Information Design

It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages. — Adam Smith, The Wealth of Nations.

What is Information Design?

Communication does not just happen in fully cooperative scenarios. In some cases, the sender can persuade the receiver by “strategically deceiving”, to increase its own expected payoff. Actually, “one quarter of gdp is persuasion.” (McCloskey & Klamer 1995).

Nontrivially, “successful partially deceiving” is a better equilibrium than “saying nothing” and “revealing all information”. Information design is the study of this persuasion. And Bayesian persuasion is a special case of information design, which consists of a sender and a receiver.

The problem of information design can be equivalently approached from various perspectives:

1. $\max_{\varphi} \mathbb{E}_{s \sim \mu_0}\left[\mathbb{E}_{\sigma \sim \varphi(\cdot\mid s)}\left[r^i(s,a^*(\sigma))\right]\right]$.
2. $\max_{\varphi} \mathbb{E}_{\mu\sim\tau}\Big[\mathbb{E}_{s\sim\mu}\big[r^i(s,a^*(\mu))\big]\Big]$, s.t. $\mathbb{E}_{\mu\sim\tau}(\mu) = \mu_0$.
3. In a two-signal case: The sender manipulates the receiver’s posterior beliefs (each corresponding to a sent signal) to find the highest intersection point of the line segment $(\mu_1 - \mu_2, \hat{v}_1 - \hat{v}_2)$ and $x = \mu_0$.
4. Denote the concavification of $\hat{v}(\mu) = \mathbb{E}_{s\sim\mu}\big[r^i(s,a^{*}(\mu))\big]$ as $f(\mu)$. The sender’s optimal expected payoff is $f(\mu_0)$.
5. The sender select an optimal Bayes correlated equilibrium given an objective function.
6. $\max_{\varphi} \mathbb{E}_{s \sim \mu_0}\left[\mathbb{E}_{\sigma \sim \varphi(\cdot\mid s)}\left[r^i(s,a^*(\sigma))\right]\right]$, s.t. Obedience.

Papers

The following part of this note is to summarize the essence of these papers:

1. Bayesian Persuasion (Kamenica & Gentzkow 2011)
2. An equilibrium perspective:
   1. Correlated Equilibrium in Games with Incomplete Information (Bergemann & Morris 2011a)
   2. Robust Predictions in Games with Incomplete Information (Bergemann & Morris 2011b)
   3. Bayes Correlated Equilibrium and the Comparison of Information Structures in Games (Bergemann & Morris 2016)
3. Surveys:
   1. Bayesian Persuasion and Information Design (Kamenica 2019)
   2. Algorithmic Information Structure Design: A Survey (Dughmi 2019)
   3. Information Design: A Unifified Perspective (Bergemann & Morris 2019)

Timing

Consider a persuasion between a sender and a receiver (named as Bayesian persuasion). The timing of it is:

1. The sender chooses a signaling scheme $\varphi$. (Given a state $s$, $\varphi(\cdot \mid s)$ is a distribution of $\sigma$.)
2. The receiver observes this $\varphi$.
3. The environment generates a state $s$.
4. A signal $\sigma$ is sampled accroding to the commited $\varphi(\cdot\mid s)$.
5. The receiver calculates its posterior belief $\mu(\cdot \mid \sigma)$, and then chooses an action $a$ that maximizes $\mathbb{E}_{\mu}(r^j)$. (Thus the term Bayesian persuasion.)
6. The sender and the receiver get rewards $r^i(s,a)$ and $r^j(s,a)$, respectively.

The core of information design is to find the optimal signaling scheme $\varphi$ for the sender and to study the nature of this problem. But the sender’s payoff is determined by the receiver’s action, so it has to use its information advantage to influence the receiver’s behavior (by designing $\varphi$), thereby indirectly optimizing its own expected payoff.

Important Examples

1. Recommendation Letter (Example 2.1 in Dughmi 2019).
2. Courtroom (The first example in Section 2.2 of Kamenica 2019).
3. Routing Game (Example 3.2 in Dughmi 2019) or Braess’s paradox.

