constraints on the distribution over sets of credences?

Here is one way to think about it. Consider the following scenario, which clearly violates minimal fidelity. Imagine a repressive state that wants to silence critical voices on climate change. One way minimal fidelity would surely be violated is if the state only allowed evidence entailing that it is more likely that global warming will not reach 1.5°C between 2030 and 2052 (i.e., $P r (P ¯ (T) < 0.5) = 1$ ) to be published. Alternatively, the state may permit non-conclusive evidence but suppress any evidence suggesting that global warming is more likely than not (i.e., $P r (P ¯ (T) > 0.5) = 0$ ). The rationale might be to avoid an epistemic environment where the only reasonable response to evidence is to favour the undesirable truth. Both scenarios surely would violate minimal fidelity. Minimal fidelity requires the possibility of evidence that unequivocally favours the true state (i.e., every credal function in your set favours the correct verdict, $P ¯ (T) > 0.5)$ . Now imagine, as before, that there is no repressive state. However, publicly funded research is scarce, while privately funded research, influenced by special interests (e.g., oil companies), dominates. With little unbiased science, we face a troubling scenario. Whenever evidence unequivocally favours one verdict, it is more likely incorrect because private organisations selectively publish favourable results. These are cases where we don’t expect voter competence to emerge. So let us exclude these cases with the following requirement on a set of credences.

Minimal Fidelity of Sets of Credences (MFSC): Formally, we say that individual i holds a convex set of credences, $[P i ¯ (T), P i ¯ (T)]$ , with $P i ¯ (T), P i ¯ (T) ∈ [0, 1]$ about the true state of the world. $P i ¯ (T)$ and $P i ¯ (T)$ are independent draws between voters satisfying $P r (P i ¯ (T) > 0.5 | X = T) > P r (P i ¯ (T) < 0.5 | X = T)$ and for $X = F, P r (P i ¯ (T) < 0.5 | X = F) > P r (P i ¯ (T) > 0.5 | X = F)$ , where $P i ¯ i (T)$ is the lower credence for individual i about the truth of statement $T$ , and $P i ¯ (T)$ is the upper credence for individual i about the truth of statement $T$ .

One key feature of MFSC is that it generalises MFC. MFSC is satisfied if it is more probable that every credence function in a voter’s credal set favours the correct verdict than that every credence function in their credal set favours the incorrect verdict. If the credal set contains only one credal function, MFSC just amounts to MFC. MFSC combines two constraints: 1. The epistemic state must allow for unequivocal favouring of the correct verdict (i.e., $P ¯ (T) > 0.5) ≠ 0$ ), excluding extreme cases like a repressive state. 2. If evidence unequivocally favours one verdict (i.e., $P ¯ (T) > 0.5$ or $P ¯ (T) < 0.5$ ), it should not systematically favour the incorrect verdict, even though it may occur by chance. This excludes cases of biased science.²³

To summarize what we have postulated so far, we no longer demand that voters ascribe a single probability to an event. However, MFSC ensures that agents’ epistemic states are more likely to unequivocally favour the correct verdict than the other way around. This excludes epistemic environments (e.g., misleading propaganda, biased science) where we don’t expect an agent to be epistemically competent.²⁴ The next section explores how rational agents make decisions using a set of credences.

3.1.3. Practical Rationality: Γ-Maximin

Imagine that you are one of the research assistants. You face the decision of advising Gyde on including one of the following statements in the briefing: (1) Global warming will reach 1.5°C between 2030 and 2052 if it continues to increase at the current rate ( $x = T$ ), or (2) Global warming will not reach 1.5°C between 2030 and 2052 if it continues to increase at the current rate ( $x = F$ ). Now, imagine reading in the report from the Intergovernmental Panel on Climate Change (IPCC) that “Global warming will likely reach 1.5°C between 2030 and 2052 if it continues to increase at the current rate.” Rather than assigning precise probabilities to each scenario, the IPCC presents intervals of probabilities. The IPCC uses standard intervals coded by calibrated language.²⁵ For example, likely indicates an assessed likelihood of 66%–100% (Allen et al. 2018). After reading the IPCC’s explanation of likely, you adopt an epistemic state that can be described by $[P ¯ (T) = 0.66, P ¯ (T) = 1]$ . As a rational agent, which principle should you follow to advise Gyde?

