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I would like your help to better understand the notion of Bayes Correlated Equilibrium (BCE) in a single-agent decision problem with uncertainty The notion of BCE is provided in this paper for a generic $N$-player game.


Consider the following single-agent choice problem under uncertainty.

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{V})$. First, let nature draw a realisation $v$ of $V$ from $P_V$. Then, let the decision maker choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the decision maker gets a payoff $u(y,v)$.

Before choosing an action, the decision maker can receive some signal to refine her prior about the state of the world.


Now, I want to use Theorem 1 in Bergemann and Morris (2016) to characterise the set of optimal strategies under minimal assumptions on the amount of information that is processed by the decision maker. To do that, I introduce the notion of one-player BCE.

A one-player BCE of the game described is a probability distribution $P_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$ such that:

1) $\forall v \in \mathcal{V}$ $$\sum_{y\in \mathcal{Y}}P_{Y,V}(y,v)=P_V(v)$$

2) $\forall y\in \mathcal{Y}$ and $\forall \tilde{y}\neq y$ $$ \sum_{v\in \mathcal{V}} [u(y,v)-u(\tilde{y},v)] \times P_{Y,V}(y,v)\geq 0 $$


Let $\mathcal{P}$ be the set of BCE. Let $$ \mathcal{Q}\equiv \{P_Y\in \Delta(\mathcal{Y}): P_Y(y)\equiv \sum_{v\in \mathcal{V}}P_{Y,V}(y,v) \text{ }\forall y \in \mathcal{Y}, P_{Y,V}\in \mathcal{P}\} $$


Questions:

1) Is $\mathcal{Q}$ convex? I guess it is because $\mathcal{P}$ is convex. Could you confirm?

2) Let $ \mathcal{B}\equiv \{b\in \mathbb{R}^{|\mathcal{Y}|}: b^Tb\leq 1\}$ be the unit ball in $\mathbb{R}^{|\mathcal{Y}|}$. Let us stack in a $|\mathcal{Y}|\times 1$ vector the image set of every probability distribution $P_{Y}\in \mathcal{Q}$. With some abuse of notation, let us still denote any such a vector by $P_{Y}\in \mathcal{Q}$. Does $$ \max_{P_Y\in \mathcal{Q}} b^T P_Y $$ exists $\forall b\in \mathcal{B}$? I guess the answer to this question depends on the features of $\mathcal{Q}$. For example, if $\mathcal{Q}$ is compact, then the maximum above exists. Could you help?

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    $\begingroup$ Some notations are undefined---$v_y$, $v_{\tilde{y}}$, $\mathcal{P}$, etc. For 1), if $\mathcal{P}$ is convex, than $\mathcal{Q}$ is also, since it's the image of a convex set under a linear map. (The map of taking marginal distribution). For 2), if $\mathcal{Q}$ is convex, then it's a linear program with convex constraint, which always have a solution. $\endgroup$ – Michael Aug 11 at 2:32
  • $\begingroup$ Thanks. Notation on $v_y, v_{\tilde{y}}$ fixed. $\mathcal{P}$ is the set of BCE. $\endgroup$ – user3285148 Aug 11 at 11:34
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    $\begingroup$ From your definition of $\mathcal{P}$, you can see that indeed it is convex. It is the intersection of two convex sets, one from each condition. $\endgroup$ – Michael Aug 12 at 3:19
  • $\begingroup$ @Michael. I dont think that your claim about 2) is correct. If convexity was sufficient than $\underset{x \in B_1(0)}{max} 1 \times x$ would have a solution as the unit open ball around zero($B_1(0)$) is convex. You need closedness at some level. $\endgroup$ – Grada Gukovic Aug 12 at 8:39
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Yes, $Q$ is convex and closed precisely because $P$ is and Q is its image under a continuous function. Therefore, $Q$ is compact.

The function you are maximizing is continuous as well, so the maximum must exist.

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