The Paternalistic Bias of Expert Advice

Transcription

1 The Paternalistic Bias of Expert Advice John Lightle June, 9 Abstract When communication from an expert to a decision maker (DM) is inherently imprecise, a Pareto improvement occurs when the expert overstates his position in order to send a strong signal in favor of a certain course of action. However, when a DM consults multiple experts, each with a di erent area of expertise, this overstatement may interfere with the ability of the DM to aggregate the information received from each expert, as overstatement reduces the informativeness of communication. Using a theoretical model of expert advice, I derive the optimal trade-o between these e ects. Because the expert in ates his message toward the policy he believes the DM would be better o choosing, I refer to this as a paternalistic bias. I determine that the bias exists in equilibrium only when an expert s information is su ciently extreme and show that the decision quality is highest when advice is solicited sequentially and publicly, with the DM ignoring all advice except the nal expert s. JEL Classi cation: D83, D7 Keywords: experts, information transmission, communication, noise, biased communication Contact Information: Florida State University, Department of Economics, 3 Collegiate Loop, Room 55, Tallahassee, FL 336. ( jlightle@fsu.edu) The author wishes to thank seminar participants at the Ohio State University Theory Workshop and the FSU Workshop on Experimental Game Theory, John Kagel, James Peck, Dan Levin, PJ Healy, Navin Kartik, and particularly Massimo Morelli, for their expert advice. All errors are my own.

2 Introduction Employing the advice of experts is an essential part of being a successful decision maker in the public arena. When facing a particularly complicated decision, a decision maker often faces constraints which make acquiring and/or processing all the relevant information ine cient or impossible. For example, consider a CEO who must decide whether or not to buyout a rival company. Analysts will attempt to explain the expected costs and bene ts of acquisition to the CEO, but the ultimate decision is his alone. Conventional wisdom suggests that, for the good of the company, analysts should be forthright with their knowledge, giving as accurate a description of the state of the world as possible. However, this paper challenges that intuition when you include an often dismissed, yet quite realistic, caveat: communication between analyst and CEO is inherently imprecise. In other words, the analyst s intended description of the state of the world is not equal to the CEO s interpretation of this message. In this case, using the model presented in this paper, I identify a novel "paternalistic bias" on the part of an advisor i.e. an endogenous bias that occurs because an advisor believes that by sending an insincere message about the state of the world, he can help the decision maker make a better choice than had he revealed his information sincerely. Expert advisors are expected to reveal their information truthfully and accurately, and may su er consequences when the information they provide appears to be inaccurate or insincere. Facing such consequences, why would an advisor choose to send an insincere message? It could be that the advisor s incentives are not aligned with the decision maker s, in which case the advisor may attempt to in uence the decision maker s action by strategically biasing his message. In a seminal analysis of strategic information transmission, Crawford and Sobel (98) demonstrate that an expert cannot credibly send precise information to a decision maker if their preferences over actions di er. However, vague communication is possible in equilibrium, assuming that the di erence between the expert s and the decision maker s desired action is not too large. In such a case, the expert chooses one of a set of messages which, given their exogenous bias, is credible to the receiver. This behavior, while not fully revealing, is not necessarily insincere. However, there is a body of empirical (Malmendier and Shanthikumar, 5) and experimental (Dickhaut, McCabe, and Mukherjee, 995; Cai and Wang, 6) evidence that biased experts typically send messages which are biased toward their preferred outcome, and with non-trivial probability those messages are believed. This behavior can be explained by the bounded rationality of receivers (Ottaviani and Squintani, 6), however, in this paper I describe a bias which emerges despite the assumption of common knowledge of rationality for all agents. In the afore mentioned studies, experts and decision makers have misaligned preferences over outcomes. Why would an expert send a biased message if he and the decision maker would prefer the same outcome ex post? Ottaviani and Sorensen (6) show that in this case an advisor may be insincere if he is also motivated with career concerns. Similarly, in a repeated game setting, an insincere message can signal to a decision maker that an advisor is of high quality, which can bene t both parties in future interactions (Grosskopf and Sarin, 9). Even in the one-shot setting where the utility functions of an advisor and a In what follows, I will only consider the case where an advisor has a message space which is as rich the space from which her information is drawn. If there is no one-to-one correspondence between information and messages, then it is not in general possible to fully communicate one s information. There are interesting examples of insincere behavior when messages are restricted, e.g. voting. Austen-Smith and Banks (996) demonstrate that even when all voters have common preferences over outcomes, sincere voting is not rational, due to the inferences made about others information. (See also, Fedderson and Pesendorfer 996, 998.) The addition of a round of deliberation before a vote takes place can eliminate non-sincere voting behavior (Coughlan, ; Gerardi and Yariv, 7), which reduces the game to one in which the message space is no longer insu cient. When perfectly informed experts with an exogenous bias advise a decision maker (Krishna and Morgan, ; Park, 5), full honesty is not sustainable in equilibrium. The equilibria of cheap talk games do not generally involve complete sincere revelation of information (Seidmann, 99; Austen-Smith, 99; Farrell and Rabin, 996).

