Mystifying but not misleading: when does political ambiguity not confuse voters?

Abstract

The purpose of political campaigns in democracies is to provide voters with information that allows them to make “correct” choices, that is, vote for the party/candidate whose proposed policy or “position” is closest to their ideal position. In a world where political talk is often ambiguous and imprecise, it then becomes important to understand whether correct choices can still be made. In this paper we identify two elements of political culture that are key to answering this question: (i) whether or not political statements satisfy a so-called “grain of truth” assumption, and (ii) whether or not politicians make statements that are comparative, that is contain information about politicians’ own positions relative to that of their adversaries. The “grain of truth” assumption means that statements, even if vague, do not completely misrepresent the true positions of the parties. We find that only when political campaigning is comparative and has a grain of truth, will voters always make choices as if they were fully informed. Therefore, the imprecision of political statements should not be a problem as long as comparative campaigning is in place.

Appendix

Proof of nondisclosure equilibrium depicted on Fig. 2, Sect. 4

We show that the described strategy profile is an equilibrium in two steps. First, under the grain of truth condition no type can imitate the strategy of another type. Second, no type of a party has an incentive to deviate. To see that note that for a party of any type the nondisclosure equilibrium payoff is exactly the same as its full revelation payoff and is the same as the full revelation payoff of any other type making the same equilibrium statement. On the $45^{\circ }$ line this payoff is equal to 0.5 for both parties, and on any downward-sloping segment the payoffs of the parties are uniquely determined by $\widehat{\lambda }=\frac{x_1+x_2}{2}$, which is the same for any $(x_1,x_2)$ on the given segment. This second observation implies that a simple set of voter out-of-equilibrium beliefs rules out incentives for deviation. For example, suppose that after a deviating statement voters are certain that the true type is a particular type in the intersection of the deviating statement and the equilibrium nondisclosure statement of the other party. As this type is one of the equilibrium pooling types to which a given deviating type belongs, the resulting deviation payoff of the party is exactly equal to its equilibrium payoff. $\square$

An example of equilibrium under comparative campaigning where pooling in a non-generic set of types is likely to misguide voters

Consider a strategy profile where all types on the downward-sloping diagonal of the $[0,1]\times [0,1]$ square pool and all the other types fully reveal both positions. All pooling types $(x_1,x_2)$ are such that $x_1=1-x_2$, that is, positions of the two parties are symmetric around 0.5, and half of all types have $x_1<x_2$ (above the $45^{\circ }$ line), while the opposite is true for the other half (below the $45^{\circ }$ line). This means that given the nondisclosure statement, all voters are indifferent between the two parties, and parties’ associated payoff is 0.5. Note that the full disclosure payoff of both parties (of any actual pooling type) is also 0.5, as then the indifferent voter is located exactly in the middle of the unit interval.

It is easy to see that the proposed strategy profile is an equilibrium. First, no type can imitate the strategy of another type under the grain of truth condition. Second, by fully revealing both positions, a party of any pooling type obtains the same payoff as in the proposed equilibrium. Therefore, if after a deviating statement, voters believe that the true type is one of the types on the downward-sloping diagonal (which belongs to the intersection of the deviating statement and the equilibrium statement of the other party), then a deviation is not gainful.

The described strategies can easily lead to inefficient choices for some or all voters. Indeed, as a half of all pooling types have $x_1<x_2$ and a half $x_1>x_2$ (each option having the same probability given the nondisclosure statement), whatever choice each voter makes in equilibrium, it is equally likely to be wrong or right. That is, voters mistakenly choose the least preferred party with probability 0.5. $\square$

Proof of Proposition 2

Consider a set of types $\Phi \subseteq [0,1]\times [0,1]$ such that $\Phi$ is not a subset of types on the upward- and downward-sloping diagonals.³¹ In the following we show that for any such set, as soon as there exists an equilibrium in which all types in $\Phi$ pool with each other, their nondisclosure does not mislead voters with probability one. That is, for a measure-one set of types in $\Phi$, voters’ equilibrium choices are the same as under full disclosure. This will then suggest that the only types in the $[0,1]\times [0,1]$ square that (a) may have incentives to pool with other types and (b) by pooling lead to a positive probability of “wrong” voters’ choices are all located on either of the two diagonals. Therefore, there does not exist an equilibrium wherein the set of types that (a) do not fully disclose and (b) make a statement inducing inefficient voters’ choices has a positive measure.