Assumptions

1. The prior $\mu_0$ of states is common knowledge.
2. The receiver is self-interested.
   • The receiver has its objective function to optimize (In my understanding it does not need to be consistent with its epected environmental reward), and the sender wants to influence the receiver’s behavior.
   • The sender’s objective function does not need to be consistent with its environmental reward, e.g., it can be social welfare $r^i+r^j$. (In this way, the sender is altruistic.)
   • Anyway, there is no limit to the goals of the sender and the receiver. They may be fully cooperative, mixed motived or adversarial.
3. The receiver’s strategy is based on Bayes’ rule (i.e. Bayesian rational). (Timing 5)
4. Commitment: The sender will honestly commit a signaling scheme to the receiver before the interaction with the receiver. (Timing 1-2)
   • It is this that makes Bayesian persuasion (or information design) different from other communication models, e.g. cheap talk, verifiable message, signaling games. (Kamenica 2019)
   • Justifications:
     • Reputation
     • The signaling schemes are common knowledge, e.g., the sender (prosecutor) can choose investigative means (forensic tests, calling witness, etc.) to partially reveal the truth, which are public to everyone. (Courtroom)
     • (Justifications vary across applications. See Section 2.2 in Kamenica 2019)
5. An analysis analogues to the revelation principle: The optimal scheme needs no more signals than the number of states of nature.
   • Proof: probability simplex + Caratheodory’s theorem. (Dughmi 2019)

Simplification and Geometric Interpretations

Reformulation

• The receiver’s strategy:
  • Given a sent $\sigma$, its posterior belief is $\mu(\cdot\mid \sigma)$.
  • Then it chooses $a^{*}(\sigma) = a^{*}(\mu(\cdot\mid\sigma)) = \arg\max_{a}\mathbb{E}_{s\sim\mu(\cdot\mid\sigma)}\left[r^j(s,a)\right]$.
  • The mapping from signals to posterior beliefs is many-to-one: All the $\sigma$ that induce the same posterior belief $\mu(\cdot\mid \sigma)$, the receiver will behave the same way (thus the payoffs of both agents).
• The distribution of posterior beliefs $\tau$
  • Every sent signal induces a specific posterior belief: Given a committed signaling scheme $\varphi$ and a sent signal $\sigma$, the receiver’s posterior belief is $\mu(\cdot\mid\sigma)$. Calculated as $\mu(s_i \mid\sigma) = \frac{\mu_0(s_i)\cdot \varphi(\sigma\mid s_i)}{\sum\limits_{s_j}\mu_0(s_j)\cdot \varphi(\sigma\mid s_j)}$.
  • Again: The mapping from signals to posterior beliefs is many-to-one.
  • A distribution of signals corresponds to a distribution of posterior beliefs: Before the signal $\sigma$ realized, the receiver can only estimate its distribution by the committed signaling scheme: $\sigma\sim p^1_{\mu_0, \varphi}(\cdot) = \sum\limits_{s}\mu_0(s)\cdot \varphi(\cdot\mid s)$. Then $\mu\sim p^2_{\mu_0, \varphi}(\cdot) = \sum\limits_{\sigma: \mu(\cdot\mid \sigma) = \mu}\sum\limits_{s} \mu_0(s)\cdot\varphi(\sigma\mid s)$.
  • A $\tau$ induced by a $\varphi$ is denoted as $\tau_{\mu_0,\varphi}$.
• The sender’s optimization problem
  • Original optimization: $\max_{\varphi} \mathbb{E}_{s\sim\mu_0}\Big[\mathbb{E}_{\sigma\sim\varphi(\cdot\mid s)}\big[r^i(s,a^*(\sigma))\big]\Big]$.
  • Reformulation:
    • An equivalent objective function: $\mathbb{E}_{\mu\sim\tau}\Big[\mathbb{E}_{s\sim\mu}\big[r^i(s,a^*(\mu))\big]\Big]$.
    • The only contraint: $\mathbb{E}_{\mu\sim\tau}(\mu) = \mu_0$. (i.e. $\tau = \tau_{\mu_0,\varphi}$. This is named as Bayes plausible. See the next subsection.)

Bayes plausible

The sender’s optimization problem can be approached from the perspective of any arbitrary distribution of induced posterior beliefs that satisfies the Bayes plausible.
Understanding this is important for grasping the concepts of geometric interpretation (concavification) in the next subsection.

If an arbitrary $\tau$ satisfies $\mathbb{E}_{\mu\sim\tau}(\mu) = \mu_0$, then this $\tau$ is Bayes plausible. (Kamenica 2019)

• What is necessary and sufficient:
  • Every $\tau_{\mu_0,\varphi}$ (a $\tau$ induced by the $\varphi$) is Bayes plausible. (It can be proved by the law of iterated expectations. Kamenica 2019)
  • If a $\tau$ is Bayes plausible, then it can be induced by a $\varphi$. (Kamenica 2019. Proved in Kamenica & Gentzkow 2011.)
• A geometric interpretation: probability simplex (Figure 2 of Dughmi 2019).
  • A point in the probability simplex represents a distribution of singals (a convex combination of signals).
  • The blue point is the prior belief. The red points are posterior beliefs.
  • Bayes plausible = The blue point is inside the simplex of the red points.