Unfortunately, there is no consensus on a single decision rule for rational decision-making with imprecise probabilities. However, fortunately, many of these rules converge when verdicts have symmetric payoffs. We take a decision rule as a stand-in for a whole class of decision rules that we may refer to as non-permissive (Mahtani 2019), which typically require a single rational decision.²⁶ One of the most prominent decision rules (Seidenfeld 2004; Troffaes 2007) is called $Γ$ -Maximin. $Γ$ -Maximin prescribes agents to adopt pessimistic expectations (Berger 1985; Gilboa and Schmeidler 1989). Given a possible decision, agents assume the probability in their set that yields the lowest expected utility. Then they compare their most pessimistic expectations for each possible decision and pick the one that gives them the best-expected utility (given their pessimism). Translating this to a formal statement, we call the lowest expected payoff we achieve by picking the most pessimistic $p$ $∈ P$ for a decision d $EU ¯ P (d)$ . We then call any decision $d ∈ D$ optimal if it maximizes $EU ¯ P$ among all decisions in $D$ . Thus, a $Γ$ -Maximin decision in $D$ is described as follows: $o p t EU ¯ P (D) : = a r g m a x d ∈ D EU ¯ P (d)$ .

Imagine that you are such a cautious reasoner. As in the simpler model, you are still motivated by honesty, and your prior utility assignments remain in place. You consider if you should advise to include verdict $T$ , then what is your most pessimistic expectation? Given your credal set $[0.66, 1]$ , you use $p (T) = 0.66$ to calculate the expected utility of advising Gyde to include verdict $T$ , resulting in an expected utility of 0.66. This is the lowest expected utility for verdict $T$ that is compatible with your epistemic state. In a second step you consider whether advising to include the opposite verdict $F$ results in the lowest expected utility compatible with your epistemic state. You use $p (T) = 1$ (i.e., $p (F) = 0$ ) to calculate the expected utility for verdict $F$ , resulting in $0$ . In the next step, you compare $0.66$ and $0$ and decide to advise Gyde to include verdict $T$ as this maximizes your expectation under a pessimistic assessment. See Figure 3 for an illustration of this.

Figure 3: X-axis: Epistemic state of voter i of statement $T$ , with left extreme denoting probability 0 and right extreme 1. Y-axis: Expected utility of the respective actions for voter i. Red circles indicate $E U i ¯ P i (d)$ for each $d$ (i.e., $T$ , $F$ ).

At this point, let us stop once more and summarize the model that we built: Every voter has an epistemic state, described by $P i ¯ (x)$ and $P i ¯ (x)$ over the truth of verdict x. These epistemic states are drawn from a distribution satisfying MFSC, defined as $P r (P i ¯ (x) > 0.5 | X = x) > P r (P i ¯ (x) < 0.5 | X = x)$ . Each voter then makes one of two possible decisions by adhering to $Γ$ -Maximin, caring only to vote correctly. Next, I will show why the assumptions chosen so far do not guarantee voter competence, and thus fall short of ensuring the epistemic benefits of voting.

3.2. A Failure of Voter Competence

Voter competence is not guaranteed under the conditions of epistemic competence, honesty, and practical rationality, as shown by a simple counterexample. Assume that for any research assistant i of Gyde, their epistemic states are drawn from a distribution satisfying the following: $P r (P i ¯ (T) > 0.5 | X = T) = .45, P r (P i ¯ (T) < 0.5 < P i ¯ (T) | X = T) = .38, P r (P i ¯ (T) < 0.5 | X = T) = .17$ . Additionally, whenever $P r (P i ¯ (T) < 0.5 < P i ¯ (T) | X = T), P i ¯ (T) = 0.33, P i ¯ (T) = 0.6$ . Whenever $P i ¯ (T) > 0.5$ , $Γ$ -Maximin will select voting for $T$ as optimal. Similarly, if $P i ¯ (T) < 0.5$ , $Γ$ -Maximin selects voting for $F$ as optimal. For $[P i ¯ (T) = 0.33, P i ¯ (T) = 0.6]$ , voting for $F$ will be the decision that gives the best-expected utility given being reasonably pessimistic i.e., maximizes the minimum EU. Thus, with a probability of $45 %$ , a voter will vote for the correct verdict, but with a probability of $55 %$ , they will vote for the wrong verdict, failing to satisfy voter competence.

Hence, letting the research assistants email them back which verdict to include might not be wise. One factor driving this result is that our decision rule $Γ$ -Maximin singles out the extreme points of the credal set. However, our definition of MFSC imposes few restrictions on these extreme points. There is a mismatch in the requirements: Epistemic competence and practical rationality do not necessarily translate into competent behavior under uncertainty, as they do under risk.

