# Public Information: Relevance or Salience?

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Model

- Nature determines the state of the world $\omega $.
- Each voter observes a private signal and the public signal (observed by everyone).
- Agents cast their votes and the collective decision d is determined according to the majority of votes.
- The true state is revealed and agents receive their payoffs.

#### Equilibrium Analysis

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**1**

**Corollary**

**1.**

- if ${s}_{i}={s}_{p}$, players’ best response is to follow both signals
- if ${s}_{i}\ne {s}_{p}$, then:
- (a)
- if $Q\le q$, always follow the private signal ($\mu =0$),
- (b)
- if $Q\in (q,{Q}^{H}]$, follow ${s}_{p}$ with probability $\mu \in (0,1)$,
- (c)
- if $Q>{Q}^{H}$ always follow the public signal ($\mu =1$).

**Proposition**

**2**

**Proof.**

## 3. Experimental Design

**Recency.**This treatment varied whether the public signal was displayed before or after the private signal.

**Asymmetric Prior.**The least salient way to convey the public message is not to show it at all. This treatment corresponds to the last 10 rounds of each session. In these rounds, an asymmetric prior was provided instead of a symmetric ($\pi =0.5)$ prior, and no public signal was displayed. Subjects were told that the computer placed the prize in the blue box with probability $\pi =0.7$ (or $\pi =0.55$, depending on the session), and in each round each subject received only a private signal. From a Bayesian standpoint, these ten rounds conveyed the same information as the previous ones: having a symmetric prior and a public signal with accuracy $Q=0.7$ is identical to having an asymmetric prior $\pi =0.7$ and no public signal. After receiving the private message, subjects were asked to vote for one of the two boxes, as in the first part of the experiment.

**Jingle.**This treatment varied the way the public signal was projected on the central screens. In the absence of this treatment, the public signal was displayed with the picture of a blue or red box (as for the private signals projected on subjects’ monitors). With the jingle treatment, the public message was projected on the central screen with a video displaying a star jumping within an empty, white box, which then became either red or blue. The video was accompanied by a striking soundtrack, and to make the jingle treatment less repetitive, the music theme varied. I used famous music pieces such as Also sprach Zarathustra by Strauss, Eye of the tiger, The final Countdown, Thrift Shop and the Game of Thrones’ soundtrack.10 I hypothesize that salience of the public signal is increasing in both recency and emphasis.

## 4. Results

#### 4.1. Subjects’ Response to Salience

**Asymmetric Prior.**Figure 3 shows the proportion of votes with the public signal (under mismatch) in the treatment with public signal calculated by aggregating over the other salience treatments vs. the treatment with asymmetric prior.

**Recency.**The recency treatment varied whether the public signal was projected on the central screens of the laboratory before or after the private signals were displayed on the subjects’ monitors. Recency effects were substantively and statistically significant. In particular, when the public signal accuracy is higher (right columns in Figure 4), there is a $16\%$ difference between the fraction of times subjects followed the public signal when it was displayed before the private ($60\%$) as opposed to after the private ($76\%$). When the public signal accuracy is lower than the private (left columns), there is a $11\%$ difference. Both differences are significant at any conventional level.

**Jingle.**For what concerns the jingle treatment effect, the direction is the one expected and it is in line with the recency treatment effect. The magnitude is smaller, as Figure 5 shows. When the public signal accuracy is higher (right columns in Figure 5), there is a $4\%$ difference between the fraction of times subjects followed the public signal when it was displayed before the private ($67\%$) as opposed to closer to the vote ($71\%$). When the public signal accuracy is lower than the private (left columns), there is a $5\%$ difference.

#### 4.2. Individual Treatment Sessions

## 5. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Online Appendix

#### Appendix A.1. Proofs—Preliminaries

#### Appendix A.2. Main Results

**Proof of**

**lemma 1.**

**Committee of size 3.**To find the values of q, Q such that it is a dominant strategy to follow the private signal, set Equation (A1) equal to Equation (A2) and $\mu =0$:

**Committee of arbitrary size.**In this case, Equation (A3) becomes

**Proof of**

**Proposition 1.**

**Committee of size 3.**In order to characterize the equilibrium mixing probability, $\mu $, we set equal the expected utilities for the two alternatives

**Committee of arbitrary size.**Consider a committee of arbitrary size N (with N odd). In order to characterize the equilibrium mixing probability, $\mu $, set

**Proof**

**of Corollary 1.**

#### Appendix A.3. Aggregate Data—Individual Sessions

**Figure A1.**Asymmetric prior treatment in individual sessions. Average fraction of votes with public signal under mismatch and $95\%$ confidence intervals. Standard errors are clustered at the individual level. The blue columns correspond to the last 10 rounds in each sessions, where the public signal content was conflated in the prior. Note: Blue lines represent the unique optimal decision in the individual treatment (follow the more precise signal).

