Decision theory is an interdisciplinary area of study that concerns mathematicians, statisticians, economists, philosophers, managers, politicians, psychologists and anyone else interested in analyses of decisions and their consequences. The basic formalism of decision theory is the payoff table, which maps mutually exclusive decisions to mutually exclusive states of nature. For example, "Decision X leads to Outcome Y", "Decision Y leads to Outcome Z", and so on. When the set of outcomes corresponding to any given decision is not known, we refer to this situation as decision under uncertainty, the field of study which dominates decision theory.
Outcomes in decision theory are usually assigned utility values. For example, from the point of view of a military planner, the death of 1000 men on the battlefield might be assigned a negative utility of 1000, and the death of 500 a negative utility of 500. Possible outcomes in a decision theory problem may be positive, negative or both. Utility assignments can be arbitrary and based on the opinions of the decision maker -- for example, the death of 1000 men might be assigned greater than twice the negative utility of the death of 500 men.
The expected utility of a decision is computed as the sum of the probability of each possible outcome multiplied by the utility of each outcome. For example, making a given decision might lead to positive utility 100 with a probability of 75%, and a negative utility of 40 with probability 25%. 75% times 100 equals positive 75. 25% times -40 equals -10. 75 minus 10 gives 65, meaning the overall expected utility of the decision is 65.
Obviously, such quantitative precision is only possible in problems in which all the numbers and probabilities are known ahead of time. This is true in certain gambling problems, like poker. Decision theory provides a number of suggestions for how to estimate complex probabilities under uncertainty, most of which are derived from Bayesian inference.
Decision theory can be normative or descriptive. Normative decision theory refers to theories about how we should make decisions if we want to maximize expected utility. Descriptive decision theory refers to theories about how we actually make decisions. Descriptive decision theories are complex, often unnecessarily so, and they help teach us the ways in which human decisions systematically go wrong. This connects to the related field of heuristics and biases, which has come into great vogue in the field of economics within the last decade.