The Impact of Disclosing Conflicts of Interest on Quality of Advice: Experimental Evidence


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1 The Impact of Disclosing Conflicts of Interest on Quality of Advice: Experimental Evidence Christoph Lex July 9, 2014 Abstract This study investigates in a laboratory experiment how disclosure of conflicts of interest affects quality of advice. The influence of different levels of incentives as well as the effect of feedback is also analyzed. The experiment is designed to reflect the relationship between an insurance agent and his customer. To best reflect such a setting, choices are made over discrete options, incentives are only partially misaligned, and customers have an outside option. This study provides policy implications for the two currently discussed regulatory measures: disclosing conflicts of interest, and limiting the amount of commissions. In addition, an alternative measure where reputation building is facilitated is analyzed. Keywords: Disclosure, conflicts of interest, insurance intermediation JEL classification codes: D03, G22, G28 Munich Risk and Insurance Center, LMU Munich, Germany,
2 1 Motivation and Purpose of the Study There are numerous examples of markets where less informed customers approach better informed experts, but receive biased advice due to incentives between both parties being (partially) misaligned These conditions potentially apply to occupations such as pharmacists, physicians or salespeople, although many first think of financial advisers and insurance agents whose compensation is typically linked to the values of the products they sell. That is, the expert has incentives to recommend products which maximize his benefits but are not necessarily optimal for the customer. Incentive schemes like this have been exploited in practice many times. In the insurance industry, commissions have therefore been constantly under discussion and are targeted by various regulatory measures. The most rigorous regulation is in force in Finland and Denmark, where commissions have been completely banned for brokers since 2005 and 2006, respectively. Other countries impose a limit on the amount of commissions available to be earned. This is the case concerning German private health insurance, where commissions may not exceed 9 monthly premiums. Additionally, it is currently being debated in Germany whether to extend cancellation liability for life insurance products from 5 to 10 years. Another approach is to disclose all commissions received, as has been the case in both Sweden and the United Kingdom since The European Commission is currently discussing this measure within the framework of the Insurance Mediation Directive (IMD) II. 1 The main focus of this directive is to increase transparency in order to reveal potential conflicts of interest to the customer. The current approach suggests that insurance agents must disclose the commissions of contracts they recommend to customers. Disclosing conflicts of interest to the customer is an intuitive and very cheap regulatory tool and therefore often suggested to reduce biases, also in other industries. 2 However, past research has led to a controversial debate as to whether disclosing conflicts of interest actually improves recommendations. The predominantly experimental literature provides evidence both 1 For the current version of the IMD II proposal see: docs/consumers/mediation/ directive_en.pdf. 2 Inderst and Ottaviani (2012), for example, mention that in the US, real estate brokers were mandated to disclose the payments they received to their customers; The Markets in Financial Instruments Directive (MiFID) requires the disclosure of commissions on retail financial products in the European Union; Similar rules have been installed by the FSA in the UK. 1
