Strategic Information Acquisition and Transmission

Transcription

1 Strategic Information Acquisition 1 Strategic Information Acquisition and Transmission By Rossella Argenziano, Sergei Severinov and Francesco Squintani (American Economic Journal: Microeconomics, August 2016) Presenters: Ben Andrews, Nicole Florack, Minyi Zhu

2 Strategic Information Acquisition 2 Outline Motivation Model Set- up Benchmarks: Direct Information Acquisition and Delegation The Communication Game Implications for Organisation Design Conclusions

3 Strategic Information Acquisition 3 Motivation Information transmission via cheaptalk Seminal paper: Crawford andsobel (Etrica, 1982) Common assumption: perfect information is exogenously given to the sender for free. Results: 1) Cheaptalk generally leads to imprecise decisions. Any information transmission, if it occurs at all, is coarse. Therefore, the principal would be strictly better off if he could somehow receive the information signal himself. 2) Because of this imprecision, the principal prefers delegation of the decision making to the agent over communication wherever communication could be informative.

4 Strategic Information Acquisition 4 Cheap talk with costly information acquisition More realistic assumptions: 1) information is costly. 2) the decision to acquire it is taken endogenously. Do the results of basic cheap talk models continue to hold under these assumptions? Results: Information acquisition and strategic communication by the agent to the principal may lead to more precise information and higher expected utility for the principal compared to centralization or delegation even if: 1) the agent is biased. 2) the agent is no more efficient than the principal at acquiring information. How does this work? Overinvestment and limited information loss

5 Strategic Information Acquisition 5 Overview of the Argenziano et al. model The model is an extension of the Crawford and Sobel (1982) cheap talk model of strategic communication between an expert (agent) and a decision maker (principal). In the Crawford and Sobel model, the agent learns about the state of the world exogenously and without bearing any cost. Whereas in this model, acquiring information is costly and how much to acquire is chosen by the agent. Information is acquired by conducting a series of trials/experiments that are informative about the state of the world. Although the process of information acquisition assumed in the paper is quite simple, the driving forces behind the results extend to other models of information acquisition.

6 Strategic Information Acquisition 6 The players and their preferences Two players: a decision maker (principal) and an expert (agent) Preferences: Principal: U " y, θ = y θ ( P s ideal: θ Agent: U * y, θ, b = y θ b ( cn A s ideal (given n): θ + b Where: y is the decision θ is the state of the world; θ ~ U[0,1] b 0 is the bias of the agent (the divergence in preferences between the agent and the principal). It is common knowledge. cn is the cost of information acquisition when n trials are performed in the following way.

7 Strategic Information Acquisition 7 How information is acquired The state of the world is initially unknown to both the principal and the agent. They have the same common knowledge prior beliefs that θ ~ U[0,1] To learn about θ, the agent undertakes trials/experiments. Each trial results in either success or failure, with probability of success equal to the true θ. The results of trials are independent, conditional onθ. The agent chooses the number of trials n to perform. Each trial has cost c > 0. The number of trials n is observable to both the principal and agent, whilst the number of successful trials is observable only to the agent.

8 Strategic Information Acquisition 8 Timing of the communication game 1. Both the agent and principal are initially uninformed about θ 2. The agent chooses the number of trials n to conduct in order to acquire information about θ (the choice of n is made once and for all) 3. The agent learns the results of the n trials 4. The agent sends a costless and unverifiable message m to the principal. 5. The principal updates beliefs about θ and then chooses y

9 Strategic Information Acquisition 9 Preliminaries for solving the model Given n and θ, and the Bernoulli feature of the trials, the number of successes k from n trials has a Binomial distribution: f k n, θ = =!?! =@?! θ? (1 θ) =@?, for 0 k n It follows that the posterior distribution of θ given k successes in n trials is distributed Beta(k + 1, n k + 1) Using the mean of the Beta distribution, the posterior expectation of θ given k successes in n trials is therefore: θh?,= : = E θ k, n = k + 1 n + 2

10 Strategic Information Acquisition 10 Benchmarks: Direct Information Acquisition and Delegation Before solving the model with communication between the agent and the principal, we describe two benchmarks: Benchmark 1: Direct Information Acquisition Here, the principal acquires information herself, before making a decision. She acts without the agent. Benchmark 2: Delegation Here, the principal delegates both the information acquisition task and the decision makingauthority to the (biased) agent. Why do we consider these benchmarks? Because once we have solved the main model (in which there is communication), we can compare the three organisational structures (communication vs delegation vs direct information acquisition.