In the next section, we address this mismatch by spelling out practical rationality and epistemic competency differently. We then explore their effects on voter competence. Unfortunately, changing practical rationality alone is not promising. However, spelling out epistemic competency differently shows greater promise for improving voter competence. Lastly, I show that reformulating these concepts is not needed to guarantee the epistemic benefits of voting. Instead, a more practical and promising solution lies in changing the voting rule.

4. Possible Escape Routes

The failure of voter competence means we cannot guarantee the correct result, even as the number of voters approaches infinity. Adding more voters could even worsen the outcome. Failure of voter competence might be common, but surely we don’t want to add to this predicament by expecting voter incompetence when agents are epistemically competent, practically rational, and honest.

This section explores three ways to restore the epistemic benefits of voting. The failure of voter competence arises from the combination of practical rationality and epistemic competence. More precisely, the underlying issue is a decision rule that puts weight on the extreme points of the credal set and a requirement on epistemic states that only indirectly puts some restrictions on these extreme points. The first route explores restoring voting competence by spelling out practical rationality differently. The second route examines changing the minimal fidelity of credal sets. The former will not turn out to be a promising route, while the latter has some promising results. Finally, the tension between practical rationality and epistemic competency can be resolved by changing the available actions for voters. Allowing abstention as part of the voting rule offers a promising third route.

4.1. Revisiting Practical Rationality: E-admissibility

Given the tension between a decision rule that focuses solely on the extreme points of the credal set and an epistemic competence requirement that imposes few restrictions on those extreme points, let us consider another contender in rational decision making with imprecise probabilities. Another class of decision rules filters the available decisions and typically singles out a set of rationally permissible decisions. We will use the decision rule e-admissibility as a stand-in for these decision rules.²⁷ E-admissibility demands that you maximize expected utility according to some probability function $p$ $∈ P$ (Good 1952; Levi 1974; Seidenfeld 2004), and thus is not specifically looking at the extreme points of the credal set. It generalizes the principle of maximizing expected utility but does not impose a complete ordering over decisions. Differences between e-admissibility and $Γ$ -Maximin occur, in principle, with the type of epistemic state (i.e., [0.3, 0.66]) we utilized to show that $Γ$ -Maximin will not result in voter competence.²⁸ We denote the set of $d ∈ D$ that maximizes expected utility according to $p$ $o p t E U p (D)$ . Now we call any decision $d ∈ D$ e-admissible if it is in $o p t E U p (D)$ for some $p$ $∈ P$ . Thus, all e-admissible decisions can be denoted by the following union:

o p t P (D) := ⋃ p ∈ P o p t E U p (D)

In our previous example, you, one of the research assistants, adopted the credal set $[0.66, 1]$ . You now reason as follows, according to any probability function $p$ in $P$ : $E U (T) > E U (F)$ and thus $o p t P (D)$ contains $T$ but does not contain $F$ . Hence, in this case, you would also advise Gyde to include verdict $T$ . See Figure 4 for an illustration of this. However, like $Γ$ -Maximin, e-admissibility does not guarantee voter competence, though for different reasons. E-admissibility is a permissive decision rule. In the cases where rationality alone does not prescribe a unique decision, picking any e-admissible verdict is acceptable for the voter. This can result in voters failing to select the correct verdict.

Figure 4: X-axis: Probability of statement $T$ (0 to 1). The dotted lines indicate the voter’s credal set. Y-axis: Expected utility of decisions; horizontal lines represent the upper and lower probabilities for $T$ . Red lines show the decision maximizing expected utility for the corresponding probabilities.

E-admissibility fails to retrieve voter competence, as shown by the same example as $Γ$ -Maximin.²⁹ E-admissibility and $Γ$ -Maximin differ only for $[P i ¯ (T) = 0.33, P i ¯ (T) = 0.6]$ . In these cases, e-admissibility allows voting for either $T$ or $F$ . Voters must vote for $T$ with a probability of $45 %$ , for $F$ with $17 %$ , and may vote either way with $38 %$ . To guarantee voter competence, it must be rationally required, not merely permissible, to vote for the correct verdict with a probability over $50 %$ .

One might object that even if different choices are permissible under e-admissibility, voters’ choice processes require further explanation. Voters may have other second-order rationality criteria that impose structure on their choices. For example, one common criterion would be to be risk-averse among the rationally permissible verdicts. Yet, as seen in the previous section, risk aversion would require them to vote for $F$ in this case, resulting in the same dilemma as with $Γ$ -Maximin.

