# Decision Theory: A (Normative) Introduction

I. Introduction

Let’s define decision theory as the study of decisions, specifically their effects on outcomes.

There are three main branches of decision theory: descriptive decision theory (how real agents make decisions), prescriptive decision theory (how real agents should make decisions), and normative decision theory (how ideal agents should make outcomes).

Since decision theory as a field is too broad to be summarized in one post, I’ll primarily focus on normative decision theory and only two-thirds of it.

Decisions under ignorance, rational choice theory, bounded rationality, prospect theory, heuristics, and the VNM axioms of rational choice all deserve separate posts.

II. Terms and Definitions

Before we begin with specific procedures in decision theory, let’s start with defining some important terms.

We can define a decision as an act or choice an agent has made and an outcome as a result of such decisions.

Utility should represent an agent’s preference over said outcomes and while may be assigned a cardinal value (such as when the agent is VNM-rational), is still a representation of ordinal preferences.

Decisions can be made under certainty, risk, or ignorance. The latter two represent when an agent is uncertain of the outcome corresponding to a decision, however, the former in contrast with the latter allows one to assign subjective probabilities to the outcomes.

In this post, only decisions under certainty and risk will be analyzed.

III. CDT, EDT, and FDT

Finally, we can now discuss three types of decision theory algorithms: Causal Decision theory (CDT), Evidential Decision Theory (EDT), and Functional Decision Theory (FDT).

To briefly define each procedure, we can say that CDT recommends choosing decisions that cause the best-expected outcome, EDT recommends choosing which decision “one would prefer to know one would have chosen”, and FDT recommends treating a decision as the output of a ﬁxed mathematical function that answers the question, “Which output of this very function would yield the best outcome?”.

Let’s give out some counterexamples to the decision algorithms. First is Newcomb’s Problem:

Omega, a being with near-perfect prediction accuracy, shows you two boxes designated A and B. You are given a choice between taking only box B or taking both boxes A and B. You know the following:

Box A is transparent and always contains a visible \$1,000.

Box B is opaque, and its content has already been set by the predictor:

If Omega has predicted that you will take both boxes A and B, then box B contains nothing.

If she has predicted that you will take only box B, then box B contains \$1,000,000.

You do not know what Omega predicted or what box B contains while making the choice.

In most iterations of Newcomb’s problem, CDT recommends two-boxing due to the dominance principle, whereas both EDT and FDT recommend one-boxing because utility is maximized given that Omega predicts you one-box.

Of course, this can differ with a change of interpretation. If Omega were to be a perfect predictor (or invoke the meta-Newcomb’s problem), then obviously CDT agents would one-box, and if, ceteris paribus, EDT agents were to take into account that two-boxing would not affect the posterior probabilistic prediction of Omega they would two-box.

Let’s move on to the second counterexample, the Smoking Lesion Problem:

Smoking is strongly correlated with lung cancer, but in the world of the Smoker’s Lesion, this correlation is understood to be the result of a common cause: a genetic lesion that tends to cause both smoking and cancer. Once we fix the presence or absence of the lesion, there is no additional correlation between smoking and cancer.

Suppose you prefer smoking regardless of whether or not you have cancer. Should you smoke?

CDT and FDT say “yes”, since smoking in this world has no causal effect on whether or not you get cancer. Naive EDT says “no”, because smoking is strongly correlated with cancer.

Standard decision recommendations can of course be altered, e.g. if CDT were to take into account Omega’say pseudo-retrocausal Bayesian updates given information about the agent (such as in meta-Newcomb’s problem), it would recommend acting like an agent who one-boxes and thus one-boxing, and if EDT were to take into account the fact that smoking alone would not affect the posterior probability of developing cancer, it would recommend smoking.

Conversely, in Naive CDT, only causal chains are considered, while in Naive EDT, only conditional probabilities are considered, while in FDT, both are considered. However, if causal chains and conditional probabilities consider one another then CDT and EDT are equivalent, causing one to choose the dominance principle (assuming probabilities are held constant) and thus two-boxing and smoking.

But if probabilities update based on the agent’s actions, such as in meta-Newcomb’s problem and Parfit’s Hitchhiker, then CDT=EDT would recommend the maximization of expected utility and thus one-boxing and paying up respectively.

I don’t think defining CDT = EDT is useful, so coming up with a new decision theory may be optimal. Some name ideas include Probabilistic, Bayesian, or Rational Decision Theory.

In summary, CDT picks out actions that cause utility maximization (e.g. dominance in both Newcomb’s problem and the Smoking Lesion problem), EDT picks out actions that maximize utility given that they were performed, and FDT prescribes decision procedures that an agent whose utility is maximized would follow, though that is not to be confused with a rational agent. Indeed, finding out what a rational agent would do is a mystery worthy of a follow-up post and further discussion and consideration.

IV. Conclusion

This post is not intended to endorse any decision theoretic procedure, rather, it is meant to give the reader an idea of the study of decision theory and hopefully its uses.

Decision theory is a broad study helpful to governments, businesses, and individuals such as me and you. And if I can introduce more people to decision theory, then that is an outcome I would certainly prefer.

If given enough support, further posts on decision theory as well as more on game theory, economics, and philosophy will be given out.

• “There are three main branches of decision theory: descriptive decision theory (how real agents make decisions), prescriptive decision theory (how real agents should make decisions), and normative decision theory (how ideal agents should make outcomes).”

This doesn’t seem right to me, I would say: an interesting way you can divide up decision theory is between descriptive decision theory (how people make decisions) and normative decision theory (how we should make decisions).

The last line of your description, “how ideal agents should make outcomes” seems especially troubling. I’m not quite sure what you are trying to say.

I think there are good parts of this post, for example, you’re hitting some interesting thought experiments. But several aspects are slightly confusing as written. For example, Newcomb’s problem isn’t (I believe) a counterexample to EDT, but that isn’t clear from your post.

• Broadly speaking, without regard to labels or choosing a specific algorithm, I think one should choose the dominance principle if probabilities are held constant and thus choose smoking, two-boxing, etc. But if probabilities were to update based on the agent’s actions such as in Parfit’s Hitchhiker, then one should pre-commit to an action that does not necessarily dominate in exchange for an outcome that is (second?) best and thus pay the biker.

We can of course play semantic slapfights and call this something like ‘Bayesian Decision Theory’ but ideas matter more than labels.