Strategic information selection
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Preker, Jurek; Karos, Dominik Working Paper Strategic information selection Center for Mathematical Economics Working Papers, No. 689 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Preker, Jurek; Karos, Dominik (2024) : Strategic information selection, Center for Mathematical Economics Working Papers, No. 689, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29883821 This Version is available at: https://hdl.handle.net/10419/289846 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
689 March 2024 Strategic Information Selection Jurek Preker and Dominik Karos Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en
Strategic Information Selection∗ Jurek Preker†Dominik Karos‡ March 28, 2024 Abstract Before choosing her action to match the state of the world, an agent observes a stream of messages generated by some unknown binary signal. The agent can either learn the underlying signal for free and update her belief accordingly or ignore the observed message and keep her prior belief. After each period the stream stops with positive probability and the final choice is made. We show that a Markovian agent with Gilboa-Schmeidler preferences learns and updates after confirming messages, but she ignores contradicting messages if her belief is sufficiently strong. Her threshold solely depends on the least precise signal. The agent has strictly higher anticipatory utility than an agent who uses every message to update. However, the latter has a higher chance to choose the correct outcome in the end. In a population of strategic agents, who only differ in their initial beliefs, polarization is inevitable. Keywords: Dynamic Decision Problem, Ambiguity, Gilboa-Schmeidler Preferences, Confirmation Bias, Polarization JEL Classification: D81, D83, D91 ∗The authors thank Chris Woolnough, Gerrit Bauch, Giorgio Ferrari, Ehud Lehrer, Fynn Louis N¨armann, and Frank Riedel as well as many seminar participants at the universities of Bielefeld, Durham, and Tel Aviv for their comments and suggestions. †Center for Mathematical Economics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany. jurek.prek[email protected], corresponding author ‡Center for Mathematical Economics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany. dominik.k[email protected] 1
1 Introduction In an average minute of December 2023, social media users sent out 360,000 tweets on X, sent 694,000 reels on Instagram via direct messages, and liked four million Facebook posts.1A user of any of these social media platforms cannot read all the articles whose headlines appear in her feed, but has to choose what headlines to follow up on and what to ignore. This makes information acquirement on social media fundamentally different from using print media. While a printed newspaper, even if read from cover to cover, will provide the reader only with a fixed number of articles, incoming news in social media form an endless stream. Thus, the information acquirement process in social media requires a pre-step: before reading and processing an article, the user needs to select what to read from a huge supply. It has been well established that this causes people to learn differently: for instance, they learn less from online than from print sources (Eveland and Dunwoody, 2002; Yang and Grabe, 2011); but at the same time they are more focused on specific topics selected by themselves rather than the news editors (Kruikemeier, Lecheler, and Boyer, 2017). Empirical studies suggest the main drivers behind information selection be attitude consistency and source credibility. S¨ulflow, Sch¨afer, and Winter (2019) investigated the impact of both and showed that attitude consistency has no impact on the time people spend looking at a headline in their news feed but increases the probability they follow up on it and actually read the underlying article. Moreover, they found that the assumed source credibility has a positive effect on both the time spent with a headline as well as on the follow-up probability.2 As social media have become increasingly important as a source of news,3understanding the mechanics behind information selection as well as its individual and social consequences, such as polarization4and collective decision making, are crucial. This arti1See Statista (2024). 2See also Chiang and Knight (2011) for the importance of credibility. 3The share of people in Germany who weekly use social media as a source of news has increased from 18 % in 2013 to 29 % in 2023, having overtaken classical print media in 2019. In the US, this number has risen from 27 % to 48 % in the same time. The effect is particularly strong in the group of young adults (Reuters Institute for the Study of Journalism, 2023). According to a study by the Pew Research Center, a majority across 19 countries agrees that social media is an effective way to raise public awareness about political or social issues (77 % agree), to change people’s minds (65 %), get elected officials to pay attention to issues (64 %), and to influence policy decisions in their own country (61 %) (Pew Research Center, 2022). 4In the above mentioned report by the Pew Research Center, 65 % of the investigated subjects say 2
cle provides a dynamic model of strategic information selection that rationalizes observed behavior and investigates its implications. We consider a decision maker who has to match an unknown binary state of the world. Before making her choice, she receives a discrete stream of binary and symmetric messages, to be thought of as articles on social media that endorse one of the two states, that are correct with some unknown probability of at least 50%. She has the option to learn this probability; in the news media example, after obtaining a headline (the message) she may click on the link, read the article, learn how well the article is researched, and infer the probability that the message of the headline was correct. If she chooses to ignore the message, her belief does not change; if she chooses to learn the underlying signal, she has to perform a Bayesian update. The underlying assumption is that people can easily ignore a headline they only see briefly, but once they have engaged with it, it cannot be unseen and will affect their belief. For her belief update she uses her previous belief, the message, and the signal. That is, we assume that the decision maker is Markovian, meaning that she does not use the entire history of previous messages and signals.5 In each period the stream of messages stops with some exogenous probability and the decision maker has to make her choice about the state; otherwise a new signal and a new message will be drawn. Maximizing her utility thus becomes a dynamic optimization problem, where with every decision she has to take into account how her belief will develop in the future. The Markovian structure gives rise to a Bellman equation whose solution is the decision maker’s anticipatory utility function.6We show that optimal strategies exist and describe their general structure. Afterwards, we focus on a decision maker with max-min preferences (cf. Gilboa and Schmeidler, 1989) and show that any optimal strategy requires her to ignore messages that oppose her belief whenever her belief is sufficiently strong. Hence, she exhibits what Stone and Wood (2018) call “non-ego-based that access to the internet and social media has made people more divided in their political opinions. Allcott, Braghieri, Eichmeyer, and Gentzkow (2020) showed that regular Facebook users who turned off the network for a month were less polarized in their attitudes afterwards. See Prior (2013) and Kubin and von Sikorski (2021) for political reviews on how (social) media drive polarization. 5There are two major reasons for this assumption: On the one hand, people see and read so many headlines and articles on social media that eventually, they cannot keep track of the entire history of their messages, signals, and beliefs. On the other hand, it allows us to condition the agent’s behavior only on her belief and the signal, independent of time and signal history, so that the analysis remains tractable. 6We call this utility anticipatory as it takes into account all possible further paths on which beliefs might change. This relates to the concept of utility from anticipation, introduced by Loewenstein (1987) and later investigated, for instance, by Caplin and Leahy (2001) and K˝oszegi (2017). 3