3 decision maker are exactly the same, an advisor may withhold information if his prior belief about the state of the world di ers from the decision maker (Che and Kartik, 9). In this paper, I depart from the standard cheap talk, sender-receiver literature by assuming identical preferences and priors for all actors, and unlike strategic voting, I allow the senders message space to be identical to their information space. Rather, the underlying source of this bias is the fact that communication between expert and decision maker is inherently noisy or vague 3. Examples of this type of information transmission abound, e.g, a doctor who cannot fully explain the risks and bene ts of a complicated medical procedure to a patient, or military advisors who cannot fully explain the consequences of a military action to a commander-in-chief. In the realm of public policy making, a decision maker is responsible for incorporating expert information into his decision, even if he is lacking in that particular area of expertise. For any number of reasons (vagueness, misunderstanding, or mistranslation) the message sent may not be the same as the message recieved. functioning society. Despite its imperfection, this type of communication is ubiquitous and vital to a Noisy communication has long been a topic of economic interest. Myerson (99) shows that imperfect communication can actually improve the welfare of a sender and receiver in a cheap talk game. Furthermore, there are a host of examples in which randomness can be utilized to obtain a better outcome than the one predicted by Crawford and Sobel (98) 4, but these examples, like all standard cheap talk games, do not have aligned incentives. The topic of noisy communication with aligned incentives was broached in a section of the investigation of noisy talk by Blume, Board, and Kawamura (7). In their model, an expert chooses a message to send to a decision maker, and there is a xed probability of error. If an error occurs, a random message from a known distribution is received by the decision maker. If no error occurs, the decision maker receives the exact message sent by the expert. In equilibrium, maximum e ciency can be achieved because the expert can almost always elicit the ex post optimal action when there is no error (by sending a message from a measure-zero subset of the message space), while the ex ante optimal action is chosen whenever there is an error. inherently imprecise. This work di ers from my model in an important way: in my model all communication is Rather than a xed probability of error, I am concerned with the situations in which the decision maker will never be as informed as an expert, but nevertheless a decision must be made. I model the actions of multiple expert advisors and a decision maker with common preferences and common priors over the state of the world. I claim that an advisor has a paternalistic bias when he biases his message toward whichever outcome he believes is most likely to be optimal ex post. This bias has the potential to reduce the probability that the decision maker makes a suboptimal decision due to the e ect of unfortunate noise. The word "paternalism" traditionally refers to the in uence of an authority on the actions of individuals, typically against their will, in order to make those individuals better o. However, when attention is restricted to the case where all agents have identical preferences, no realistic action taken by the authority in this case the expert advisors would ever contradict the will of the decision maker. Therefore, even though the traditional usage is not perfectly suited to the model presented here, the essence of paternalism is still captured by the advisors manipulation of the transmitted information for the decision maker s own good. In a contemporaneous work with similar goals as the one presented here, Blume and Board (9) show that noisy communication can lead interior-type advisors to distort their messages, however, this distortion 3 The model presented in this paper can equally be interpreted as incorporating information processing constraints of the decision maker, in a reduced form way. Rather than inherently noisy communication, the decision maker could recieve a great deal of information and only be able to process some of it. For a review of the literature on information processing constraints within the theory of organizations, see Van Zandt (999). 4 See for example, Krishna and Morgan (4), and Forges (986). 3

4 is due to strategic considerations rather than paternalistic ones. When they consider the case where all incentives are aligned, they mention that the advisor s problem becomes trivial as the advisor di erentiates the messages sent by the possible types as much as possible in order to make the decision maker as informed as possible, a process known as maximal di erentiation. In cases such as this, we may ask the following question: if the advisor knows better than the decision maker about the true state of the world, why is the decision not simply delegated to the advisor? 5 than the former decision maker, a Pareto improvement will occur. If the advisor is given the right to make the decision, rather However, the model which I present has multiple advisors, each perfectly informed about a single dimension of a multi-dimensional state of the world, making ex-ante delegation to any one advisor suboptimal. In other words, while the decision maker is unable to perfectly interpret the messages of advisors, that decision maker still plays a necessary role as an information aggregator. It is in this regard that this paper brings a novel approach to the study of noisy information transmission. In order to justify using real numbers as both the realized state of the world and the messages about the state of the world (rather than english words and phrases which are more easily subjected to imprecise interpretations) consider the following scenario: a large company must decide which of two markets to enter, Market A or Market B. Each market is comprised of, regions, where each region is either pro table or unpro table. The Board of Directors of the company will decide in a week which market to enter. If the Board selects the market which is more pro table, the company will survive and all involved will experience a high payo ; if the less pro table market is chosen, the company faces the threat of extinction from a rival, giving each agent a low payo. Now consider the actions of two experts, one for Market A and another for Market B, both of whom work for the company and receive the same high or low payo that the company receives. Each expert is perfectly informed of the pro tability of every region in his respective market. However, the Board of Directors will only consider information that has been professionally veri ed. Moreover, each expert only has the resources to professionally verify the pro tability of out of the, regions of their respective markets. expert chooses which regions information will be veri ed and sent to the board, and in doing so chooses what proportion of pro table regions the board will receive from the expert. The board faces an information processing constraint that prevents the accurate aggregation of information from all regions. Each Instead, the board creates a forecast for each market, equal to the actual proportion of pro table regions sent by the advisor plus some random unbiased noise that occurs perhaps because of errors of judgment by the board members. In this setting, it makes sense to think of the experts as sending a single message to the board: the proportion of pro table regions. to report the true proportion. Consequentially, the expert is faced with the decision of whether or not Henceforth, I refer to the expert s strategy as sincere if the message sent is representative of her true information (e.g. the proportion of good regions in the message is equal to that of the entire market), and I refer to the strategy as paternalistic if the expert sends a message which is biased in favor of whichever option the expert believes is more likely. 6 In order to clarify the origins of a paternalistic bias, in section I describe a benchmark model of noisy communicaction with a single expert. Because the 5 When the payo -dependent state of the world has a single dimension, delegation to exogenously biased experts may be optimal (Li and Suen, 4), and clearly such delegation is also optimal for unbiased experts. 6 Notice that outright lying by the experts is not possible, since all information is professionally veri ed. However, the expert can "mislead" the board by sending them a nonrepresentative sample of regions. Thus, being sincere is not the same as choosing not to lie, and the advisor s decision will not be a ected by a preference for telling the truth. See Kartik, Ottaviani, and Squintani (7) and Kartik (9) for an analysis of strategic communication where misrepresentation of information by senders is costly and/or some receivers are credulous. 4