Another objection is that assuming mere rationality is not the most charitable approach. Typical voter behavior is relevant for applying the CJT. Voters may randomize uniformly if rationality does not dictate a choice. Imagine choosing between two equally good bottles of water at the supermarket; you consistently pick the one at eye level. The arrangement influences your choice without changing your preferences. Display height is irrelevant if one bottle is sparkling and the other still. Similarly, we might say that if you are undecided between policies you are more likely to choose whichever verdict is listed first on the ballot. If, for instance, the status quo verdict is always listed first, voters may vote for the status quo verdict more often. If the status quo is often wrong (e.g., based on outdated science), this creates a non-random selection of verdicts without requiring second-order criteria or irrationality. Establishing that voters randomize over rationally permissible actions is not trivial. Hence, changing the decision rule will not affect voter competence, suggesting a need to reconsider assumptions about MFSC.

4.2. Revisiting the Relation of Epistemic States to the Truth: Symmetric Minimal Fidelity of Sets of Credence Functions

Since altering practical rationality is unpromising, let us turn to epistemic competency. One might argue that our requirement on credal sets is too weak to qualify as minimal. MFSC ignores the asymmetry in the spread of possible verdict values (e.g., $P ¯ (T) = 0.51 vs. P ¯ (T) = 0.9$ ). To address this, we could consider how strongly a verdict is supported. This means imposing restrictions directly on the extreme points of the credal set. What minimal restrictions on credal sets respect the epistemic importance of these extreme points?

Imagine again the repressive state. They aim to create an environment where the strongest credences compatible with the epistemic state deny climate change. That is, the state censors evidence where the strongest credence compatible with the evidence favours the undesirable truth (i.e., $P r (P ¯ (T) > 1 – P ¯ (T) | X = T) = 0$ ). Now imagine that instead of the repressive state, we have predominantly private companies conducting research with an interest in evidence where the strongest signals favour the incorrect verdict. Science is again biased, systematically producing evidence where the strongest signals favour the incorrect verdict ( $P r (P ¯ (T) > 1 – P ¯ (T) | X = T) < P r (P ¯ (T) ≤ 1 – P ¯ (T) | X = T)$ ). This suggests another intuition about what constitutes minimal fidelity in the case of credal sets. Let us exclude these cases.

Symmetric Minimal Fidelity of Sets of Credence Functions (SMFSC): $P r (P ¯ (T) > 1 – P ¯ (T) | X = T) > P r (P ¯ (T) ≤ 1 – P ¯ (T) | X = T)$ (and similarly for $X = F$ ). SMFSC captures the intuition that the correct verdict is more often associated with the highest probability than the incorrect one. SMFSC ensures voter competence for $Γ$ -Maximin but not for e-admissibility (see proofs in the appendix). The reasons for this are outlined below.

4.2.1. Voter Competence under SMFSC with Γ-Maximin

SMFSC is not merely a condition under which $Γ$ -Maximin guarantees voter competence. SMFSC is the minimal epistemic condition that you, as a cautious reasoner with credal sets, need to fulfill to ensure competent voting. The intuition is straightforward. When outcomes are symmetrical in severity (i.e., being correct or incorrect is equally good or bad for both verdicts), $Γ$ -Maximin selects the verdict with the highest credence compatible with favouring it. By SMFSC, it is guaranteed that for any voter i, $P r (P i ¯ (T) > 1 – P i ¯ (T)) > 0.5$ . By $Γ$ -Maximin, it is guaranteed that for any voter i, $P r (E U i ¯ P i (T) > E U i ¯ P i (F)) > 0.5$ . Thus, it follows that $P r (V i = T) > 0.5$ for voter i. Thus, voters are guaranteed to be competent.

This is good news. This means that rethinking our requirements on credal sets ensures ambiguity does not hinder the epistemic benefits of voting. We can advise Gyde to pursue their approach to the briefing. However, if rationality permits permissive decision rules like e-admissibility, voters might not be competent under SMFSC (see proof in the appendix).