cognitive dissonance”7: media consumers avoid contradicting information “not because this threatens their egos, but simply because uncertainty is unsettling”. Ignoring free information is obviously not optimal in the framework of Blackwell (1951, 1953). However, we show that it is optimal in our setting, rationalizing human behavior in social media.8In particular, when comparing our strategic decision maker with a Bayesian decision maker, who updates his belief after every round, we find that she obtains a strictly higher anticipatory utility, while her probability to match the true state of the world is strictly lower. Surely, this model describes social media only in a very simplified way. For instance, we assume that the observed messages are independent of the user’s belief, and we ignore algorithms, echo chambers, as well as the targeted use of fake news by political agents. Yet, even without all these or any other form of homophily in social media networks9we obtain quite extreme polarization results where society only becomes more divided over time. The rest of the paper is organized as follows. In Section 2, we present further related literature. Section 3develops the model, discusses its assumptions, and derives general existence and uniqueness results. In Section 4, we derive optimal strategies in the absence of ambiguity, while Section 5covers agents with Gilboa-Schmeidler preferences under ambiguity. In Section 6, we compare our strategic decision maker with one who updates after every message he observes. Social consequences in terms of polarization and collective decision making are covered in Section 7. Section 8provides several ways to extend the model and Section 9concludes the paper. Proofs are relegated to the appendix. 2 Further Related Literature A large class of models in which agents do not fully exploit the available information focuses on limited cognitive abilities. For instance, the observed tendency to favor and search for information that reinforces prior beliefs while ignoring opposing information 7The psychological concept of cognitive dissonance has been popularized by Festinger (1957): Uncertainty about what to believe and facing a trade-off between feeling validated and wanting to know the truth makes people feel uncomfortable. 8Social media are not the only place where information might be ignored: even some reader of this paper might reject our model and put it aside without letting it affect their belief about human behavior. 9See Aiello et al. (2012) for a statistical analysis of homphily in social media. 4
is known as selective exposure.10 Lord, Ross, and Lepper (1979) demonstrate that when people with strong opinions on controversial issues—in that case, the death penalty in the USA—are given equally convincing evidence for both positions, they become even stronger in their opinions. So, different pieces of information favoring contrary states of the world do not neutralize, but people pick what they want to hear, interpret it as confirming evidence, and ignore the rest. This confirmation bias11 is not what drives our results, as our decision maker correctly processes all information that she uses, but deliberately chooses to ignore some of it. This distinction relates to the psychological distinction between automatic and controlled processes (Schneider and Shiffrin, 1977). Among the models of strategic information selection, an important strand of the literature, following Sims (2003), focuses on rational inattention where agents only obtain partial information due to cost. (See Ma´ckowiak, Matˇejka, and Wiederholt (2023) for a review.) In contrast, information in our model comes at no cost and may still be rejected. In many models agents choose ex ante how to gather information, that is, they only obtain the actual message after their decision. Machina (1989) gives a stylized example of such behavior based on the Allais Paradox, where non-expected utility maximizers choose not to obtain information. Carrillo and Mariotti (2000) present a decision maker with time-inconsistent preferences who decides to acquire only incomplete information about a future externality. In Suen (2004), agents observe coarsened signals and decide themselves on a coarsening rule. Similarly, decision makers in Che and Mierendorff (2019) have access to different biased sources and choose the optimal one. In contrast, our decision maker first obtains a message and then chooses what to do with it. This brings our model closer to Chen (2022), who considers agents with different models at hand who update their beliefs according to the one that best supports their bias, and to Fryer, Harms, and Jackson (2019) where agents deal with “ambiguous” signals that can be used in different ways.12 Yet, our model is distinct in several key features. First, agents can decide to entirely discard and ignore a message. This brings us closer to Allahverdyan and Galstyan (2014), where an agent does not react to advice (sent by another agent who wants to persuade the former one) that conflicts with her prior conviction. Second, a decision maker with Gilboa-Schmeidler preferences will not 10Hart et al. (2009) provide a psychological meta-study on that subject. 11See Nickerson (1998) for a psychological overview as well as Rabin and Schrag (1999). 12Note that their use of “ambiguous” differs from our terminology as we refer to ambiguity as missing knowledge about probabilities. 5
operate under the assumption that the message has been generated by the signal that fits best, but rather by the signal that provides her with the lowest utility. Third, even if there is no ambiguity at all, that is, if the signal is known, some of the messages will be ignored. The concept of motivated beliefs might serve as a bridge between our model and the literature on ex-ante information design. According to Epley and Gilovich (2016), agents do not directly choose their beliefs, but “their motivations guide what information they consider, resulting in favorable conclusions that seem mandated by the available evidence”. Even though our agent acts ex post, she follows precisely this approach: if she expects a more favorable conclusion from learning the signal and interpreting the message, she does so; otherwise she discards the message entirely. The good news-bad news effect or optimism bias—experimentally confirmed by Eil and Rao (2011) and axiomatically characterized by Bracha and Brown (2012)—prescribes people to respect the strength of a signal when the news is favorable. In Brunnermeier and Parker (2005), agents choose subjective probabilities such that their current well-being is maximized, at the cost of possibly taking a wrong decision afterwards. This effect differs from our model as our states are neutral. However, agents in our setup are happier, the more convinced they are of any state; being unsure about the true state of the world makes them uncomfortable. Signal ambiguity as a reason for information selection has been described by Gentzkow and Shapiro (2006). The main argument is that a decision maker who does not like what she reads can conveniently tell herself that the article must be poorly researched and can be safely ignored. Ambiguity as a driver of polarization has been modeled by Glaeser and Sunstein (2013). In their model, people believe different parts of the information they get. In Baliga, Hanany, and Klibanoff (2013), polarization occurs as a result of ambiguity aversion. This is not the case here: even though we consider extremely ambiguity averse decision makers, our results about polarization would remain true even if there were no ambiguity at all. 3 Optimal Information Selection An agent has to match the correct state of the world from the state space Ω = {0,1}. That is, the choice space is C= Ω and her von Neumann-Morgenstern utility function 6