So, suppose that there exists an equilibrium in which all types in set $\Phi$ pool. Pooling requires the existence of at least two different types in $\Phi$. Let us denote them by $(x_1, x_2)$ and $(y_1,y_2)$. Moreover, since $\Phi$ is not a subset of the upward- and downward-sloping diagonals, there exists at least one type in $\Phi$—say $(y_1,y_2)$— that does not belong to either of the diagonals, so that $y_1\ne y_2$ and $y_1\ne 1-y_2$.

Note that the payoff of each party of any type in $\Phi$ is equal to the payoff of this party in the full disclosure equilibrium. This follows from the fact that the sum of paries’ payoffs is equal to one in any equilibrium, and each party knows and can fully reveal both positions. Indeed, if one of the parties obtained a payoff that is lower than under full disclosure, then it would have an incentive to deviate by revealing both positions precisely.

Now, given this and given that parties’ payoffs are uniquely determined by $\widehat{\lambda }$ (whenever it is well-defined) and by whether the party is (perceived as) left or right, it must be that for any type in $\Phi$ where parties’ relative positions with respect to each other are the same as their relative perceived positions, the indifferent voter is the same as under a given equilibrium pooling statement. Moreover, as the value of $\widehat{\lambda }$ under the equilibrium pooling statement is uniquely defined for all types in $\Phi$, the probability measure of all other types in $\Phi$—where the positions of parties are reversed and the indifferent voter is equal to $1-\widehat{\lambda }$,—must be zero in $\Phi$. Thus, we obtain that for any type in a measure-one set of $\Phi$ (i) the position of the indifferent voter is the same under a given equilibrium pooling statement and under full disclosure, and (ii) parties’ relative positions with respect to each other are the same as their relative perceived positions (owing to the equilibrium pooling statement). The two conditions imply that with probability one nondisclosure by types in $\Phi$ does not mislead voters and the equilibrium is ex-post efficient.

It remains to consider the case where the indifferent voter is not well-defined, as this is the only case in which the above argument does not go through. Below we show that since our set of pooling types $\Phi$ includes $(y_1,y_2)$, where parties’ positions are neither equal nor symmetric around 0.5, this situation is not an equilibrium. This would then contradict the definition of set $\Phi$, and thereby, conclude the proof.

Consider two possibilities in turn: first, where the indifferent voter is not well-defined for the equilibrium pooling statement, and then, where the indifferent voter is not well-defined when one of the pooling types in $\Phi$ fully discloses. Suppose the former is true. Then the equilibrium payoff to a party of any type in $\Phi$ is either 0.5 (when all voters are indifferent between the two parties) or zero or one (if no voter is indifferent). In either case, it is easy to see that one of the parties of type $(y_1,y_2)$ would strictly benefit from deviating to full disclosure. Now, suppose that the latter is true: the indifferent voter is not well-defined when one of the pooling types in $\Phi$ fully discloses. This can occur only when the positions of the two parties of that type are equal, so that all voters are indifferent between parties 1 and 2. Then the full revelation payoff of both parties of this type is 0.5 and given the equality of the full revelation payoff and equilibrium nondisclosure payoff, the equilibrium nondisclosure payoff of this—and of any other type in $\Phi$—is also equal to 0.5. But this implies that the party of type $(y_1,y_2)$ whose location is closer to 0.5 can benefit from deviating to full disclosure. Hence, the situation where the indifferent voter is not well-defined—either for the equilibrium pooling statement or for one of the pooling types in $\Phi$—is not an equilibrium.

