Robustness

In this section, we discuss the robustness of the persuasion result in Theorem 2. In particular, we ask: can the sender persuade the voters even when he does not know the exact details of the environment? How “stable” is the equilibrium? Are there other equilibria?

1.4.6.1 Robustness: Detail-Freeness

In this section, we show that in order to persuade the voters, the signal structure does not need to be finely tuned to the details of the environment. Suppose that the prior and the preference distribution are such that

|Φ(0)−¹

2|>|Φ(Pr(α))−¹

2|, (1.49)

|Φ(1)−¹

2|>|Φ(Pr(α))−¹

2|; (1.50)

therefore, when the citizens vote optimally given their beliefs, the election is closer to being tied when they they are uninformed and hold the prior belief relative to when know the state.

Proposition 1. Taker=1and(x,y)∈_{0, 1}². For any prior and preference distri-bution satisfying (1.49) and (1.50), there is a sequence of equilibria σ_n^∗_n∈

Ngiven the sequence of signal structures(πn^x,r,y)n∈N such that

limn→∞Pr α|piv,a;σ^∗_n=x, (1.51) limn→∞Pr α|piv,z;σ^∗_n=Pr(α), (1.52) limn→∞Pr α|piv,b;σ^∗_n=y. (1.53) The proposition implies that the sender can implement any policy using a single signal structure that works uniformly across the large set of priors and preference distributions satisfying (1.49) and (1.50). For example, the constant policyAis im-plemented by choosingx =y=1, which leads to an equilibrium in whichAhas a vote shareΦ(1)as the election becomes large.

The proof is in the appendix in Section 1.B.4. The basic idea is that, given this signal, the vote shares are close toΦ(Pr(α))in statesα2andβ2. Hence, by assump-tions (1.49) and (1.50), if voters behave according to the posteriorsxandyin states

α1andβ1, the election is closer to being tied inα2andβ2than inα1 andβ1. Thus, just as before, conditional on being pivotal, voters with signalsaandbbelieve that they are in statesα₂ andβ₂, and—interpreting their signals conditional on these substates—their critical posteriors are as given in the proposition.

A similar argument implies that the signal structure from Lemma 1 is also ef-fective when the actual environment is slightly different: When the prior and Φis slightly different from the one used to calculate(x,r,y), then there is still an equi-librium close-by with critical beliefs that are close toµ_α,ˆ_{r, and} µ_β, provided that vote shares at the critical beliefs imply that the election is still closer to being tied in statesα2andβ2than in statesα1 andβ1.

Random Signal Quality. Note that the signal from Proposition 1 matches the de-scription in the introduction. In particular, we can swap the timining in the descrip-tion of the signal. Rather than choosing the “quality” of the signal after the state of nature has realized, one can first choose randomly whether the signal is “revealing”

or “obfuscating” and then, if it is revealing, send a signal corresponding to the real-ized state of nature to all voters (as in substatesα1 andβ1), and, if it is obfuscating, send the signalszorbinαandzorainβ (as in substatesα2 andβ2whenx=0 andy =1).

1.4.6.2 Robustness: Basin of Attraction

We show that, for a large set of initial strategies, an iterated best response leads quickly to the “manipulated equilibrium” of Theorem 2 described earlier.

Let(µ_α,µ_β)be any pair of beliefs withΦ(µ_α)6= ¹₂ andΦ(µ_β)6= ¹₂. By Lemma 1, there is a sequence of information structures(π^x,r,yn )_n∈Nand equilibria(σ^∗_n)_n∈Nthat implements the pair of beliefs asn→ ∞, in the sense that, with probability close to1, almost all voters will have such beliefs conditional on being pivotal. Hence, by choosing (µα,µ_β) appropriately, a sender can implement any desired policy. The next result shows that, for almost any strategyσ, the twice iterated best response is arbitrarily close to σ^∗_n whennis large, in the sense that the posteriors conditional on being tied are close to(µ_α,µ_β).

