How to Make an Action Attractive

1. Problem Statement and Motivation

The paper addresses a fundamental problem in robust paternalism and mechanism design: How can a policymaker (or designer) modify a desired action $a$ into a new action $\hat{a}$ such that $\hat{a}$ is guaranteed to be at least as attractive as $a$ relative to a competing alternative $b$, regardless of the decision-maker's (DM) subjective beliefs about uncertainty or their specific risk preferences?

• Context: Policymakers often want to steer agents toward a specific behavior (e.g., vaccination, retirement saving, quitting smoking) but lack reliable knowledge of the population's beliefs (priors) or utility functions (risk aversion levels).
• The Challenge: A modification that appears to be an improvement under one set of beliefs or risk preferences may backfire under another. The authors seek conditions that ensure robustness across all possible beliefs and all increasing concave utility functions (risk aversion).
• Formal Question: Under what conditions is $\hat{a}$ $b$-superior to $a$? That is, for every belief $\mu$ and every utility $u \in \mathcal{U}$, if $E_\mu[u(a)] \geq E_\mu[u(b)]$, then $E_\mu[u(\hat{a})] \geq E_\mu[u(b)]$.

2. Methodology and Model

The authors employ a formal expected utility framework with the following structure:

• State Space: A compact, metrizable topological space $\Theta$ with Borel $\sigma$-algebra.
• Actions: Two actions, $a$ and $b$, are bounded Borel-measurable functions mapping states to monetary payoffs ($a_\theta, b_\theta \in \mathbb{R}$).
• Decision Maker: An Expected Utility maximizer with a von Neumann-Morgenstern utility function $u: \mathbb{R} \to \mathbb{R}$.
• Uncertainty: The designer does not know the DM's belief $\mu \in \Delta(\Theta)$ or the specific form of $u$, only that $u$ belongs to a class $\mathcal{U}$.
  • Primary Class ($\mathcal{U}_{RA}$): Strictly increasing and weakly concave (risk-averse) utilities.
  • Secondary Class ($\mathcal{U}_{\uparrow}$): Strictly increasing utilities (no risk aversion assumption).
• Modification: The designer modifies $a$ to $\hat{a}$ by altering the state-dependent payoff profile (e.g., via subsidies, transfers, or contract redesign).

3. Key Contributions and Results

A. Main Result: Robust Paternalism with Risk Aversion (Theorem 3.1)

The paper's central contribution is the characterization of $\hat{a}$ when the DM is risk-averse.

• Theorem 3.1: Action $\hat{a}$ is $b$-superior to $a$ over all beliefs and all risk-averse utilities if and only if:
  1. State-wise Improvement in Favorable States: $\hat{a}_\theta > b_\theta$ for all states $\theta$ where $a_\theta > b_\theta$ (denoted as set $A$).
  2. Dominance of a Mixture: $\hat{a}$ weakly dominates a convex combination (mixture) of $a$ and $b$. Specifically, there exists a $\lambda \in [0,1]$ such that for all $\theta$:
     $$\hat{a}_\theta \geq \lambda a_\theta + (1-\lambda)b_\theta$$
• Intuition: If a risk-averse agent prefers $a$ to $b$, they must also prefer any mixture of $a$ and $b$ to $b$ (by the independence axiom and risk aversion). Therefore, if $\hat{a}$ dominates such a mixture, it is strictly more attractive than $b$.
• Counter-intuitive Insight: To make an action more attractive to a heterogeneous, risk-averse population, the designer must sometimes increase the riskiness of the action. If the alternative $b$ is riskier than $a$, a robust improvement $\hat{a}$ must "move $a$ toward $b$" (become riskier) rather than simply smoothing payoffs. Simply "insuring" $a$ (making it less risky) may fail if $b$ is the riskier option.

