 So, here is an alternative definition of Bayesian games that is essentially identical mathematically, but presented differently. It's based on types, or more fully epistemic types, and the type of the agent is supposed to capture everything that's private information to the agent. So, if you look at the first definition of Bayesian games that we saw having to do with certainty about types and the common prior, then the type of the agent was her private signal, that is the information set in which the chosen game lies, as well as everything that emanates from it, namely her beliefs about what the possible information of the other agents are, and information of the other agents about herself, and so forth. So, all of that is folded into the notion of a type. So, that's mathematically very convenient, packing all this information into a type. Formally speaking, then the Bayesian game is defined as follows. It's defined as this tuple that is as follows. We have a set of agents. We have the actions available to the agent, so now we don't have sets of games, we have very directly the actions available to the agent. And now we have the type, this abstract mathematical object that captures the private information of the agents. So, we have a type for each agent, and we have a common prior, as in the first definition of games, we have a common prior, but now it's not over games, it's over types. So, each agent has a type, and that prior is common, the type is chosen according to a probability distribution that's commonly known by all the agents, and each agent knows their own type. Therefore, they also have a posterior about the type of the other agents, and beliefs about what the other agents might believe it about their own type, and so on and so forth. This is the type of the agent, and we have the utility function now depend not only of the actions taken by the agents, but of their type. That's the formal definition. Again, it's mathematically in some sense very simple, but the intuition is complicated because the notion of a type packs into it a lot of things. So, let's see it in action. Consider this game that we saw when we discussed the first definition of Bayesian games. Again, we had four possible games being played, chosen at random by nature according to this prior, and we had the private signals that the information sets that the agents found themselves in. Here is the type perspective on this. So, what are the actions available to agents? Very simply, the row agent has the up or down actions, and the column agents have the left to right actions. The payoff, however, will depend on their type. So, let's look for example what happened here. What is the payoff when the agent, the row agent plays up and the column player plays lift? Well, that depends. If the type of the agents is this one, what is the type? The type corresponds now to this information that they have. And the type of the second player is this. Well, what's the payoff then? Well, the payoff corresponds now to this cell right there is what happens when they play up and lift. And so you get 2 and 0 when the types are as they are. Let's take some other random example here. Let's clear the slide. Let's take some rather example here, clearing the slide. And let us look for example at down and lift when the types are these. Well, what are the types? So the type is this one right here. So this is the information available to the first agent. The second agent has this information available to her, which means that this right here is the game being played. And what is dL? dL means that we're playing down and lift. So it would be this one. And therefore the payoff will be correspondingly 0 and 0. So you can look at other examples and figure out what the type-based formulation means by just taking some random row here and figuring out why the row is the way it is. The last thing to say here before we move on to analysis is that in this particular example, by fixing the type you ended up with a very specific game. And this is a complicated topic where, in fact, if you wanted to map it to uncertainty over games, you may not have a unique game and you need to look at the set of games and the expectation there. But I'm just flagging this as a topic. What we discussed will give you a good handle on the two formulations of Bayesian games. The explicit listing of games and a common prior over them and a partition structure for the agents or the type-based formulation.