**Figure A2.**Recency effects in individual sessions. Average fraction of votes with public signal under mismatch and $95\%$ confidence intervals. Standard errors are clustered at the individual level. Blue lines represent the unique optimal decision in the individual treatment (follow the more precise signal).

**Figure A3.**Jingle effects in individual sessions. Average fraction of votes with public signal under mismatch and $95\%$ confidence intervals. Standard errors are clustered at the individual level. Blue lines represent the unique optimal decision in the individual treatment (follow the more precise signal).

**Table A1.**Jingle and recency effects in individual sessions. The dependent variable is a dummy variable equal to 1 when public and private signals differ, and the subject votes according to the public signal, 0 otherwise. The variable Jingle is a dummy variable equal to 1 when the public information is displayed with a salient video, and the variable Public last is a dummy variable equal to 1 when the public signal is displayed before the private signal. Column (3) shows that when controlling for order effects, the effect of the jingle is not significant anymore. Standard errors are clustered at the individual level in parenthesis. * corresponds to $p<0.1$, ** to $p<0.05$, and *** to $p<0.01$.

Vote Public | |||
---|---|---|---|

(1) | (2) | (3) | |

Jingle | 0.048 ** | 0.050 | |

(0.023) | (0.033) | ||

Public Last | 0.055 *** | 0.057 ** | |

(0.020) | (0.023) | ||

Jingle * Public Last | −0.005 | ||

(0.047) | |||

Observations | 2414 | 2414 | 2414 |

#### Appendix A.4. Individual Data

**Figure A4.**Proportion of times each individual voted according to the public signal in the first four sessions (with group task). The left image plots values for sessions where $Q>q$, the right one $Q<q$. Different colors correspond to different sessions.

**Figure A5.**Proportion of times each individual voted according to the public signal in sessions 5–10 (with individual task). The left image plots values for sessions where $Q>q$, the right one $Q<q$. Different colors correspond to different sessions.

**Figure A6.**Asymmetric Prior (group task): Proportion of times each individual voted according to the public signal. The left image plots values for sessions where $Q>q$, the right one $Q<q$. Red dots correspond to the public signal delivered, blue dots to public signal conflated in the prior.

**Figure A7.**Jingle effect (group sessions). Proportion of times each individual voted according to the public signal. The left image plots values for sessions where $Q>q$, the right one $Q<q$. Blue dots correspond to the public signal displayed with the jingle.

**Figure A8.**Jingle effect (individual sessions). Proportion of times each individual voted according to the publice signal. The left image plots values for sessions where $Q>q$, the right one $Q<q$. Blue dots correspond to the public signal displayed with the jingle.

**Figure A9.**

**No Learning.**This figure plot aggregate votes as a function of time (experimental rounds. Subjects’ behavior does not approach theoretical predictions.)

#### Appendix A.5. Experimental Instructions

- 70 experimental dollars if the group guessed the correct box;
- 10 experimental dollars if the group guessed the wrong box.

- The number of votes for the blue box cast by your group;
- The number of votes for the red box cast by your group;
- The box selected by majority in your group;
- The outcome of the period: that is, whether the group decision was correct or not;
- The earnings for the round.

