This is a linkpost for Money Pumps for Agents Who Are Ambiguity Averse or Ambiguity Seeking by Adam Elga and Johan Gustafsson, which was added to PhilPapers on 9 March 2026. Below is a summary from Claude Opus 4.8 High. Adam said "I did not see any problems when I took a quick look at the summary". I also think the summary is accurate based on my read of the article. I used the following prompt. "Hi. Make an in-depth summary of the paper "Money Pumps for Agents Who Are Ambiguity Averse or Ambiguity Seeking", which I send attached".
The central claim
Adam Elga and Johan Gustafsson argue that, under very general conditions, any agent who is ambiguity-sensitive—whether ambiguity-averse or ambiguity-seeking—is vulnerable to a money pump: a sequence of choices through which the agent gives up something for no gain, ending up worse off than they could have been for free. Since vulnerability to a money pump is widely taken to be a mark of irrationality, the conclusion is that ambiguity-sensitivity is irrational.
The paper positions itself as an advance over the existing literature in three respects. It is more general, resting on only minimal assumptions about the target agent. It avoids the standard reliance on a strong form of backward induction [context]—in particular, it does not assume that a rational agent expects to keep choosing rationally even after having made an irrational choice. And it is deliberately simple.
The phenomenon: ambiguity
The setup distinguishes two kinds of evidence. Unambiguous evidence comes with known chances—radioactive decay rates, a well-studied medical procedure with a documented 50% success rate, a demonstrably fair coin. Ambiguous evidence does not—an esoteric physical process whose governing chances are unknown, a never-attempted procedure with no statistics, the question of extraterrestrial life where it is unclear how to weigh the evidence.
An agent is ambiguity-sensitive if they care about this difference. The canonical illustration is ambiguity aversion: offered an unambiguous medical procedure with a known 50% success rate versus two novel procedures (with no known chances, but where exactly one of the two would succeed), the ambiguity-averse agent prefers the unambiguous one. The parallel two-urn case (an urn known to be half red, versus an urn of unknown composition) makes the same point: the agent prefers to bet on the urn whose composition they know.
Characterizing ambiguity-sensitivity (Section 3)
A key methodological move is to state everything in terms of preferences over acts rather than probabilities. This sidesteps any commitment to how ambiguous beliefs ought to be represented (interval-valued probabilities, sets of probability functions, second-order probabilities, etc.), which is exactly the territory where rival theories disagree.
The agent's preference relation (≽) is assumed transitive and complete over Savage acts (functions from states of the world to outcomes). Let H be a less ambiguous proposition and U a more ambiguous one. Writing E for a bet that pays the better outcome a if E is true and the worse outcome b otherwise, the authors define ambiguity-sensitivity by two conditions holding for some outcomes a ≻ b:
- (1) Betting on H is preferred to betting on U.
- (2) Betting against H is preferred to betting against U.
In other words, the agent favors bets resolved by the less ambiguous proposition, in both directions. (If H were the more ambiguous proposition, the same pattern would express ambiguity-seeking.)
The strength of the paper lies in how undemanding this characterization is. It is compatible with every non-trivial proposition being somewhat ambiguous; with an agent being averse for some pairs and seeking for others; and it imposes none of the structural conditions that earlier results required. It does not assume Seidenfeld's lottery structure, the sure-thing principle [context], Epstein and Le Breton's modified Savage axioms, biseparability, or backward induction. The authors don't claim (1) and (2) are necessary for ambiguity-sensitivity; their weaker claim is that if any interesting form of ambiguity-sensitivity is rationally permissible, then so is having preferences that satisfy (1) and (2).
Step one: ambiguity-sensitivity forces a preference reversal (Section 4)
An anticipated preference reversal is a situation where the agent strictly prefers X to Y now, yet would weakly prefer Y to X upon learning E and would also weakly prefer Y to X upon learning not-E. (Either way the news goes, the ranking flips—something the agent can foresee in advance.)
The argument needs a few mild assumptions: preferences stay transitive and complete after learning; preferences over outcomes don't drift over time; and the agent knows all this. Two definitions do the real work. Two acts agree on a proposition E if they yield identical outcomes at every state compatible with E. A proposition is null if any two acts that agree on its negation are ranked as equally good (null propositions are treated as negligible).