3 for against a quality increasing effect. Additionally, there is research showing that disclosure has neither a positive nor negative effect. However, these studies deviate in various aspects from an insurance intermediary setting, such as the choice over continuous instead of discrete options and a lacking outside option for the customer. The contribution of this experimental study is to resemble the essential characteristics of an insurance agentcustomer relationship and thus to provide insight into the effectiveness of disclosure in the insurance intermediary setting. Besides disclosure and no disclosure treatments, feedback treatments were conducted to allow for reputation building. All treatments were conducted under two different incentive schemes for the agents: one in which the agents could earn about the same as the buyers and one where the agents could earn substantially more than the buyers. The goal of these treatments is to test whether customers are deterred by the relatively high compensation agents receive and whether limiting commissions affects the quality of advice. The three main research questions to be answered are: First, does disclosure lead to better quality of advice in an insurance intermediary setting? Second, does the amount of commissions influence the agent s quality of advice and/or are customers deterred by high commission payments to the agent? Third, are feedback mechanisms a potential alternative to increase market efficiency? This study therefore provides insight into three potential regulatory measures: disclosure of agents compensation, limiting commission payments, and establishing reputational/feedback mechanisms. The paper proceeds as follows. The next section provides a detailed literature review and identifies the research gap we aim to fill. Section 3 presents the experimental design. In section 4 the underlying game structure is analyzed and theoretical predictions derived. Results of the experiment are presented in section 5 and discussed in section 6. Section 7 concludes. 2 Literature Review and Research Gap The literature on the disclosure of incentive conflicts has been primarily stimulated by Cain, Loewenstein, and Moore (2005). They find that disclosing conflicts of interest leads to more biased recommendations and worse outcomes for the customers. At first sight, strategic con 2
4 siderations are, on the one hand, the reason for these counterintuitive results. As soon as a customer knows about an adviser s incentives and discounts the advice of the expert more, the adviser anticipates this and may even exaggerate more to begin with. Secondly, Cain, Loewenstein, and Moore (2005) find moral reasons to be important. Advisers report feeling morally licensed to exaggerate their recommendations when customers know about their incentives and could react to this behavior. Disclosure apparently increases discounting by customers, but not enough to offset the increase in the bias of the received advice. Customers are worse off when conflicts of interest are disclosed than when they are not, and advisers earn more money with disclosure than without disclosure. The study by Cain, Loewenstein, and Moore (2005) was not conducted in a computer laboratory. In their experiment, customers estimate the value of a jar full of coins which they are only shown for a short period of time. Advisers who have superior knowledge about the value give recommendations to the estimators, but might face misaligned incentives. Results have been replicated by Cain, Loewenstein, and Moore (2011) in a more controlled setting where subjects are asymmetrically informed about house prices. Koch and Schmidt (2010) also find the same results in a computer lab experiment where different players are given different information about the outcome of a random number. In addition, Koch and Schmidt (2010) show that experience improves the quality of advice and that reputation building is restrained by the disclosure of conflicts of interest. In order to find situations in which disclosure actually improves the outcome for advisees, Sah, Loewenstein, and Cain (2013) conduct different disclosure treatments. For example, the disclosure information is revealed by an external source instead of the adviser himself. In another treatment, customers are given some time before they make their decision, which they make privately and without being influenced by the agent. These variations positively influence the value of disclosure. Church and Kuang (2009) allow customers to sanction advisers for bad recommendations which also fosters the efficacy of disclosure. In their study, subjects decide on how much to invest in a project. The subjects are illinformed about the outcome of the projects, but receive advice from biased experts. This brief literature review shows that, depending on the underlying setup, the effect of disclosure can go in either direction. The selected setup depends crucially on the subject of 3
