# Tag Info

No, there is not. Consider a game with two players, Ann and Bob. Both choose such vectors with entries $0$ or $1$ of the form $(a_1,a_2,\ldots,a_J)$ or $(b_1,b_2,\ldots,b_J)$, respectively. If $\sum_{i=1}^J a_i+b_i$ is odd, Ann wins and Bob loses. If the number is even, Ann loses and Bob wins. Clearly, one of them loses in every profile of pure strategies, ...
Condition 3 can be written in the following way: $a^\ast \in B$ iff for all $i$ and all $j$ $$a^\ast_{i,j} = 1 \iff u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = 0\}, a^\ast_{-i})$$ As there are no ties, this can be written as: $$a^\ast_{i,j} = 1 \to u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = ... 2 The concept of the BCE from their 2016 paper is similar to what you have. I think Bergemann and Morris' intuitive explanation is valuable so I'll paraphrase it here. Each player in the game has a decision rule that chooses an action, y, dependent on the state of the world V, and the player's information set, which we'll call S. This information set ... 2 \begin{array}{|c|c|c|} \hline &L&R\\\hline T&1,1&0,0\\\hline B&0,0&0,0\\\hline \end{array} In the game above, there are two pure strategy Nash equilibria: (T,L) is an equilibrium in weakly dominant strategies; (B,R) is an equilibrium in weakly dominated strategies. Noting that "dominant" and "dominated" are two different words,... 1 Question 1 Yes, the BCE induced by a completely informative information structure will look like this. This is true even though there are other ways to represent fully informative information structures. Think of T as labels. A fully informative information structure should use each element of T to label only one state of the world. That way, when the ... 1 Think about an auction, where the designer is selling good and trying to sell it to the person that values it the most while collecting as much revenue as possible. A direct mechanism means that the seller asks buyers for how much they value the good and based on that decides who gets the good and how much they pay. Suppose that the designer uses a second-... 1 You have specialized the definition of BCE in two dimensions: there is only one player, and the player has no private information. If you want to allow for private information you can let the player have some signal \pi:\mathcal{V}\rightarrow\Delta(T_i) And let the decision rule P_{\mathcal{Y},\mathcal{T},\mathcal{V}}\in\Delta(\mathcal{Y}\times \mathcal{... 1 I'm not an expert in epistemic game theory but let me see if I can help with an example. Suppose the state space is \Omega = \{1,2,3,4,5\}, and there are two players with information partitions$$ \mathcal{P}_1 = \left\{\{1,2,3\},\{4,5\} \right\} \\ \mathcal{P}_2 = \left\{\{1,2\},\{3,4\},\{5\}\right\}  Suppose the true state is $\omega = 5$. Let us ...