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The following model setup is from the paper Dynamic Information Provision: Rewarding the Past and Guiding the Future by Ian Ball. It generalizes both the ideas of strategic information transmission of Vincent P. Crawford and Joel Sobel and Dirk Bergemann and Stephen Morris in information design.

$\textbf{The Model}$

We assume a infinite horizon continuous time model of two players, a sender (S) and a receiver (R). At each point in time $t\in [0,\infty)$, the state $\omega_t\in\mathbf{R}$ is realized and the receiver chooses an action $\alpha_t\in\mathbf{R}$. Flow payoffs for the receiver and sender, respectively, are given by \begin{equation}\tag{*}u_{R}\left(\alpha_t,\omega_t\right)=-(\alpha_t-\omega_t)^2,\quad\quad\quad u_{S}\left(\alpha_t,\omega_t\right)=-(\alpha_t-\omega_t-\beta)^2\end{equation}

The receiver wants to match his action with the state, but the sender wants the receiver to shift his action above the state by $\beta$. Without loss, assume $\beta>0$. The bias $\beta$ parametrizes the preference misalignment between the sender and receiver. The initial state $\omega_0$ is normally distributed with mean $\mu_0$ and variance $\sigma_0^2$. The state then evolves according to the stochastic differential equation

$$\tag{**}d\omega_t=\kappa\omega_tdt+\sigma dZ_t$$

where the driving process $\{Z_t\}_{t≥0}$ is a standard Brownian motion, independent of the initial state $\omega_0$. The variance rate $\sigma^2$ is strictly positive. The percentage drift $\kappa$ has unrestricted sign. If $\kappa < 0$, then the state follows a mean-reverting Ornstein–Uhlenbeck process. If $\kappa = 0$, then $\omega_t − \omega_0 = \sigma Z_t$, so the state follows a Brownian motion with zero drift.15 If $\kappa > 0$, then the state process is explosive. The solution of $(**)$ is

$$\omega_t=\omega_0 e^{-\kappa t} +\sigma\int_{0}^te^{\kappa(s-t)}dZ_s$$

The effect of $\kappa$ can be seen in the formula for the conditional expectation: For $s > t ≥ 0$, we have $E[\omega_s|\omega_t] =e^{\kappa(s−t)}\omega_t$.

The state process is common knowledge, but the state realizations are observed only by the sender. The sender also observes the receiver’s actions. Before observing anything, the sender first commits to a dynamic information policy $S$, which specifies a signal space $\mathbb{S}$ and assigns a signal distribution to each history $h_t = (\omega_{\tau}, s_{\tau}, a_{\tau})_{\tau<t}$ of past states, signals, and and actions.

Once the sender commits to an information policy, the receiver faces a sequential decision problem. At each time t, after observing the signals sent up to time t, the receiver updates his beliefs and chooses a (pure) action. The sender’s signals are the receiver’s only source of information about the state. In particular, the receiver does not observe past flow payoffs. Otherwise, from his own action history and utility history, the receiver could pin down the current state realization to at most two values.

The receiver’s action choices over time determine a stochastic process $A = \{A_t\}_{t≥0}$, called a decision rule as in Bergemann and Morris (2016). The sender and receiver have a common discount rate $r$. Assume $r > 2\kappa$. This ensures that taking a constant action yields a finite expected utility for both players. The payoffs from a decision rule $A$ are given by the expected discounted accumulated flow utilities

\begin{equation}\tag{***}u_{R}\left(\alpha_t,\omega_t\right)=-\mathbb{E}\left[\int_0^{\infty}re^{-rt}(A_t-\omega_t)^2dt\right],\quad u_{S}\left(\alpha_t,\omega_t\right)=-\left[\int_0^{\infty}re^{-rt}(A_t-\omega_t-\beta)^2dt\right]\end{equation}

Given an information policy $S$, the receiver chooses his actions over time to maximize his expected utility. More formally, the receiver chooses from among all decision rules $A$ that are compatible with $S$ in the sense that, at each time $t$, the sender’s signals up to time $t$ provide sufficient information for the receiver to set his action equal to $A_t$. A decision rule $A$ is a best response to an information policy $S$ if $A$ maximizes the receiver’s utility over all decision rules compatible with $S$. Anticipating the receiver’s best response, the sender chooses an information policy to maximize her expected utility. Formally, the sender’s problem is to maximize $u_S(A)$ over all pairs $(S, A)$ with the property that $A$ is a best response to $S$.

$\textbf{Questions}$

  1. By reading this part, I can not understand how to write down the maximization problems of both the sender and the receiver. The rest of the paper is not making it easier for me. So how can someone write down the problems of the agents and solve them?

  2. Why is the receiver myopic? Could we change the assumption to make her forward looking in the sense that she monitors her past actions (pure or mixed)?

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    $\begingroup$ Have not read these papers, but I will bet right now is that the answer to 2. is "because the math is much easier with this assumption", or possibly "otherwise zero/infinitely many equilibria exist". $\endgroup$
    – Giskard
    Sep 21, 2022 at 15:55

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