Understanding the notations in Bayesian game definition

Question

I am having trouble understanding the definition of a Bayesian game based on the following definition from class. I would appreciate it if you could explain the notations and overall meaning for point 2, 3, 4, and 5. I am sure it is pretty simple once you know it, but they make no sense to me whatsoever. Thank you, and let me know if you need anything else.

Answer 1

1. Set of players, I guess this is quite clear. As an example, take player $A$ and $B$
2. Actions. This just tells you what the two players can do. For example $A$ can choose $U$(p) or $D$(own) and $B$ can choose $L$(eft) or $R$(ight).
3. Types: A type is attached to a player. For example, player $A$ can be of two different types, say $A_1$ or $A_2$. The type that player $A$ has is usually known by the player herself, but usually not known by the other players (why we need a probability distribution). Similarly assume that player 2 can have two types $B_1$ or $B_2$. Again, one normally assume that player 2 knows his own type but $A$ does not know the type of player $B$.
4. A belief over the type combinations, which in our case gives probabilities that nature draws the particular types for $A$ and $B$. In our example, this gives 4 numbers that add up to 1. For example $\pi(A_1, B_1) = 1/8, \pi(A_2, B_1) = 2/8, \pi(A_1, B_2) = 3/8, \pi(A_2, B_2)= 4/8$. From these beliefs we can determine the posterior beliefs using Bayes theorem. Assume, for instance, that player $A$ knows she is of type $A_1$ then the probability that she beliefs player 2 to be of type $B_1$ is given by: $$ \pi(B_1|A_1) = \frac{\pi(A_1, B_1)}{\pi(A_1, B_1) + \pi(A_1, B_2)} = \frac{1/8}{4/8} = \frac{1}{4}. $$
5. A payoff function $u$ that specifies for every type and all possible combination of actions and combinations of types the payoff the particular type will receive. In our case, this gives 16 numbers. For example, the number $u_{A_1}(U,L, B_2)$ gives the payoff of player $A$ when she is of type $A_1$, when she chooses the action $U$, when she faces a type $B_2$ adversary and when this adversary chooses $L$.

A strategy of a certain player specifies for each type an action to play. For example, one strategy for player $B$ could be to choose $L$ if he is of type $B_1$ and to choose $R$ when he is of type $B_2$. Player $B$ has therefore 4 possible strategies $(L, L)$, $(L,R)$, $(R,L)$ and $(R,R)$ where each tuple specifies to do when he is of type $B_1$ and what to do if he is of type $B_2$.

Player $A$ also has 4 strategies $(U,U)$, $(U,D)$, $(D,U)$, $(D,D)$. For example, the strategy $(U,D)$ says to choose $U$ if she is of type $A_1$ and to choose $D$ if she is of type $A_2$ also.

In total there are 16 possible strategy profiles (4 for $A$ times 4 for $B$). We could summarize this profile by quadruples, for example $((L, R), (U, D))$

Each such strategy profile gives an expected payoff for each player type. For example, the expected payoff of $((L,R), (U,D))$ for player type $A_1$ is then given by: $$ V_{A_1}((L,R),(U,D)) = \pi(B_1|A_1) u_{A_1}(U, L, B_1) + \pi(B_2|A_1) u_{A_1}(U,R,B_2). $$ The first term gives the probability of facing a type $B_1$ adversary times the payoff this generates (i.e. $A_1$ plays $U$ and $B_1$ plays $L$). The second term gives the probability of facing a type $B_2$ adversary and receiving the payoff this generates (i.e. when $A_1$ plays $U$ and $B_2$ plays $R$).

These payoffs, in term, determine the best responses (as the strategies of a particular player type that maximize expected payoff given the strategies of the other player), and this, in turn, gives the definition of a (Bayesian) Nash equilibrium as the strategy profiles which are mutual best responses.