Consider The “Cheap Talk” Model We Saw In Class

Consider the “cheap talk” model we saw in class

Consider the “cheap talk” model we saw in class. Suppose everything else is the same as in class, but now we have two agents who announce their reports simultaneously. Suppose both agents have the same bias b > 0. (a) Is there still an uninformative equilibrium? Explain why or why not. (b) Suppose the government tells the experts “I will always set the policy equal to the lower of your two signals”. Is truth-telling now an equilibrium? Explain why or why not. (c) Suppose the government just gives up, and says “I’ll implement whatever policy you tell me if you agree, otherwise I’ll set it randomly.” Is this better or worse than the two-signal equilibrium we saw in class? Explain. Total for Question 1: .

Paper For Above instruction

The classic “cheap talk” model discusses communication between strategic agents where messages may be non-binding and potentially biased. In the scenario with two agents reporting simultaneously with identical positive bias b > 0, the existence of an uninformative equilibrium depends on the strategic incentives of the agents. Typically, in the standard model, the equilibrium where messages are completely uninformative is maintained when agents have no incentive to deceive or misreport, especially if their biases are symmetric and known to the receiver.

In the original framework, the uninformative equilibrium arises because agents' reports do not influence the outcome and they lack incentives to deviate from their truth-telling strategy. However, when the reports are made simultaneously and both agents share the same bias, the strategic landscape changes. Each agent knows that reporting truthfully may not lead to an outcome aligned with their bias, and since the government commits to choosing the lower signal, the agents have incentives to modify their reports strategically. As a result, the uninformative equilibrium can still exist, especially if agents anticipate their reports being ignored or manipulated without credible commitment.

When the government commits to always setting the policy to the lower of the two signals, truth-telling may become an equilibrium under certain conditions. If agents believe that honesty will result in their signals being reflected, and if their bias is aligned or can be mitigated by the government's policy rule, then truthful reporting can be an equilibrium. However, since both agents are biased positively and report simultaneously, they might have incentives to lower their reports artificially to influence the government's policy, thus potentially undermining full truthfulness.

The final scenario considers the government relinquishing control and simply implementing whatever the agents agree upon, or choosing policies at random if no consensus exists. This approach generally worsens the strategic environment because it provides no incentives for honest communication, possibly leading to misreporting or strategic manipulation. Compared to the equilibrium with the government choosing the lower signal, this laissez-faire approach is more problematic because it does not incentivize truthful communication or accurate signals, likely resulting in less efficient policies and greater misalignment with true preferences or information.

In the herding model we saw in class, suppose there were two types of experts: skilled and unskilled

In the herding model, suppose there are two types of experts: skilled with accuracy q_s = 0.80 and unskilled with accuracy q_u = 0.55. The key question is how many consecutive reports of G (good) are necessary to initiate a herd for each type. Since skilled experts are more accurate, fewer of their consistent reports are needed to trigger herding. Specifically, for the skilled experts, a smaller number n_s of consecutive G reports is sufficient to outweigh prior beliefs and convince others to follow G. Conversely, unskilled experts need a larger number n_u of reports because their signals are less reliable.

The precise thresholds depend on the prior probabilities and the Bayesian updating process. Typically, in the standard herding model, the number of reports needed to start a herd is derived by comparing the likelihood ratios. For skilled experts, it generally takes fewer than 3-4 consecutive G reports, while for unskilled experts, it might require 5 or more, depending on the prior and the accuracy parameters. These thresholds denote when the cumulative evidence becomes strong enough to persuade even skeptical observers that the event G is likely true.

When considering which group to consult first, as a government or decision-maker, it is advantageous to hear from the skilled experts initially. Since they provide more accurate signals, their opinions carry more weight in updating beliefs about the true state of the world. Listening to skilled experts first enhances the reliability of the information, thereby reducing the risk of herd manipulation based on less credible signals from unskilled experts. In essence, prioritizing skilled experts minimizes the chance of herding based on inaccurate or misleading reports.