5 expert knows which policy is best, using language in ation he can almost always convince the DM to choose the optimal policy. The fact that the true state of the world is not precisely communicated is irrelevant. However, when there are multiple experts, each with a di erent area of expertise, the lack of precision of the expert s message is detrimental because it hinders the DM s ability to aggregate the information obtained from all the experts. For example, if Market A and Market B are both highly pro table, the board of directors is interested in just how pro table each market is; it is not su cient to learn whether or not each market is pro table overall. In this work, I nd that there is an interior solution to the expert s problem of how much to bias a message to a DM, corresponding to the optimal trade o between the bene ts of language in ation and the detriment from loss of precision. For the remainder of this paper I will say that experts have a paternalistic bias only when they in ate their messages to the DM while facing a non-trivial cost to this behavior because the precision of the communication matters. In section 3 of the paper, I show that such a paternalistic bias emerges in a perfect Bayesian equilibrium of a model where advice is given sequentially and publicly. Section 4 adds a restriction to the model of section 3 which forces the Board of Directors to enter the market with the best forecast. Section 5 shows that a paternalistic bias still exists in equilibrium when advice is given simultaneously. Section 6 compares the welfare of the three models. Section 7 summarizes results, discusses applications of the current model, and provides an alternative interpretation of the results which sheds light on an observed behavioral phenomenon: the biased recall of information. Several proofs can be found in the appendix. The Benchmark Model In this section I will consider only the case where there is one advisor, one decision maker (DM), and a unidimensional state of the world, R. For simplicity, I assume that there is a common prior over the state of the world, F (), and after is realized, the advisor is perfectly informed of the value of. We can think of this advisor as an expert, whose knowledge is speci c to his eld of expertise, yet vital to the otherwise uninformed DM. The advisor chooses a message, m R, to be sent to the DM. However, because communication is inherently noisy, the message received by the DM is not in general equal to the message sent. Instead, the DM observes a noisy signal, x, which is equal to m +, where is drawn from a known distribution, G(), which does not depend on the choice of m, and satis es the following: E [] = Var [] = < After the DM observes the noisy signal, x = m +, the DM chooses one of two policy options, = H, or = L. In this regard I am departing from the standard strategic communication literature, where there are typically a continuum of actions. The reason for this departure is that I want to focus on the case where a DM must choose between given alternatives and does not need to know the state of the world exactly in order to make the "optimal" decision. where The advisor and the DM have the same utility function, u(j) 8 >< ; if = H and E[] u(j) = ; if = L and E[] >: ; otherwise 5

6 therefore, all parties want to match a high state of the world to a high policy, and low state of the world to a low policy. Claim (Language In ation Can Eliminate Error) The advisor and DM can agree upon a strategy for sending messages which makes the probability of choosing the sub-optimal policy arbitrarily small. words, they can make the probability of error less than ", for any " (; ). In other Proof. Consider any " (; ). Then the advisor can choose a message which depends on in the following way: m() = ( E[] + ", for E[] E[] ", for < E[] The DM can choose a policy of = H whenever x E[], and = L otherwise. Therefore, an error can occur only when jj ". Since has a mean of zero, and variance of, using Chebyshev s inequality, we know that as desired. Pr jj " < " " This type of language in ation will always be possible when there is noise added to the message which is independent of the message sent and has a xed nite variance, and the message space is unbounded. By a similar argument, it follows that an advisor can always communicate with a DM with arbitrarily high precision when these two conditions are satis ed. Since I want to preserve the fact that advisors have knowledge based on idiosyncratic expertise, which can never fully be transmitted to a DM, I must restrict attention to bounded message spaces. In order to preserve the fact that message spaces and information spaces should be identical, the information space must be bounded as well. Without loss of generality, advisors in the model presented in the next section will choose a message to be sent from the unit interval. However, consistent with the observations of Blume and Board (9), the benchmark model of this section is still trivial when the message space is bounded, as the advisor maximially di erentiates his type by sending the highest possible or lowest possible message. In the next section, we will see that the solution is not trivial when there are multiple advisors, as this sort of maximal di erentiation may interfere with the ability of the DM to aggregate information from all of his advisors. 3 Sequential Public Advice The following is an abstract model based on the motiving example given in the introduction. Rather than having experts know the proportion of pro table regions in a nite set, they know a real number which corresponds to the proportion of pro table regions in an arbitrarily large set. The bounded information processing of the board is not explicitly modeled. Instead, a reduced form treatment of the process which creates imperfect communication is introduced as uniformly distributed noise added to the message sent by an advisor. This noise has a mean of zero, and therefore does not itself introduce any bias in the transmission of information from experts to the Board of Directors. In this model, the Board meets rst to develop a forecast for Market A, and then meets to develop a forecast for Market B. The forecast for Market A is made public and the expert corresponding to Market B can consider this information when deciding on a message to send. 6