5 interest. The studies mentioned so far characterize a situation between an analyst and an investor. One feature which makes the situation specific and distinct from the relationship between an insurance agent and his customer is the continuous form of advice. Insurance agents generally provide information on different contracts and recommend one of them. Then the customer has to make a discrete choice. Another typical characteristic is an outside option, i.e. the advisee has the opportunity to walk away without choosing one of the contracts. Chung and Harbaugh (2012), Ismayilov and Potters (2013), and Angelova and Regner (2013), among others, study the quality of advice in situations where subjects make discrete choices. Chung and Harbaugh (2012) look at the advice of biased and unbiased agents. In their study, advisees are differently informed about the incentive structure of the agents. They either know what the agent s incentives are or are uncertain whether they deal with a biased or unbiased agent. Customers in this study have an outside option. The potential threat that a customer does not follow the agent s advice, but instead takes the outside option, serves as a disciplining device for the agent since he would not receive any profits in this case. The authors find that lack of transparency about experts incentives leads to an increase in lying by experts, i.e. disclosure leads to higher quality of advice. The results of Ismayilov and Potters (2013) suggest that disclosing advisers interests neither impairs nor improves recommendations. In this study an outside option is missing as a disciplining device. One limitation of these two studies is that experts can deceive by telling the truth as only two options can be chosen. For the two option case, Sutter (2009) shows that an expert is able to fool the customer into selecting his favorable option by recommending the alternative option, which would actually be optimal for the customer. This problem can be dealt with by adding a third option, as has for example been done in the study by Angelova and Regner (2013), who find that better advice is provided when customers make voluntary payments to the adviser. However, they do not analyze the effects of disclosure. In summary, the existing research draws a diverse picture of the effect of disclosure of conflicts of interest. These heterogeneous results can be related to the specific setups of the individual studies. The focus of this study is to create a setup which resembles the situation of an insurance agent and his customer. Therefore, discrete options instead of continuous choices 4
6 are implemented and incentives are only partially misaligned. Additionally, the customer has an outside option of not buying a contract from the agent. As this is the least favored situation for the agent, it serves as a disciplining device. In addition, it rules out the problem of deception through telling the truth, because the customer has more than two choices. A random draw at the beginning of the experiment decides which option is best for the buyer. This produces situations where incentives between the agent and the customer are aligned or misaligned with equal probability. 3 Experimental Design Our game setup is based on the senderreceiver game by Gneezy (2005). We add two important features to replicate the insurance agentcustomer relationship. Two players, player A (agent) and player B (buyer) decide between two contracts, 1 and 2. One of the contracts provides player B with a high payoff, the other with a low payoff. The contract which is preferable to player B will be randomly drawn with equal probability at the beginning of the game. Therefore, either state S1 or state S2 realizes. Table 1 shows the payoff scheme for both players. Only player A knows which of the states realizes and therefore knows which contract provides a higher payoff to player B. Player B knows that one of the two contracts yields a higher payoff, but not which contract it is. If state S1 is drawn, there is a conflict of interest, Table 1: Agent s and buyer s payoff schemes contract state S1 state S2 follow not follow follow not follow low incentive 1 (6,10) (0,7) (6,3) (0,7) 2 (10,3) (0,7) (10,10) (0,7) high incentive 1 (18,10) (0,7) (18,3) (0,7) 2 (30,3) (0,7) (30,10) (0,7) Note: Values in brackets denote the epayoffs of both players: (player A, player B). as player B prefers contract 1 and player A prefers contract 2. In state S2, incentives are aligned, as both players prefer contract 2. Player A sends one of the two messages to player B: Message m = 1: Contract 1 will earn you more money than contract 2. 5