Consider the game in Figure 1, where x ∈ [0, 1]

(a) There exists a Pure Bayesian Nash Equilibrium (PBNE) where S sets x = 0 if, in equilibrium, Player S believes that choosing x = 0 maximizes their expected payoff given Player W's beliefs. Typically, if W's expectations and incentives align, setting x = 0 could be justified, but this depends on the payoff structure and the strategic beliefs of W.

(b) A PBNE where both players set the same x is plausible when symmetry and identical objectives exist, or when players' incentives are aligned so that sharing a common strategy is optimal. Such equilibria often involve both players choosing a particular x that balances their payoffs or maximizes mutual benefits under the belief system.

(c) A PBNE where S sets x = 1 exists if, given W's beliefs and the strategic environment, Player S's best response is to choose x = 1. The existence of such an equilibrium depends on the payoff structure and whether the incentives favor the highest or lowest value of x.

(d) A better PBNE than in (c) might involve mixed strategies or alternative x values that improve expected payoffs for the players, or more stable equilibria that incorporate beliefs and payoff adjustments. The assessment of whether this is indeed better relies on the specific payoff matrix and strategic considerations.

Consider the game in Figure 2, with the described modifications

(a) To find all three NE, especially including the Markov Stable Nash Equilibrium (MSNE), we analyze the best responses in the matrix. The pure strategies (S,S) and (NS,NS) are often equilibria, and mixed strategies can also constitute NE. Expected payoffs are computed based on the strategies’ outcomes, with (S,S) likely yielding (0,0) if both cooperate, and (NS,NS) yielding (6,6) if neither cooperates.

(b) When a third-party observer reveals which cell occurs but not the specific action, the game transforms into a game tree with imperfect information. The observer's signal informs each player's perception, and the game involves information sets that group nodes accordingly, leading to new strategy considerations based on partial information.

(c) Following the strategy “Do what nature tells me to do” constitutes a Perfect Bayesian Equilibrium if players believe that nature's signals are truthful and consistent with equilibrium strategies. This approach can be justified if the signals are accurate and credible, aligning incentives and beliefs properly.

(d) The ex ante payoff in this PBNE is calculated based on the strategies where players follow natural signals and anticipate the other doing the same. Comparing this to earlier results shows whether strategic adaptation or information misalignment has led to higher or lower expected payoffs.

Credit Rationing Model

The credit rationing model involves a bank providing loans to a large number of firms, each with uncertain project outcomes. Firms are either safe or risky, with the safe being more probable but earning less, and risky less probable but potentially more profitable. The bank has a total fund α and sets a contract D, requiring repayment only if the project succeeds.

(a) If D > Rs, safe firms are unlikely to borrow because their expected return does not justify the loan if the repayment exceeds their project value. Risky firms might still borrow if their expected return is higher than D, assuming their probability of success exceeds a certain threshold. The bank would set D to attract the most profitable risk profile, typically D Rs to exclude safe firms that would not benefit from the loan.

(b) In equilibrium, the bank’s belief about the type of firm accepting the loan depends on D. If D is set high but still attractive to risky firms, the bank expects the firms borrowing to be mostly risky. Conversely, lower D might attract safer firms. Profit calculations involve the difference between D and the project's return, adjusted for the probability of success.

(c) To maximize profits, the bank sets D to balance attracting risky firms while ensuring repayment probability is maximized. The optimal D usually falls just below Rr, and expected profit equals the product of the fraction of risky firms accepted and the profit margin.

(d) If D ≤ Rs, safe firms would accept, but risky firms may not, especially if their expected return dips below D, discouraging risky borrowing. The bank’s profit analysis then shifts, likely reducing overall profits, and the optimal D would be recalibrated accordingly.

(e) As β approaches zero, meaning very few safe firms exist, the bank's profit is driven mainly by risky firms. In this case, the bank would favor D close to Rr, maximizing profit from risky projects. Profitability depends on the risk-return trade-off, and when safe firms are negligible, the profit profile simplifies.

(f) Not all risky firms can find financing if the bank’s D is set too high, excluding some risky projects or if the total funds α are insufficient to cover all willing risky firms. This scenario introduces market inefficiencies, leading to adverse selection or credit rationing, which could be problematic for optimal risk allocation and economic efficiency.

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