So here's 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 Game that we saw having to do with certainty about types of 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 the possible information of the other agents are, information of the other agents, like out self and so forth. So, all of that is folded into the notion of a type. So that's mathematically very convenient packing all of 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 agents, so now we don't have sets of games. We have, very directly, the actions available to the agents. And now we have the type. This abstract mathematical object, that carry, that captures the private information of the agents. So we have a type for each agent. And we have a common prior as be, as in the first definition of games. We have a common prior. But now, it's not over games. It's over types. So, agent, each agent have a type. And that prior is, is common. The type is chosen according to a probability distribution that's commonly known by all the agents. And, each agent knows there own type. Therefore, they also have a posterior about the type of the other agents, and belief's about what the other agent's 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 on the action's taken by the agent's but on their type. That's the formal definition. Again it's mathematically in substance 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 matrix according to this prior. And we've all, and we had the private signals. the, 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 action, actions and the column agents had the left to right actions. The payoff, however, will depend on their type. So let's, let's look for example what happened here. So what is the payoff when the agent, the row agent, plays up and the column player plays left? Well, that depends. If the type, of the agents is this one well 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 path then? Well the path corresponds now to this cell right there. Is what happens when they play up and left. And so you get two and zero when the types are as they are Let's. Take some other random example here. Let's clear the slide. Let's take some random example here, clearing the slide and let us look for example at down and left. When the types are these. Well, what is the what are the, what are the, what are the types? So the type is, is, is this one right here. So the, 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 payoffs will be corresponding to the zero and zero. 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 t hing 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, on uncertainty over games may not have unique game and you need to look at the set of game then the expectation there a,h but I'm just flying this as a topic what we discuss would give you a good handle on the two formulations of Bayesian Games. The exclusive from listing of games and at common prior over them and a partition structure for the agents or the type based formulation.
