Divinity Criterion

The divinity criterion (also called D1) is an equilibrium refinement for signaling games and other dynamic games of incomplete information, introduced by Jeffrey Banks and Joel Sobel in 1987. It strengthens the intuitive criterion by imposing a systematic ordering on how off-path beliefs should be formed after observing unexpected deviations.

Where the intuitive criterion asks only whether a deviation could be profitable for some type, the divinity criterion asks which type is most likely to benefit. It formalizes the intuition that if the set of receiver responses that make a deviation profitable for type θ is strictly smaller than the set that makes it profitable for type θ′, then θ′ is "more divine" — more likely to be the source of the deviation — and the receiver should assign zero probability to θ.

Formal Definition

Consider a signaling game with a sender of type θ ∈ Θ and a receiver who observes a message m. For each type θ, define D(θ, m) as the set of receiver best responses (mixed strategies over actions) that would make sending m profitable for type θ relative to the equilibrium payoff.

A type θ is eliminated by D1 at message m if there exists another type θ′ such that:

D(θ, m) ⊂ D(θ′, m)

That is, θ is eliminated whenever there is another type whose set of beneficial responses strictly contains θ's set. Under the divinity criterion, the receiver's off-path belief after observing m must assign probability zero to all eliminated types.

This is stronger than the intuitive criterion. The intuitive criterion eliminates a type only if it could never benefit from the deviation — if D(θ, m) = ∅. The divinity criterion eliminates types even when they could benefit, provided some other type could benefit from more responses.

Relationship to Other Refinements

The divinity criterion sits in a hierarchy of belief-based refinements between perfect Bayesian equilibrium (which imposes no restrictions on off-path beliefs) and universal divinity (D2, which is stronger still).

PBE: No restriction on off-path beliefs.
Sequential equilibrium: Beliefs must be consistent with trembles.
Intuitive Criterion: Eliminate types that could never benefit.
Divinity (D1): Eliminate types that are "less likely" to benefit (strict subset).
Universal Divinity (D2): Eliminate types whose beneficial-response set is a subset of every other type's.

The hierarchy is strict: every equilibrium satisfying D2 satisfies D1; every equilibrium satisfying D1 satisfies the intuitive criterion; and every equilibrium satisfying the intuitive criterion is a sequential equilibrium (which is a PBE). But the converses do not hold.

The divinity criterion is also connected to the Never-a-Weak-Best-Response condition and to the broader strategic stability framework of Kohlberg and Mertens (1986). These later approaches shift attention from restrictions on beliefs to restrictions on strategies, but they capture similar intuitions about which equilibria are robust to small perturbations.

Applications and Controversies

In the canonical Spence signaling model, divinity often selects the same equilibrium as the intuitive criterion: the efficient separating equilibrium. But in richer games — particularly those with more than two types or multidimensional signals — divinity and the intuitive criterion can disagree. Divinity is especially powerful in games where types differ in how much they value the receiver's response, not just in the cost of sending the signal.

The criterion has been applied to industrial organization (limit pricing, predation), financial markets (security design), and political economy (campaign signaling). In each case, it sharpens predictions by eliminating pooling equilibria that the intuitive criterion might permit.

But divinity is not without controversy. Like all refinement programs, it relies on a counterfactual story about what players believe after events that never occur in equilibrium. The "more divine" type is not more divine because of any empirical regularity; it is more divine because the mathematician has defined the set containment relation in a particular way. Critics argue that this is a formal sleight of hand: the criterion imports normative content under the guise of positive analysis.

Moreover, experimental evidence casts doubt on whether human subjects reason in ways consistent with D1. When deviations occur in laboratory signaling games, subjects' belief revisions are often heuristic, context-dependent, and driven by salience rather than by systematic comparison of beneficial-response sets.

The divinity criterion is the refinement that game theorists reach for when the intuitive criterion proves too weak. But the difference between "could never benefit" and "benefits from fewer responses" is a difference in degree, not in kind — and the slide from one to the other is the slide from intuition to artifice. D1 does not capture what real people believe after surprising events; it captures what theorists believe real people should believe. The criterion is not a discovery about human reasoning; it is a convention about mathematical elegance. And like all such conventions, its authority rests not on evidence but on consensus.