Definition of information structure in an incomplete information game

Question

I try to understand the model in this paper, pages 48-49. (Bergemann, D., & Morris, S. (2019). Information design: A unified perspective. Journal of Economic Literature, 57(1), 44–95. https://doi.org/10.1257/jel.20181489) They define an incomplete game (basic game) along with an information structure, where the "basic game" is defined by:

• There are a finite number of players, each denoted by $i$,
• A finite set of actions for each $i$, denoted $A_i$, and action profile is denoted by $A$
• A set of payoff states of the world, denoted $\Theta$, and $\theta$ denotes a specific state from $\Theta$
• A payoff function for each $i$, $u_i : A \times \Theta \to \mathbb{R}$
• A probability distribution with full support over $\Theta$, denoted $\psi\in \Delta(\Theta)$

Then the basic game is $G=((A_i,u_i), \Theta, \psi)$. On the other hand, the "information structure" $S$ is defined by:

• A finite set of types for each $i$, denoted $t_i \in T_i$, and type profile is denoted by $T$
• A type distribution $\pi : \Theta \to \Delta(T)$

Then they define $S= ((T_i), \pi)$ and they combine these two to define the general incomplete information game: $(G,S)$.

My only background on the topic is the course that I took last semester, named "Economics of Information". There, in all of the models we considered, ranging from contract theory to mechanism design to information design, we only defined "types", and we regarded them as the only variable that creates the uncertainty. So, I guess what we did was setting the "set of payoff states of the world" equivalent to "set of private informations (types) of players". But in this paper, the terminology is different, so I am having trouble with that. We always considered the distribution of types through some cumulative distribution function, often a simple one such as uniform distribution. We solved the various problems by taking expectations over types, using the cdf and the related pdf. But in the paper, the types are distributed according to some vague function $\pi$, which is defined on the set of states of the world. My first guess is that in the models we considered in my course, we did not differentiate between "the states of the worlds", and simply used some cdf in the place of $\pi$ in the paper.

What I understand from this is that after the true state $\theta$ of the world is realized, $\pi(\theta)$ is available to the players, and it assigns to a probability for each possible type profile $t= (t_1, ..., t_n) \in T$? So, each player, including the mechanism designer, knows the probability distribution of types once they observe the true state $\theta$? If my interpretation is correct, can we say that the paper adds one more layer of uncertainty on top of the uncertainty of types? Because just as knowing the cdf does not mean knowing the types of others, knowing $\pi(\theta)$ does not mean that the uncertainty is resolved. All of these comments are done with my initial assumption that "$\pi(\theta)$ serves like a cdf over the private information of players." and I doubt that this is a correct interpretation.

Last one: What is the interpretation of "payoff states of the world"? I mean why do we need it, we can draw similar results from various models where we define the uncertainty only by "types", private informations for players. Why they define two different objects like $\Theta$ and $T$?

Answer 1

In the usual formulation of games of incomplete information and in mechanism design, types represent both everything that enters a player's payoff function that is not an action and the information available to players. In information design, one wants to study the effects of changing only the information.

One can illustrate the problem with the game-theoretically trivial case of the introductory example of [Kamenica, Emir, and Matthew Gentzkow. "Bayesian persuasion." American Economic Review 101.6 (2011): 2590-2615.]

There is a single player (!), a judge who is tasked to either convict ($G$) or not convict ($N$) some defendant. So $A=\{G,N\}$ The defendant is either guilty ($G$) or not guilty ($N$). We take these to be the payoff-relevant states of nature: $\Theta=\{G,N\}$. The judge thinks that the defendant is a priori guilty with a probability of $0.3$. This gives $\psi(G)=0.3$ and $\psi(N)=0.7$. The judge wants to convict guilty defendants and not convict innocent ones. Getting the decision right, gives a payoff of $1$, getting it wrong $0$. So $u(a,\theta)=1$ if $a=\theta$ and $u(a,\theta)=0$ if $a\neq\theta$. This gives us the basic game.

But we also want to model what happens if the judge receives more information. It turns out that two signals are enough here, so we can take $T=\{g,i\}$. A signal $\pi:\Theta\to\Delta(T)$ specifies for both states the conditional distribution over signals. One can show (for example, using the methods Bergemann and Morris present) that a signal that maximizes the conviction probability sets $\pi(g\mid G)=1$ and $\pi(g|N)=3/7$. Suppose the judge receives the signal $g$. This happens with probability $0.3\cdot 1+0.7\cdot 3/7$. The conditional probability of the defendant being guilty then is $0.3/(0.3+0.7\cdot3/7)$. If we take this information structure as fixed, we could define the judge's payoff for each action to be the expected payoff given the signal (=type), and the judge will behave in the same way. But then we could not vary the information.

Note that Bergemann and Morris assume that there are finitely many states of nature, so a full support probability distribution can be given by simply listing the probability of each state, as done here. Probability distributions given by cumulative distribution function must be defined on the real line ($n$-dimensions work too), but $\Theta$ can be abstract as here. A probability distribution coming from a density (with respect to Lebesgue measure) can never have finite support. So all these issues about probabilities are red herrings here.