v:C×Ω→ {0,1}is given by v(c, ω) = 1 if c=ωand v(c, ω) = 0 otherwise. To keep notation simple, we identify any belief Λ ∈∆ (Ω) she might entertain with λ= Λ(1) ∈[0,1]. With such a belief her expected utilities from choosing 0 or 1 are E[v(0,·)] = 1 ·(1 −λ)+0·λ= 1 −λ, and E[v(1,·)] = 1 ·λ+ 0 ·(1 −λ) = λ, respectively. Thus, her indirect expected utility at belief λis u(λ):= max {λ, 1−λ}. Prior to her choice the agent obtains information, which is modeled as a discrete stream of symmetric binary signals, that is, maps ζ: Ω →∆ (Ω) with ζ(1|1) = ζ(0|0), and realized messages m∈ {0,1}. We assume without loss of generality that ζ(1|1) ≥1 2, so that any signal ζcan be identified with z=ζ(1|1) ∈1 2,1.13 Let Z⊆1 2,1be a compact set of signals. At each period t∈N, the signal stream ends with some (exogenous) probability 1 −pand the agent has to make her decision. With probability p, the stream continues. In this case a signal zt∈Zis generated, in a potentially unknown way, and afterwards a message mt∈ {0,1}realizes according to zt, that is, Pzt(mt= 0|ω= 0) = Pzt(mt= 1|ω= 1) = zt.14 The decision maker only observes mt. For any signal zthe expected probabilities that 0 and 1 realize are q0 z(λ):=λ(1 −z) + (1 −λ)zand q1 z(λ):=λz + (1 −λ) (1 −z), respectively. Define for each z∈1 2,1two functions gz, g−1 z: [0,1] →[0,1] by g−1 z(λ):=λ(1 −z) λ(1 −z) + (1 −λ)zand gz(λ):=λz λz + (1 −λ) (1 −z)(1) for z∈1 2,1, as well as g−1 1≡0 and g1≡1.15 Surely, if the agent knew ztand mt, then her posterior beliefs in case of a message of 0 or 1 were λt=g−1 zt(λt−1) or λt=gzt(λt−1), 13The triple (Ω, ζ (·|0) , ζ (·|1)) is a symmetric binary Blackwell experiment, see Blackwell (1951,1953). 14The signals (zt)t≥1can be correlated. Yet, given the signals, the probability measures (Pzt)t≥1are independent. 15The latter definitions coincide with (1) for λ∈[0,1) or λ∈(0,1], respectively. The values of g−1 1(1) and g1(0) are irrelevant for the agent’s optimal decision: if she is certain that the true state is 0 (or 1, respectively) and receives a message stating the opposite, she will consider it impossible that the signal is perfectly accurate. 7
0z 0.5 λ 0.5 1 Figure 2: The graphs of z7→ λ∗(z) (solid line) and z7→ 1−z(dashed line) Corollary 4.2. The function U∗ {z}is continuous, satisfies U∗ {z}(1 −λ) = U∗ {z}(λ)for all λ∈[0,1] as well as U∗ {z}(0) = U∗ {z}(1) = 1, and is strictly decreasing on [0, λ∗(z)]. Moreover, U∗ {z}(gz(λ)) < U∗ {z}g−1 z(λ)for λ∈0,1 2and U∗ {z}(gz(λ)) > U∗ {z}g−1 z(λ)for λ∈1 2,1. Theorem 4.1 shows that even without ambiguity there are beliefs where it is optimal for a strategic decision maker to ignore contradicting messages. This region, denoted by I(z) = [0, λ∗(z)] ∪[1 −λ∗(z),1] increases as zdecreases. Corollary 4.3. Let z, z0∈1 2,1. Then I(z)⊆I(z0)if and only if z≥z0. The relation between zand λ∗(z) is depicted in Figure 2. As can be seen there (and analytically verified), λ∗(z)>1−zfor all z∈1 2,1. 5 Information Selection under Ambiguity In this section we shall return to the decision problem when zis not fixed, but stems from some compact set Z⊆1 2,1.23 We first derive the optimal strategy of a decision maker 23We exclude 1 2∈Zto avoid some tedious case distinctions. 14
with Gilboa-Schmeider preferences. Afterwards we consider an alternative approach in which the decision maker is unaware that she might reassess her view on possible signals after having received a message. 5.1 Optimal Strategies Our main finding is that a Gilboa-Schmeidler decision maker who faces a compact set Zof signals will behave exactly as if she knew that every message were generated by z= min Z. Theorem 5.1. Let Z⊆1 2,1be compact and and let H:F × P → Rbe the GilboaSchmeidler aggregator in (2). Let z= min Z. Then a strategy is optimal given Zif and only if it is optimal given {z}. In particular, U∗ Z=U∗ {z}. A Gilboa-Schmeidler decision maker facing signal set Zis behaviorally indistinguishable from a decision maker who knows that zis the only possible signal. Moreover, such an agent always follows up on messages that confirm her prior belief but ignores contradictory information if her prior is sufficiently strong. The respective threshold λ∗only depends on the least precise signal that is deemed possible. One might be inclined to argue that Theorem 5.1 is not surprising: after all a GilboaSchmeidler decision maker is pessimistic and should, hence, presume the message stem from the “worst” possible signal. We shall briefly illustrate that this line of thought is incorrect though as there is a difference between the “worst” and the “least precise” signal. Example 5.2. Let the signal set be Z=8 10,9 10such that U∗ Z=U∗ {8 10 }and λ∗8 10=1 3. Let the agent’s belief be λ=1 20 and suppose that message 1 realizes. Updating with respect to signal z=8 10 leads to an updated belief of g8 10 1 20=4 23, while updating with respect to z=9 10 would lead to g9 10 2 10=9 28. As 0 <4 23 <9 28 <1 3and U∗ {8 10 }is strictly decreasing in this area by Corollary 4.2, the more precise signal would lead to a lower utility. The reason why the constellation in the previous example does not affect our finding in Theorem 5.1 is that all beliefs where the higher signal would lead to a lower utility lie in the area I(z) where the strategic decision maker does not update at all. 15
5.2 Naive Decision Makers Recall that at the beginning of any period the decision maker faces two sources of uncertainty: uncertainty about what message she will observe and uncertainty about the signal that will have generated that message. The definition of the anticipatory utility function in (3) made the implicit assumption that she evaluates these two sources independently of one another and that she is aware of this independence at the beginning of the period. Alternatively, she might be unaware that her preferences could require her to reassess what constitutes a worst case scenario. The following example illustrates the issue. Example 5.3. Recall Example 5.2 with Z=8 10,9 10. At the prior of λ=1 20 <1 3= λ∗8 10, the worst that could happen to the decision maker is observing message 1. Thus, ex ante, she will evaluate her situation under the presumption that z=8 10 as for this z the probability of observing message 1 is maximal. Upon observing message 1, however, the worst case for her is that the message was generated by signal z=9 10 as seen above. The strategic decision maker in (3) takes this into account and uses two different values of zat the two different stages. The decision maker we considered so far was aware that she might have a change of heart along the way, reflected by the use of Bm Z,σm(λ)(λ) as the set of possible beliefs for the inner aggregator in (3). A decision maker who is not aware of this will have an anticipatory utility function that satisfies ˆ U(λ) = (1 −p)u(λ) + pH hq0 ·(λ)Hhˆ U, B0 {·},σ0(λ)(λ)i+q1 ·(λ)Hhˆ U, B1 {·},σ1(λ)(λ)i, Zi. (9) We call a decision maker with such an anticipatory utility function naive, as opposed to sophisticated, which would refer to the decision maker in (3), following a similar distinction made by O’Donoghue and Rabin (1999) in the context of time-inconsistent preferences. Showing that Proposition 3.5 carries over to the naive decision maker, i.e., that there exists a unique naive anticipatory utility function for each σ∈ S, henceforth denoted by ˆ Uσ Z, can be done analogously to the previous analysis and is omitted. Example 5.4. Recall Example 3.4 with strategy ρdefined by ρ(λ) = (l, l) for all λ∈ [0,1]. The anticipatory utility function of a naive Gilboa-Schmeidler decision is the unique 16