7 3. The Model There are two expert advisors, indexed by i f; g, and a single decision maker (DM). The true state of the world is given by a two dimensional vector found in the unit square: = ( ; ) [; ] [; ] with common knowledge of a uniform prior over possible values of, so that F ( i ) = i, for i [; ]. Advisor i f; g has perfect knowledge of i. The timing of the game is as follows: advisor chooses a message m [; ] to send to the decision maker (DM). The DM is not able to observe the original message, but instead receives a noisy signal, x, from advisor, where nature selects x according to 7 x U[m ; m + ] for an exogenous ; 4. Thus we can interpret as a measure of the noisiness of the communication between advisor and DM. The signal, x, is public and is observed not only by the DM but by advisor. Advisor, knowing and x, chooses m [; ] to send to the DM, who receives x, where x U[m ; m + ] After observing x = (x ; x ), the DM chooses a policy fh; Lg. The utilities received by advisor, advisor, and the DM are identical and depend only on the state of the world and policy chosen: 8 >< ; if = H and + u(j) = ; if = L and + >: ; otherwise In other words, all three agents receive a utility of if the chosen policy is H when the two components of the state of the world sum to more than their expected value (a high state of the world), and they also receive a utility of if the chosen policy is L when the components sum to less than their expected value (a low state of the world). The utility received is if the correct "match" is not made. 8 All agents are expected utility maximizers. The solution concept for this model will be that of perfect Bayesian equilibria (PBE). De nition A perfect Bayesian equilibrium of the game with sequential public advice is a triplet, (m ( ); m ( jx ) ; (x ; x )), giving the strategies of advisor, advisor, and the DM, respectively, along with the shared posterior beliefs, (g ( jx ); g ( jx ; x )) such that each player s strategy maximizes utility given his beliefs, and beliefs are consistent with equilibrium strategies according to Bayes rule. In this sense, an advisor s type is his private information. Because a multitude of PBE exist (in particular the well-known babbling equilibrium), we restrict attention to the most informative (and hence 7 In section 6, I discuss the implications of x being equal to the proportion of good regions taken from a random sample of the regions sent by advisor. This implies a non-uniform distribution of x. For tractability, I assume here x is uniformly distributed, and interpret the noise involved as subjective information processing error rather than sampling error. 8 To relate the utility function u(j) to the motivating example, let be the proportion of good regions in Market A, and let represent the proportion of good regions in Market B. Then u(j) is whenever the more pro table market is chosen. 7

8 welfare superior) perfect Bayesian equilibrium. 3. Policy Decision of DM An important observation of the structure of this game is that while the DM knows (x ; x ), advisor knows (x ; ). Because x is equal to m ( ) with added noise, and m () is known in equilibrium, it follows that the DM s information is a garbling of advisor s. Claim (Quasi-Delegation of Decision Maker to Advisor ) In the most informative perfect Bayesian equilibrium of the game with sequential public advice, the decision maker ignores the signal of advisor, and chooses a policy based only on the information provided by advisor. Proof. By Blackwell s (953) theorem, advisor will make a weakly better decision under uncertainty than the DM because his information is superior (the DM s information is a garbling of advisor s). Since the utilities are identical for all agents, all agents weakly prefer the outcome of the truncated game where the decision is delegated to advisor. Therefore, the most informative PBE involves the DM choosing the policy suggested by the signal received from advisor, as desired. Given this proposition, we need only consider PBE of the full game where the DM "quasi-delegates" the decision to advisor. This quasi-delegation occurs as follows: advisor sends m = if he selects policy L, m = if he selects policy H. The DM ignores the signal x and can unambiguously choose the policy selected by advisor, because < 4. Speci cally, ( H; if x > (x ; x ) = L; if x The use of the term "quasi-delegation" is justi ed in the sense that the actual outcome is identical to the scenario where the DM delegates the decision to advisor. 3.3 Strategy of Advisor Considering only those PBE with quasi-delegation, we only need to determine for which values of (x ; ) will advisor select policy H. In a PBE, advisor knows the message function of advisor, m (), the prior distribution of, and must use Bayes rule to construct his posterior beliefs of given x. De nition A message function of advisor i, m i (), is a function which fully describes the strategy of advisor i in the following way: given information i, advisor i will send a message of m i ( i ) to the DM. Claim 3 (Binary Message of Advisor ) In an informative PBE of the sequential public advice game where m () is the continuously increasing message function of advisor, advisor s message function is given by (, if > E [ jx ] m ( jx ) =, if < E [ jx ] where E [ jx ] = m (min fx + ; g) + m (max fx ; g) where m () is the inverse message function of advisor with the provision that m () = and m () =. 8

