Origins of Decision Theory
Humans have always made decisions, but the systematic study of how decisions should be made, and how they actually are made, is surprisingly recent. For most of human history, decision-making was guided by custom, intuition, religious authority, or the counsel of experienced elders. The idea that there might be a formal science of decision-making, a set of mathematical principles governing how a rational agent should choose among alternatives under conditions of uncertainty, did not emerge until the seventeenth and eighteenth centuries, and did not achieve its mature form until the mid-twentieth century.
The intellectual journey from the first attempts to formalize reasoning about chance in the gambling houses of Renaissance Europe to the sophisticated decision-theoretic frameworks used today in artificial intelligence, military strategy, medical diagnosis, financial engineering, and public policy is a story of remarkable creativity, fierce intellectual debate, and enduring tension between how people should decide and how they actually do. This tension, between normative decision theory (what rationality requires) and descriptive decision theory (what people actually do), has driven the field's development and continues to generate its most productive controversies.
Understanding the origins of decision theory matters beyond academic curiosity. The frameworks that emerged from this intellectual history shape how governments evaluate policies, how doctors weigh treatment options, how investors price assets, how engineers design systems, and how artificial intelligence systems are built to make choices on our behalf. The assumptions embedded in these frameworks, assumptions about rationality, probability, utility, and information, have consequences that affect billions of lives. Knowing where these assumptions came from, what problems they were designed to solve, and what alternatives were considered and rejected provides essential context for evaluating whether those assumptions are appropriate for the decisions we face today.
The Prehistory: Probability and the Problem of Uncertainty
Decision theory could not exist without probability theory, because most interesting decisions involve uncertainty. And probability theory, despite seeming like an obvious mathematical concept, emerged remarkably late in the history of mathematics.
Gambling and the Birth of Probability
The mathematical study of probability began in 1654 with a famous correspondence between Blaise Pascal and Pierre de Fermat, prompted by a question from the gambler Antoine Gombaud, Chevalier de Meré. Gombaud wanted to know how to fairly divide the stakes of a game that was interrupted before completion. This seemingly trivial question about gambling forced Pascal and Fermat to develop the first rigorous mathematical treatment of uncertain outcomes.
Pascal and Fermat's key insight was that uncertain future outcomes could be analyzed through systematic enumeration of all possible outcomes and their relative frequencies. If a fair coin is tossed twice, there are four equally likely outcomes (HH, HT, TH, TT). If you need to know the probability of getting at least one head, you count the favorable outcomes (3) and divide by the total (4), giving 3/4. This seems obvious now, but before Pascal and Fermat, even sophisticated thinkers struggled with problems involving chance because they lacked a mathematical framework for reasoning about uncertainty.
Pascal himself extended probabilistic reasoning beyond gambling in his famous "Pascal's Wager," perhaps the first explicit use of decision-theoretic reasoning. Pascal argued that a rational person should believe in God because the expected value of belief (infinite reward if God exists, minimal cost if God doesn't) exceeds the expected value of disbelief (no reward if God exists, no cost if God doesn't). The wager is philosophically problematic for many reasons, but its historical significance lies in applying mathematical reasoning about uncertain outcomes to a practical decision, combining probability (the uncertain existence of God) with utility (the value of the outcomes) to determine rational action.
Jacob Bernoulli and the Law of Large Numbers
Jacob Bernoulli's Ars Conjectandi (The Art of Conjecturing), published posthumously in 1713, established the first rigorous foundations of probability theory. Bernoulli proved the law of large numbers, which states that as the number of observations increases, the observed frequency of an event converges to its true probability. This theorem provided the mathematical link between the abstract concept of probability and the observable world of repeated events, giving probability theory its empirical grounding.
Bernoulli's work also began to consider the application of probability to practical decision-making beyond gambling. He discussed the evaluation of evidence in legal proceedings, the assessment of risk in commerce, and the general problem of making rational judgments under uncertainty. His nephew, Daniel Bernoulli, would take this practical orientation much further.