4.2.2. Failure of Voter Competence under SMFSC with E-admissibility

Roughly, the failure of voter competence is driven by the fact that e-admissibility does not take into account how strongly some probability $p$ favours one verdict, but merely which verdicts are favoured by some $p$ in your credal set. In fact, the minimal epistemic requirement under e-admissibility for guaranteeing voter competence would be that $P r (P ¯ (1) > 0.5) > 0.5$ (see full proof in the appendix). SMFSC does not guarantee that epistemic states ever unequivocally favour one verdict,³⁰ yet this is required to single out one rationally required action under e-admissibility. However, combining e-admissibility as a first-order criterion with $Γ$ -Maximin as a second-order criterion restores voter competence. Although e-admissibility can, in principle, exclude actions chosen by $Γ$ -Maximin, this does not occur in our scenarios. When a credal set unequivocally favours a verdict, e-admissibility and $Γ$ -Maximin agree. Since e-admissibility does not exclude any actions at $p (T) = 0.5$ , both actions are optimal. Thus, the action selected by $Γ$ -Maximin as a first-order criterion remains unchanged when applied as a second-order criterion after e-admissibility.

There is, however, a third route to resolve the tension between epistemic competency and practical rationality. Next, we will expand our model to account for the possibility that voters may abstain. If voters can abstain, it becomes possible to guarantee that voting is epistemically beneficial with MFSC and $Γ$ -Maximin.

4.3. Revisiting Voter Competence and Majority Rule

Let us consider voter competence differently from how it is defined in the CJT. Instead, let us focus on whether a voter is more likely to vote for the correct verdict than for the incorrect verdict. Let us call this concept generalized voter competence. With only two ballot choices, both definitions of voter competence coincide. However, they diverge if abstention is included. When determining the outcome of an election, abstentions are disregarded. The outcome $O$ is $O = T$ if there are more voters with $V i = T$ than $V i = F$ , $O = F$ if there are more voters with $V i = F$ than $V i = T$ , and $O = {}$ in the event of a tie.

Formally, the difference is as follows. Given that $X = T$ , generalized voter competence holds that $P r (V i = T | X = T) > P r (V i = F | X = T)$ , whereas voter competence holds that $P r (V i = T | X = T) = c$ with $c > 0.5$ . The rationale for changing the voting rule is that voting is a form of expressing attitudes. If a voting rule allows a more fine-grained expression of attitudes, a voter is not forced to guess incorrectly. In other words, this approach reduces information loss and distortion, allowing voters to express ambiguity to some degree. However, to conclude that rational voters will use this opportunity in a way that will guarantee the epistemic benefits of voting, more assumptions are needed. A voting rule that allows more nuanced expression does not guarantee voters will use it. Let us begin by introducing a third verdict to the decision space, $D$ . $D$ now includes the verdict to abstain, denoted by $A$ . In the second step, we need to decide what utility a voter may get from abstaining. Surely, voting for the correct verdict should be better than avoiding a wrong choice. Furthermore, avoiding a wrong choice should be better than making one. Thus, we are left with a modeling choice between $– 1 < U i (V i = A | X = x) < 1$ . Let us fix the utility to $0$ (i.e., $U i (V i = A | X = x) = 0$ ).

4.3.1. General Voter Competence under MFSC with Γ-Maximin

Abstention, as outlined above, does guarantee the epistemic benefits of voting (see full proof in the appendix). There are two key features that drive this result. The utility of abstaining is not outcome-dependent, so its expected utility is independent of the credence function. Second, the expected utility is always $0$ . This means that whenever the credal set contains the credence $p (T) = 0.5$ , the extreme points of the credal set become irrelevant for decision making. $Γ$ -Maximin will always pick to abstain when a credal set does not unequivocally favour one verdict over the other. Yet, since MFSC requires that there is some probability of epistemic states unequivocally favouring one verdict, voters will not always abstain. Additionally, from MFSC, it also follows that we expect voters to vote for the correct verdict when they do not abstain. If voters face severe uncertainty, their method of expressing epistemic states needs to be more fine-grained. It is somewhat surprising that having merely three possible ways to express your epistemic states (V=T, V=F, V=A) suffices for us to retrieve the epistemic benefits of voting. Given a certain decision rule, the epistemic requirements on the credal sets can be lessened by changing the voting rule.

4.3.2. Failure of General Voter Competence under MFSC with E-admissibility

Abstaining will not guarantee general voter competence if voters can freely choose between e-admissible actions. The reason for this is quite simple: e-admissibility excludes actions that do not maximize expected utility under any credal function in your credal set. However, if the credal set contains $p (T) = 0.5$ , voting for verdict $T$ , verdict $F$ , or abstaining is rationally permissible. Hence, the same counterexample from the previous sections illustrates the failure of general voter competence.³¹ Adding $Γ$ -Maximin as a second-order criterion under SMFSC would also retrieve general voter competence. This means that in the example of our research assistants, even if only MFSC is satisfied, Gyde can divide the work among their research assistants. It might be best to allow their research assistants to email that their evidence is insufficient for advice. If Gyde knows their research assistants are cautious reasoners, this procedure guarantees the benefits of voting. If not, it at least avoids worsening the outcome. While Gyde cannot control the epistemic environment, they can adjust the voting procedure to improve the chances of a good epistemic outcome.