9 Proof. Using Bayes rule, the posterior probability density of given x and m (), and prior density f( ) =, is g( jx ) = = = Z 8 >< >: 8 >< >: f(x j )f( ) f(x j b )f( b )d b ( )()[m ( )() (minfx+;g) m (maxfx ;g)], for m (min fx + ; g); m (max fx ; g), otherwise [m (minfx+;g) m (maxfx ;g)], for m (min fx + ; g); m (max fx ; g), otherwise where m () is the inverse message function of advisor, which exists assuming that the message function is continuous and increasing. The bracketed expression in the denominator represents the length of the interval of -space for which a signal of x is possible. Because the DM relies solely on advisor s signal, advisor s message function is in the form of a cuto strategy. Given, advisor will select H if and only if the expected value of conditional on receiving x is greater than. Given the posterior density is constant with respect to E [ jx ] = = Z m (minfx+;g) m (maxfx ;g) b m (minfx + ; g) m (maxfx ; g) db m (minfx + ; g) m (maxfx ; g) m (minfx + ; g) m (maxfx ; g) = m (minfx + ; g) + m (maxfx ; g) Because g( jx ) is constant, the distribution of is equal on both sides of E [ jx ], thus H is more likely to be the optimal policy if and only if E [ jx ] >!, so advisor uses the strategy of m ( jx ) = (, if > E [ jx ], if < E [ jx ] as desired. 3.4 Strategy of Advisor Advisor s equilibrium strategy will be a message function m () which maximizes the expected utility of sending the message m ( ) given the information. 9

10 Proposition The following describes the strategy of advisor in the most informative PBE of the game with sequential public advice:. (Sincere Transmission of Moderate Information) If advisor s information,, is such that (; ), then advisor sends a sincere message to the DM, or m ( ) =. (Paternalistic Transmission of Extreme Information) If advisor s information,, is such that <, then advisor biases his message downward in the following way m ( ) =, for (; ) m ( ) =, for (; ) Similarly, if >, then messages are biased upward such that m ( ) =, for ( ; ) m ( ) = +, for ( ; ) Proof. See appendix. This proposition completes the description of the most informative perfect Bayesian equilibrium of the game with sequential public advice. The DM quasi-delegates the decision to advisor by ignoring advisor s signal. Advisor knows that her signal will be ignored, and in fact, her motivation becomes to send the DM a signal (knowing it will be observed by advisor ) which maximizes the chance advisor will make the proper selection. Proposition establishes a paternalistic bias in its simplest form. Advisor has not been delegated to make a decision, yet if his information is within of either endpoint, he does not truthfully report his information, but sends a biased message in an attempt to improve the decision that will be made. 3.5 Intuition of Paternalistic Bias The intuition behind the paternalistic bias comes from the fact that an advisor has better information about a dimension of the state of the world than the DM will have (even after receiving a signal from that advisor), and therefore this advisor may choose to in uence the decision of the DM beyond simply sending a message equal to his private information. For example, say that advisor knows that is very high, and therefore = H is most likely to be the optimal policy. If advisor is sincere, there are two types of error: (i) a downward shock to his message induced the DM to select L when really H was optimal, and (ii) an upward shock induced a selection of H when L was optimal. Now consider the marginal e ect of advisor biasing his message upward. The probability of the rst type of error (downward shock) occuring is reduced, while the probability of the second type of error (upward shock) is increased. However, the marginal reduction of the rst type of error is greater than the marginal increase of the second type, which means a paternalistic bias increases utility. The intuition behind this e ect is as follows. The distortion of a high shock on the receiver s expectation of the sender s information is dampened by the fact that the receiver knows there is only so high the actual information can be. In other words an upward shock does not a ect E[ jx ] as much as a downward shock. If is high, advisor can reduce the average di erence between E[ jx ] and by sending a higher

11 message. Because a policy error only occurs when falls between E[ jx ] and, utility is maximized when the di erence between E[ jx ] and is minimized. I refer to this e ect as a boundary e ect, since it occurs due to the boundedness of. Given the motivating example, the boundedness of is justi ed by the fact that even if all regions are pro table, there is a limit to the amount of pro t that can be earned in each region. The boundary e ect described here deals only with the information and messages sent of one advisor, regardless of the actions of the other. A second intuition for the paternalistic bias is identical to the intuition behind the language in ation result of the benchmark model with an unbounded message space. In the benchmark model, the message space is stretched, or in ated, so that potential negative e ects of noisy communication disappear. With multiple advisors and a bounded message space, there is still a bene t from stretching the message space, which stems from the fact that di erent information has di erent qualities, or strengths. Let the strength of the information be the probability that advisor would correctly predict the optimal policy, or +. In this model, the DM would like to follow the recommendation of whichever advisor has the strongest information. By paternalistically biasing his information, an advisor can help to distinguish strong information from weak, and reduce the probability that a weaker piece of information appears to dominate stronger information. I refer to this as the information strength e ect. In a sense, the information strength e ect is due to tacit coordination between the advisors that stronger information will be paternalistically biased. Why does the e ect not create a maximally di erentiated message strategy? The answer is due to the existence of a countervailing e ect of the paternalistic bias which harms the decision process. I refer to this this as the information dilution e ect, which is once again the result of a bounded message space. When advisors paternalistically bias their information, the DM faces a more di cult signal extraction problem because there are more values of which result in extreme messages sent. This dilutes the informative content of an extreme signal, and the reduced precision of the DM s inference about the state of the world results in worse decision quality. The interesting result of this section is that for the model of sequential public advice, the information dilution e ect perfectly negates the information strength e ect for all (; ), and in equilibrium, advisors are sincere over these values. 3.6 Comparative Statics To begin a discussion of the e ects of the noise parameter,, on paternalistic bias, we need a measure of the bias. De nition 3 The paternalistic bias of advisor i when his information is i is given by b i ( i ), where b i ( i ) = ( m i ( i ) i, for i > i m i ( i ), for i If b i ( i ) >, advisor i is said to be paternalistic with the information i. sincere. If b i ( i ) =, advisor i is Claim 4 In the most informative PBE of the game with sequential public advice, (i) db i ( i ) d, with strict inequality for i (; ) [ ( ; ) and (ii) lim b i( i ) =, for all i [; ]!