The St. Petersburg Paradox and the Birth of Utility Theory
The conceptual foundations of decision theory crystallized around a single paradox that revealed the inadequacy of the most obvious approach to decision under risk.
The Paradox
In 1713, Nicolas Bernoulli (another member of the mathematically prolific Bernoulli family) posed a problem that became known as the St. Petersburg paradox. Consider a game in which a fair coin is tossed repeatedly until it lands heads. If the first heads appears on the nth toss, you receive 2^n dollars. So if heads appears on the first toss, you get $2. If heads first appears on the second toss, you get $4. Third toss, $8. Fourth toss, $16. And so on.
The expected value of this game, the probability-weighted average of all possible payoffs, is infinite. The probability of heads on the first toss is 1/2, and the payoff is $2, so the expected value from this outcome is $1. The probability of first heads on the second toss is 1/4, and the payoff is $4, so the expected value from this outcome is also $1. Each possible outcome contributes $1 to the expected value, and there are infinitely many possible outcomes, so the total expected value is 1 + 1 + 1 + ... = infinity.
If rational decision-making simply means maximizing expected value, then a rational agent should be willing to pay any finite amount to play this game. But no reasonable person would pay more than a modest amount, perhaps $10 or $20. The paradox is that the mathematically "rational" decision (pay any price) conflicts dramatically with the intuitively sensible decision (pay only a small amount).
Daniel Bernoulli's Solution: Diminishing Marginal Utility
Daniel Bernoulli proposed a solution in 1738 that introduced one of the most important concepts in economics and decision theory: diminishing marginal utility. Bernoulli argued that the value of money to a person is not proportional to the amount of money but to the logarithm of the amount. The first $1,000 you receive has much greater value to you than the additional $1,000 received after you already have $100,000. By replacing monetary value with utility (subjective value), the expected utility of the St. Petersburg game becomes finite, and the paradox dissolves.
Bernoulli's insight had implications far beyond the paradox. It provided a psychological explanation for risk aversion: because utility is a concave function of wealth (each additional dollar provides less additional utility than the previous one), a certain outcome is preferred to a gamble with the same expected value. Receiving $50 for certain provides more utility than a 50/50 chance of receiving $100 or $0, because the utility lost by going from $50 to $0 is greater than the utility gained by going from $50 to $100. This explained why people buy insurance (accepting a small certain loss to avoid a large uncertain loss) and why investors demand higher returns for riskier investments.
The Twentieth-Century Formalization
The informal insights of Pascal, the Bernoullis, and their successors were transformed into a rigorous mathematical framework in the twentieth century through three landmark contributions that, together, defined the structure of modern decision theory.
Von Neumann and Morgenstern: Axiomatic Utility Theory
In 1944, mathematician John von Neumann and economist Oskar Morgenstern published Theory of Games and Economic Behavior, which laid the foundations for both game theory and modern utility theory. Their contribution to decision theory was the axiomatic derivation of expected utility.
Von Neumann and Morgenstern showed that if an agent's preferences over gambles satisfy four axioms, then the agent behaves as if assigning numerical utilities to outcomes and choosing the option with the highest expected utility. The four axioms are:
Completeness: For any two options A and B, the agent either prefers A, prefers B, or is indifferent between them. No options are incomparable.
Transitivity: If the agent prefers A to B and B to C, then the agent prefers A to C. Preferences are consistent.
Continuity: If the agent prefers A to B to C, then there exists some probability p such that the agent is indifferent between B for certain and a gamble giving A with probability p and C with probability 1-p. Preferences vary smoothly, without abrupt jumps.
Independence: If the agent prefers A to B, then the agent also prefers a gamble that gives A with probability p and C with probability 1-p to a gamble that gives B with probability p and C with probability 1-p. Preferences between two options are not affected by what other options are mixed in.