11 Strategic Information Acquisition 11 Benchmark 1: Direct Information Acquisition The principal acquires information herself and then makes a decision based on the information acquired (no role for the agent) I.e. the principal chooses how many trials to conduct herself and bears the costs Because she conducts the trials herself, she observes the number of successes, k Hence, after learning the number of successes, the decision taken would be simply the posterior expectation of θ given k successes in n trials (recall from above): y MNOPQR *QS = E θ k, n = θh?,= = k + 1 n + 2

12 Strategic Information Acquisition 12 Benchmark 1: Direct Information Acquisition The expected utility of the principal when she performs n trials herself is: E (θh?,= θ) ( n cn = T U =V( cn ( ) where θh?,= : = E θ k, n =?VT =V( The principal chooses the number of trials n to maximise her payoff (given by ) Bearing in mind that n must be discrete, the optimal number of trials n c can be characterised by n 1 c = max {n: 6 n + 2 cn > 1 c(n 1)} 6 n Intuition: keep increasing n until the net marginal benefit (the increase in ( )) stops being positive.

13 Strategic Information Acquisition 13 Benchmark 2: Delegation Under delegation, both information acquisition and the decision making responsibility are delegated to the (biased) agent. What decision will the agent make after observing the results of n trials? The agent s ideal point is θ + b. But he does not receive perfect information about θ. So after observing the results of the trials, his best estimate of his ideal point and the decision he would make is: y MP_P` = E θ + b k, n = E θ k, n + b = θh?,= + b = k + 1 n b So, the agent s expected payoff (for a given n) is: E (θh?,= + b θ b) ( n cn = E θh?,= θ ( n cn N.B., this is the same as the principal s expected payoff under direct information acquisition, for a given n

14 Strategic Information Acquisition 14 Benchmark 2: Delegation Proposition. Under delegation, the agent will perform the same number of trials as the principal would choose if she were to acquire information directly. Intuition: The agent chooses n to maximise his expected payoff given by. But because the objective function here is the same as the principal s expected payoff under direct information acquisition ( ), the optimal choice of n is the same in both benchmarks. Hence, under delegation, the agent will choose to perform n c trials.

15 Solving the communication game: PBE First let s suppose n is exogenously imposed. Then the PBE would be described by the agent s partition of {0,1,, n} into messages about k, that is, for some integers, k T, k (,, k c@t 0,1,, n, p T = 0,, k T, p ( = k T + 1,, k (,...,p c = k c@t + 1,, n and k p N m = m N the principal s belief about the smallest set to contain k, which we denote p. the principal s decision variable y, which may respond to the message m = m N where the strategies are sequentially rational and the beliefs are consistent. Now, in order for the agent to choose the optimal n, it must be that we have specified for each n 0,1,2,, the agent s communication partition and for each possible message from the partition, the principal s beliefs and decision variable (remark: helpful to think of n as different subgames) Ben, Nicole and Minyi Strategic Information Acquisition 15

16 Strategic Information Acquisition 16 PBE analysis: The principal s optimal decision Proposition: For any n and for any message m, given the principal s most precise belief about k, i.e. k p, the principal s only optimal strategy is to set y p = E θ n, k p. P chooses y to maximize E u " y, θ n, k p where u " y, θ = y θ (. Hence the principal s optimal action is y p = E θ n, k p which he can calculate using E θ n, k =?VT =V(.

