Introduction
Expected Utility Theory (EUT) stands as a cornerstone of decision-making under uncertainty, widely applied in economics, psychology, and philosophy of mathematics. Developed by John von Neumann and Oskar Morgenstern in their seminal work (von Neumann and Morgenstern, 1944), EUT posits that rational agents make decisions by maximising the expected value of their utility, a numerical representation of their preferences over outcomes. However, the incorporation of unbounded utility functions—those that can theoretically take infinite values—raises significant conceptual and practical challenges for EUT. This essay explores whether EUT is compatible with unbounded utility functions and whether rational agents should be required to have bounded utility functions. By examining the theoretical underpinnings of EUT, the problems posed by unbounded utilities (notably through paradoxes like the St. Petersburg Paradox), and the normative implications for rational agency, this essay argues that while unbounded utility functions are technically compatible with EUT, they introduce issues that may necessitate bounded utilities for practical and normative coherence in rational decision-making.
The Framework of Expected Utility Theory
Expected Utility Theory provides a formal framework for rational decision-making under risk. According to EUT, a rational agent evaluates uncertain prospects by calculating the weighted average of the utility of each possible outcome, with weights determined by the probability of each outcome occurring (von Neumann and Morgenstern, 1944). Mathematically, for a prospect with outcomes \(x_1, x_2, …, x_n\) and respective probabilities \(p_1, p_2, …, p_n\), the expected utility \(EU\) is given by:
[
EU = \sum_{i=1}^{n} p_i \cdot u(x_i)
]
where (u(x_i)) represents the utility of outcome (x_i). The axioms of EUT, including completeness, transitivity, independence, and continuity, ensure that a utility function exists to represent an agent’s preferences consistently. Crucially, EUT does not inherently restrict utility functions to being bounded; in principle, utilities can grow without limit as outcomes become increasingly desirable. However, as we shall see, the compatibility of unbounded utilities with EUT becomes problematic in certain scenarios, raising questions about their theoretical and practical validity.
Unbounded Utility Functions and the St. Petersburg Paradox
One of the most enduring challenges to EUT with unbounded utility functions is the St. Petersburg Paradox, first introduced by Daniel Bernoulli in 1738. In this thought experiment, a player is offered a game with a fair coin toss: if the first heads appears on the \(n\)-th toss, the player wins \(2^n\) dollars. The expected monetary value of this game is infinite, calculated as:
[
EMV = \frac{1}{2} \cdot 2 + \frac{1}{4} \cdot 4 + \frac{1}{8} \cdot 8 + … = 1 + 1 + 1 + … = \infty
]
If utility is proportional to money (i.e., unbounded), EUT suggests that a rational agent should be willing to pay an infinite amount to play this game, which contradicts intuitive notions of reasonable behaviour (Bernoulli, 1738, as cited in Savage, 1972). Bernoulli proposed a solution through the concept of diminishing marginal utility, suggesting that utility increases logarithmically with wealth, thus rendering the expected utility finite and the game’s value reasonable. However, this solution implies a bounded or at least sub-linear utility function, casting doubt on the compatibility of unbounded utilities with rational decision-making under EUT. Indeed, the paradox illustrates that unbounded utilities can lead to counterintuitive or impractical prescriptions for action, challenging the descriptive and normative force of EUT.
Compatibility of Unbounded Utilities with EUT
Despite such paradoxes, it is worth noting that EUT, as a formal framework, does not explicitly prohibit unbounded utility functions. The axioms of EUT focus on the consistency of preferences rather than the magnitude of utility values (von Neumann and Morgenstern, 1944). Mathematically, an unbounded utility function can still satisfy these axioms, provided preferences remain well-ordered and probabilities are appropriately defined. For instance, an agent might assign utility \(u(x) = x\) for monetary outcomes \(x\), without theoretical contradiction within EUT’s structure. Furthermore, some argue that unbounded utilities are necessary to capture preferences in extreme scenarios, such as life-and-death decisions, where outcomes might be valued beyond any finite bound (Harsanyi, 1955).
However, practical issues arise when expected utilities become infinite, as in the St. Petersburg Paradox. In such cases, EUT struggles to provide meaningful decision rules, as comparing infinite expected utilities becomes impossible. This suggests that while unbounded utilities are technically compatible with EUT’s formal structure, their application often undermines the theory’s usefulness as a guide for rational choice. This tension highlights a limitation in EUT’s applicability rather than a fundamental incompatibility, but it is a significant concern nonetheless.
Should Rational Agents Have Bounded Utility Functions?
Turning to the normative question, there are compelling arguments for requiring rational agents to have bounded utility functions. First, bounded utilities avoid the decision-theoretic absurdities illustrated by paradoxes like St. Petersburg. By ensuring that utilities remain finite, expected utilities are comparable, enabling clearer decision-making (Savage, 1972). Second, bounded utilities align more closely with psychological realism; human agents typically exhibit diminishing marginal utility for most goods, particularly wealth, as Bernoulli suggested. Experimental evidence from behavioural economics supports this, showing that individuals’ willingness to pay or risk diminishes as stakes increase beyond certain thresholds (Kahneman and Tversky, 1979).
On the other hand, imposing bounded utilities as a requirement for rationality may be overly restrictive. Some outcomes—such as avoiding catastrophic risks or achieving profound personal goals—might reasonably be assigned unbounded value by an agent. Forcing boundedness could misrepresent genuine preferences, violating the principle of descriptive accuracy in utility theory (Harsanyi, 1955). Moreover, in mathematical terms, boundedness is not a necessary condition for rationality as defined by EUT’s axioms. Thus, while bounded utilities often facilitate practical decision-making, mandating them risks oversimplifying the complexity of human values and preferences.
Implications and Alternative Approaches
The debate over unbounded utilities in EUT also invites consideration of alternative decision theories. For instance, Prospect Theory, developed by Kahneman and Tversky (1979), incorporates psychological insights into decision-making, such as loss aversion, and often implicitly assumes bounded valuations of outcomes. Such frameworks may offer more realistic models of rational behaviour without grappling with the infinities posed by unbounded utilities. Additionally, some philosophers and economists advocate for lexicographic preferences or non-standard probability measures to address paradoxes, though these approaches remain less mainstream (Fishburn, 1971).
Conclusion
In conclusion, Expected Utility Theory is technically compatible with unbounded utility functions, as its formal structure does not preclude infinite utilities. However, the practical and normative challenges posed by unbounded utilities, exemplified by the St. Petersburg Paradox, suggest significant limitations in their application. While unbounded utilities might capture extreme preferences, they often lead to decision-theoretic absurdities that undermine EUT’s utility as a prescriptive tool. On the normative question, requiring rational agents to have bounded utility functions offers practical advantages, ensuring comparability and aligning with psychological evidence of diminishing marginal utility. Nevertheless, such a requirement may fail to fully accommodate the breadth of human values. Ultimately, the tension between theoretical flexibility and practical coherence in EUT reflects broader questions in the philosophy of mathematics and decision theory about how best to model rationality—a topic that warrants continued exploration and debate.
References
- Fishburn, P. C. (1971) Utility Theory for Decision Making. Wiley.
- Harsanyi, J. C. (1955) Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility. Journal of Political Economy, 63(4), pp. 309-321.
- Kahneman, D. and Tversky, A. (1979) Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47(2), pp. 263-291.
- Savage, L. J. (1972) The Foundations of Statistics. Dover Publications.
- von Neumann, J. and Morgenstern, O. (1944) Theory of Games and Economic Behavior. Princeton University Press.
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