A new paper by Federico Echenique and Farzad Pourbabaee explores a surprising question in economics: when people share risk efficiently, how likely is it that they can improve the deal after a shock hits the economy?
Their answer is striking:
As the number of possible states of the world grows, the chance of finding a mutually beneficial renegotiation falls exponentially—basically to zero.
What problem are they studying?
Imagine a group of risk-averse people (or firms) who already made a good risk-sharing agreement. Then something changes—a shock hits. This shock creates aggregate uncertainty (system-wide uncertainty), and they consider renegotiating the original agreement.
But there’s a catch: renegotiation is only worth doing if everyone gets at least a minimum improvement (think transaction costs, coordination costs, or simply requiring a meaningful gain).
The paper asks: How often does such an improvement actually exist?
The big result: complexity kills renegotiation chances
The authors show that when the economy has a large number of states (many possible outcomes, scenarios, assets, contingencies), the probability of finding a new deal that improves welfare shrinks exponentially with the number of states.
This is not just “rare.” It becomes extremely unlikely.
And here’s the surprising part:
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This happens regardless of how risk-averse people are
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It still happens even when there seems to be “room” for trade in low dimensions
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It persists under different economic setups (individual improvements, collective improvements, equilibrium settings)
In plain English: the more complex the uncertainty, the harder it becomes to find a win-win renegotiation—even if everyone is rational and risk-sharing is the goal.
Why this is counterintuitive
Normally, we might think:
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“If people are only mildly risk-averse, they should be flexible.”
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“If a shock creates opportunities, someone should benefit.”
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“With many agents, at least one person should be willing to absorb the shock.”
The paper says: not in high dimensions.
In high-dimensional spaces, geometry behaves differently. The authors rely on deep results from probability and geometry (concentration of measure and isoperimetric inequalities), which imply a “shape doesn’t matter” effect.
That means the exact shape of preferences (how curved utility functions are, etc.) matters much less than we would expect. The geometry of high dimensions dominates.
The three main contributions (simplified)
1) Individual welfare improvement becomes extremely unlikely
Even if you ask a weaker question—“Can at least one agent be made better off by the shock?”—the answer still becomes “almost never” as the number of states grows.
This is true even if there are many agents. The probability bound does not depend on the number of agents in the favorable way one might hope.
2) Collective welfare-improving renegotiation also collapses
The paper then studies the stronger question: can the group renegotiate after an aggregate shock so that everyone is better off (by a minimum margin)?
Again, the probability drops exponentially as the state space grows.
This result also connects to classic welfare economics concepts like the Scitovsky contour and Pareto improvements.
3) Ambiguity and multiple priors: improving trade requires “small” belief sets
In a two-agent model with multiple priors (think ambiguity-averse agents, not just standard expected utility), the authors show something even more subtle:
If there is room for Pareto-improving trade, then at least one agent must have a very small set of prior beliefs.
In practical terms, one side must be close to ambiguity-neutral (less ambiguity-averse) for improvement to be possible.
This gives a high-dimensional, quantitative version of a known intuition in economics: ambiguity aversion reduces trade.
Why this matters outside theory
This paper has implications for:
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Financial markets (many assets = many states)
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Insurance and reinsurance
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Contract renegotiation under uncertainty
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Macroeconomic policy design
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Mechanism design in complex environments
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Behavioral economics of ambiguity aversion
The message is uncomfortable but powerful:
In very complex systems, “there should be a better deal” is often mathematically false in practice.
Even when improvements exist in low-dimensional examples or intuitively “feel” possible, high-dimensional uncertainty can make them vanish.