17 Strategic Information Acquisition 17 PBE analysis: The agent s incentive compatible partition in an informative subgame Proposition: Consider a communication partition of {0,1,, n} into sets p T, p (, p c where p T = 0,, k T, p ( = k T + 1,, k (,...,p c = k c@t + 1,, n. Suppose the principal believes that k p N m = m N. Then this partition that the principal believes that agent is using is incentive compatible (and therefore the principal s belief can be consistent) if and only if for all i = 1,2,, N 1, 4b n p NVT p N 4b n In contrast to Crawford- Sobel model, full revelation is incentive compatible if and only if 4b n

18 PBE analysis: Summing up the subgames Principal Agent Utility function y θ ( y θ + b ( cn Realized utility given k p Ex ante expected utility E θ n, k p θ ( E θ n, k p θ + b ( cn E Var θ n, k p E Var θ n, k p b ( cn Ben, Nicole and Minyi Strategic Information Acquisition 18

19 Strategic Information Acquisition 19 PBE analysis: The threat of babbling equilibria The principal s ability to observe and therefore respond to the level of trials the agent chooses can lead to equilibria in which the agent overinvests in information acquisition (i.e. chooses n > n c ). Suppose for all n n c, P believes that A s message m about k is babbling. Then for all n n c, a babbling subgame equilibrium is played. Note that if the agent chooses n n c, it must be that the agent choses n = 0 because for any n n c, the principal is just going to ignore any information the agent acquires so there s no point in paying to acquire it.

20 Strategic Information Acquisition 20 PBE analysis: The optimal number of trials Consider the agent s choice between choosing n = 0 and n = n c + 1. Suppose that the agent s bias b is sufficiently small relative to n = n c + 1 so that the full revealing partition is incentive compatible in the n = n c + 1 subgame. A s ex ante expected utility in Any given subgame E Var θ n, k p b ( cn The n c + 1 fully revealing subgame E Var θ n c + 1, k b ( c n c + 1 The 0 trial babbling subgame E Var θ b ( The delegation case E Var θ n c, k cn c When c is small (so n c is large), the payoff from the n = n c + 1 subgame is very close to the first best, while the payoff from the n = 0 subgame is very far away so the agent prefers the full revelation n = n c + 1 subgame to the n = 0 subgame.

21 Strategic Information Acquisition 21 PBE analysis: The optimal number of trials Consider the agent s choice between choosing n = 0 and n = n c + 1. Suppose that the agent s bias b is sufficiently small relative to n = n c + 1 so that the full revealing partition is incentive compatible in the n = n c + 1 subgame. A s ex ante expected utility in Any given subgame The n c + 1 fully revealing subgame E Var θ n c + 1, k b ( c n c + 1 The 0 trial babbling subgame The delegation case E Var θ n c, k cn c When c is small (so n c is large), the payoff from the n = n c + 1 subgame is very close to the first best, while the payoff from the n = 0 subgame is very far away so the agent prefers the full revelation n = n c + 1 subgame to the n = 0 subgame.

22 Strategic Information Acquisition 22 PBE analysis: The optimal number of trials Consider the agent s choice between choosing n = 0 and n = n c + 1. Suppose that the agent s bias b is sufficiently small relative to n = n c + 1 so that the full revealing partition is incentive compatible in the n = n c + 1 subgame. A s ex ante expected utility in Any given subgame The n c + 1 fully revealing subgame The 0 trial babbling subgame E Var θ b ( The delegation case E Var θ n c, k cn c When c is small (so n c is large), the payoff from the n = n c + 1 subgame is very close to the first best, while the payoff from the n = 0 subgame is very far away so the agent prefers the full revelation n = n c + 1 subgame to the n = 0 subgame.

23 Strategic Information Acquisition 23 PBE analysis: The optimal number of trials Consider the agent s choice between choosing n = 0 and n = n c + 1. Suppose that the agent s bias b is sufficiently small relative to n = n c + 1 so that the full revealing partition is incentive compatible in the n = n c + 1 subgame. A s ex ante expected utility in Any given subgame The n c + 1 fully revealing subgame E Var θ n c + 1, k b ( c n c + 1 The 0 trial babbling subgame E Var θ b ( The delegation case E Var θ n c, k cn c When c is small (so n c is large), the payoff from the n = n c + 1 subgame is very close to the first best, while the payoff from the n = 0 subgame is very far away so the agent prefers the full revelation n = n c + 1 subgame to the n = 0 subgame.