B. Result without Risk Aversion (Theorem 3.3)

If the designer cannot assume the DM is risk-averse (only that they prefer more money to less):

• Theorem 3.3: The conditions become much stricter. $\hat{a}$ is $b$-superior if and only if:
  1. $\hat{a}_\theta > b_\theta$ for all $\theta \in A$.
  2. $\hat{a}$ state-wise dominates either $a$ or $b$.
• Implication: Without the structure provided by risk aversion (concavity), only unambiguous dominance improvements work. Any "rotation" of the payoff profile that helps some agents but hurts others (even slightly) fails the robustness test.

C. Known Beliefs (Section 5.1)

When the designer knows the DM's belief $\mu$ but not their utility:

• The problem reduces to a revealed preference exercise using Fishburn's (1975) results.
• Result: $\hat{a}$ improves upon $a$ if and only if the lottery induced by $\hat{a}$ satisfies a specific stochastic dominance condition relative to a mixture of the lotteries induced by $a$ and $b$. This allows for First-Order Stochastic Dominance (FOSD) or Second-Order Stochastic Dominance (SOSD) depending on the utility class, but the condition is linear in the space of lotteries.

D. Multiple Alternatives (Section 5.2)

When comparing $a$ against a set of alternatives $B$ (not just a single $b$):

• Proposition 5.3: Under "Rich" state spaces and "Single-Peaked" preferences, $\hat{a}$ is superior to $a$ against all $b \in B$ if and only if $\hat{a}$ dominates a mixture of $a$ and each $b \in B$ individually.
• This suggests that pairwise robustness conditions aggregate naturally under specific structural assumptions.

E. Endogenous Information Acquisition (Section 6)

The authors explore a scenario where the DM can choose to acquire information before acting (Rational Inattention).

• Two States: If $\hat{a}$ dominates $a$ or $b$ (state-wise), it is selected more frequently.
• Three or More States: Dominance is no longer sufficient. The paper provides a counter-example where $\hat{a}$ state-wise dominates $a$, yet the DM acquires information in a way that leads to $\hat{a}$ being chosen less frequently than $a$.
• Mechanism: Changing the payoff structure alters the geometry of the value function, changing the optimal information acquisition strategy (the set of posteriors). A "better" action can induce a learning strategy that makes the agent less likely to choose it compared to the original action.

4. Applications

The paper illustrates the theoretical results with four applications:

1. Smoking Cessation: To robustly encourage quitting among risk-averse smokers with unknown beliefs, a policy must make the "failure" state (relapse) less painful (insurance-like) rather than penalizing it. Penalizing relapse (making the payoff more negative) only works for optimistic, low-risk-aversion types, failing the robustness test.
2. Political Competition: To win against an opponent $b$, a party should not just move to the center arbitrarily. The candidate $\hat{a}$ must be a mixture of the original candidate $a$ and the opponent $b$. This formalizes the "move to the center" strategy as a convex combination of the status quo and the opponent.
3. Bilateral Trade: For a trade agreement to remain robustly acceptable to both a buyer and seller with unknown risk preferences, the new transfer scheme must be a convex combination of the old transfer and the asset's value.
4. Insurance: A new insurance contract is robustly acceptable only if it is a convex combination of the status quo and the uninsured state. Interestingly, this can imply that a "better" contract exposes the consumer to more risk if the alternative (no insurance) is extremely risky.

5. Significance

• Theoretical Advancement: The paper bridges the gap between stochastic dominance (which applies to lotteries) and state-dependent payoffs (which are non-random). It provides the first complete characterization of robust action modification under uncertainty about both beliefs and utilities.
• Policy Design: It offers a rigorous "checklist" for policymakers. It warns that intuitive improvements (like smoothing payoffs or adding penalties) can fail if the population is heterogeneous in risk aversion.
• Strategic Insight: The finding that robust improvements often require "moving toward the opponent" (making the action riskier if the alternative is riskier) challenges standard intuition in product design and political strategy.
• Information Acquisition: The final section highlights a critical limitation of dominance in dynamic settings, showing that even unambiguous improvements can be detrimental when agents endogenously choose how much to learn.

In summary, Pease and Whitmeyer provide a rigorous framework for robust paternalism, demonstrating that the only modifications guaranteed to work across all risk-averse agents and beliefs are those that make the desired action a "dominant mixture" of itself and the alternative.