Home > An introduction to decision theory > A problem with naive decision theory

A problem with naive decision theory

This is part of the sequence, “An introduction to decision theory”. It is not designed to make sense as  a stand alone post.

You watch the creature hesitate, processing the best data it can gather about the environment around it. You hold your breath. The last ten years of your life have been spent as part of the team programming this artificial creature – so simple on the face of it, yet it took such a level of complexity to equal even this simple achievement of evolution. The creature is now face with its first decision: one patch of food is more tempting but less sheltered from predators. Another is less tempting but well sheltered. In this setup, predators are so prevalent that the sensible decision is to go for the more sheltered food.

Finally, the agent makes its decision. Seconds later it is caught by a predator. You sigh and download the log. Time to figure out what went wrong. You see it straight away. Before the creature could make its decision, it first needed to build up a utility table. You had expected it to build up the following table.

Predators present Predators absent
Substantial patch -10 5
Less substantial patch 2 2

Instead, it had built up the following utility table:

Death Survival
Substantial patch -10 5
Less substantial patch -10 2

There’s nothing wrong with this second utility table. Dying is always worth -10. Dying in either patch of food is worth the same disutility. Survival is indeed worth more in the substantial patch of food because it’s a preferable location. However, the point is that the creature is more likely to survive if it heads for the less substantial patch. In other words, the probability of the world state depends on the decision.

However, if we look at the formula for expected utility that we discussed last week, we can see that the probability of the state of the world doesn’t take into account the decision at all.

Expected \ Utility (Decision) =\sum_{i}Probability(WorldState_{i})\times Utility(WorldState_{i}\ \wedge \ Decision )

The term to look at is:

Probability(WorldState_{i})

This fails to take into account the fact that the decision can influence the world state (which patch the creature choses influences whether it is likely to survive). This equation is unable to handle this model of reality. So what equation should you reprogram into your artificial creature? This issue is the focus of one of the principle debates in decision theory.

Historically, the debate was between two main decision theories: evidential decision theory and causal decision theory. Broadly, evidential decision theory says that an agent should ask what evidence a decision provides about the world state. So the creature heading toward the sheltered patch of food would provide evidence that the creature is more likely to survive. Causal decision theory, on the other hand, says that an agent should ask what the causal influence of the decision is on the world state. So heading for the sheltered patch causes the creature to be more likely to survive.

The next few posts will outline both of these theories in more detail.

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