I don't understand the following match equation for A (number 2), especially the subscripts. This is within the context of a Bayesian game (which is fine if you don't know, but I thought it might help explain for some of you who know economics). Thanks a lot!
I pasted the entire definiton for Bayesian so you can understand the context.
$A$ is the cartesian product of $n$ players' simultaneous "actions", that is, $a\in A$ is an $n$-tuple containing the action ("pure action profile") of each player. Similarly $t\in T$ is an $n$-tuple containing the type of player each player is. Each ordered-set of player-types has a probability $\pi(t)$ of occurring.
$$u_{i_k}:\left((a_{i_1},a_{i_2},\cdots,a_{i_n}),(t_{i_1},t_{i_2},\cdots,t_{i_n})\right)\mapsto y_{i_k}$$
where $y_{i_k}=u_{i_k}(a,t)$ is the payoff received by player $i_k$ from that situation.
Everywhere they wrote $i$ without subscript they could've written $i_k$ for clarity's sake