The brilliance of this approach was that it derived expected utility maximization as a theorem rather than assuming it as a postulate. If your preferences satisfy these reasonable-seeming axioms, then you are mathematically guaranteed to be behaving as if maximizing expected utility. This gave the rational actor model an axiomatic foundation that seemed nearly irrefutable: who would want to have incomplete, intransitive, discontinuous, or non-independent preferences?
Savage: Subjective Probability and Decision Theory
Leonard Savage extended von Neumann and Morgenstern's framework in his 1954 book The Foundations of Statistics. While von Neumann and Morgenstern assumed that probabilities were objectively given (as in a fair coin toss or a roulette wheel), Savage showed that both utilities and probabilities could be derived from an agent's preferences.
Savage's framework dealt with decisions where the probabilities of different states of the world are not objectively known, which describes most real-world decisions. When deciding whether to carry an umbrella, you do not know the objective probability of rain; you have a subjective degree of belief that it will rain. Savage showed that if an agent's preferences over actions satisfy a set of axioms (similar to but more elaborate than von Neumann and Morgenstern's), then the agent behaves as if assigning subjective probabilities to states and expected utility to actions.
This was a profound result because it unified probability and utility into a single decision-theoretic framework. A rational agent, in Savage's sense, simultaneously has consistent beliefs (represented by subjective probabilities) and consistent values (represented by utilities), and makes choices that maximize expected utility given those beliefs and values. The framework provided the theoretical foundation for Bayesian decision theory, which became the dominant normative framework for decision-making under uncertainty.
The Role of World War II: Operations Research
While theorists were developing axiomatic foundations, World War II generated enormous practical demand for systematic decision-making methods. Operations research (OR), which emerged from military efforts to optimize resource allocation, logistics, and strategy, pushed theoretical developments toward practical application.
British and American military scientists developed quantitative methods for deciding how to deploy radar installations, how to configure convoy formations to minimize submarine losses, how to allocate bombing resources among multiple targets, and how to schedule aircraft maintenance to maximize fleet availability. These were genuine decision problems involving uncertainty, multiple objectives, and scarce resources, exactly the problems that decision theory was designed to address.
The success of wartime operations research led to its rapid adoption in civilian contexts after the war. Companies like the RAND Corporation, founded in 1948 as a military think tank, employed mathematicians, economists, and engineers to apply decision-theoretic methods to problems ranging from nuclear strategy to urban planning. The Ford Foundation funded the creation of management science programs at leading business schools. Linear programming, dynamic programming, queuing theory, and other optimization methods developed during and after the war became standard tools of business management and government planning.
The Split: Normative Versus Descriptive Decision Theory
By the 1960s, decision theory had achieved an impressive mathematical structure. But a fundamental tension had been building within the field from the beginning: the tension between how rational agents should decide and how actual human beings do decide. This tension eventually split the field into two distinct branches.
The Difference Between Normative and Descriptive
Normative decision theory prescribes how decisions should be made if the goal is to satisfy the axioms of rationality. It says: if you want to be consistent, coherent, and immune to exploitation by money pumps (schemes that extract money from someone with intransitive preferences), then you should maximize expected utility. Normative theory does not describe what people do; it describes what rationality requires.
Descriptive decision theory studies how people actually make decisions. It uses experiments, surveys, and observation to discover the patterns, regularities, and systematic deviations that characterize real human decision-making. Descriptive theory does not prescribe what people should do; it describes what they actually do.
The gap between normative and descriptive theory is the gap between the rational actor model and real human behavior. And that gap, as decades of research have shown, is enormous.
The Allais Paradox: The First Major Challenge
The first major empirical challenge to expected utility theory came from French economist Maurice Allais in 1953. Allais presented a pair of choice problems to a group of sophisticated decision-makers at a conference in Paris and showed that the majority violated the independence axiom, one of the cornerstones of expected utility theory.
In the first choice problem, people chose between:
- Option A: $1 million with certainty
- Option B: 89% chance of $1 million, 10% chance of $5 million, 1% chance of nothing
Most people chose A, preferring the certain $1 million to the gamble.