7 Message m = 2: Contract 2 will earn you more money than contract 1. Player B receives the message and follows the advice or takes the outside option which gives him a certain payoff of 7 and a payoff of 0 to the agent. 3 As it is the worst outcome for player A, it serves as a disciplining device in the game as well as in the real life relationship between an insurance intermediary and his customer. In total, six different treatments are conducted: low incentive  disclosure, low incentive  no disclosure, high incentive  disclosure, high incentive  no disclosure, low incentive  feedback  no disclosure, and high incentive  feedback  no disclosure. The main goal of the first four treatments is to determine whether disclosure of conflicts of interest increases the quality of advice from a biased agent. As usually uninformed individuals or people with limited knowledge about insurance approach insurance agents to get reliable and sound advice, it is crucial to analyze whether disclosure affects the advice of the agent. Therefore, the main focus in these treatments is on the agent s decision. A currently discussed and in some markets already implemented measure is to limit the amount of commissions. A comparison of the high and low incentive treatments allows to distinguish the quality of advice between these treatments as well as customer reactions to different earning prospects of the agents. Finally, the feedback treatments (which were only conducted without disclosure) serve as a suggestion for an alternative regulatory measure. 4 Theoretical Predictions In this section, Nash equilibria of the underlying games are derived in order to make theoretical predictions. Individuals are assumed to be risk neutral, i.e. are only interested in maximizing their expected payoff. Table 2: Strategies of Player A State S1 State S2 m=1 p 11 p 12 = 1 p 22 m=2 p 21 = 1 p 11 p 22 3 Taking the option which was not recommended is not possible. 6
8 Player A has the option (see Table 2) to send the truthful messages with probability p 11 or p 22. Alternatively, he can send the deceiving messages with probabilities p 21 = 1 p 11 or p 12 = 1 p 22. On the other hand, player B (see Table 3) can follow a received message with probability q 11 or q 12. Alternatively, he can choose not to follow the advice with counter probabilities q 21 = 1 q 11 or q 22 = 1 p 12, which means taking the outside option. This would yield a payoff of 7 to player B and 0 to player A. Table 3: Strategies of Player B m=1 m=2 follow q 11 q 12 not follow q 21 = 1 q 11 q 22 = 1 q 12 A Nash equilibrium requires mutually best responses. In the following, Nash equilibria are derived, first for a general form of the game, then for the specific payoffs which are actually used in the experiment. 4.1 Low incentive treatment, disclosure In this treatment of the experiment, the payoffs of player A are disclosed to player B. Player A receives payoff a for contract 1 and b for contract 2, with b > a. Player B receives the payoff x for contract 1 (2) in state 1 (2) and y for contract 2 (1) in state 1 (2), with x > y. If player B chooses the outside option, which yields z to him, player A ends up with a payoff of 0. The outside option satisfies the following condition: z > 1 2 (x + y). This assumption makes sure that a risk neutral player B has to rely on the informational value of player A s message. Table 4 shows the payoff scheme for both players. Player A and B s strategies are P and Q, Table 4: Agent s and buyer s payoff schemes contract state S1 state S2 follow not follow follow not follow 1 (a,x) (0,z) (a,y) (0,z) 2 (b,y) (0,z) (b,x) (0,z) Note: Values in brackets represent the payoffs of both players: (player A, player B). respectively: 7