#### Appendix A.6. Last Ten Rounds

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1. | The article can be found here: https://www.mirror.co.uk/news/uk-news/who-legend-roger-daltrey-hes-8252353. There were several other instances of this kind. Google searches for celebrities and Brexit peaked in the week before the referendum, and many web pages displayed long lists of Brexit’s supporters. |

2. | |

3. | |

4. | This challenge is posed by Reference [8], who suggests that to isolate framing mechanisms, one would need to study the effect of completely uninformative events. |

5. | Assume that voters cannot abstain, and that there is no cost of casting votes. |

6. | For instance, consider a committee of five members where each member observes a private signal and everyone observes the same public signal. One equilibrium is that one member of the committee votes with her private signal, when this disagrees with the public, and the other four members vote with (against) the public signal. In another equilibrium, two committee members vote with their private signals, which disagree with the public, and the other three members of the committee vote with (against) the public signal. |

7. | This is true for the first four sessions. As displayed in Table 1 and explained later, in sessions 5–10 there were no committees and subjects performed an individual task. |

8. | The position of the vote buttons was randomly shuffled in each round. |

9. | Reference [2] formalize this assumption in a model where decision makers overweight states that draw their attention by comparing payoff magnitude in different lotteries. This experimental treatment manipulates salience by changing the way information is presented, rather than payoffs. |

10. | All the videos are available upon request. |

11. | Administering the salience treatments within subjects was a natural choice to get more data points under the time and budget constraints. |

12. | One possibility is that subjects realize that others do not play the symmetric responsive equilibrium, and best respond to that. For instance, if a subject realizes that some other committee members always play the conformist equilibrium in favor of the public signal, then she would best respond by voting for the public signal less than what prescribed by the symmetric responsive equilibrium. In each session, the average vote with the public signal (when signals do not match) is close to $60\%$. A best-response to this would be to vote with the public signal less than $37\%$ of the time (the symmetric responsive equilibrium prediction). The data show that subjects follow the public signal more than what these best replies to experimental data predict. |

13. | It might be that some subjects are playing the conformist equilibria and others are mixing. With Figure 1 we would not be able to disentagle between the two behaviors. |

14. | KV only analyze the case where $Q>q$, with slightly different parameters and committee size. |

15. | I also performed a Kolmogorov–Smirnov test to compare the two samples of voting for the public signal when it is provided before or after the private. Although we cannot reject that the two distribution are the same (with a p-value of $0.15$), the number of observation is too small to rely on this result, and graphically showing the ECDFs provides much better evidence. |

16. | As Figure A4 and Figure A5 in Appendix A show, there are no session-specific effects: individual votes are homogeneous across different sessions. |

17. | All the salience treatment effects in the individual task sessions are shown in Appendix A. |

18. | These are the instructions for sessions with group task and high public signal accuracy. The other instructions and the pictures displayed during the instruction period are available upon request. |

**Figure 1.**

**Responsiveness of vote to signals’ precision**: Average fraction of votes with public signal in sessions with committee decisions, and associated $95\%$ confidence intervals. Standard errors are clustered at the individual level. In the left plot, the public signal and the private signal disagree, and red lines represent the symmetric responsive equilibrium prediction for $\mu $, the probability of voting for the public signal under mismatch. In the right plot, the signals agree, and blue lines represent the unique optimal decision when the two signals agree.

**Figure 2.**

**Individual deviations from equilibria under mismatch**: Red lines represent the symmetric equilibrium predictions for $\mu $, when the public and private signal disagree. Blue lines represent the conformist equilibrium prediction. When $Q>q$ (upper picture), the number of subjects who vote more with the public signal is greater than when $Q<q$ (lower picture).

**Figure 3.**

**Asymmetric prior treatment**: Average fraction of votes with public signal under mismatch, and $95\%$ confidence intervals. Standard errors are clustered at the individual level. Dark columns correspond to rounds where the public signal was provided. Light columns correspond to the last 10 rounds in each sessions, where the public signal content was conflated into the prior. Note: Horizontal lines represent the symmetric responsive equilibrium prediction for $\mu $.

**Figure 4.**

**Recency effect**: Average fraction of votes with public signal under mismatch, and $95\%$ confidence intervals. Standard errors are clustered at the individual level. Dark columns correspond to rounds where the public signal is displayed first. Horizontal lines represent symmetric responsive equilibrium predictions for $\mu $.

**Figure 5.**

**Jingle effect**: Average fraction of votes with public signal under mismatch, and $95\%$ confidence intervals. Standard errors are clustered at the individual level. Dark columns correspond to rounds where the public signal is displayed with the jingle. Horizontal lines represent symmetric responsive equilibrium predictions for $\mu $.