solution of ˆ Uρ Z(λ) = (1 −p)u(λ) + pmin z∈Znq0 z(λ)ˆ Uρ Zg−1 z(λ)+q1 z(λ)ˆ Uρ Z(gz(λ))o for all λ∈[0,1]. Since Bm {z},σm(λ)(λ) is a singleton set for all z∈Zand all σ∈ S, the naive GilboaSchmeidler decision maker in (9) presumes only one expression take its minimum, while their sophisticated counterpart in (3) expects the minimum over all possible signals twice—she is being pessimistic “once more”. Hence, it is not surprising that the anticipatory utility of a naive decision maker is higher.24 Lemma 5.5. Let H:F ×P → Rbe the Gilboa-Schmeidler aggregator in (2). For any compact set Z⊆1 2,1and any σ∈ S, it holds that Uσ Z(λ)≤ˆ Uσ Z(λ)for all λ∈[0,1]. For the naive decision maker, the optimal utility at prior λis ˆ U∗ Z(λ) = sup σ∈S ˆ Uσ Z(λ).(10) We show that in terms of optimal strategies and optimal anticipatory utilities there is no difference between a naive and a sophisticated decision maker. Theorem 5.6. Let Z⊆1 2,1be compact and and let H:F × P → Rbe the GilboaSchmeidler aggregator in (2). Let z= min Z. Then ˆ U∗ Z=U∗ Z, and ˆ U∗ Z=ˆ Uσ Zif and only if σsatisfies (8) for z=z. While Example 5.3 demonstrates that naive and sophisticated Gilboa-Schmeidler decision makers might evaluate some situations differently, Theorem 5.6 shows that these differences have no effect on the agents’ behavior or her anticipatory utility function. In particular, ˆ U∗ Z=U∗ Z=U∗ {z}=ˆ U∗ {z}, so that no matter whether she is sophisticated or naive, a Gilboa-Schmeidler decision maker will always act and feel as if she knew that z were the only possible signal. 24We are not the first to highlight this conflict between being happy and overthinking uncertainty. It has been beautifully illustrated, for instance, in Woody Allen’s “Annie Hall”. But there is also empirical evidence supporting the other direction: that happy people appear less clever (cf. Barasch, Levine, and Schweitzer, 2016). 17
6 Benchmark Comparison We have derived the optimal behavior of a decision maker who can strategically ignore information. We shall call such a decision maker strategic to distinguish her from a Bayesian decision maker, who mechanically updates his belief after every signal. In this section, we shall contrast the two with respect to their anticipatory utility as well as their success probabilities, i.e., the probabilities with which they make the correct choice after the stream of messages stops. 6.1 Ignorance is a Bliss Recall that a Bayesian decision maker uses strategy ρwith ρ(λ) = (l, l) for all λ∈ [0,1]. He will learn zafter every message and update his belief accordingly. Both for the sophisticated and the naive decision maker there are, by Proposition 3.5, unique anticipatory utility functions Uρ Zand ˆ Uρ Z, respectively. Moreover, U∗ Z(λ)≥Uρ Z(λ) and ˆ U∗ Z(λ)≥ˆ Uρ Z(λ) for all λ∈[0,1] by definition. We show that these inequalities are strict for all λ∈(0,1). Theorem 6.1. Let Z⊆1 2,1be compact. Then U∗ Z(λ)> Uρ Z(λ)and ˆ U∗ Z(λ)>ˆ Uρ Z(λ) for all λ∈(0,1). For any non-degenerate belief, the anticipatory utility of a strategic decision maker is higher than that of a Bayesian decision maker. That is, strategic ignorance makes people happier, as long as the stream of signals continues. 6.2 Final Outcome We shall turn to the probabilities with which the strategic and the Bayesian decision maker choose correctly at the end of the message stream. Throughout the subsection, we assume that the agents’ prior belief λis correct and that they are facing a known signal z∈1 2,1. The ex-ante probability that a decision maker with initial belief λand strategy σmakes the correct choice is denoted by rσ z(λ).25 25At this point, we are a bit inaccurate as we do not formally define the concerning probability space. This space would have to contain the correct state of the world ω, the number of signals T, and for each T, the sequence of messages (mt)T t=1. Defining such a space is not very difficult; however, it requires some cumbersome notation and does not provide additional insight. 18
As the optimal strategy for the strategic decision maker is not unique, we will focus on σ∗∗ z, which is defined by σ∗∗ z(λ) = (l, i) for λ∈[0, λ∗(z)] , (l, l) for λ∈(λ∗(z),1−λ∗(z)) , (i, l) for λ∈[1 −λ∗(z),1] . (11) By Theorem 4.1, this strategy is optimal. Observe that if T≥1, i.e., if there is at least one message, then λ1∈I(z), so that for t≥2 the strategic decision maker will not learn upon a contradicting message. Thus, the agent exhibits an extreme primacy effect and her final choice is determined at latest after the first message. This allows us to easily derive the probability that her choice is correct. Proposition 6.2. For initial belief λ∈[0,1], the probability for a correct decision under σ∗∗ zis rσ∗∗ z z(λ) = u(λ)if λ∈I(z), pz + (1 −p)u(λ)otherwise. An immediate consequence of this proposition is that the ex-ante probability of a correct choice is increasing in z. While it does not change for “extreme” initial beliefs, it strictly increases for “moderate” beliefs. Corollary 6.3. Let 1≥z > z0≥1 2. Then rσ∗∗ z z(λ) = rσ∗∗ z0 z0(λ)for all λ∈I(z)and rσ∗∗ z z(λ)> rσ∗∗ z0 z0(λ)for all λ∈(λ∗(z),1−λ∗(z)). Finding the ex-ante probability that a Bayesian decision maker chooses correctly is more complex, as it does not solely depend on the first message. Its precise formula for each λ∈(0,1) is derived within the proof of the next theorem. Theorem 6.4. For any λ∈(0,1) and z∈1 2,1it holds that rρ z(λ)> rσ∗∗ z z(λ). Theorem 6.4 shows that strategic news selection lowers the decision maker’s long-run welfare as it decreases her probability to choose correctly in the end. This contrasts with Theorem 6.1 which states that as long as the stream of signals continues, the strategic 19
decision maker is happier than the Bayesian decision maker. Thus, while strategic ignorance increases instantaneous well-being, it is detrimental in the long run if the initial belief is correct. 7 Social Outcome Consider a population Nof decision makers with initial beliefs (λi)i∈Nwho face a stream of signals drawn from some compact set Z⊆1 2,1and who select their information according to σ∗∗ zas in (11). By Theorems 4.1 and 5.6 it holds that λi t∈I(z) for all t≥1 and all i∈N. Thus, already after the first signal, polarization is inevitable and extreme: There will only be “extremists”, who never look at messages contradicting their worldview. The “political center” (λ∗(z),1−λ∗(z)) is empty and will stay so forever. Polarization even increases over time. After the first message, all beliefs will be in [0, λ∗(z)] (“left-extremists”) or in [1 −λ∗(z),1] (“right-extremists”). Subsequently, the beliefs of all left-extremists will (for z6= 1) monotonically decrease and approach 0, while the beliefs of all right-wing extremists will monotonically increase towards 1. In particular, for large T, all beliefs will be close to 0 or 1. The set of decision makers who choose the correct outcome depends on the initial distribution of priors and Z. Corollaries 4.3 and 6.3 imply that this set increases as z increases. Hence, more people will make a correct choice if the least precise information source becomes more precise. 8 Extensions Two extensions of our model seem straightforward: using aggregators other than GilboaSchmeider, and using a non-binary state space. Neither of them is trivial and general results about the structure of optimal strategies remain unclear. 8.1 Aggregation of Second Order Beliefs If the decision maker knows the distribution of signals, she can exploit this additional information. Modifying Definition 3.2, one can then define an aggregator as a map H:F(D)×∆(D)→Rthat satisfies: 20
