Maximin Safety: When Failing to Lose is Preferable to Trying to Win

We present a new decision rule, \emph{maximin safety}, that seeks to maintain a large margin from the worst outcome, in much the same way minimax regret seeks to minimize distance from the best. We argue that maximin safety is valuable both descriptively and normatively. Descriptively, maximin safety explains the well-known \emph{decoy effect}, in which the introduction of a dominated option changes preferences among the other options. Normatively, we provide an axiomatization that characterizes preferences induced by maximin safety, and show that maximin safety shares much of the same behavioral basis with minimax regret.

💡 Research Summary

The paper introduces a novel decision rule called “maximin safety,” which seeks to maximize the distance from the worst possible outcome, mirroring the way minimax regret minimizes distance from the best outcome. The authors begin by pointing out the shortcomings of expected‑utility maximization in settings of ambiguous (Knightian) uncertainty, citing classic paradoxes such as Ellsberg’s. While alternative robust criteria—maximin utility (optimizing the worst‑case payoff) and minimax regret (optimizing the worst‑case regret)—have been proposed, both satisfy a “independence of dominated alternatives” (IDA) property: adding a dominated (decoy) option to the menu does not affect the ranking of the original alternatives. Empirical work, however, shows that human decision‑makers are highly sensitive to decoys, a phenomenon known as the decoy (or asymmetric dominance) effect, which cannot be captured by the existing rules.

Maximin safety is defined formally. For a finite set of states S, a set of outcomes X with utility function U, and a feasible menu M of acts, the safety of an act a in state s is
 safety(a,s) = U(a,s) – min_{a′∈M} U(a′,s).
Thus safety measures how far the act’s payoff lies above the worst payoff that could be obtained in that state. The overall safety of a is the worst‑case safety across states: safety(a) = min_{s∈S} safety(a,s). The decision rule selects the act with the highest safety value. This definition makes safety the dual of regret: regret measures the shortfall from the best possible payoff, safety measures the surplus over the worst possible payoff.

The authors illustrate the rule with two intuitive examples. In a camera‑purchase scenario, two cameras (travel and sports) are evaluated across two possible market conditions (safari, World Cup). Without a decoy, both cameras have safety zero because each is the worst in some state. Adding a dominated “decoy” camera eliminates the worst‑case for the travel camera, giving it a positive safety and making it the preferred choice. This mirrors experimental findings where the presence of a decoy raises the selection share of the dominated alternative from 36 % to 46 %. A second example involves hunters choosing running speeds on wet versus dry roads; the “hustle” option never becomes the slowest, so its safety is positive, reflecting a natural desire to stay ahead of the slowest competitor.

The paper then compares maximin safety to existing criteria. Like minimax regret, safety is a min‑max principle, but it is menu‑dependent and deliberately violates IDA, thereby accounting for decoy effects. The authors provide an axiomatic characterization. Their axioms include monotonicity of safety (higher utility yields higher safety), a non‑symmetric treatment of dominated alternatives (adding a dominated act can raise the safety of other acts), preservation of the minimum safety across states, and continuity. Together these axioms uniquely identify the maximin‑safety preference ordering, distinguishing it from both maximin utility and minimax regret.

A unifying framework is also proposed. By introducing an “anchoring function” that interpolates between utility, regret, and safety via a parameter λ, the authors show that each classic rule can be recovered as a special case (λ = 0 for pure utility, λ = 1 for regret, λ = −1 for safety). This suggests a broader family of robust decision criteria that can be tuned to the decision‑maker’s attitude toward best‑case versus worst‑case considerations.

The paper acknowledges several limitations. The analysis assumes a finite state space and deterministic utilities; extensions to continuous or infinite state spaces, stochastic utilities, and multi‑attribute outcomes remain open. Moreover, while safety maximization captures the decoy effect, its relationship to conventional notions of risk aversion or loss aversion requires further empirical validation. Finally, the axioms, though mathematically elegant, are abstract and need behavioral experiments to confirm their psychological plausibility.

In summary, “maximin safety” offers a theoretically grounded, axiomatized decision rule that explains why dominated alternatives can sway human choices. It complements existing robust criteria, provides a clear formal definition, and opens avenues for future research on menu‑dependent preferences, continuous uncertainty, and experimental testing of safety‑oriented behavior.