9 P = p 11 1 p 22 1 p 11 p 22 Q = q 11 q 12 1 q 11 1 q 12 In order to find mutually best responses, player B s optimal response to player A s strategy P is regarded first: Player A s expected payoff, given state S1 occurs is: E(A S1) = ap 11 q 11 + b(1 p 11 )q 12 (1) Player A s expected payoff, given state S2 occurs is: E(A S2) = ap 12 q 11 + b(1 p 12 )q 12 (2) Player A optimizes over the choice of p 11 and p 12, therefore the following four cases must be distinguished: aq 11 > bq 12, aq 11 < bq 12, aq 11 = bq 12, and q 11 = q 12 = 0. Case 1: aq 11 > bq 12 Can it be a best response for player B to set q 11 and q 12 so that aq 11 > bq 12 is fulfilled? In this case, player A would set p 11 = p 12 = 1; i.e. player A always sends message m=1 which is uninformative to player B. Player B s expected payoff then becomes: E(B) = 1 2 [ (p11 (xq 11 + (1 q 11 )z) ) + ( (1 p 11 )(yq 12 + (1 q 12 )z))+ + (p 12 (yq 11 + (1 q 11 )z) ) + ( (1 p 12 )(xq 12 + (1 q 12 )z)) ] = = 1 [( 1(xq11 + (1 q 11 )z) ) + ( 0(yq 12 + (1 q 12 )z) ) ( 1(yq 11 + (1 q 11 )z) ) + ( 0(xq 12 + (1 q 12 )z) )] ( ) 1 = z + q 11 2 (x + y) z 8
10 Since 1 2 (x + y) < z q 11 = 0. This contradicts with aq 11 > bq 12, i.e. this case cannot be an equilibrium. Case 2: aq 11 < bq 12 Can it be a best response for player B to set q 11 and q 12 so that aq 11 < bq 12 is fulfilled? In this case, player A would set p 11 = p 12 = 0; i.e. player A always sends message m=2 which is uninformative to player B. Player B s expected payoff becomes: E(B) = 1 [( p11 (xq 11 + (1 q 11 )z) ) + ( (1 p 11 )(yq 12 + (1 q 12 )z) ) + 2 ( p12 (yq 11 + (1 q 11 )z) ) + ( (1 p 12 )(xq 12 + (1 q 12 )z) )] = 1 [( 0(xq11 + (1 q 11 )z) ) + ( 1(yq 12 + (1 q 12 )z) ) + 2 ( 0(yq11 + (1 q 11 )z) ) + ( 1(xq 12 + (1 q 12 )z) )] = z + q 12 ( 1 2 (y + x) z ) Since 1 2 y x < z q 12 = 0 which contradicts with aq 11 < bq 12 No equilibrium in this case. Case 3: aq 11 = bq 12 In this case, player A is indifferent between sending message m = 1 and m = 2. P keeps the general form: P = p 11 p 12 = 1 p 22 p 21 = 1 p 11 p 22 As P contains values 0 in every row, all conditional probabilities are defined: p i is the probability that message m = i is sent, i.e. p 1 = 1 2 (p 11 +p 12 ) and p 2 = 1 2 (2 p 11 p 12 ). Player 9
11 B s expected payoff given he receives message m = 1 becomes: E(B m = 1) = 1 2p 1 ( p11 (xq 11 + (1 q 11 )z) ) + ( (p 12 )(yq 11 + (1 q 11 )z) ) = = z zq 11 + yq 11 + (x y)p 11 q 11 = 2p ( 1 ) (x y)p11 = z + q 11 + y z 2p 1 } {{ } V Player B s expected payoff given he receives message m = 2 becomes: E(B m = 2) = 1 2p 2 ( p21 (yq 12 + (1 q 12 )z) ) + ( (p 22 )(xq 12 + (1 q 12 )z) ) = = z zq 12 + yq 12 + (x y)p 22 q 12 = 2p ( 2 ) (x y)p22 = z + q 12 + y z 2p 2 } {{ } W All 9 possible combinations of V 0 and W 0 are potential candidates for equilibria and are checked in the following. Cases i), ii) and iii): W > 0 and V 0 Whenever W > 0, player B always accepts the recommendation m = 2 of player A, i.e. sets q 12 = 1. However, this contradicts with the basic assumption of case 3: aq 11 = bq 12, since b > a. Furthermore, this can also not be an equilibrium, because given q 12 = 1, player A would always send message m = 2. Case iv): W = 0 and V > 0 In this case, player B is indifferent whether to follow or not follow message m = 2 and player B always follows message m = 1, i.e. q 11 = 1. The acceptance probability q 12 can assume any value. In order to satisfy the basic condition of case 3 (aq 11 = bq 12 ), this is q 12 = a b. From 10
12 V > 0 follows: (x y)p 11 2p 1 + y z > 0 (x y)p 11 > (z y)(p 11 + p 12 ) p 11 x yp 11 > zp 11 yp 11 + zp 12 yp 12 p 11 (x z) > z zp 22 y + yp 22 zp 22 yp 22 > z y + p 11 (z x) p 22 > 1 + p 11 z x z y (3) W = 0 implies: (x y)p 22 + y z = 0 2p 2 p 22 = z y x z + p y z 11 x z (4) Inserting (4) in (3) gives: z y x z + p y z 11 x z > 1 + p z x 11 z y (x z) 2 (z y) 2 p 11 > x + y 2z (z y)(x z) x z } {{ } <0 p 11 < (z y)(x + y 2z) (x z) 2 (z y) 2 ( = 5 ) 8 Using (4), p 22 becomes: p 22 > 1 (2z x y)(x z) (z y) 2 (x z) 2 Figure 1 shows the area of possible values for p 11 and p 22. ( = 5 ) 8 To sum up, this case can be an equilibrium, where player A sends the correct messages with probabilities p 11 < 5 8 and p 22 > 5 8. Player B follows these messages with probabilities q 11 = 1 and q 12 =