Consistency. For all x∈Dand the point measure δxon xit holds that H(U, δx) = U(x). Monotonicity. For all φ∈∆(D) and U, V :D→Rwith U|supp(φ)≥V|supp(φ)it holds that H(U, φ)≥H(V, φ). If U|supp(φ)≥V|supp(φ)+εfor some ε > 0 it holds that H(U, φ)> H(V, φ). For instance, an ambiguity neutral decision maker with smooth preferences in the sense of Klibanoff, Marinacci, and Mukerji (2005) with expected utility function Uand second order belief F∈∆(D) uses the aggregator H[U, F] = EF[U].(12) Let F∈∆1 2,1be a probability measure over the set of signals and let Fm λdenote the probability measure on Zconditional on the prior belief being λ∈(0,1) and the message being m. That is, for any measurable subset Z0⊆Z, Fm λ(Z0) = RZ0qm z(λ)F(dz) RZqm z(λ)F(dz). Moreover, define four maps ψλ,l :Z→[0,1], z 7→ g−1 z(λ), ψλ,i :Z→[0,1], z 7→ λ, ϕλ,l :Z→[0,1], z 7→ gz(λ), ϕλ,i :Z→[0,1], z 7→ λ and push-forward measures G0 F,a(λ):=F0 λ◦ψ−1 λ,a and G1 F,a(λ):=F1 λ◦ϕ−1 λ,a. Then G0 F,l(λ) is the probability measure over posterior beliefs given prior λand observed message 0 before the generated signal is learned. The measure G1 F,l(λ) is defined accordingly for observed message 1. The anticipatory utility function of a sophisticated decision maker with aggregator His then defined by Uσ F(λ) = (1 −p)u(λ) + pH q0 ·(λ)HUσ F, G0 F,σ0(λ)(λ)+q1 ·(λ)HUσ F, G1 F,σ1(λ)(λ), F and the anticipatory utility function of a naive decision maker is defined by ˆ Uσ F(λ) = (1 −p)u(λ) + pH hq0 ·(λ)Hhˆ Uσ F, G0 δ·,σ0(λ)(λ)i+q1 ·(λ)Hhˆ Uσ F, G1 δ·,σ1(λ)(λ)i, Fi. It is easy to see that Proposition 3.5 carries over and that the aggregator in (12) satisfies 21
its premise. Hence, both the sophisticated and the naive ambiguity neutral decision maker with smooth preferences have a unique anticipatory utility function which are fixed points of the functionals Tσ F,ˆ Tσ F:F([0,1]) → F ([0,1]), defined by Tσ FU(λ) = (1 −p)u(λ) + pH q0 ·(λ)HU, G0 F,σ0(λ)(λ)+q1 ·(λ)HU, G1 F,σ1(λ)(λ), F, ˆ Tσ FU(λ) = (1 −p)u(λ) + pH q0 ·(λ)HU, G0 δ·,σ0(λ)(λ)+q1 ·(λ)HU, G1 δ·,σ1(λ)(λ), F. We show that these functionals coincide. This contrasts with Example 5.3 which shows that such a statement fails to hold in the case of Gilboa-Schmeidler preferences. Lemma 8.1. Let Hbe defined as in (12). Then Tσ FU(λ) = ˆ Tσ FU(λ)for any F∈ ∆1 2,1,σ∈ S,U∈ F, and λ∈[0,1]. The previous lemma might not be entirely surprising as the sophisticated decision maker essentially takes an expected value of a conditional expected value, while the naive decision maker take the expected value only once. In contrast, it might not be true for a decision maker who is not ambiguity neutral. An immediate consequence is that the anticipatory utility functions coincide for the sophisticated and the naive decision maker. Corollary 8.2. Let Hbe defined as in (12). For all F∈∆1 2,1and all σ∈ S, it holds that Uσ F=ˆ Uσ F. Hence, in case of ambiguity neutral preferences, we do not have to distinguish between sophisticated and naive decision makers. The existence of optimal strategies can be derived as in Proposition 3.6, and they are identical for both types of decision makers, so that U∗ F(λ) = supσ∈S Uσ F(λ) = supσ∈S ˆ Uσ F(λ) = ˆ U∗ F(λ) for all λ. What remains unclear, however, is whether optimal strategies have a similar threshold structure as those of the Gilboa-Schmeidler decision maker. In the following example, a numerical analysis suggests that they might. Example 8.3. Recall Example 5.2 where Z=8 10,9 10and let Fbe the uniform distribution on Z. Suppose the decision maker has smooth preferences and is ambiguity neutral. The anticipatory utility function U∗ Fis depicted by the solid line in Figure 3. The function λ7→ EF1 λ[U∗ Z(g·(λ))] is depicted by the dotted line. The two functions have a unique intersection, denoted λ∗(F). The optimal strategies require that the agent ignores any 1-message for λ<λ∗(F) and updates after any 1-message for λ>λ∗(F). Numerically, we obtain that λ∗(F)≈0.2823 ∈1 4,1 3=λ∗9 10, λ∗8 10. 22
0z 0.5 0.5 1 ⇤(z) Figure 2: The graph of z7! ⇤(z) U⇤ z(gz()) >U ⇤ zg1 z()for 21 2,1.473 474 Theorem 4.1 proves that even without ambiguity there will be beliefs at which a decision475 maker will ignore contradicting messages. This region, denoted by I(z)=[0,⇤(z)] [476 [1 ⇤(z),1] increases as zdecreases.477 Corollary 4.3. Let z,z021 2,1⇤. Then I(z)✓I(z0)if and only if zz0.478 Proof. It is sufficient to show that ⇤(z)⇤(z0)ifandonlyifzz0. But this follows479 from the observation that ⇤(z)=z1+p(1z)z 2z1, which is strictly decreasing on 1 2,1⇤.⌅480 The relation between zand ⇤(z) is depicted in Figure 2. Note that ⇤(z)>1zfor all481 z21 2,1,482 5 Information Selection under Ambiguity483 In this section we shall return to the decision problem when zis not fixed, but stems from484 some compact set Z✓1 2,1⇤.28 We first derive the optimal strategy of a decision maker485 with Gilboa-Schmeider preferences. Afterwards we consider an alternative approach in486 28We exclude 1 22Zto avoid some tedious case distinctions. 19 0z 0.5 0.5 1 ⇤(z) Figure 2: The graph of z7! ⇤(z) U⇤ z(gz()) >U ⇤ zg1 z()for 21 2,1.473 474 Theorem 4.1 proves that even without ambiguity there will be beliefs at which a decision475 maker will ignore contradicting messages. This region, denoted by I(z)=[0,⇤(z)] [476 [1 ⇤(z),1] increases as zdecreases.477 Corollary 4.3. Let z,z021 2,1⇤. Then I(z)✓I(z0)if and only if zz0.478 Proof. It is sufficient to show that ⇤(z)⇤(z0)ifandonlyifzz0. But this follows479 from the observation that ⇤(z)=z1+p(1z)z 2z1, which is strictly decreasing on 1 2,1⇤.⌅480 The relation between zand ⇤(z) is depicted in Figure 2. Note that ⇤(z)>1zfor all481 z21 2,1,482 5 Information Selection under Ambiguity483 In this section we shall return to the decision problem when zis not fixed, but stems from484 some compact set Z✓1 2,1⇤.28 We first derive the optimal strategy of a decision maker485 with Gilboa-Schmeider preferences. Afterwards we consider an alternative approach in486 28We exclude 1 22Zto avoid some tedious case distinctions. 19 0z 0.5 0.5 1 ⇤(z) Figure 2: The graph of z7! ⇤(z) U⇤ z(gz()) >U ⇤ zg1 z()for 21 2,1.473 474 Theorem 4.1 proves that even without ambiguity there will be beliefs at which a decision475 maker will ignore contradicting messages. This region, denoted by I(z)=[0,⇤(z)] [476 [1 ⇤(z),1] increases as zdecreases.477 Corollary 4.3. Let z,z021 2,1⇤. Then I(z)✓I(z0)if and only if zz0.478 Proof. It is sufficient to show that ⇤(z)⇤(z0)ifandonlyifzz0. But this follows479 from the observation that ⇤(z)=z1+p(1z)z 2z1, which is strictly decreasing on 1 2,1⇤.⌅480 The relation between zand ⇤(z) is depicted in Figure 2. Note that ⇤(z)>1zfor all481 z21 2,1,482 5 Information Selection under Ambiguity483 In this section we shall return to the decision problem when zis not fixed, but stems from484 some compact set Z✓1 2,1⇤.28 We first derive the optimal strategy of a decision maker485 with Gilboa-Schmeider preferences. Afterwards we consider an alternative approach in486 28We exclude 1 22Zto avoid some tedious case distinctions. 