In the second choice problem, people chose between:
- Option C: 11% chance of $1 million, 89% chance of nothing
- Option D: 10% chance of $5 million, 90% chance of nothing
Most people chose D, preferring the slightly lower probability but much larger potential payoff.
The combination of choosing A and D violates the independence axiom. Allais demonstrated this to several of the assembled decision theorists, including Savage himself, who initially made the "irrational" choice pattern and was disturbed to discover that he had violated his own axioms. The Allais paradox showed that even highly sophisticated, mathematically trained decision-makers systematically violated the axioms of expected utility theory, not through ignorance or carelessness, but because something about the axioms failed to capture a genuine feature of human preferences.
The feature that Allais identified was what he called the "certainty effect": people place special value on outcomes that are certain versus merely probable, a value that is not captured by the linear probability weighting assumed by expected utility theory. This insight would later be incorporated into Kahneman and Tversky's prospect theory, which explicitly modeled the nonlinear weighting of probabilities.
The Ellsberg Paradox: Ambiguity Aversion
A second major challenge came from Daniel Ellsberg (later famous for leaking the Pentagon Papers) in 1961. Ellsberg showed that people's preferences violate Savage's axioms when they face ambiguity, meaning uncertainty about the probabilities themselves.
In Ellsberg's experiment, subjects are presented with an urn containing 90 balls. They know that 30 balls are red, and the remaining 60 are some unknown mixture of black and yellow. They are asked to choose between bets, and their choices consistently reveal a preference for known over unknown probabilities, even when this preference violates the axioms of subjective expected utility theory.
Ellsberg's paradox demonstrated ambiguity aversion: people dislike situations where they don't even know the probabilities, above and beyond their dislike of risk (known probabilities of loss). This distinction between risk and ambiguity (sometimes called "Knightian uncertainty" after economist Frank Knight, who distinguished them in 1921) has proved important in understanding behavior in domains from financial markets (investors demand higher returns for ambiguous investments) to medical decision-making (patients prefer treatments with well-documented outcomes over treatments with uncertain evidence).
Prospect Theory: The Descriptive Revolution
The most influential alternative to expected utility theory emerged in 1979 when Daniel Kahneman and Amos Tversky published "Prospect Theory: An Analysis of Decision Under Risk" in Econometrica. Prospect theory did not merely identify anomalies in expected utility theory; it proposed a comprehensive alternative model of how people actually make risky decisions.
How Prospect Theory Challenged Classical Decision Theory
Prospect theory departed from expected utility theory in three fundamental ways.
First, reference dependence: prospect theory evaluated outcomes as gains and losses relative to a reference point, not as absolute levels of wealth. Expected utility theory evaluated outcomes in terms of total wealth: a decision that results in a final wealth of $100,000 has the same utility regardless of whether the person started with $50,000 (a gain of $50,000) or $200,000 (a loss of $100,000). Prospect theory recognized that these situations feel completely different psychologically and produce different choices.
Second, loss aversion: the value function in prospect theory was steeper for losses than for gains, reflecting the empirical finding that people feel the pain of a loss approximately twice as intensely as the pleasure of an equivalent gain. This asymmetry explained the endowment effect (people value things they own more than identical things they don't own), the status quo bias (people prefer to keep what they have), and the disposition effect in investing (people hold losing investments too long and sell winners too early).
Third, probability weighting: prospect theory replaced the linear probability weighting of expected utility theory with a nonlinear weighting function that overweights small probabilities and underweights large probabilities. This explained both the certainty effect identified by Allais (the discontinuous jump in value between a certain outcome and an almost-certain outcome) and the simultaneous purchase of insurance (overweighting the small probability of catastrophic loss) and lottery tickets (overweighting the small probability of a large gain).
Prospect theory's success was not merely empirical. It was also explanatory. The theory did not merely describe departures from expected utility theory; it explained them in terms of psychologically plausible mechanisms. Reference dependence reflects the brain's orientation toward detecting changes rather than absolute levels. Loss aversion reflects the evolutionary advantage of reacting more strongly to threats than to opportunities. Probability weighting reflects the psychological salience of certainty and the difficulty of discriminating between similar probability levels.