Concavification

Does the sender benefit from persuasion? What is an optimal mechanism?

This technique is best described with a two-signal example (Figure 1 of Kamenica & Gentzkow 2011, Kamenica 2019):

• Assume that there are $n$ states. Then a posterior belief $\mu$ is a point in $\mathbb{R}^{n-1}$. (2011)
• The values of $x$-axis are in $\mathbb{R}^{n-1}$. Each represents a posterior belief $\mu$. (2011)
• A $\mu$ induces an expected payoff of the sender $\hat{v}(\mu) = \mathbb{E}_{s\sim\mu}\big[r^i(s,a^{*}(\mu))\big]$. The black line in the figure denotes this function. (2011,2019)
• $\sigma_1$ and $\sigma_2$ induce posterior beliefs $\mu_1(\cdot\mid\sigma_1)$ and $\mu_2(\cdot\mid\sigma_2)$ respectively. These two $\mu$ correspond to two values on the $x$-axis, and indicate two expected payoff of the sender $\hat{v}_1$ and $\hat{v}_2$. (2019)
• $\mu_1$ and $\mu_2$ are almost arbitrary: The distribution of $\mu_1$ and $\mu_2$ should be Bayes plausible, i.e., their expectation should be $\mu_0$.
  • Assume that the distribution of $\mu$ is $\tau = (k, 1-k)$. Then $k\cdot\mu_1+(1-k)\cdot\mu_2 = \mu_0$. Thus if $\mu_{i}$ is on the left side of $\mu_0$ on the $x$-axis, then $\mu_{1-i}$ is on the right side. Unless they both coincide with $\mu_0$.
  • Also, $\mathbb{E}_{\mu\sim\tau}(\hat{v}(\mu)) = k\cdot \hat{v}_1+(1-k)\cdot\hat{v}_2$, which is the sender’s expected payoff (obejective). It means that: When the receiver’s posterior beliefs are $\mu_1,\mu_2$ (after receiving $\sigma_1, \sigma_2$ respectively), the sender will gets $\mathbb{E}_{\mu\sim\tau}(\hat{v}(\mu))$.
  • Since the coordinates are proportional, connecting $(\mu_1, \hat{v}_1)$ and $(\mu_2, \hat{v}_2)$ to get a line segment $l$, $(\mu_0, \mathbb{E}_{\mu\sim\tau}(\hat{v}(\mu)))$ is on $l$. In this way, the sender needs to manipulate the receiver’s posterior beliefs to find the highest intersection point.
• A concavification of $\hat{v}(\mu)$ is the smallest concave function everywhere greater than $\hat{v}(\mu)$. (The red line in Figure 1 of 2011, 2019)
  • The concavification of $\hat{v}(\mu)$ evaluated at $\mu_0$ equals $\max\set{y\mid(\mu_0, y)\in co(\hat{v})}$, where $co(\hat{v})$ denotes the convex hull of the graph of $\hat{v}$, i.e., the light blue region in Figure 1 of 2011.
• In this case, a signaling scheme with more than two signals cannot improve the sender’s expected payoff. (Kamenica 2019)

Corollaries and propositions

(Kamenica & Gentzkow 2011):

