Betting under Common Beliefs: The Effect of Probability Weighting

Abstract: This paper examines the impact of introducing a Rank-Dependent Utility (RDU) agent into a von Neumann-Morgenstern (vNM) pure-exchange economy with no aggregate uncertainty. In the absence of the RDU agent, the classical theory predicts that Pareto-optimal allocations are full-insurance, or no-betting, allocations. We show how the probability weighting function of the RDU agent, seen as a proxy for probabilistic risk aversion that is not captured by marginal utility of wealth, can lead to Pareto optima characterized by endogenous betting, despite common baseline beliefs. Such endogenous betting at an optimum leads to uncertainty-generating trade arising purely from heterogeneity in the perception of risk, rather than in beliefs. Our results formalize the intuitive understanding that probability weighting can act as an endogenous source of belief heterogeneity, and provide a new behavioral foundation for the coexistence of common beliefs and speculative behavior, in an environment with no initial aggregate uncertainty. Interpreting the RDU agent's nonlinear weighting function as an ``internality'' prompts the question of whether a social planner should intervene. We show how a benevolent social planner can nudge the RDU agent to behave closer to a vNM agent, through costly statistical or financial education, thereby (partially) restoring the optimality of full-insurance allocations.

• The paper shows that one rank-dependent utility agent can generate Pareto-efficient betting alongside expected-utility agents with identical beliefs, overturning the classical no-betting result.
• The quantile-based characterization links full insurance and endogenous uncertainty to the convex envelope of the probability distortion, with non-convex weighting producing mixed payoffs and betting regions.
• The paper finds that nudging toward linear probability weighting is worthwhile only above a distortion threshold, with calibrated effort reaching 6.57%–9.25% of aggregate wealth in severe cases.

This paper by Beißner, Boonen, and Ghossoub studies pure-exchange economies under uncertainty in which a single Rank-Dependent Utility (RDU) maximizer interacts with von Neumann–Morgenstern (vNM) expected-utility (EU) agents, all sharing a common baseline probability measure $P$ on a nonatomic state space, and facing a constant aggregate endowment $\mathtt{w}$. The central contribution is to show that the RDU agent's nonlinear probability weighting function—an instance of probabilistic risk aversion not captured by marginal utility—can generate Pareto-efficient allocations involving endogenous betting, thereby overturning the classical no-betting theorem that holds when all agents are EU maximizers with common beliefs (2602.24194).

Motivation and relation to prior work

Under EU theory, risk-averse agents with common beliefs never engage in uncertainty-generating trade at Pareto optima: full-insurance (no-betting) allocations characterize efficiency (Billot et al.; Rigotti, Shannon, and Strzalecki). Since subjective probabilities in the Savage/de Finetti framework are preference-derived, heterogeneous preferences should generically induce heterogeneous beliefs and hence widespread betting—yet real economies exhibit far less speculation than this predicts. Billot et al. attributed the discrepancy to Ellsberg-type behavior, analyzing maxmin expected utility and Choquet expected utility. The present paper instead attributes it to Allais-type behavior, extending the authors' earlier two-agent RDU analysis ("(No-)Betting Pareto-Optima under Rank-Dependent Utility," Mathematics of Operations Research 2024) to an economy with one RDU agent and $n-1$ EU agents.

Model and characterization of Pareto optima

The economy has $n>1$ agents; agent 1 maximizes $U_1(X)=\int u_1(X)\,dT\circ P$ (a Choquet integral with weighting function $T$), agents $2,\dots,n$ are EU maximizers, all utilities are strictly increasing and strictly concave, and $T$ is twice differentiable with $T(0)=0$, $T(1)=1$. Efficiency is studied via a weighted-utilitarian problem whose solutions are Pareto optimal (and, if $\mathtt{w}$0 is convex, every Pareto optimum is such a solution).

The key technical device is a quantile reformulation: comonotonicity of efficient allocations allows the inner problem over the EU agents to be solved pointwise via the Borch rule, yielding a representative EU agent with aggregate utility $\mathtt{w}$1. The main characterization states that an allocation is optimal if and only if

$\mathtt{w}$2

where $\mathtt{w}$3, $\mathtt{w}$4, and $\mathtt{w}$5 is the convex envelope of the conjugate distortion $\mathtt{w}$6. Two corollaries frame the economics:

• Classical case recovered: if $\mathtt{w}$7 is convex, then $\mathtt{w}$8 and Pareto-optimal allocations are exactly full-insurance allocations.
• Betting emerges: if $\mathtt{w}$9 is non-convex, $n-1$0 is affine precisely on the set where $n-1$1; the probability mass of this event measures endogenous uncertainty—the betting component of the optimum. Probability weighting thus acts as an endogenous source of belief heterogeneity even though baseline beliefs coincide.