The crucial principle is Learning Elimination: once a non-null E is learned, the agent's comparison between any two acts depends only on those acts' outcomes at states compatible with E. States ruled out by the evidence drop out of consideration. (Illustrated with a 1-to-500 lottery: if you learn the number is either 5 or 18, only the payoffs at 5 and 18 can matter.) The authors stress this is not Savage's sure-thing principle (P2)—Learning Elimination governs preferences after learning, whereas P2 constrains initial preferences. This distinction is important, because many ambiguity-sensitivity sympathizers reject P2, treating Ellsberg's cases as counterexamples to it; the argument would be dialectically useless if it smuggled P2 back in.
The proof introduces the proposition S = "H and U have the same truth value" (equivalently, H if and only if U). One first shows S is non-null (if it were null, conditions (1) and (2) would be contradicted). The engine of the proof is that the bets are linked through S in a revealing way: conditional on S, a bet on H and a bet on U coincide as acts (and likewise the two bets against); conditional on not-S, betting on H coincides with betting against U (and betting on U with betting against H). By completeness, the agent must rank the conditional comparison one way or the other, and Learning Elimination then propagates that ranking. Whichever way it falls, one of the two unconditional strict preferences—(1) or (2)—ends up contradicted by weak conditional preferences in both branches. That is exactly a preference reversal. So a reversal is unavoidable for any ambiguity-sensitive agent.
Step two: a preference reversal yields a money pump (Section 5)
To turn the reversal into exploitation, the paper adds machinery about sourings. A souring X⁻ of an act X is just like X but certainly slightly worse along some dimension the agent cares about (e.g., minus a token penalty). Two plausible principles govern these: the Souring Principle (an act is always preferred to any souring of it) and Souring Continuity (if X ≻ Y, a souring X⁻ exists with X ≻ X⁻ ≻ Y). Together they say a small penalty can shave an act's value by an arbitrarily small amount.
On the behavioral side, the agent is assumed to choose rationally at every node reachable without any prior irrational choice, and to be sophisticated: able to predict their own future rational choices and to factor those predictions into present decisions. (Appendix C extends the result to naive and myopic choosers.) Notably, this is a restricted form of backward induction—it does not require the agent to assume they'd choose rationally at nodes only reachable via irrational choices, which sidesteps a standard objection (Binmore) to backward-induction arguments. The decision problem is "BI-terminating," so the weaker assumption suffices.
The pump itself: start the agent holding a soured act X⁻, where (from the reversal) Y is strictly preferred to X⁻ both after learning E and after learning not-E, yet a doubly-soured X⁻⁻ is still strictly preferred to Y. At node 1, the agent may either pay a small penalty (swap X⁻ for the worse X⁻⁻ and exit) or decline, proceed to learn whether E holds, and then at node 2 (E true) or node 3 (E false) choose between keeping X⁻ and switching to Y.
Reasoning it through: a sophisticated agent who declines at node 1 foresees that, once E resolves, they will rationally switch to Y (since Y is preferred to X⁻ on both branches). So declining leads to Y. But X⁻⁻ is preferred to Y—so the agent pays the penalty at node 1, taking X⁻⁻. The upshot is that the agent ends up with X⁻⁻ when they could simply have held onto X⁻ for free. They have paid for nothing. That is the money pump.
Conclusion and significance
The headline result is that ambiguity-sensitive agents, under extremely general conditions, are money-pumpable, and so—if exploitability signals irrationality—ambiguity-sensitivity is irrational. This is framed as a direct reply to Ellsberg, who insisted that ambiguity-averse behavior need not be intransitive or amount to "throwing away utility" through dominated choices. Elga and Gustafsson reply that, under very general conditions, such agents do throw away utility.
The authors are appropriately cautious about what remains to be shown (footnote 32): a full case for irrationality would also need a converse result (that suitable non-ambiguity-sensitive preferences are not money-pumpable) and would have to engage the broader objections to inferring irrationality from exploitability.
The appendices
Appendix A surveys the prior literature the result builds on and generalizes (Seidenfeld 1988, Machina 1989, Hammond 1988 on consequentialism, Al-Najjar & Weinstein 2009 on naive choosers, Epstein & Le Breton 1993 on dynamic consistency, White 2009 on dilation). Appendix B reruns the reversal proof under a weakened condition—replacing the strict (1) with a weak "betting on H is at least as good as betting on U"—at the cost of one extra assumption, Strong Dominance. Appendix C adapts the money pump to naive choosers (who follow whatever plan currently looks best, ignoring whether they'll stick to it) and myopic choosers (who treat each trade as if it were their last), using an additional Sweetening Continuity assumption.