5. Conclusion

In many vital areas of collective decision-making, we face severe uncertainty and often rely on ambiguous evidence (e.g., climate science). As a result, rational agents’ epistemic states reflect the ambiguity of the evidence in such decision-making scenarios. Unfortunately, our assessments of collective decision-making procedures seldom do the reality justice.

This paper demonstrated a mismatch between scenarios where a simple majority vote seems to guarantee epistemic benefits and when the conditions for such a guarantee actually hold. More precisely, I showed that in contrast to situations of risk, voter competence is not guaranteed in situations of ambiguity, even if we assume practical rationality, honesty, and epistemic competency. The results depend on how practical rationality, honesty, and epistemic competency are defined together. Whether voting competence holds depends on the interplay of all three assumptions. The first conclusion of this paper is that considering different kinds of uncertainty significantly affects the conditions required for achieving voting competence.

The second conclusion is that collective decision-making rules that perform well in situations of risk might not apply to decision-making under ambiguous prospects. If ambiguity is the norm rather than the exception, we must reconsider and revise the rules for collective decision-making that perform well epistemically. As a start to this research project, I proposed extending the ballot choices and showed that including abstention can retrieve similar results to the CJT. The intuition behind this proposal was that procedures designed without ambiguity in mind force more precision than is warranted. The novel aspect of this result was not that abstention allows for a more fine-grained description of voters’ attitudes but how rationality guarantees its use.

Notes

For notable exceptions, see Dietrich (2008); Dietrich and Spiekermann (2013). ⮭
It suffices if individuals are, on average, competent. ⮭
This set-up is similar to the example used by Steele (2012). ⮭
This way of presenting CJT is inspired by List and Spiekermann (2016). ⮭
This means that, conditional on the truth of the proposition (i.e., whether $X = T$ or $X = F$ ), the knowledge of some voters’ votes provides no additional information regarding the votes of others. ⮭
This assumption merely serves to simplify our calculations later but is not needed for any substantial results. ⮭
Notable exceptions include Austen-Smith and Banks (1996) and Bradley and Thompson (2012). ⮭
I thank an anonymous referee for suggesting this framing. ⮭
I will use “credence” and “probability function” interchangeably when referring to an agent’s epistemic state. ⮭
Because of this relationship, we will use only $p (T)$ for the rest of the paper. ⮭
A voting outcome may be seen as the result of aggregating the private information of voters (see Feddersen and Pesendorfer 1997). ⮭
We leave open how credences come about. However, one natural interpretation is that voters base their credences on evidence that satisfies similar constraints. A very simple underlying model would be that voters adopt expert opinions that reflect the evidence. Alternatively, a slightly more complicated model might combine priors and new evidence such that the above constraints are satisfied. This can be achieved in multiple ways by imposing stricter constraints on either priors or new evidence. ⮭
An alternative interpretation would be that this condition shields us from trivial failures of voter competence, where voters consist only of anti-science agents who anti-update on scientific evidence. ⮭
In Austen-Smith and Banks (1996), the first argument stands for the election outcome. However, this can lead to “noble lies” (see Bright 2017). ⮭
Yet, as Bradley and Thompson (2012) also show, the unweighted majority rule is not necessarily the best-performing rule. ⮭
Ambiguous evidence is also a subject of discussion in other sciences. See, e.g., (Augustin et al. 2014; Bradley et al. 2014; Budescu and Wallsten 1995; Camerer and Weber 1992). ⮭
In Ellsberg’s paradox, agents are presented with two urns containing red and black balls: One urn has a known distribution (e.g., 50 red and 50 black), while the other urn has an unknown distribution. The participants are then presented with a series of bets, revealing an inconsistency with subjective expected utility theory that is now often interpreted as ambiguity aversion. ⮭
Note that this set may only contain one element $p$ . Thus, this generalizes precise probabilities. ⮭
A convex set is a set that includes all linear combinations of all members x, y in the set, such that $(1 – w) x + wy$ , with $w ∈ [0, 1]$ . ⮭
This is the simplest form of imprecise probabilities. Lower and upper probabilities cannot fully capture all the nuances of imprecise probabilities. Thus, usually, the literature refers to other frameworks, such as lower and upper previsions or lower and upper envelopes. However, for two mutually exclusive verdicts, the differences between frameworks are negligible. Thus, we simplify our analysis by sticking to the cruder notion of lower and upper probabilities. ⮭
We use $P ¯ (T)$ and $P ¯ (T)$ as a shortcut for $P ¯ (X = T)$ and $P ¯ (X = T)$ . ⮭
Note that this implies, by the rationality constraints laid out above, that $[P ¯ (F) = 0, P ¯ (F) = 0.4]$ . ⮭
Note that to satisfy this without constraint 1, a weak inequality would suffice: $P r (P i ¯ (T) > 0.5 | X = T) ≥ P r (P i ¯ (T) < 0.5 | X = T)$ . The strict inequality in MFSC ensures assumption 1 by ensuring that $P r (P ¯ (T) > 0.5) ≠ 0$ . ⮭
If you believe MFSC does not make agents epistemically competent, it should not be surprising that voter competence fails under MFSC. However, MFSC is worth considering, as it can still guarantee the epistemic benefits of voting, even if voters’ epistemic states satisfy only MFSC (see §5.3). In this case, the results show that the epistemic benefits of majority voting can still be guaranteed, even without epistemically competent voters. ⮭
For a more detailed discussion of uncertainty presentation in the IPCC, see Bradley et al. (2017) and Dethier (2023). ⮭
With symmetric payoffs, the verdict with the best expected payoff also maximizes worst-case expectations. Thus, rules based on either coincide. ⮭
For example, decision rules such as interval dominance or maximality. For more details about these decision rules and why they are at least as permissive as e-admissibility in our example, see Troffaes (2007). ⮭
E-admissibility may exclude the $Γ$ -Maximin decision, because that decision may not maximize expected utility according to any $p ∈ P .$ ⮭
We assumed: $P r (P i ¯ (T) > 0.5 | X = T) = .45, P r (P i ¯ (T) < 0.5 < P i ¯ (T) | X = T) = .38$ , and $P r (P i ¯ (T) < 0.5 | X = T) = .17$ . Additionally, whenever $P r (P i ¯ (T) < 0.5 < P i ¯ (T) | X = T), P i ¯ (T) = 0.33$ and $P i ¯ (T) = 0.6$ . ⮭
The same counterexample from the previous section demonstrates this. ⮭
To check that this allows for a failure of general voter competence is left to the reader. ⮭