12 Proof. For (i), the message, and hence paternalistic bias, of advisor is constant with respect to unless i (; ) [ ( ; ) in which case b i ( i ) = ( i, for i (; ) i +, for i ( ; ) and dbi(i) d =, as desired. For (ii), it is su cient to observe that the set of information which leads to a paternalistic bias vanishes in a smooth way as goes to, as the endpoints of the interval are linear in. This claim establishes that the paternalistic bias is not explosive, but continuously grows as the noise parameter increases. It also con rms the well-known fact that truth-telling is an equilibrium of an information aggregation game when communication is perfect and incentives are aligned. 4 Sequential Public Advice without Quasi-Delegation Returning to the motivitating example from the introduction, in the model I present here the Board of Directors is under the close scrutiny of the company s shareholders. In particular, the shareholders demand that the forecasts for Market A and Market B (both of which are public information) be regarded equally, and the company must enter whichever market has the best forecast. Likewise, the directors on the Board do not want to risk losing their jobs by making the wrong move when the forecasts accurately predicted the correct move. 9 Therefore, the DM of this model cannot simply ignore a signal (e ectively delegating the decision to the other expert), but must choose the option suggested by both signals. Formally, the model is updated to include a utility of for the DM if the correct match was not made when the signals suggest it would. Notice that the preferences over policy outcomes of the DM and advisors are still aligned. 4. Policy Decision of the DM As the DM must regard both signals equally and risks a large loss if he goes against the matching implied by the sum of those signals, we will restrict attention to equilibria where the strategy of the DM is given by 4. Strategy of Advisor (x ; x ) = ( H; if x + x > L; if x + x Since there is no quasi-delegation possible, advisor will choose a message, m, to send to the DM which minimizes the probability of error, given (x ; ). In a PBE, advisor also knows (x ; x ), described above, m (), and uses Bayes rule to nd g( jx ). Claim 5 (Binary Message of Advisor with no quasi-delegation) In an informative PBE of the sequential public advice game with no quasi-delegation, where m () is the continuously increasing message function of advisor, advisor s message function is m ( jx ) = (, if > E [ jx ], if < E [ jx ] 9 An alternative interpretation of this model is that the Board of Directors gives the information to yet another party, who is unable to distinguish between the two signals x and x.

13 where E [ jx ] = m (min fx + ; g) + m (max fx ; g) where m () is the inverse message function of advisor with the provision that m () = and m () =. Proof. See appendix. Even when quasi-delegation is impossible, advisor chooses to bias his message to one extreme or the other for every value of. In a sense, this strategy represents the most severe form of paternalism an advisor can implement because advisor decides for the DM what the best policy is, and biases his message entirely toward one of the endpoints even if he only slightly believes one policy to be better. The only reason this strategy is not a paternalistic bias in the strictest sense is that the most critical step of information aggregation (that of advisor, and not that of the DM) occurs before advisor s message is sent. In this sense, rather than a paternalistic bias, advisor s behavior is an attempt to force the DM to unwittingly delegate the decision to himself. 4.3 Strategy of Advisor Proposition The following describes the strategy of advisor in the most informative PBE of the game with sequential public advice and no quasi-delegation:. (Sincere Transmission of Moderate Information) If advisor s information,, is such that (; ), then advisor sends a sincere message to the DM, or m ( ) =. (Paternalistic Transmission of Extreme Information) If advisor s information,, is such that <, then advisor biases his message downward in the following way m ( ) =, for ; 3 m ( ) = (3 ), for + ; 3 Similarly, if >, then messages are biased upward such that m ( ) =, for 3 ; m ( ) = m ( ), for ; 3 Proof. See appendix. This proposition establishes a paternalistic bias from advisor in the game without quasi-delegation. It also completes the description of the most informative PBE of this game. The DM, in order to avoid making the wrong choice when the forecasts predict correctly, will choose H if the sum of the signals is greater than. Advisor sends a purely paternalistic message to the DM, according to whichever policy he believes is more likely to be best. As before, advisor is sincere given moderate information, but biases more extreme information paternalistically. The comparative statics with respect to are similar to that 3