19 Thus,652 ˆ U⇤ Z()= ˆ TZ,⇤ˆ U⇤ Z()>ˆ TZ,⌧ˆ U⌧ Z()= ˆ U⌧ Z()653 654 as required. ⌅655 For any non-degenerate belief, the anticipatory utility of a strategic decision maker is656 higher than that of an agent who always mechanically updates her belief. That is, strategic657 ignorance makes people happier, as long as the stream of signals continues.658 6.2 Final Outcome659 We have seen thus far that until the end of the signal stream the strategic decision maker660 is happier than the Bayesian decision maker. We shall now look at what happens when661 there are no new signals and a choice c2{0,1}has to be taken. The question we shall662 answer is: who will make the correct choice with higher probability?663 Throughout the subsection, we will assume that the agents are facing a known signal664 z21 2,1, i.e., we are in the situation of Section 4. Moreover, the initial belief is correct,665 that is, Nature draws the true state of the world according to . The decision maker666 then receives a stream of messages mt,t2{1,...,T}whose length Tis geometrically667 distributed with parameter p.668 Consider the strategy ⇤⇤ zdefined by669 ⇤⇤ z()=8 > > > < > > > : (l,i)for2[0,⇤(z)] , (l,l)for2(⇤(z),1⇤(z)) , (i, l)for2[1 ⇤(z),1] . (21)670 671 By Theorem 4.1, this strategy is optimal. Observe that if T1, i.e., if there is at least672 one message, then 12I(z), so that the strategic decision maker will never learn upon673 any contradicting message. Thus, the agent exhibits an extreme primacy e↵ect and will674 have made her choice at latest after the first message. This allows us to easily derive the675 probability that her choice is correct.676 Proposition 6.2. For initial belief 2[0,1], the probability for a correct decision under677 26 Thus,652 ˆ U⇤ Z()= ˆ TZ,⇤ˆ U⇤ Z()>ˆ TZ,⌧ˆ U⌧ Z()= ˆ U⌧ Z()653 654 as required. ⌅655 For any non-degenerate belief, the anticipatory utility of a strategic decision maker is656 higher than that of an agent who always mechanically updates her belief. That is, strategic657 ignorance makes people happier, as long as the stream of signals continues.658 6.2 Final Outcome659 We have seen thus far that until the end of the signal stream the strategic decision maker660 is happier than the Bayesian decision maker. We shall now look at what happens when661 there are no new signals and a choice c2{0,1}has to be taken. The question we shall662 answer is: who will make the correct choice with higher probability?663 Throughout the subsection, we will assume that the agents are facing a known signal664 z21 2,1, i.e., we are in the situation of Section 4. Moreover, the initial belief is correct,665 that is, Nature draws the true state of the world according to . The decision maker666 then receives a stream of messages mt,t2{1,...,T}whose length Tis geometrically667 distributed with parameter p.668 Consider the strategy ⇤⇤ zdefined by669 ⇤⇤ z()=8 > > > < > > > : (l,i)for2[0,⇤(z)] , (l,l)for2(⇤(z),1⇤(z)) , (i, l)for2[1 ⇤(z),1] . (21)670 671 By Theorem 4.1, this strategy is optimal. Observe that if T1, i.e., if there is at least672 one message, then 12I(z), so that the strategic decision maker will never learn upon673 any contradicting message. Thus, the agent exhibits an extreme primacy e↵ect and will674 have made her choice at latest after the first message. This allows us to easily derive the675 probability that her choice is correct.676 Proposition 6.2. For initial belief 2[0,1], the probability for a correct decision under677 26 Thus,652 ˆ U⇤ Z()= ˆ TZ,⇤ˆ U⇤ Z()>ˆ TZ,⌧ˆ U⌧ Z()= ˆ U⌧ Z()653 654 as required. ⌅655 For any non-degenerate belief, the anticipatory utility of a strategic decision maker is656 higher than that of an agent who always mechanically updates her belief. That is, strategic657 ignorance makes people happier, as long as the stream of signals continues.658 6.2 Final Outcome659 We have seen thus far that until the end of the signal stream the strategic decision maker660 is happier than the Bayesian decision maker. We shall now look at what happens when661 there are no new signals and a choice c2{0,1}has to be taken. The question we shall662 answer is: who will make the correct choice with higher probability?663 Throughout the subsection, we will assume that the agents are facing a known signal664 z21 2,1, i.e., we are in the situation of Section 4. Moreover, the initial belief is correct,665 that is, Nature draws the true state of the world according to . The decision maker666 then receives a stream of messages mt,t2{1,...,T}whose length Tis geometrically667 distributed with parameter p.668 Consider the strategy ⇤⇤ zdefined by669 ⇤⇤ z()=8 > > > < > > > : (l,i)for2[0,⇤(z)] , (l,l)for2(⇤(z),1⇤(z)) , (i, l)for2[1 ⇤(z),1] . (21)670 671 By Theorem 4.1, this strategy is optimal. Observe that if T1, i.e., if there is at least672 one message, then 12I(z), so that the strategic decision maker will never learn upon673 any contradicting message. Thus, the agent exhibits an extreme primacy e↵ect and will674 have made her choice at latest after the first message. This allows us to easily derive the675 probability that her choice is correct.676 Proposition 6.2. For initial belief 2[0,1], the probability for a correct decision under677 26 maker with smooth preferences have a unique anticipatory utility function that is achieved as a fixed point of the functionals T F,ˆ T F:F([0,1]) !F([0,1]), defined by T FU()=(1p)u()+pH ⇥q0 ·()H⇥U, G0 F,0()()⇤+q1 ·()H⇥U, G1 F,1()()⇤,F⇤, ˆ T FU()=(1p)u()+pH ⇥q0 ·()H⇥U, G0 ·,0()()⇤+q1 ·()H⇥U, G1 ·,1()()⇤,F⇤. We show that these functionals coincide. This contrasts with Example 5.3 which shows that such a statement fails to hold in the case of Gilboa-Schmeidler preferences. Lemma 8.1. Then for any U2F([0,1]) and 2[0,1],T FU()= ˆ T FU(). It directly follows from Lemma 8.1 that in case of ambiguity neutral smooth preferences, the anticipatory utility functions coincide for the sophisticated and the naive decision maker. Corollary 8.2. For each 2S,U F=ˆ U F. Hence, in case of ambiguity neutral preferences, we do not have to distinguish between sophisticated and naive decision makers. Thus, the optimal anticipatory utilities for the naive and the sophisticated decision maker coincide, i.e., U⇤ F()=sup 2SU F()= sup2Sˆ U F()= ˆ U⇤ F()forall. Analogously to Proposition 3.5, one can show that an optimal strategy exists. In general, it is not clear whether optimal strategies have a threshold structure. In the following example, a numerical analysis suggests that they do. Example 8.3. Recall Example 5.2 where Z=8 10,9 10 and let Fbe the uniform distribution on Z. Suppose the decision maker has smooth preferences and is ambiguity neutral. The anticipatory utility function U⇤ Fis depicted by the solid line in Figure 3. The function EF[U⇤ Z(gz())] is depicted by the dotted line. The two functions have a unique intersection, denoted ⇤(F). The optimal strategies require that the agent ignores any 1-message for <⇤(F) and updates after any 1-message for >⇤(F). Numerically, we obtain that ⇤(F)⇡0.2803 21 4,1 3=⇤9 10,⇤8 10.⇤ 8.2 More than Two States The model can be generalized to more than two symmetric states, a message space that coincides with the state space, and a set of symmetric signals that generate with probability z1 na correct message and with probability 1z n1each other message. While the 22 Figure 3: Anticipatory utility function for ambiguity neutral smooth preferences. 