Acknowledgements

I thank Liam Kofi Bright, Richard Bradley, Anna Mahtani, Lukas Beck, Johanna Thoma, Shira Ahissar, Matteo Michelini, James Michelson, Gyde Wollesen, and the reviewers for their valuable comments and support.

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Appendix

A: MFC, Honesty, and EU-maximisation Guarantee Voter Competence

Proof. Assume, without loss of generality, that 1 is the true state of the world. We know that if $p i (T) = 0.5$ then $E U i (V i = T) = 0$ and $E U i (V i = F) = F$ . Hence if $p i (T) > 0.5$ then $E U i (V i = T) > E U i (V i = F)$ . We know from MFC that if $X = T$ , $P r (p i (T) > 0.5 | X = T) > 0.5$ . Thus, this simple model guarantees voter competence. □

B: Under SMFSC, Γ-Maximin Guarantees Voter Competence

Proof. Without loss of generality, assume that $X = T$ . If any voter i uses $Γ$ -Maximin we know that for $D = {T, F}$ , for any voter i, whenever $E U i ¯ P i (T) > E U i ¯ P i (F)$ , then $V i = T$ . Since $P (T)$ is a convex set of probabilities, we only need to use the extreme points $P i ¯ (T)$ and $P i ¯ (T)$ to find $E U i ¯ P i (T)$ and $E U i ¯ P i (F)$ . In combination with the utility assumption that for any voter i, $U i (T, T) = U i (F, F) = 1$ , and $U i (T, F) = U i (F, T) = – 1$ , we have:

E U i ¯ P i (T) = min {P i ¯ (T) – (1 – P i ¯ (T)), P i ¯ (T) – (1 – P i ¯ (T))}

E U i ¯ P i (F) = min {– P i ¯ (T) + (1 – P i ¯ (T)), – P i ¯ (T) + (1 – P i ¯ (T))}