• Corollary 1: The sender benefits from persuasion if and only if there exists a Bayes plausible distribution $\tau$ of posterior belief $\mu$ such that $\mathbb{E}_{\mu\sim\tau}\Big[\mathbb{E}_{s\sim\mu}\big[r^i(s,a^{*}(\mu))\big]\Big] > \mathbb{E}_{s\sim\mu_0}\big[r^i(s,a^{*}(\mu_0))\big]$. The sender’s optimal expected payoff is $\max_{\varphi} \mathbb{E}_{\mu\sim\tau}\Big[\mathbb{E}_{s\sim\mu}\big[r^i(s,a^*(\mu))\big]\Big]$, s.t. $\mathbb{E}_{\mu\sim\tau}(\mu) = \mu_0$.
• Corollary 2: The sender benefits from persuasion if and only if $f(\mu_0)>\hat{v}(\mu_0)$, where $\hat{v}(\mu) = \mathbb{E}_{s\sim\mu}\big[r^i(s,a^{*}(\mu))\big]$, and $f(\mu)$ is the concavification of $\hat{v}(\mu)$. The sender’s optimal expected payoff is $f(\mu_0)$.
• Proposition 2: If $\hat{v}(\mu)$ is concave, the sender does not benefit from persuasion for any prior $\mu_0$. If $\hat{v}(\mu)$ is convex and not concave, the sender benefits from persuasion for every prior $\mu_0$.
• Proposition 3: If there is no information the sender would share, the sender does not benefit from persuasion.
  • “There is information the sender would share” is defined as $\exists \mu$, s.t. $\hat{v}(\mu)> \sum\limits_{s} \mu(s)\cdot r^j(s, a^{*}(\mu_0))$.
  • In other words, there must exist a $\mu$ such that, if the sender had private information that led it induces $\mu$, it would prefer to share this information with the receiver rather than have the receiver act based on $\mu_0$.
  • Otherwise, the sender does not have the motivation to share its information.
• Proposition 4: If there is information the sender would share, and the receiver’s preference is discrete at the prior, the sender benefits from persuasion. (sufficient)
  • “The receiver’s preference is discrete at belief $\mu$” is defined as: If there is an $\epsilon>0$ s.t. $\forall a\ne a^{*}(\mu)$, $\sum\limits_{s}\mu(s)\cdot r^j(s, a^{*}(\mu)) > \sum\limits_{s} \mu(s)\cdot r^j(s, a)+\epsilon$.
  • In other words, the receiver’s expected payoff from its preferred action $a^{*}(\mu)$ is bounded away from its expected payoff from any other action.
  • Otherwise, the sender does not have the ability to influence the receiver.
• Proposition 5: If $A$ is finite, the receiver’s preference is discrete at the prior generically.
• Proposition 7: An optimal mechanism exists.
• Proposition 8: For any prior,
  • if $\hat{v}(\mu)$ is (strictly) concave, no disclosure is (uniquely) optimal;
    • no disclosure: $\mu = \mu_0$.
  • if $\hat{v}(\mu)$ is (strictly) convex, full disclosure is (uniquely) optimal;
    • full disclosure: $\mu(\cdot\mid\sigma)$ is degenerate for all $\sigma$ sent in equilibirum. $\mu$ is degenerate if there is $s$ s.t. $\mu(s) = 1$.
  • if $\hat{v}(\mu)$ is convex and not concave, strong disclosure is (uniquely) optimal.
    • strong disclosure: $\mu$ is at the boundary of $\Delta(s)$ for all $\sigma$ sent in equilibirum. (?)

(Dughmi 2019):

• If reward functions are identical (i.e. $r^i = r^j$), then the sender’s objective function is convex. The optimal signaling scheme is to reveal all the information.
• If $(r^i+r^j)(s,a) = k, \forall s,a$, where $k \in \mathbb{R}$, then the sender’s objective function is concave. The optimal signaling scheme is to reveal nothing (In this case, $\mu = \mu_0$).

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An Equilibrium Perspective

(Bergemann & Morris 2016)

Settings in my understanding

This paper studies the persuasion of a group of receivers with private channels.

• A game of incomplete information can be decomposed into a basic game and an observation structure (or information structure).
  • The basic game defines the set of actions, the set of payoff states, the payoff functions, and the common prior over the payoff states.
  • The observation structure refers to the function of partial observations that the players receive in the game.
• An incomplete information game $\mathcal{G} = (G, M)$.
  • A basic game $G = ((A^j, r^j)_{j=1}^{J}, \mu_0)$.
    • $A = \prod_{j} A^j$.
    • $r^j: S\times A \to \mathbb{R}$.
    • $\mu_0 \in \Delta_{++}(S)$.
  • An observation structure $M = ((O^j)_{j=1}^{J}, q)$.
    • Partial observation (type) $O = \prod_{j} O^j$.
    • Emission function $q: S \to \Delta(O)$.
    • A possible (and natural) observation structure is the null observation structure in which each player’s set of ovservations (tyes) $O^j$ is a singleton, $O^j = \set{t^j}$.
• A signaling scheme (decision rule) $\varphi: S\times O \to \Delta(A)$.
  • There is an omniscient mediator, who first observes the sampled $s\sim\mu_0$ and the sampled $o\sim q(\cdot\mid s)$, and then picks actons and privately announces $a^j$ to each receiver $j$.
• A policy (behavioral strategy) for receiver $j$ in $(G, M)$ is $\pi^j: O^j \to \Delta(A^j)$.

Obedience & BCE

A decision rule $\varphi$ is obedient for $(G, M)$ if, for each $j = 1, \ldots, J$, $o^j\in O^j$, and $a^j\in A^j$, we have

\[\begin{aligned} & \sum\limits_{s, o^{-j}, a^{-j}} \mu_0(s) \cdot q\Big((o^j, o^{-j})\mid s \Big) \cdot \varphi\Big( (a^j, a^{-j})\mid s, (o^j, o^{-j}) \Big) \cdot r^j\Big(s, (a^j, a^{-j})\Big) \ge \\ & \sum\limits_{s, o^{-j}, a^{-j}} \mu_0(s) \cdot q\Big((o^j, o^{-j})\mid s \Big) \cdot \varphi\Big( (a^j, a^{-j})\mid s, (o^j, o^{-j}) \Big) \cdot r^j\Big(s, (a^{j\prime}, a^{-j})\Big) \end{aligned}\]

for all $a^{j\prime}\in A^j$.