**Figure 6.**

**Individual average treatment effect of recency.**Each dot represents the proportion of rounds an individual votes with the public signal under mismatch. Dark dots correspond to when the public signal is more recent, light dots when the private is more recent. Horizontal lines represent the symmetric responsive equilibrium predictions for $\mu $. Recency has a positive, constant effect across different subjects.

**Figure 7.**

**Individual task vs. group task**. Average fraction of votes with public signal, with $95\%$ confidence intervals. Standard errors are clustered at the individual level. Horizontal dotted lines represent symmetric responsive equilibrium predictions for $\mu $. Solid lines represent the unique optimal decision in the individual treatment (follow the more precise signal).

**Figure 8.**

**Recency effect (individual sessions)**. Each dot represents the proportion of rounds an individual votes with the public signal. Dark dots correspond to when the public signal is more recent, light dots when the private is more recent. The left image plots values for sessions where $Q>q$, the right one $Q<q$. Horizontal lines represent the symmetric responsive equilibrium predictions for $\mu $. Recency has a positive, constant effect across different subjects.

**Table 1.**Summary of experimental sessions. The accuracy of the private signal was set to $q=0.6$ throughout the whole experiment. A total of 75 subjects were assigned to a “group” condition, and divided in committees of size five. The other 82 subjects were assigned to an “individual” condition.

Session | Q | Committee, Size | # Rounds | # Subjects |
---|---|---|---|---|

${s}_{1}$ | $0.7$ | Yes, 5 | 70 | 20 |

${s}_{2}$ | $0.55$ | Yes, 5 | 70 | 20 |

${s}_{3}$ | $0.7$ | Yes, 5 | 70 | 20 |

${s}_{4}$ | $0.55$ | Yes, 5 | 70 | 15 |

${s}_{5}$ | $0.7$ | No | 70 | 13 |

${s}_{6}$ | $0.55$ | No | 70 | 17 |

${s}_{7}$ | $0.7$ | No | 70 | 10 |

${s}_{8}$ | $0.55$ | No | 70 | 16 |

${s}_{9}$ | $0.7$ | No | 70 | 13 |

${s}_{10}$ | $0.55$ | No | 70 | 13 |

**Table 2.**Factorial design for every session (for both $Q>q$ and $Q\le q$). The row values are associated to the recency treatment (i.e., whether the private signal was displayed before or after the public). The column values are associated with the jingle treatment. The values within the matrix display the number of rounds for each interaction.

$\mathit{Jingle}$ | $\mathit{No}\phantom{\rule{3.33333pt}{0ex}}\mathit{Jingle}$ | |
---|---|---|

Asymmetric prior | - | 10 rounds |

Public first | 8 rounds | 22 rounds |

Private first | 8 rounds | 22 rounds |

**Table 3.**OLS regression. The dependent variable is a dummy variable equal to 1 when public and private signals differ, and the subject votes according to the public signal, 0 otherwise. The variable Jingle is a dummy variable equal to 1 when the public information is displayed with a salient video, and the variable Public last is a dummy variable equal to 1 when the public signal is displayed after the private signal. Column (3) shows that, when controlling for order effects, the effect of the jingle remains significant, but the magnitude of recency is higher. Standard errors are clustered at the individual level in parenthesis. * corresponds to $p<0.1$ and *** to $p<0.01$.

Vote Public | |||
---|---|---|---|

(1) | (2) | (3) | |

Jingle | 0.047 * | 0.059 * | |

(0.024) | (0.034) | ||

Public Last | 0.131 *** | 0.138 *** | |

(0.022) | (0.026) | ||

Jingle * Public Last | −0.025 | ||

(0.049) | |||

Observations | 2068 | 2068 | 2068 |

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**MDPI and ACS Style**

Invernizzi, G.M. Public Information: Relevance or Salience? *Games* **2020**, *11*, 4.
https://doi.org/10.3390/g11010004

**AMA Style**

Invernizzi GM. Public Information: Relevance or Salience? *Games*. 2020; 11(1):4.
https://doi.org/10.3390/g11010004

**Chicago/Turabian Style**

Invernizzi, Giovanna M. 2020. "Public Information: Relevance or Salience?" *Games* 11, no. 1: 4.
https://doi.org/10.3390/g11010004