First, let us define the twice iterated best response: take any belief pand the strategy σ^p that is optimal given these beliefs. Then,σ^ρ(σp)is the best response to σ^p and is optimal given the beliefs

ρ¹(p)=ρ(σ^p), (1.54)

whereρ(σ^p)is the vector of the posteriors conditional on the pivotal event and the signals. In the same way,σ^ρ(σρ1(p)) is the best response toσ^ρ1(p) (so it is the twice iterated best response toσ^p) and is optimal given the beliefs

ρ²(p)=ρ(σ^ρ1(p)). (1.55)

1.4 Monopolistic Persuasion | 27 Proposition 2 shows that for almost anyp, we have|ρ²(p)− µα,ˆr,µβ

|< when nis sufficiently large. This means that the twice iterated best response is arbitrarily close to the manipulated equilibriumσ_n^∗since the equilibrium is consistent with the beliefρ(σ^∗_n)≈(µα,ˆr,µ_β); see (1.13).

Proposition 2. Take any beliefs(µα,µβ)∈_[0, 1]²withΦ(µα)6= ¹₂ andΦ(µβ)6= ¹₂ and the corresponding information structures(π^x,r,y_n )n∈Nfrom Lemma 1.

For anyδ >0, there is someB⊂_[0, 1]³with Lebesgue-measure of at least1−δ and some¯_n∈_Nsuch that, for alln≥¯_n,

∀p∈B:|ρ²(p)− µα,ˆr,µβ

|< δ. (1.56)

The proof is in Section 1.B.5 in the Appendix. The proof also implies that, for

“almost any” strategyσ—even those that are not optimal given some beliefp—the twice iterated best reply is arbitrarily close to the manipulated equilibriumσ^∗_nwhen n is large, where the genericity requirement is with respect to the induced vote shares; see condition (1.102), replacingσ^pbyσ.

Simple Reasoning.Proposition 2 illustrates that a simple reasoning underlies the manipulated equilibriumσ^∗_n. The result loosely relates to the concepts of level k-thinking and level-k-implementability (De Clippel, Saran, and Serrano (2019)).

The theorem implies that, for almost any strategy (a “behavioral anchor”), the strate-gies that are consistent with level-2-thinking are close to the manipulated equilib-rium. In this sense, any state-dependent target policy(x(α),x(β))∈_{A,B}²is level-2-implementable.12

1.4.6.3 Other Equilibria

Proposition 2 shows that the basin of attraction of the iterated best response of an arbitrarily small neighborhood of the manipulated equilibria consists of almost all strategies whennis large enough. However, this still leaves open the possibility that there are other equilibria, such that if we begin exactly at such a strategy profile, the best response dynamic stays there. In the working paper version, Heese and Lauermann (2019, Theorem 4),13 we show that this is indeed the case. There exists another equilibrium and that equilibrium is not “manipulated” but implements the full information outcome asn→ ∞. We restate the result here:

12. De Clippel, Saran, and Serrano (2019) consider a different notion of level-2-implementability that demand that there issomebehavioral anchor such thatanyprofile of strategies that are level-1-consistent or level-2-level-1-consistent for this anchor implement a given social choice function. Here, almost any strategy can be such an anchor.

13. The working paper is publicly available here https://ideas.repec.org/p/bon/boncrc/

crctr224_2019_128.html.

Theorem 3. Let Φbe strictly increasing. For all information structures(πn^x,r,y)n∈N

with(x,r,y)∈(0, 1)³, there exists an equilibrium sequence(σ_n^∗)n∈N for which the full information outcome is elected asn→ ∞,

limn→∞Pr(A|α;σ^∗_n,π_n,n)=1, limn→∞Pr(B|β;σ^∗_n,π_n,n)=1.

Intuition.Note that the signalπ_n almost always sends an (almost) perfectly re-vealing signal whennis large. Hence, there is a sequence of strategies (e.g. given by sincere voting) for which the full-information outcome is elected asn→ ∞. The question is if such a sequence of strategies can be an equilibrium sequence. The the-orem shows that, whenever Φis monotone, the answer is yes. This is easy to see in the extreme case when voters have a common typet, and, hence, have common interests. A result of McLennan (1998) shows that, with common interest, the util-ity maximizing symmetry strategy is a symmetric equilibrium. Hence, for this case, the existence of a sequence of strategies that yields the full-information outcome immediately implies the existence of an equilibrium sequence that yields it as well.