2.1 Possible Worlds

Most representations of uncertainty (certainly all the ones considered in this book) start with a set of possible worlds, sometimes called states or elementary outcomes. Intuitively, these are the worlds or outcomes that an agent considers possible. For example, when tossing a die, it seems reasonable to consider six possible worlds, one for each of the ways that the die could land. This can be represented by a set W consisting of six possible worlds, {w₁, …, w₆}; the world w_i is the one where the die lands i, for i = 1, …, 6. (The set W is often called a sample space in probability texts.)

For the purposes of this book, the objects that are known (or considered likely or possible or probable) are events (or propositions). Formally, an event or proposition is just a set of possible worlds. For example, an event like "the die landed on an even number" would correspond to the set {w₂, w₄, w₆}. If the agent's uncertainty involves weather, then there might be an event like "it is sunny in San Francisco", which corresponds to the set of possible worlds where it is sunny in San Francisco.

The picture is that in the background there is a large set of possible worlds (all the possible outcomes); of these, the agent considers some subset possible. The set of worlds that an agent considers possible can be viewed as a qualitative measure of her uncertainty. The more worlds she considers possible, the more uncertain she is as to the true state of affairs, and the less she knows. This is a very coarse-grained representation of uncertainty. No facilities have yet been provided for comparing the likelihood of one world to that of another. In later sections, I consider a number of ways of doing this. Yet, even at this level of granularity, it is possible to talk about knowledge and possibility.

Given a set W of possible worlds, suppose that an agent's uncertainty is represented by a set W′ ⊆ W. The agent considers U possible if U ∩ W′ ≠ ∅; that is, if there is a world that the agent considers possible which is in U. If U is the event corresponding to "it is sunny in San Francisco", then the agent considers it possible that it is sunny in San Francisco if the agent considers at least one world possible where it is sunny in San Francisco. An agent knows U if W′ ⊆ U. Roughly speaking, the agent knows U if in all worlds the agent considers possible, U holds. Put another way, an agent knows U if the agent does not consider U (the complement of U) possible.

What an agent knows depends to some extent on how the possible worlds are chosen and the way they are represented. Choosing the appropriate set of possible worlds can sometimes be quite nontrivial. There can be a great deal of subjective judgment involved in deciding which worlds to include and which to exclude, and at what level of detail to model a world. Consider again the case of throwing a fair die. I took the set of possible worlds in that case to consist of six worlds, one for each possible way the die might have landed. Note that there are (at least) two major assumptions being made here. The first is that all that matters is how the die lands. If, for example, the moods of the gods can have a significant impact on the outcome (if the gods are in a favorable mood, then the die will never land on 1), then the gods' moods should be part of the description of a possible world. More realistically, perhaps, if it is possible that the die is not fair, then its possible bias should be part of the description of a possible world. (This becomes particularly relevant when more quantitative notions of uncertainty are considered.) There will be a possible world corresponding to each possible (bias, outcome) pair. The second assumption being made is that the only outcomes possible are 1, …, 6. While this may seem reasonable, my experience playing games involving dice with my children in their room, which has a relatively deep pile carpet, is that a die can land on its edge. Excluding this possibility from the set of possible worlds amounts to saying that this cannot happen.

Things get even more complicated when there is more than one agent in the picture. Suppose, for example, that there are two agents, Alice and Bob, who are watching a die being tossed and have different information about the outcome. Then the description of a world has to include, not just what actually happens (the die landed on 3), but what Alice considers possible and what Bob considers possible. For example, if Alice got a quick glimpse of the die and so was able to see that it had at least four dots showing, then Alice would consider the worlds {w₄, w₅, w₆} possible. In another world, Alice might consider a different set of worlds possible. Similarly for Bob. For the next few chapters, I focus on the single-agent case. However, the case of multiple agents is discussed in depth in Chapter 6.

The choice of the set of possible worlds encodes many of the assumptions the modeler is making about the domain. It is an issue that is not one that is typically discussed in texts on probability (or other approaches to modeling uncertainty), and it deserves more care than it is usually given. Of course, there is not necessarily a single "right" set of possible worlds to use. For example, even if the modeler thinks that there is a small possibility that the coin is not fair or that it will land on its edge, it might make sense to ignore these possibilities in order to get a simpler, but still quite useful, model of the situation. In Sections 4.4 and 6.3, I give some tools that may help a modeler in deciding on an appropriate set of possible worlds in a disciplined way. But even with these tools, deciding which possible worlds to consider often remains a difficult task (which, by and large, is not really discussed any further in this book, since I have nothing to say about it beyond what I have just said).

Important assumption For the most part in this book, I assume that the set W of possible worlds is finite. This simplifies the exposition. Most of the results stated in the book hold with almost no change if W is infinite; I try to make it clear when this is not the case.