The definitions of obedience in (Bergemann & Morris 2011a, 2011b) is a bit different from this.

A decision rule $\varphi$ is a Bayes correlated equilibrium (BCE) of $(G,M)$ if it is obedient for $(G,M)$.

Degenerated BCE

1. There is complete information, i.e., if $S$ is a singleton: this definition reduces to the definition of correlated equilibrium (CE) for a complete information game. (Aumann 1987)

2. If $M$ is the null observation function (a singleton), then this is essentially the “universal Bayesian solution” of (Forges 1993).

   A decision rule $\varphi$ is obedient for $(G)$ if, for each $j = 1, \ldots, J$ and $a^j\in A^j$, we have

   \[\begin{aligned} & \sum\limits_{s, a^{-j}} \mu_0(s) \cdot \varphi\Big( (a^j, a^{-j})\mid s \Big) \cdot r^j\Big(s, (a^j, a^{-j})\Big) \ge \\ & \sum\limits_{s, a^{-j}} \mu_0(s) \cdot \varphi\Big( (a^j, a^{-j})\mid s \Big) \cdot r^j\Big(s, (a^{j\prime}, a^{-j})\Big) \end{aligned}\]

   for all $a^{j\prime}\in A^j$.

3. If $M$ is the null observation function, and if there is only one receiver, then this definition reduces to behavior in the concavification problem of (Aumann et al. 1995) and the Bayesian persuasion of (Kamenica & Gentzkow 2011).

   A decision rule $\varphi$ is obedient for $(G)$ if, for each $a\in A$, we have

   \[\sum\limits_{s} \mu_0(s) \cdot \varphi( a\mid s ) \cdot \Big( r^j(s, a) - r^j(s, a') \Big) \ge 0\]

   for all $a^{\prime}\in A$.

Why is it called obedience? Why do the receivers follow the recommendation?

(Bergemann & Morris 2019)

\[\begin{aligned} & \Leftrightarrow \sum\limits_{s} \frac{\mu_0(s) \cdot \varphi( a\mid s )} { \sum\limits_{s'}\mu_0(s') \cdot \varphi( a\mid s')} \cdot \Big( r^j(s, a) - r^j(s, a') \Big) \ge 0 , \forall a'\in A.\\ & \Leftrightarrow \sum\limits_{s} \mu(s\mid a) \cdot \Big( r^j(s, a) - r^j(s, a') \Big) \ge 0 , \forall a'\in A. \end{aligned}\]

BNE

A strategy profile (joint policy) $\pi$ is a Bayes Nash equilibrium (BNE) of $(G, M)$ if for each $j = 1 , \ldots, J$, $\sigma^j\in\Sigma^j$, and $a^j\in A^j$ with $\pi^j(a^j \mid o^j) > 0$, we have

\[\begin{aligned} & \sum\limits_{s, o^{-j}, a^{-j}} \mu_0(s) \cdot q\Big((o^j, o^{-j})\mid s \Big) \cdot \Bigg( \prod_{j \ne k} \pi^k(a^k \mid o^k) \Bigg) \cdot r^j\Big(s, (a^j, a^{-j})\Big) \ge \\ & \sum\limits_{s, o^{-j}, a^{-j}} \mu_0(s) \cdot q\Big((o^j, o^{-j})\mid s \Big) \cdot \Bigg( \prod_{j \ne k} \pi^k(a^k \mid o^k) \Bigg) \cdot r^j\Big(s, (a^{j\prime}, a^{-j})\Big) \end{aligned}\]

for all $a^{j\prime}\in A^j$.

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The following part has not been finished yet. One may check my writing schedule.

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Extensions

1. Multiple senders
2. Multiple receivers
3. Dynamic environment
4. Others:
   1. The receiver has private information

An Ethical Justification

I do not think information design is immoral. Information is a kind of property of the sender, and it is legal for it to profit from its information. (Unless the utilized information is a public resource.)
Furthermore, in those cases where the sender can improve its own expected payoff through information design, the receiver’s payoff is not worse than that of the sender not reveal information at all.
Nevertheless, practice use of information design should take the sender’s objective function into some serious consideration.

Disclaimer: The above content is summarized from the mentioned papers. Corresponding links or references have been provided.