14 of the rst model. 4.4 E ects of Quasi-Delegation on Paternalism In the game without quasi-delegation, the signal x a ects the decision of the DM not only indirectly through in uencing advisor, but also directly in that x + x determines the policy chosen. paternalistic bias of advisor depends on whether or not quasi-delegation can occur. For this reason, the Claim 6 For any, the paternalistic bias of advisor in the most informative PBE of the sequential advice game where quasi-delegation is allowed, b D ( ), is greater than or equal to the bias of advisor in the most informative PBE when quasi-delegation is not allowed, b ND ( ), with strict inequality for some. Proof. For [; ], b D ( ) = b ND ( ) =. For [; ], b D ( ) = which is the maximum possible bias, therefore b D ( ) b ND ( ). Thus we only need to check (; ), which means that b D ( ) = m D ( ) = while on the same interval, [; ], thus b ND ( ) = m ND ( ) = (3 ) + b D ( ) b ND ( ) = = = 4 + which must be positive since the denominator is obviously positive and the numerator is positive, since 4 > because we are only considering <. Therefore b D ( ) > b ND ( ) on (; ). Because of symmetry, a similar argument applies for > and bd ( ) b ND ( ) for all as desired. The above claim establishes that the paternalistic bias (weakly) diminishes when quasi-delegation is not allowed to occur. This result could be seen as counter-intuitive because a restriction on the DM actually induces less of a bias in the messages sent to him. A deeper look reveals that the reason the paternalistic bias decreases has to do with the inability of advisor to de nitively decide the policy. Because advisor is purely paternalistic (sends a binary message), the DM will pick against advisor only when the signal received from advisor is extreme. advisor s attempt at self-delegation fails. Furthermore, the more extreme advisor s message, the more likely Therefore, when advisor decides on a message, he must take into account the additional e ect that biasing his message has on increasing the chances that the DM picks against advisor, who is generally more informed than he. It is this tendency which explains the reduction in the equilibrium level of paternalistic bias when the DM is required to choose the policy suggested by both signals (or is unable to distinguish them.) in Section 5. The welfare consequences of quasi-delegation will be discussed 4

15 5 Simultaneous Advice Returning again to the motiviting example from the introduction, in this section I consider the case where both advisors send their reports to the Board of Directors simultaneously, and the Board formulates a forecast for Market A and Market B at the same time. Equivalently, this scenario describes the situation where the Board of Directors keeps the report of the rst advisor private, leaving the second advisor uninformed of the message sent or signal received from the rst advisor. Both advisors are ex-ante identical in this case, and their incentives are still aligned with the DM. Formally, the model is updated so that the DM receives x and x simultaneously, and advisor s message can only depend on his information, or m ( ). 5. Policy Decision Due to the symmetry of the state space, the policy space, and the two advisors, PBE with the DM strategy of ( H; if x + x > (x ; x ) = L; if x + x will yield expected payo s as large as any other. This is because in any symmetric PBE, any distortions upward from one advisor are exactly mirrored by distortions downward from the other advisor, and x + x will be greater than one if and only if the conditional expectations of and sum to more than one. 5. Strategy of Advisors As before, advisors and choose a message function which gives a message for each value of information the advisor may have. messages m ( ) and m ( ). Given the DM strategy, it is possible to calculate the probability that = H given Claim 7 If advisor and send messages m ( ) and m ( ), then the probability that policy H is chosen is given by 8 >< Pr( = H) = where z = z(m ( ); m ( )) is de ned as Proof. See appendix. >: ; if z + z + z ; if z + z z ; if z ; if z z := m ( ) + m ( ) Considering that the signals x and x received by the DM are uniformly distributed and centered around the original messages sent by advisors, it follows that the DM becomes more likely to choose H than L whenever the sum of the original messages is greater than. The variable z is a su cient statistic for determining the probability with which policy H is chosen because it measures how far beyond the cuto point of is the sum of the messages, relative to the size of the largest possible error,. 5

16 Given message functions m () and m (), we can unambiguously calculate the common expected utility of all agents. Speci cally, E [u()jm (); m ()] = Z Z Z Pr( = H)dF ( )df ( ) + Z Pr( = L)dF ( )df ( ) where Pr( = H) is given in Claim 7. message function, given the other advisor s choice of message function. Notice that each advisor knows the expected utility of choosing a 5.3 Existence of Symmetric Perfect Bayesian Equilibrium As in the case with sequential advice, we will use the concept of perfect Bayesian equilibrium to predict the strategies of advisors. In this section, I prove that there exists a symmetric PBE of the game with simultaneous advice, where the DM chooses H if and only if x + x >, which ensures that the PBE will be informative. Before proving the existence of a PBE, I need to de ne some terms and establish two useful lemmas. Lemma (M; d) is a convex, compact, metric space, where M is the set of functions from [; ] to [; ], and d() is the distance function on M implied by the supremum norm, or d(m; m ) = sup fm() [;] m ()g Proof. See appendix. This lemma establishes a metric space over the set of possible message functions. The distance function implies a natural topology, where topological de nitions such as convexity and compactness can be applied. De nition 4 The best response correspondence is a mapping, B : M! M, such that B(m()) is the set of best responses to the message function m(), or B(m()) = arg max m () E [u()jm (); m(); ()] Notice that the symmetry of the advisors implies that the best response correspondence itself does not depend on the identity of the advisor whose message function is taken as given. Lemma Given the optimal DM strategy (), (i) the set B(m()) is non-empty and convex for all m() M, and (ii) the correspondence B : M! M has a closed graph, using the supremum distance function d(). Proof. See appendix. The graph of the best response correspondence is de ned as the subset of M M such that the second component is a best response to the rst. This graph is closed if the limit of any convergent sequence of best responses of functions is a best response to the limit of the original sequence of message functions. Using the supremum distance, this must be the case because best responses functions can be broken down into a series of maximization problems, each one taking information i as given. As the distance between two functions goes to zero, so does the di erence between the messages for any i, and the arguments of the isolated maximization problem for i also converge to a point that must maximize utility. 6