Comparative statics of endogenous uncertainty

For S-shaped and inverse S-shaped distortions with inflection point $n-1$2, the paper derives the convex envelope in closed form through a tangent point $n-1$3, giving the mass of the full-insurance event:

Distortion $n-1$4	Full-insurance mass $n-1$5
Inverse S-shaped	$n-1$6
Linear or convex	$n-1$7
S-shaped	$n-1$8

Efficient payoffs have mixed distributions: an atom at the full-insurance value plus an absolutely continuous component whose density is characterized explicitly as $n-1$9.

For Prelec weighting $n>1$0, the tangent point admits a closed form for $n>1$1: $n>1$2. A notable—and counter-intuitive—comparative static emerges: in the inverse S-shaped region ($n>1$3), increasing the nonlinearity of the weighting function increases the probability of the full-insurance event, so more distortion can coexist with more insurance; the same pattern holds for Tversky–Kahneman weighting. Under concave $n>1$4, there is no atom at all ($n>1$5, heavy betting); under convex $n>1$6, full insurance prevails.

With CARA utilities ($n>1$7), optimal allocations take the explicit form $n>1$8 plus deterministic zero-sum side payments governed by the welfare weights $n>1$9. In the calibrated three-agent example ($U_1(X)=\int u_1(X)\,dT\circ P$0, $U_1(X)=\int u_1(X)\,dT\circ P$1, $U_1(X)=\int u_1(X)\,dT\circ P$2), roughly $U_1(X)=\int u_1(X)\,dT\circ P$3 of the probability mass lies on full insurance; the RDU agent's payoff is positively skewed while the EU agents bear negatively skewed exposures. Switching from $U_1(X)=\int u_1(X)\,dT\circ P$4 to $U_1(X)=\int u_1(X)\,dT\circ P$5 flips the sign of skewness for every agent: the RDU agent moves from "buyer" of a bet (small chance of large gains) to "seller" (small chance of large losses). For Hurwicz expected utility weighting (Bleichrodt, Han, Grant, Yang), the full-insurance probability increases with the ambiguity index $U_1(X)=\int u_1(X)\,dT\circ P$6, though less steeply when perceived ambiguity $U_1(X)=\int u_1(X)\,dT\circ P$7 is large; e.g., with $U_1(X)=\int u_1(X)\,dT\circ P$8, $U_1(X)=\int u_1(X)\,dT\circ P$9. Welfare evaluation via cash-additive certainty equivalents shows that the sum of certainty equivalents is positive whenever $T$0 and grows as $T$1 moves away from 1, so by the Kaldor–Hicks criterion betting combined with suitable side payments can make all agents better off.

Nudging the RDU agent

Interpreting nonlinear probability weighting as an "internality"—a within-person externality in the sense of Herrnstein et al.—the paper asks whether a benevolent planner following libertarian-paternalistic principles should intervene. The planner exerts costly effort $T$2, reducing aggregate wealth to $T$3 and mixing the weighting function toward linearity via $T$4. Two useful structural results follow: the nudged functional satisfies $T$5, and the nudged convex envelope is affine in $T$6: $T$7. The planner's first-order condition equates the marginal welfare gain from de-distorting against the marginal cost borne by the EU agent through reduced aggregate wealth.

In the CARA/Prelec numerical illustration with $T$8 and $T$9, the optimal effort is $2,\dots,n$0 of aggregate wealth when $2,\dots,n$1 ($2,\dots,n$2), rising to $2,\dots,n$3 for $2,\dots,n$4; crucially, $2,\dots,n$5 whenever $2,\dots,n$6. This threshold behavior implies the planner tolerates a nontrivial amount of endogenous uncertainty before expending resources, intervening only when probability distortions are sufficiently pronounced relative to the cost of education.

Limitations and open questions

Several caveats are explicit in the paper. First, the welfare analysis requires taking a stand on the divergence between experienced and decision utility; treating the weighting function as an internality is an interpretive choice, and the planner's criterion embeds it rather than deriving it. Second, the comparative statics and closed forms rely on specific families (Prelec, Tversky–Kahneman, Hurwicz-type weightings) and on exponential utility for fully explicit allocations; general characterizations are distributional but less transparent. Third, the nudging technology is assumed to act linearly on $2,\dots,n$7 via a concave $2,\dots,n$8 with prescribed curvature properties, and only partial restoration of full insurance is achieved—full insurance is recovered exactly only as $2,\dots,n$9. Open questions include equilibrium (rather than Pareto-optimum) analysis with market-clearing prices, and the extension to heterogeneous baseline beliefs combined with heterogeneous weighting functions.

Conclusion

The paper establishes, within a tractable quantile-based framework, that a single RDU agent suffices to break the classical equivalence between common beliefs and no betting: probability weighting generates well-defined regions of Pareto-efficient speculative trade, with the geometry of the convex envelope of the conjugate distortion determining exactly where full insurance ends and betting begins. The closed-form characterizations—for tangent points, payoff densities, and optimal nudging expenditures—make the mechanism quantitative rather than merely qualitative, and the planner's analysis shows that correcting the induced internality is worthwhile only when distortions exceed an identifiable threshold.