8.2 More than Two States The model can be generalized to more than two symmetric states, a message space that coincides with the state space, and a set of symmetric signals that generate with probability z≥1 na correct message and with probability 1−z n−1each other message. While the existence of optimal strategies can be proven as before, it remains unclear whether they will have the same threshold structure, as the proof of Theorem 4.1, in particular part 2b, does not generalize to more than two states. 9 Conclusion This paper investigates the behavior of a decision maker who receives a stream of messages of known or unknown quality about the true state of the world. At each time period, she faces the decision whether to learn how the message has been generated and update her belief accordingly, or to discard the message and stick to her previous conviction. This “free disposal of information” is inspired by the way users selectively read news articles posted on social media platforms. If there is no ambiguity and under the assumption that the decision maker is Markovian, she will discard contradictory messages if her belief is sufficiently strong. This remains true under ambiguity if she has Gilboa-Schmeidler preferences, where the respective belief threshold depends solely on the worst possible signal. When comparing her behavior to a decision maker who updates after each message, 23
Let U∈ C∗ z. We show that TU satisfies (18), so let 1 −λ∗≤λ < λ0≤1. Then, by (18), it holds that U(λ0)> U(λ) and, by the strict monotonicity of gz, we have U(gz(λ0)) > U (gz(λ)) as well as U(gz(λ0)) > U(λ0). Hence, we can apply Lemma A.1 with α=q1 z(λ0), β=q1 z(λ), x0=U(gz(λ0)), y0=U(λ0), x=U(gz(λ)), and y=U(λ) and get q1 z(λ0)U(gz(λ0)) + q0 z(λ0)U(λ0)> q1 z(λ)U(gz(λ)) + q0 z(λ)U(λ) where we also used 1 −q0 z(·) = q1 z(·). Together with the definition of σwe have (TU)(λ0)−(TU)(λ) = (1 −p)[u(λ0)−u(λ)] +pq1 z(λ0)U(gz(λ0)) + q0 z(λ0)U(λ0)−q1 z(λ)U(gz(λ)) −q0 z(λ)U(λ) >(1 −p)[u(λ0)−u(λ)] = (1 −p)(λ0−λ) as claimed. We show that Tpreserves (19). For all λ∈[0,1] define (˜ TU)(λ):= q0 z(λ)U(g−1 z(λ)) + q1 z(λ)U(λ), λ ∈[0, λ∗], q0(λ)U(g−1 z(λ)) + q1 z(λ)U(gz(λ)) , λ ∈(λ∗,1−λ∗), q0 z(λ)U(λ) + q1 z(λ)U(gz(λ)) , λ ∈[1 −λ∗,1] , and observe that (TU)(λ) = (1 −p)u(λ) + p(˜ TU)(λ). Thus, (TU) (gz(λ)) −(TU)(λ) = (1 −p) [u(gz(λ)) −u(λ)] + ph(˜ TU) (gz(λ)) −(˜ TU)(λ)i for all λ. Hence, in order to prove (19), it is sufficient to show that ( ˜ TU) (gz(λ)) ≥(˜ TU)(λ) for all λ∈[λ∗,1). 1. First, let λ∈[1 −λ∗,1). Then, since z > 1 2, it holds that gz(λ)∈(1 −λ∗,1], so that (˜ TU) (gz(λ)) = q1 z(g(λ)) U(gz(gz(λ))) + q0 z(gz(λ)) U(gz(λ)) ≥q1 z(λ)U(gz(λ)) + q0 z(λ)U(λ) = ( ˜ TU) (λ) 30
where the inequality follows from (15), using that U(gz(gz(λ))) ≥U(gz(λ)) ≥U(λ) by the monotonicity of Uon [1 −λ∗,1], and q1 z(gz(λ)) ≥q1 z(λ) by the monotonicity of q1 z. 2. Second, let λ∈(λ∗,1−λ∗). Then, g−1 z(λ)∈[0, λ∗) and gz(λ)∈(1 −λ∗,1]. As Uis strictly monotone in these areas, U(gz(λ)) is increasing, and U(g−1 z(λ)) is decreasing in λ. By Lemma 3.1, it holds that gz1 2= 1 −g−1 z1 2, so that by the symmetry of U, Ugz1 2=U1−g−1 z1 2=Ug−1 z1 2. Consequently, U(gz(λ)) > U g−1 z(λ)for λ∈1 2,1−λ∗and U(gz(λ)) < U g−1 z(λ)for λ∈λ∗,1 2.(20) (a) Suppose first that λ∈1 2,1−λ∗. Then U(gz(gz(λ))) ≥U(gz(λ)) ≥U(g−1 z(λ)) and therefore (˜ TU) (gz(λ)) = q1 z(gz(λ)) U(gz(gz(λ))) + q0 z(gz(λ)) U(gz(λ)) ≥U(gz(λ)) ≥q1 z(λ)U(gz(λ)) + q0 z(λ)Ug−1 z(λ) = ( ˜ TU) (λ). (b) Finally, let λ∈λ∗,1 2. Since U(gz(gz(λ∗))) = U(gz(1 −λ∗)) = U1−g−1 z(λ∗)=Ug−1 z(λ∗), the monotonicity of Uon [0, λ∗) and (1 −λ∗,1] together with (20) imply that U(gz(gz(λ))) ≥Ug−1 z(λ)≥U(gz(λ)) . Thus, using that q1(gz(λ)) >0 and q1 z(λ) = q0 z(1 −λ)> q0 z(gz(λ)) for λ > λ∗, we obtain (˜ TU) (gz(λ)) = q1 z(gz(λ)) U(gz(gz(λ))) + q0 z(gz(λ)) U(gz(λ)) =q1 z(gz(λ)) U(gz(gz(λ))) + q1 z(λ) + q0 z(gz(λ)) −q1 z(λ)U(gz(λ)) 31
≥q1 z(gz(λ)) Ug−1 z(λ)+q1 z(λ)U(gz(λ)) +q0 z(gz(λ)) −q1 z(λ)Ug−1 z(λ) =q1 z(λ)U(gz(λ)) + 1−q1 z(λ)Ug−1 z(λ) = ( ˜ TU) (λ) as required. Let U∈ C∗ z. By (19) it holds that U(gz(λ)) > U(λ) for all λ∈(λ∗,1).(21) Let λ∈(1 −λ∗,1). Then g−1 z(λ)∈(λ∗,1), so that we find with (21) U(λ) = Ugzg−1 z(λ)> U g−1 z(λ)for all λ∈(1 −λ∗,1).(22) Moreover, using the symmetry of Uin (16), we obtain by (19), or similar arguments as above Ug−1 z(λ)> U(λ) for all λ∈(0,1−λ∗),(23) U(λ)> U (gz(λ)) for all λ∈(0, λ∗).(24) Consider now the functional operator Sz:F([0,1]) → F ([0,1]) that is defined by (SzU) (λ) = (1 −p)u(λ) + pq0 z(λ) max U(λ), U g−1 z(λ)+q1 z(λ) max (U(λ), U (gz(λ))) for all U∈ F ([0,1]) and all λ∈[0,1]. One can easily check that Szis a contraction on F([0,1]), so Szhas a unique fixed point, denoted by ˜ U. Let ˜σbe a fixed but arbitrary strategy that satisfies (8). By Equations (21)–(24) we have for all U∈ C∗ zthat SzU= T˜σ {z}U∈ C∗ z, which implies ˜ U∈ C∗ z. Moreover U∗ {z}=Tσ∗ {z}U∗ {z}=SzU∗ {z}for any σ∗that satisfies (6), so that U∗ {z}=˜ U∈ C∗ z. This implies that a strategy σsatisfies (8) if and only if it satisfies (6), that is, if and only if it is optimal. Proof of Corollary 4.3.It is sufficient to show that λ∗(z)≤λ∗(z0) if and only if z≥z0. But this follows from the observation that λ∗(z) = z−1+√(1−z)z 2z−1, which is strictly decreasing 32
on 1 2,1. Proof of Theorem 5.1.If Z={1}, there is nothing to show. Hence, let Z6={1}, so that z < 1. Let σ∗be some optimal strategy given Z, and let σbe some optimal strategy given {z}. It is sufficient to show the following four (in-)equalities, as they imply the claim: Uσ {z}(λ) = U∗ {z}(λ)≥Uσ∗ {z}(λ)≥U∗ Z(λ)≥Uσ Z(λ) = Uσ {z}(λ).(25) The first equality follows from Theorem 4.1 and the first inequality by the definition of U∗ {z}. For the second inequality note that H[U, Z]≤H[U, {z}] for every U∈ F ([0,1]) by (2). Hence, Tσ∗ ZU(λ)≤Tσ∗ {z}U(λ) for all U∈ F ([0,1]) by the monotonicity of H, so that Tσ∗ ZnU(λ)≤Tσ∗ {z}nU(λ) for all n∈N. Thus, U∗ Z(λ) = limnTσ∗ ZnU(λ)≤ limnTσ∗ {z}nU(λ) = Uσ∗ {z}(λ). The third inequality is satisfied by the definition of U∗ Zin (5). For the last equality it is sufficient to show that Uσ {z}is a fixed point of Tσ Z. So, without loss of generality let λ≤1 2, and recall that Uσ {z}∈ C∗ z. By Theorem 4.1,σis optimal for Uσ {z}in the sense of Equation (6). Thus, HUσ {z}, B0 Z,σ0(λ)(λ)=Uσ {z}g−1 z(λ)and HUσ {z}, B1 Z,σ1(λ)(λ)= max Uσ {z}(gz(λ)) , Uσ {z}(λ) by the monotonicity properties of Uσ {z}in Corollary 4.2. Thus, again by Corollary 4.2, HUσ {z}, B0 Z,σ0(λ)=Uσ {z}g−1 z(λ)≥max Uσ {z}(gz(λ)) , Uσ {z}(λ)=HUσ {z}, B1 Z,σ1(λ). As q0 ·(λ) is weakly increasing in z(recall that λ≤1 2) this means that Hq0 ·(λ)HUσ {z}, B0 Z,σ0(λ)(λ)+q1 ·(λ)HUσ {z}, B1 Z,σ1(λ)(λ), Z =q0 z(λ)HUσ {z}, B0 Z,σ0(λ)(λ)+q1 z(λ)HUσ {z}, B1 Z,σ1(λ)(λ) =Hq0 ·(λ)HUσ {z}, B0 Z,σ0(λ)(λ)+q1 ·(λ)HUσ {z}, B1 Z,σ1(λ)(λ),{z}. Hence, it holds that Tσ ZUσ {z}(λ) = Tσ {z}Uσ {z}(λ) = Uσ {z}(λ). 33
Thus, Uσ {z}is a fixed point of Tσ Zas claimed. Proof of Lemma 5.5.Let Tσ Zbe as defined in (13). Analogously, let ˆ Tσ Z:F([0,1]) → F([0,1]) be defined by ˆ Tσ ZU(λ) = (1 −p)u(λ) + pH q0 ·(λ)HU, B0 {·},σ0(λ)(λ)+q1 ·(λ)HU, B1 {·},σ1(λ)(λ), Z for all λ∈[0,1]. For all z∈Zand any function U∈ F ([0,1]), HU, B0 Z,σ0(λ)(λ)≤HU, B0 {z},σ0(λ)(λ)and HU, B1 Z,σ1(λ)(λ)≤HU, B1 {z},σ1(λ)(λ). By the monotonicity of H,Tσ ZU(λ)≤ˆ Tσ ZU(λ) for any function Uand all λ∈[0,1]. Thus, as in the proof of Theorem 5.1,Uσ Z(λ) = limnTσ ZnU(λ)≤limnˆ Tσ ZnU(λ) = ˆ Uσ Z(λ). Proof of Theorem 5.6.If Z={1}, there is nothing left to show, so let Z6={1}, so that z < 1. Suppose first that σsatisfies (8). In order to show that U∗ Z=ˆ U∗ Z=ˆ Uσ Zit is sufficient to show that ˆ U∗ Z(λ)≤ˆ Uσ {z}(λ) = Uσ {z}(λ) = Uσ Z(λ)≤ˆ Uσ Z(λ)≤ˆ U∗ Z(λ) (26) for all λ∈[0,1]. As z∈Z,His the Gilboa-Schmeidler aggregator, and σis optimal given {z}, one can use the same arguments as in the proofs of Theorem 5.1 and Lemma 5.5 to show that ˆ Uσ0 Z(λ)≤ˆ Uσ0 {z}(λ)≤ˆ Uσ {z}(λ) for all σ0∈ S. Taking the supremum over all σ0∈ S yields the first inequality. For the singleton set {z}, there is no difference between sophisticated and naive decision makers, as (3) and (9) coincide in this case. Hence, we arrive at the first equality. The second equality holds by Theorem 5.1, and the second inequality by Lemma 5.5. The last inequality directly follows from the definition of ˆ U∗ Zin (10). It is left to show that ˆ U∗ Z>ˆ Uσ0 Zfor any σ0∈ S that does not satisfy (8). Assume that there is σ0∈ S with ˆ U∗ Z=ˆ Uσ0 Zthat does not satisfy (8). Then ˆ Uσ0 Z(λ) = ˆ U∗ Z(λ) = U∗ {z}(λ)> Uσ0 {z}(λ) = ˆ Uσ0 {z}(λ)≥ˆ Uσ0 Z(λ), where the first equality follows from the assumption, the second one from (26), the strict inequality from Theorem 4.1, the third equality holds as for the singleton {z}Equations (3) and (9) coincide, and the weak inequality follows from the same arguments as above. But the overall inequality is impossible. 34