Decision Theory and Artificial Intelligence
The relationship between decision theory and artificial intelligence has been deeply intertwined since both fields emerged in the mid-twentieth century. Early AI pioneers like John McCarthy, Marvin Minsky, and Herbert Simon drew explicitly on decision-theoretic frameworks, and modern AI systems are built on foundations that trace directly to von Neumann, Savage, and their successors.
Utility Functions and Rational Agents
Modern AI systems, particularly those based on the rational agent paradigm described in Stuart Russell and Peter Norvig's influential textbook Artificial Intelligence: A Modern Approach, are explicitly designed as expected utility maximizers. An AI agent perceives the state of its environment, evaluates possible actions according to a utility function, and selects the action that maximizes expected utility given its beliefs about the current state and the consequences of its actions.
This design choice makes AI agents normatively rational in Savage's sense: they have consistent beliefs (represented by probability distributions), consistent values (represented by utility functions), and make choices that maximize expected utility. The question of whether this is a good design for AI systems depends on whether the normative theory is appropriate for the domains in which AI operates, a question that the descriptive critiques of decision theory make more urgent.
Bayesian Reasoning in AI
Bayesian decision theory, which combines Savage's subjective probability framework with von Neumann and Morgenstern's utility theory, provides the default reasoning framework for many AI systems. Bayesian networks, Bayesian classifiers, and Bayesian reinforcement learning all use probability theory to represent uncertainty and utility theory to evaluate outcomes.
The Bayesian framework's strength is its mathematical elegance and theoretical coherence. Its weakness is computational: in complex real-world environments, exact Bayesian reasoning is often computationally intractable, requiring approximations that may sacrifice some of the framework's theoretical guarantees. This computational challenge echoes Simon's bounded rationality argument: even an artificial agent with perfect information may lack the computational resources to fully optimize, and may need to satisfice using heuristics that approximate rational behavior.
AI Decision-Making and the Alignment Problem
As AI systems become more capable and are deployed in higher-stakes domains (autonomous vehicles, medical diagnosis, financial trading, military operations), the foundations of decision theory become directly relevant to safety and ethics. The alignment problem, ensuring that AI systems pursue goals that are aligned with human values, is fundamentally a decision-theoretic problem: how do you specify a utility function that captures what humans actually want?
The descriptive findings of behavioral economics suggest that this is harder than it might seem. Human preferences are reference-dependent, loss-averse, context-sensitive, and sometimes inconsistent. Translating these messy human preferences into the clean utility functions that AI systems require is an unsolved problem that lies at the intersection of decision theory, philosophy, and artificial intelligence.
| Era | Development | Key Figures | Significance |
|---|---|---|---|
| 1654 | Probability correspondence | Pascal, Fermat | Mathematical reasoning about chance |
| 1738 | Diminishing marginal utility | Daniel Bernoulli | Resolved St. Petersburg paradox; utility vs. money |
| 1921 | Risk vs. uncertainty distinction | Frank Knight | Not all uncertainty involves known probabilities |
| 1944 | Axiomatic utility theory | Von Neumann, Morgenstern | Mathematical foundation for rational choice |
| 1953 | Allais paradox | Maurice Allais | First systematic violation of expected utility |
| 1954 | Subjective expected utility | Leonard Savage | Unified probability and utility from preferences |
| 1961 | Ellsberg paradox | Daniel Ellsberg | Ambiguity aversion violates subjective expected utility |
| 1979 | Prospect theory | Kahneman, Tversky | Comprehensive descriptive alternative to expected utility |
| 1990s-present | AI decision systems | Russell, Norvig, et al. | Decision theory embedded in artificial agents |
The Enduring Questions
Decision theory continues to grapple with several fundamental questions that have resisted resolution since the field's earliest days.