Because $P i ¯ (T) ≤ P i ¯ (T)$ , we can write:

E U i ¯ P i (T) = P i ¯ (T) – (1 – P i ¯ (T))

E U i ¯ P i (F) = – P i ¯ (T) + (1 – P i ¯ (T))

Thus, for any voter i, if $P i ¯ (T) > 1 – P i ¯ (T)$ , it follows $E U i ¯ P i (T) > E U i ¯ P i (F)$ and it follows that $V i = T$ . By SMFSC, for any voter i, $P r (P i ¯ (T) > 1 – P i ¯ (T)) > 0.5$ , thus for any voter i, $P r (E U i ¯ P i (T) > E U i ¯ P i (F)) > 0.5$ and it follows that $P r (V i = T) > 0.5$ . Therefore, voters are guaranteed to be competent. □

C: Under SMFSC, E-admissibility Does Not Guarantee Voter Competence

Proof. Without loss of generality, assume that $X = T$ . Suppose every voter $i$ uses E-admissibility. Then, for $D = {T, F}$ , the following holds: for any voter $i$ , if

∀ p i (T) ∈ P i (T), E U p i (T) (T) > E U p i (T) (F),

then $V i = T$ .

Moreover, for $p i (T) = 0.5$ , $E U p i (T) (T) = E U p i (T) (F) = 0$ , and thus $V i = F$ or $V i = T$ . Also, for $p i (T) < 0.5$ , $E U p i (T) (T) < E U p i (T) (F)$ , and thus $V i = F$ . Furthermore, for $p i (T) > 0.5$ , $E U p i (T) (T) > E U p i (T) (F)$ , and thus $V i = T$ . Since for every $p i (T) ∈ P i (T)$ , it is the case that $p i (T) ≥ P i ¯ (T)$ , and it follows that if $E U P i ¯ (T) > 0$ for every voter i, then $V i = T$ . However, if $E U P i ¯ (T) ≤ 0$ , it may be that $V i = F$ . Thus, if and only if $P i ¯ (T) > 0.5$ , it is guaranteed that $V i = T$ . It follows that to guarantee $P r (V i = T) > 0.5$ , $P r (P i ¯ (T) > 0.5) > 0.5 .$ Yet, SMFSC only guarantees $P r (P i ¯ (T) > 1 – P i ¯ (T)) > 0.5$ , which is compatible with $P r (P i ¯ (T) > 0.5) = 0 .$ Thus, voter competence under SMFSC is not guaranteed. □

D: Under MFSC, if Voters Can Abstain, Γ-Maximin Can Retrieve Epistemic Benefits of Voting

Proof. Assume without loss of generality that $X = T$ . Then, if every voter i uses $Γ$ -Maximin for $D = {T, F, A}$ , then $V i = T$ , whenever $E U i ¯ P i (T) > E U i ¯ P i (F)$ and $E U i ¯ P i (T) > E U i ¯ P i (A)$ . Similarly, if $E U i ¯ P i (T) (F) > E U i ¯ P i (T)$ and $E U i ¯ P i (F) > E U i ¯ P i (A)$ , then for any voter i, $V i = F .$ We already know that

E U i ¯ P i (T) = P i ¯ (T) – (1 – P i ¯ (T))

E U i ¯ P i (F) = – P i ¯ (T) + (1 – P i ¯ (T))

E U i ¯ P i (A) = 0

Now consider that $P i ¯ (T) – (1 – P i ¯ (T)) > 0$ only if $P i ¯ (T) > 0.5$ . Similarly, $– P i ¯ (T) + (1 – P i ¯ (T)) > 0$ only if $P i ¯ (T) < 0.5$ . Thus the following holds:

o p t E U i ¯ P i (D) = {{T} if P i ¯ (T) > 0.5 {F} if P i ¯ (T) < 0.5 {F, T, A} if P i ¯ (T) = P i ¯ (T) = 0.5 {A} otherwise

MFSC guarantees that $P r (P i ¯ (T) > 0.5) > P r (P i ¯ (T) < 0.5)$ , and it follows that $P r (V i = T) > P r (V i = F)$ , and thus generalized voter competence is guaranteed (i.e., $c > 0.5$ ). Abstaining does not increase the votes for verdict $T$ and also not for verdict $F$ . Thus, as long as $P r (V i = T) > P r (V i = F)$ , it will be the case that by the law of large numbers, as the number of voters approaches infinity, almost surely $∑ V i = T > ∑ V i = F$ , and thus the election outcome is almost surely $T$ . □