Proof of Theorem 6.1.As Uρ Z(λ)≤ˆ Uρ Z(λ) by Lemma 5.5 and U∗ Z(λ) = ˆ U∗ Z(λ) by Theorem 5.6, it suffices to show that ˆ Uρ Z(λ)<ˆ U∗ Z(λ). So, let σ∗be an optimal strategy given Z. We first prove the claim for λ∈(0, λ∗(z)) ∪(1 −λ∗(z),1). Without loss of generality let λ∈(0, λ∗(z)). Then Hhq0 ·(λ)Hhˆ U∗ Z, B0 {·},ρ0(λ)(λ)i+q1 ·(λ)Hhˆ U∗ Z, B1 {·},ρ1(λ)(λ)i, Zi = min z∈Zq0 z(λ)ˆ U∗ Zg−1 z(λ)+q1 z(λ)ˆ U∗ Z(gz(λ)) ≤q0 z(λ)ˆ U∗ Zg−1 z(λ)+q1 z(λ)ˆ U∗ Z(gz(λ)) < q0 z(λ)ˆ U∗ Zg−1 z(λ)+q1 z(λ)ˆ U∗ Z(λ) =Hhq0 ·(λ)Hhˆ U∗ Z, B0 {·},σ∗ 0(λ)(λ)i+q1 ·(λ)Hhˆ U∗ Z, B1 {·},σ∗ 1(λ)(λ)i, Zi. Thus, ˆ Uρ Z(λ) = ˆ Tρ Zˆ Uρ Z(λ)≤ˆ Tρ Zˆ U∗ Z(λ)<ˆ Tσ∗ Zˆ U∗ Z(λ) = ˆ U∗ Z(λ), where we used the calculation above in the first and the monotonicity of ˆ Tρ Zin the second inequality. Suppose next that λ∈[λ∗(z),1−λ∗(z)]. Then g−1 z(λ)∈(0, λ∗(z)] and gz(λ)∈ [1 −λ∗(z),1) and at least one of them lies in the respective open interval. By the definition of ˆ U∗ Zit holds that ˆ U∗ Zg−1 z(λ)≥ˆ Uρ Zg−1 z(λ)and ˆ U∗ Z(gz(λ)) ≥ˆ Uρ Z(gz(λ)), and, by the first part of the proof, at least one of these inequalities is strict. Hence, Hhq0 ·(λ)Hhˆ Uρ Z, B0 {·},ρ0(λ)(λ)i+q1 ·(λ)Hhˆ Uρ Z, B1 {·},ρ1(λ)(λ)i, Zi = min z∈Zq0 z(λ)ˆ Uρ Zg−1 z(λ)+q1 z(λ)ˆ Uρ Z(gz(λ)) ≤q0 z(λ)ˆ Uρ Zg−1 z(λ)+q1 z(λ)ˆ Uρ Z(gz(λ)) < q0 z(λ)ˆ U∗ Zg−1 z(λ)+q1 z(λ)ˆ U∗ Z(gz(λ)) =Hhq0 ·(λ)Hhˆ U∗ Z, B0 {·},σ∗ 0(λ)(λ)i+q1 ·(λ)Hhˆ U∗ Z, B1 {·},σ∗ 1(λ)(λ)i, Zi. Thus, ˆ Uρ Z(λ) = ˆ TZ,ρ ˆ Uρ Z(λ)<ˆ Tσ∗ Zˆ U∗ Z(λ) = ˆ U∗ Z(λ) 35
as required. Proof of Proposition 6.2.The first part is clear as a strategic decision maker with λ∈I(z) will not update her belief at all and choose the option that is optimal given her (correct) prior. If λ /∈I(z), with probability 1−pshe will obtain no message at all and follow her prior, and with probability pshe will obtain a message and update her belief. Since necessarily λ1∈I(z), she will make a correct choice if and only if the message was correct, which happens with probability zand is independent of p. Proof of Corollary 6.3.For λ∈I(z) and λ∈(λ∗(z0),1−λ∗(z0)) the claim follows immediately from Proposition 6.2. Without loss of generality let λ∈(λ∗(z), λ∗(z0)). Since λ > λ∗(z)≥1−z, we have rσ∗∗ z z(λ) = pz + (1 −p)(1 −λ)>1−λ=rσ∗∗ z0 z0(λ) as required. Proof of Theorem 6.4.We assume without loss of generality that λ≤1 2. For any `∈N denote by g` zthe `-fold application of gz, and by g−` zthe `-fold application of g−1 z. For any history hT(ρ) = (mt, z, λt)T t=0 let M1=|{t:mt= 1}| denote the number of generated 1-messages, and M0=T−M1the number of generated 0-messages. Since gzand g−1 z commute, we have that λT=gM1−M0 z(λ). Let G`=g−` z1 2, g−(`−1) z1 2iand observe that a Bayesian decision maker with prior λ∈G`who faces history hT(ρ) will have final belief λT>1 2if and only if M1−M0≥l, and he will have final belief λT=1 2if and only if λ=g−(`−1) z1 2and M1−M0=l−1. For any T∈N0and κ∈Zdenote by Z[T, κ] the probability that at least κcorrect messages have been observed, conditional on Tmessages having been generated altogether, that is, W[T, κ] = 0 if κ > T, PT k=κT kzk(1 −z)T−kif 0 ≤κ≤T, 1 if κ < 0. 36
The probability that the number of correct messages exceeds the number of incorrect messages by at least `, conditional on Tmessages being generated, is WT, T+`+1 2. Similarly, the probability that the number of incorrect messages does not exceed the number of correct messages by (strictly) more than `−1, conditional on Tmessages being generated, is WT, T−` 2+ 1. Hence, R` z(λ) = λ∞ X T=0 (1 −p)pTWT, T+`+1 2+ (1 −λ)∞ X T=0 (1 −p)pTWT, T−` 2+ 1 denotes the ex-ante probability that either the state is 1 and the number of 1-messages exceeds the number of 0-messages by at least `or the state is 0 and the number of 1messages does not exceed the number of 0-messages by more than `−1. If λ∈int G`, this is exactly the ex-ante probability that the Bayesian decision maker will make a correct choice. We therefore define `(λ) to be the unique integer with λ∈G`and define Rz(λ) = R`(λ) z(λ). If λ∈int G`for some `∈N, then rρ z(λ) = Rz(λ). The map Rz:0,1 2→[0,1] is continuous in the interior of G`for all `∈N. We prove that R` zg−` z1 2=R`+1 zg−` z1 2for all `∈Nas this implies that Rzis continuous everywhere. To this end, we first show that g−` z1 2z`=1−g−` z1 2(1 −z)`for all `∈N0.(27) Indeed, this is surely true for `= 0. Suppose (27) is true for some `∈N0. Then g−(`+1) z1 2z`+1 =g−` z1 2z`(1 −z)z g−` z1 2(1 −z) + 1−g−` z1 2z =1−g−` z1 2(1 −z)`(1 −z)z g−` z1 2(1 −z) + 1−g−` z1 2z = 1−g−` z1 2(1 −z) g−` z1 2(1 −z) + 1−g−` z1 2z!(1 −z)`+1 =1−g−(`+1) z1 2(1 −z)`+1, 37
as required. Next observe that WT, T+`+2 2−WT, T+`+1 2= 0 if T−`is odd, −T T−` 2zT−` 2(1 −z)T+` 2if T−`is even, and WT, T−`−1 2+ 1−WT, T−` 2+ 1= 0 if T−`is odd, T T+` 2zT+` 2(1 −z)T−` 2if T−`is even. Since T T+` 2=T T−` 2, this implies R`+1 zg−` z1 2−R` zg−` z1 2 =∞ X T=0 (1 −p)pT 1 {T+`even}T T+` 2zT−` 2(1 −z)T−` 2·−g−` z1 2z`+ (1 −g−` z1 2)(1 −z)` = 0 by (27). Hence, Rzis continuous. Recall that rρ z(λ) = Rz(λ) for all λ∈0,1 2with λ6=g−` z1 2for all `∈N0. We next show that this is also true if λ=g−` z1 2for some `∈N0. To this end note that if the state is 1 and the decision maker observes exactly `more correct than incorrect messages, he will choose either action with equal probability, and the same is true if the state is 0 and he observes exactly `more incorrect than correct messages. Thus, the ex-ante probability of a correct choice is g−` z1 2"∞ X T=0 (1 −p)pTWT, T+` 2+ 1+1 2 1 {T+`even}T T+` 2zT+` 2(1 −z)T−` 2# +1−g−` z1 2"∞ X T=0 (1 −p)pTWT, T−` 2+ 1+1 2 1 {T−`even}T T−` 2zT−` 2(1 −z)T+` 2# =g−` z1 2"∞ X T=0 (1 −p)pTWT, T+`+1 2−1 2 1 {T+`even}T T+` 2zT+` 2(1 −z)T−` 2# +1−g−` z1 2"∞ X T=0 (1 −p)pTWT, T−` 2+ 1+1 2 1 {T−`even}T T−` 2zT−` 2(1 −z)T+` 2# =R` zg−` z1 2+1 2 ∞ X T=0 (1 −p)pT 1 {T+`even}T T+` 2zT−` 2(1 −z)T−` 2−g−` z1 2z`+1−g−` z1 2(1 −z)` 38
=R` zg−` z1 2 by (27). Hence, rρ z(λ) = Rz(λ) for all λ∈0,1 2. We next show that rρ z(λ)> rσ∗∗ z z(λ) for λ > λ∗(z). Since λ>λ∗(z)>1−z=g−1 z1 2 it holds that `(λ) = 1. Define for all T∈N0 fT z(λ) = λW T, T+2 2+ (1 −λ)WT, T+1 2 (28) and note that rρ z(λ) = Rz(λ) = R1 z(λ) = P∞ T=0(1 −p)pTfT z(λ). It is straightforward that f0 z(λ) = 1 −λ,f1 z(λ) = z, and f2 z(λ) = 2(1 −λ)(1 −z)z+z2≥z. We show that fT+1 z(λ)≥fT z(λ)> z for all T≥3. To this end, observe first that fT Z(λ) = WT, T+1 2if Tis odd, WT, T 2−λT T 2zT 2(1 −z)T 2if Tis even. Next, one finds that for all 1 ≤κ≤T (1 −z)W[T, κ] + zW [T, κ −1] = T X k=κT kzk(1 −z)T−k+1 + T X k=κ−1T kzk+1(1 −z)T−k = T X k=κT kzk(1 −z)T−k+1 + T+1 X k=κT k−1zk(1 −z)T−k+1 = T X k=κT+ 1 kzk(1 −z)T−k+1 +T TzT+1 =W[T+ 1, κ]. Suppose first that Tis even. Then fT+1 z(λ) = WT+ 1,T+2 2 = (1 −z)WT, T+2 2+zW T, T 2 = (1 −z)WT, T 2−T T 2zT 2(1 −z)T 2+zW T, T 2 =WT, T 2−(1 −z)T T 2zT 2(1 −z)T 2 39