13 Figure 1: Graphical illustration of possible equilibria Case v): W < 0 and V = 0 In this case, player B is indifferent whether to follow or not follow message m = 1 and player B always rejects message m = 2, i.e. q 12 = 0. This is, however, inconsistent with the basic assumption of case 3: aq 11 = bq 12 and b > a. Case vii): W = 0 and V < 0 In this case, player B is indifferent whether to follow or not follow message m = 2 and player B always rejects message m = 1, i.e. q 11 = 0. From V < 0 and W = 0 follows: q 11 = q 11 = 0, i.e. player B always takes the outside option. Please see case xi) for the response of player A that satisfies an equilibrium. 12
14 Case viii): W = 0 and V = 0 In this case, player B is indifferent whether to follow or not follow message m = 1 and m = 2. From V = 0 and W = 0 follows: and z y x z + p y z 11 x z = 1 + p z x 11 z y (2z x y)(z y) p 11 = (z y) 2 (x z) 2 (= 5 8 ) p 22 = 1 + Figure 1 depicts results for this case graphically. (2z x y)(z x) (z y) 2 (x z) 2 (= 5 8 ) Case ix): W < 0 and V < 0 In this case, player B would never follow message m = 1 or m = 2, i.e. would set q 11 = q 12 = 0. Given player B never follows a message, setting P so that V < 0 and W < 0 is a best answer of player A. V < 0 implies: 8p 11 2p 1 5 < 0 8p 11 < 5(p 11 1 p 22 ) p 22 < p 11 W < 0 implies: 8p 22 2p 2 5 < 0 p 22 < p 11 Figure 1 depicts the area of possible values for p 11 and p 22 graphically. 13
15 4.2 Low incentive treatment, no disclosure In the no disclosure treatment, what player B knows about the payoffs of player A is crucial. It seems natural that player B has the information that one of the contracts gives a higher return to the agent, but does not know which one it is. Therefore, the prior belief of player B is that player A receives the high payoff of b for contract 1 and the low payoff of a for contract 2 with equal probability, or vice versa. Since player B does not know that player A always receives the high payoff for contract 2, four different combinations must be distinguished. Table 5 shows these combinations, which are denoted as state 1 through 4. From the perspective of Table 5: Payoffs combinations from the perspective of player B state S1 state S2 state S3 state S4 contract f nf f nf f nf f nf 1 (a,x) (0,z) (a,y) (0,z) (b,x) (0,z) (b,y) (0,z) 2 (b,y) (0,z) (b,x) (0,z) (a,y) (0,z) (a,x) (0,z) Note: Values in brackets represent the payoffs of both players: (player A, player B). player B, player A has the following strategies: P = p 11 p 12 p 13 p 14 p 21 p 22 p 23 p 24 As before, player B has the option to follow or not follow message 1 and 2: Q = q 11 q 12 1 q 11 1 q 12 What is the best response of player B given player A s P? E(A S1) = ap 11 q 11 + b(1 p 11 )q 12 E(A S2) = ap 12 q 11 + b(1 p 12 )q 12 E(A S3) = bp 13 q 11 + a(1 p 13 )q 12 E(A S4) = bp 14 q 11 + a(1 p 14 )q 12 14
16 The following six cases must now be distinguished: Figure 2: Six possible cases of q 11 and q 12 Case 1: aq 11 < bq 12 and bq 11 < aq 12 In this case, player A s best response is p 21 = p 22 = p 23 = p 24 = 1 or P = Always sending message m = 2 is not informative for player B. His expected payoff becomes: E(B) = 1 4 [p 11(xq 11 + (1 q 11 )z) + (1 p 11 )(yq 12 + (1 q 12 )z) + +p 12 (yq 11 + (1 q 11 )z) + (1 p 12 )(xq 12 + (1 q 12 )z)+ +p 13 (xq 11 + (1 q 11 )z) + (1 p 13 )(yq 12 + (1 q 12 )z)+ +p 14 (yq 11 + (1 q 11 )z) + (1 p 14 )(xq 12 + (1 q 12 )z)] = = z + q 12 ( 1 (x + y) z) } 2 {{ } <0 Since 1 2 (x + y) z < 0 q 12 = 0. However, this contradicts the assumptions aq 11 < bq 12 and bq 11 < aq 12, i.e., this cannot be an equilibrium. 15
17 Case 2: aq 11 < bq 12 and bq 11 = aq 12 In this case, player A s best response is p 21 = p 22 = 1 and p 13, p 14 can assume any value, i.e.: P = 0 0 p 13 p p 13 1 p 14 Given this strategy P, the probabilities for seeing message 1 and 2 become: p 1 = P rob(m = 1) = 1 4 (p 13 + p 14 ) p 2 = P rob(m = 2) = 1 4 (2 + (1 p 13) + (1 p 14 )) = (p 13 + p 14 ) = 1 p 1 Player B s expected payoff conditional on receiving message m = 1 becomes: E(B) = 1 4p 1 (p 11 (xq 11 + (1 q 11 )z) + p 12 (yq 11 + (1 q 11 )z)+ +p 13 (xq 11 + (1 q 11 )z) + p 14 (yq 11 + (1 q 11 )z)) = ( ) x y = z + q 11 p 13 + y z 4p 1 } {{ } M Player B s expected payoff conditional on receiving message m = 2 becomes: E(B) = 1 4p 2 (p 21 (yq 12 + (1 q 12 )z) + p 22 (xq 12 + (1 q 12 )z)+ +p 23 (yq 12 + (1 q 12 )z) + p 24 (xq 12 + (1 q 12 )z)) = ( = z + q 12 (1 + p 24 ) x y ) + y z 4p 2 } {{ } N As above (see cases V and W under disclosure), all possible combinations for M 0 and N 0 must be tested. The cases where either M or N are < 0 cannot be equilibria. In these cases, player B s best response would be to set the according q 11 or q 12 = 0, which 16