Thus, we obtain that for any set of pooling types $\Phi$, the corresponding nondisclosure equilibrium is ex-post efficient. $\square$

Proof of Proposition 3

First, notice that no type of a party can or has incentives to imitate the strategy of another type. Even if the grain of truth condition allows a party to make an equilibrium statement of another type, this imitation will be detected by voters as their beliefs about the type are formed based on statements of both parties and the other party still makes an equilibrium statement. For example, if $x_{1}\in \Phi$ ,  but $x_{2}\notin \Phi$ , party 1 could imitate a type with both positions in $\Phi$ by making a statement $S^{*}=\Phi$. However, since party 2 reveals its own position precisely and this position lies outside $\Phi$, voters, who know the equilibrium strategies, deduce that party 1 has deviated. Similarly, a party of a type with both positions in $\Phi$ can fully disclose its own position imitating the equilibrium statement of a type where the position of that party (but not the position of the adversary) is in $\Phi$. But given that the statement of the other party is $\Phi$, voters deduce that both parties have positions in $\Phi$ and it is the first party that deviated. Finally, a party of a type with at least one of the positions outside $\Phi$ can imitate the strategy of another such type—if this party’s position is the same for both types. But given that the other party fully reveals its position, the imitating party cannot succeed in pretending to be of the other type.

Next, we construct a system of voter out-of-equilibrium beliefs such that given these beliefs, no deviation is profitable. Suppose that after observing $S_i=\Phi$ and $S_j\ne \Phi$ voters assign probability one to such positions in $S_i\bigcap S_j$ where party j is at least as far from 0.5 as her adversary, that is, where $|x_j-0.5|\ge |x_i-0.5|$.³² And if voters observe $S_i=\{x_i\}$ and $S_j\ne \{x_j\}$, then they assign probability one to party j being located at such $y\in S_j$ where the distance from party j to 0.5 is the largest among all locations in $S_j$, that is, where the full revelation payoff of party j, given position $x_i$ of the adversary, is minimized.

Given such out-of-equilibrium beliefs, no type of a party has an incentive to deviate from the proposed equilibrium strategy. Clearly a party of type $(x_1,x_2)$ such that $x_1\in \Phi$ and $x_2\in \Phi$ has no incentives to deviate since its equilibrium payoff, given the symmetry of the statements, is 0.5, while any deviation payoff is lower or equal than 0.5. A party of type $(x_1,x_2)$ such that either $x_1$ or $x_2$ or both positions do not belong to $\Phi$ has no incentives to deviate either. If party j deviates to some admissible statement $S_j\ne \{x_j\}$, then the subsequent choice of voters will be as if the true position of party j is y for sure, and thus the deviation payoff of party j is equal to its full revelation payoff based on its own position being y and the position of the adversary being $x_i$. As $x_j\in S_j$, too, this payoff does not exceed the party’s full revelation payoff based on the true positions. $\square$

Proof of Proposition 4

Consider a strategy profile where at least one of the parties chooses a position different from 0.5 and parties make statements that either fully disclose both positions or don’t. Then irrespective of the payoffs associated with this strategy, at least one of the parties can deviate and earn a strictly larger share of votes. To do that, a party can simply move its position $\varepsilon$-close (and in the direction of the median voter) to the position of the adversary for arbitrary small $\varepsilon$ and then reveal both parties’ positions precisely. We then obtain the median voter result: the only pair of positions for which the described deviation does not guarantee a higher payoff for either party is (0.5, 0.5). Knowing this, voters can deduce (and believe) that, even if the equilibrium statements are fuzzy, both parties are located at 0.5, so that indeed, neither party can benefit from deviation. $\square$

Proof of Proposition 6

Note that for any equilibrium strategies of the two parties, the following holds: (i) the sum of parties’ payoffs is equal to one, and (ii) each party is free to make a comparative statement and reveal both parties’ positions precisely. This means that all equilibria, with or without comparative political campaigning, must be payoff equivalent to the full-disclosure equilibrium, as otherwise one of the parties of some type would have an incentive to deviate and fully disclose both positions. Then, given that and given the monotonic functional dependence of parties’ payoffs on the position of the indifferent voter, we obtain that in any nondisclosure equilibrium, the position of the indifferent voter for any or almost any nondisclosing type must be the same as under the equilibrium nondisclosure statement. Furthermore, the relative positions of parties with respect to each other must be the same as their relative perceived positions (implied by the nondisclosing statements). This logic does not apply only when the indifferent voter is not well-defined, which in equilibrium can occur only if parties’ positions are either the same or exactly symmetric around the median voter. This was shown in the proof of Proposition 2. Thus, owing to the same argument as in that proof, generically, any equilibrium is ex-post efficient. $\square$