Some Thoughts About Utility Theory in Artificial Intelligence Part I
Utility functions are one of the elements of artificial intelligence(AI) solutions that are frequently mentioned but seldom discussed in details in AI articles. That basic AI theory has become an essential element of modern AI solutions. In some context, we could generalize the complete spectrum of AI applications as scenarios that involve a utility function that needs to be maximized by a rational agent. Before venturing that far, we should answer a more basic question: What is a utility function?
Utility functions are a product of Utility Theory which is one of the disciplines that helps to address the challenges of building knowledge under uncertainty. Utility theory is often combined with probabilistic theory to create what we know as decision-theoretic agents. Conceptually, a decision-theoretic agent is an AI program that can make rational decisions based on what it believes and what it wants. Sounds rational, right? :)
Ok, let’s get a bit more practical. In many AI scenarios, agents don’t have the luxury of operating in an environment in which they know the final outcome of every possible state. Those agents operate under certain degree of uncertainly and need to rely on probabilities to quantify the outcome of possible states. That probabilistic function is what we call Utility Functions.
Diving Into Utility Theory and MEU
Utility Theory is the discipline that lays out the foundation to create and evaluate Utility Functions. Typically, Utility Theory uses the notion of Expected Utility (EU) as a value that represents the average utility of all possible outcomes of a state, weighted by the probability that the outcome occurs. The other key concept of Utility Theory is known as the Principle of Maximum Utility(MEU) which states that any rational agent should choose to maximize the agent’s EU.
The principle of MEU seems like an obvious way to make decisions until you start digging into it and run into all sorts of interesting questions. Why using the average utility after all? Why not to try to minimize the loss instead of maximizing utility? There are dozens of similar questions that challenge the principle of MEU. However, in order to validate the principle of MEU, we should go back to the laws of Utility Theory.
Utility Theory Axioms
There are six fundamental axioms that setup the foundation of Utility Theory. In order to explain those, let’s pick a scenario in which you are having dinner at a restaurant and you are trying to decide between a salmon or a chicken dish. There are many factors that go into that simple decision. Which dish goes better with this gorgeous Montrachet we just ordered? How about the dessert? How would I feel if the chicken is overcooked? Is the salmon’s portion too small?…Hopefully you got the point.
Utility theory assigns a probability to each one of those possible states that try to orchestrate decisions based on that. However, those decisions are governed by a group of six fundamental axioms: Orderability, Transitivity, Continuity, Substitutability, Monotonicity and Decomposability. Together these six axioms help to enforce the principle of MEU. I will deep dive into each one of these axioms in the second part of this essay.