Is rationality a normative ideal or a practical requirement? The axioms of expected utility theory define a concept of rationality that many thoughtful people violate in systematic ways. Does this mean that the axioms need revision, or that people need correction? The debate between Kahneman's view (people are systematically irrational and need help) and Gigerenzer's view (people are ecologically rational, using heuristics well-adapted to their environments) reflects a deep disagreement about the nature and standards of rational decision-making.
How should decision theory handle uncertainty about probabilities? Savage's framework assumes that agents can assign subjective probabilities to all relevant events. But in many real-world situations, we face deep uncertainty where even subjective probability assignment seems arbitrary. Climate change scenarios, pandemic risks, financial system collapses, and existential risks from artificial intelligence all involve uncertainties that resist quantification. Alternative frameworks, including maximin (choose the action whose worst-case outcome is least bad), minimax regret (minimize the maximum difference between your outcome and the best outcome you could have achieved), and robust decision-making (find strategies that perform adequately across a wide range of scenarios), have been proposed for these situations.
How should decision theory incorporate ethical considerations? Classical decision theory is agnostic about values: it tells you how to efficiently pursue whatever goals you have, but it says nothing about what goals you should have. The integration of ethics with decision theory, particularly in contexts like medical allocation, criminal sentencing, and AI governance, remains a fundamental challenge.
What is the relationship between individual and collective decision-making? Decision theory was primarily developed for individual agents, but many of the most important decisions in the world are made by groups, organizations, and institutions. Arrow's impossibility theorem (1951) showed that no voting system can satisfy all reasonable fairness criteria simultaneously, revealing fundamental tensions in collective decision-making that individual decision theory does not address.
These questions ensure that decision theory remains a living, evolving field rather than a settled body of doctrine. The mathematical frameworks developed over three centuries provide indispensable tools for analyzing decisions, but the limitations of those frameworks, revealed through paradoxes, experiments, and practical failures, ensure that the search for better theories continues.
References and Further Reading
Von Neumann, J. & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press. https://press.princeton.edu/books/paperback/9780691130613/theory-of-games-and-economic-behavior
Savage, L. J. (1954). The Foundations of Statistics. John Wiley & Sons. https://doi.org/10.1002/nav.3800010316
Kahneman, D. & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-291. https://doi.org/10.2307/1914185
Bernoulli, D. (1738/1954). Exposition of a new theory on the measurement of risk. Econometrica, 22(1), 23-36. https://doi.org/10.2307/1909829
Allais, M. (1953). Le comportement de l'homme rationnel devant le risque. Econometrica, 21(4), 503-546. https://doi.org/10.2307/1907921
Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75(4), 643-669. https://doi.org/10.2307/1884324
Knight, F. H. (1921). Risk, Uncertainty, and Profit. Houghton Mifflin. https://oll.libertyfund.org/titles/knight-risk-uncertainty-and-profit
Russell, S. & Norvig, P. (2020). Artificial Intelligence: A Modern Approach (4th ed.). Pearson. https://aima.cs.berkeley.edu/
Arrow, K. J. (1951). Social Choice and Individual Values. John Wiley & Sons. https://cowles.yale.edu/arrow-social-choice-and-individual-values
Simon, H. A. (1955). A behavioral model of rational choice. Quarterly Journal of Economics, 69(1), 99-118. https://doi.org/10.2307/1884852
Hacking, I. (2006). The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction and Statistical Inference (2nd ed.). Cambridge University Press. https://doi.org/10.1017/CBO9780511817557
Peterson, M. (2017). An Introduction to Decision Theory (2nd ed.). Cambridge University Press. https://doi.org/10.1017/9781316585061
Gilboa, I. (2009). Theory of Decision Under Uncertainty. Cambridge University Press. https://doi.org/10.1017/CBO9780511840203
Resnik, M. D. (1987). Choices: An Introduction to Decision Theory. University of Minnesota Press. https://www.upress.umn.edu/book-division/books/choices