24 Strategic Information Acquisition 24 PBE analysis: The optimal number of trials Then, if we have (1) b sufficiently small to allow for full revelation and (2) the cost c of a trial sufficiently small so that the agent prefers the full revelation n = n c + 1 subgame to the n = 0 babbling subgame, the principal can induce the agent to overinvest in information acquisition so that the principal has more precise information when choosing her action than she would if she had acquired the information herself. Moreover, under these conditions, the principal receives strictly higher expected utility by allowing the agent to acquire information. More generally, it can be shown that this remains true even when full revelation is not possible, as long as the loss of information in the informative, overinvestment subgame is not too large so that the precision gains from overinvestment outweigh precision losses from coarse information transmission.

25 Strategic Information Acquisition 25 Conclusions Decisions based on a biased expert s advice may be more precise than optimal choices based on direct information acquisition, even if the expert is not more efficient than the principal at acquiring information. Intuition for organisation design: whenthe conflict of interests withinthe organisationis small: 1) Communication- based organisation outperform delegation as well as direct information acquisition by the principal. 2) Partnerships are better off dividing the information acquisition and decision making among the partners, rather than assigning both tasks to a single partner.

26 Strategic Information Acquisition 26 Future Research Questions What if the decision- maker is able to subsidize the expert s information acquisition cost? What if the expert acquires the information covertly but has an option to verifiably disclose the amountof information that he acquired?

27 Strategic Information Acquisition 27 Appendices

28 Appendix: Derivation of ( ) Intuition: the number of successes (k) is a random variable, so to find the ex ante expected payoff, you need to take expectations over k and θ

29 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT The first inequality makes an agent who observes k = k N (the highest integer in p N ) willing to report m N rather than m NVT. The agent s optimal point is y = E θ n, k N + b The principal s response to m N is y p N = E θ n, k p N E θ n, k N The principal s response to m NVT is y p NVT = E θ n, k p NVT > E θ n, k N

30 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT The first inequality makes an agent who observes k = k N (the highest integer in p N ) willing to report m N rather than m NVT. The agent s optimal point is y = E θ n, k N + b The principal s response to m N is y p N = E θ n, k p N E θ n, k N The principal s response to m NVT is y p NVT = E θ n, k p NVT > E θ n, k N

31 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT The first inequality makes an agent who observes k = k N (the highest integer in p N ) willing to report m N rather than m NVT. The agent s optimal point is y = E θ n, k N + b The principal s response to m N is y p N = E θ n, k p N E θ n, k N The principal s response to m NVT is y p NVT = E θ n, k p NVT > E θ n, k N

32 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT The second inequality makes an agent who observes k = k N + 1 (the smallest integer in p NVT ) willing to report m NVT rather than m N. The agent s optimal point is y = E θ n, k N b The principal s response to m NVT is y p NVT = E θ n, k p NVT E θ n, k N + 1. The principal s response to m N is y = E θ n, k p N < E θ n, k N + 1.

33 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT The second inequality makes an agent who observes k = k N + 1 (the smallest integer in p NVT ) willing to report m NVT rather than m N. The agent s optimal point is y = E θ n, k N b The principal s response to m NVT is y p NVT = E θ n, k p NVT E θ n, k N + 1. The principal s response to m N is y = E θ n, k p N < E θ n, k N + 1.

34 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT The second inequality makes an agent who observes k = k N + 1 (the smallest integer in p NVT ) willing to report m NVT rather than m N. The agent s optimal point is y = E θ n, k N b The principal s response to m NVT is y p NVT = E θ n, k p NVT E θ n, k N + 1. The principal s response to m N is y = E θ n, k p N < E θ n, k N + 1.

35 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT

36 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT

37 Recall: The condition for the partition to be incentive compatible is 4b n p NVT p N 4b n where p N = k N@T + 1,, k N and p NVT = k N + 1,, k NVT