18 would contradict the basic assumption of case 2: aq 11 < bq 12 and bq 11 = aq 12. Figure 3 illustrates the remaining possible cases and makes it easy to see that also no other case can be a cooperative equilibrium. The cases where M > 0 and N 0 imply that q 11 = 1, which Figure 3: Graphical illustration for case 2 with disclosure contradicts the second condition of case 2: bq 11 = aq 12, since b > a. The case where M = 0 and N 0 lie outside the borders of the probability space as at least p 13 > 1). Therefore, no cooperative equilibrium can be maintained under case 2. 17
19 Case 3: aq 11 < bq 12 and bq 11 > aq 12 Given player B sets the acceptance probabilities like this, player A would react by setting the following P : P = Given this strategy P, the probabilities for seeing message 1 and 2 become: p 1 = P rob(m = 1) = 1 4 (p 13 + p 14 ) = 1 2 p 2 = P rob(m = 2) = 1 4 (p 21 + p 22 ) = 1 2 Player B s expected payoff conditional on message m = 1 is: E(B m = 1) = 1 2 (p 13(xq 11 + (1 q 11 )z) + p 14 (yq 11 + (1 q 11 )z)+). = z + zq 11 + yq 11 + q 11 x y 4p 1 = = z + q 11 ( 1 2 (x + y) z ) } {{ } <0 Player B would set q 11 = 0 which contradicts bq 11 > aq 12, i.e., this case cannot be an equilibrium. Case 4: aq 11 = bq 12 and bq 11 > aq 12 In this case, player A s best response would be p 13 = p 14 = 1 and p 11, p 12 can assume any value or: P = p 11 p p 11 1 p
20 Given this strategy P, the probabilities for seeing message 1 and 2 become: p 1 = P rob(m = 1) = 1 4 (p 11 + p ) p 2 = P rob(m = 2) = 1 4 (p 21 + p 22 ) Player B s expected payoff conditional on receiving message m = 1 becomes: E(B) = 1 4p 1 (p 11 (xq 11 + (1 q 11 )z) + p 12 (yq 11 + (1 q 11 )z)+ +p 13 (xq 11 + (1 q 11 )z) + p 14 (yq 11 + (1 q 11 )z)) = ( ) x y = z + q p 11 + y z 4p 1 } {{ } O Player B s expected payoff conditional on receiving message m = 2 becomes: E(B) = 1 4p 2 (p 21 (yq 12 + (1 q 12 )z) + p 22 (xq 12 + (1 q 12 )z)+ +p 23 (yq 12 + (1 q 12 )z) + p 24 (xq 12 + (1 q 12 )z)) = ( = z + q 12 (p 22 ) x y ) + y z 4p 2 } {{ } P As above (cases V and W under disclosure), all possible combinations for O 0 and P 0 must be tested. The cases where either O or P < 0 cannot be an equilibrium. In these cases, player B s best response would be to set the according q 11 or q 12 = 0, which would contradict the basic assumption of case 4: aq 11 = bq 12 and bq 11 > aq 12. Figure 4 illustrates the remaining possible cases and makes it easy to see that also no other case can be a (cooperative) equilibrium. The cases where O > 0 and P 0 imply that q 11 = q 12 = 1, which contradicts the first condition of case 4: aq 11 = bq 12. The cases where O = 0 and P > 0 imply that q 12 = 1. This also contradicts the first condition of case 4 (aq 11 = bq 12 ), since b > a. The remaining case, where P = O = 0, is the interception of the two lines of Figure 4. However, this interception lies outside the borders of probabilities (p 22 > 1). Therefore, no cooperative 19
21 Figure 4: Graphical illustration for case 4 without disclosure equilibrium can be maintained under case 4. Case 5: aq 11 > bq 12 and bq 11 > aq 12 In this case, player A s best response is p 11 = p 12 = p 13 = p 4 = 1 or P =
22 Always sending message m = 1 is not informative for player B. Analog to case 1, player B s expected payoff becomes: E(B) = z + q 11 ( 1 2 (x + y) z ) } {{ } <0 Since 1 2 (x+y) z < 0 q 11 = 0. This contradicts the assumptions aq 11 > bq 12 and bq 11 > aq 12, i.e. this case cannot be an equilibrium. Case 6: q 11 = q 12 = 0 Given player B never follows any recommendation, player A is indifferent to any strategy P. This can be a (noncooperative) equilibrium if player A uses a strategy P, so that player B does not want to deviate from his strategy q 11 = q 12 = 0. P = p 11 p 12 p 13 p 14 1 p 11 1 p 12 1 p 13 1 p 14 The probabilities for message 1 and 2 are: p 1 = P rob(m = 1) = 1 4 (p 11 + p 12 + p 13 + p 14 ) p 2 = P rob(m = 2) = 1 4 ((1 p 11) + (1 p 12 ) + (1 p 13 ) + (1 p 14 )) = = (p 11 + p 12 + p 13 + p 14 ) = 1 p 1 21
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