17 Proposition 3 There exists a symmetric informative perfect Bayesian equilibrium of the game with simultaneous advice. Proof. Kakutani s (94) xed point theorem generalized to Hausdor spaces (which include all metric spaces) can be used to show that there exists a message function which is included in the set of best responses to itself. This represents a symmetric PBE along with the optimal cuto strategy by the DM and beliefs formulated using Bayes Rule. The theorem states that if M is a non-empty, compact, and convex subset of a metric space, the correspondence B : M! M has a xed point as long as long as the graph of B is closed and the set B(m()) is always non-empty and convex. By lemmas and, the extension of Kakutani s xed point theorem implies that m() B(m()) for some m(). 5.4 Best Response to Sincere Advisor The previous section establishes the existence of a PBE. In order to show that this equilibrium cannot involve sincerity, we now restrict our attention to the game where advisor is bound to be sincere, and calculate the best response of advisor. This calculation will give some insight into the shape of the symmetric PBE message function. Additionally, the sincerity constraint has a number of useful interpretations. For example, advisor could be a public agency whose advice is scrutinized by the general public, and not just the DM. Because of reputation concerns, this agency could be constrained to sending messages as close to the truth as possible. Alternatively, we can interpret this new setting as the case where there is no advisor, but the DM himself receives a private signal uniformly distributed around the true value of, or x U[ ; + ], which is equivalent to receiving sincere (noisy) advice. Claim 8 The following describes advisor s best response to a sincere advisor when the DM uses the strategy (x ; x ) = H i x + x > :. (Sincere Transmission of Moderate Information) If [; ], then advisor sends a sincere message m ( ) =. (Paternalistic Transmission of Extreme Information) If <, then advisor biases his messages downward, such that m ( ) = q + 3 h m ( ) =, for ; + 6 p 8, for p ; while if > advisor biases his message upward, such that m ( ) = h( ) + p i ( ) 3 + 6( ) 8, p for ; p i m ( ) =, for ; Proof. See appendix. This claims establishes that a paternalistic bias is the best response to a sincere advisor, only when his information is su ciently extreme. It also establishes that sincerity cannot be an equilibrium to the game 7

18 with simultaneous advice. 5.5 Characterization of Equilibrium Strategy of Advisors The best response correlation on the set of message functions has a xed point. A logical way to learn more about this xed point is to iterate the best response operation, starting with a natural choice: the sincere message function. paternalistic bias at the extremes. Claim 8 shows that the rst iteration entails sincerity for moderate information and a If we can show that the best response to a message function has some characteristics in common with the original function, then iteration of the best response correspondence will yield a xed point with those characteristics. Claim 9 There exists a PBE of the game with simultaneous advice where the DM chooses H if and only if x + x >, and both advisors use the message function m() where m( i ) < i, for i ; m( i ) > i, for i ; or in other words, the paternalistic bias b( i ) is positive for all i except the midpoint and two endpoints of the interval from which is drawn. Proof. See appendix. This claim characterizes the shape of the equilibrium message functions, and establishes a clear paternalistic bias. The previous two models presented in this paper conclude that advisors bias their messages only when their information is su ciently extreme, or equivalently, when their beliefs are strong that a certain policy is the correct one. advisors are almost everywhere paternalistic. The simultaneous advice model delivers a qualitatively di erent prediction, that The intuition behind this result lies in the fact that each advisor responds to the other s paternalistic strategy with a paternalistic bias over an even wider range, continuing until the bias is ubiquitous. The information strength e ect drives the paternalistic bias over the whole range because the advisors are ex-ante symmetric, unlike in the sequential case. However, the quantitative di erence between sequential and simultaneous advice toward the midpoint is small because the magnitude of the paternalistic bias asymptotically shrinks to zero as the information approaches the center in the simultaneous case. 6 Welfare Comparison Going back to the motivating example, the Board of Directors may have some interest in determining how to best solicit the advice of the two experts. The Board must choose whether to have the advisors report simultaneously or sequentially, and whether or not they will allow shareholders to know the forecasts. The Board wants simply to maximize expected welfare, which is equal to the ex-ante probability the best policy is chosen. Proposition 4 Let W refer to the ex-ante welfare of sequential advice, W refer to that of sequential advice with no quasi-delegation, and W 3 refer to that of simultaneous advice. Then for all ; 4 W > W > W 3 8