Reverse Bayesianism: A Generalization

2.1 Preliminaries

We briefly restate the framework of Karni and Vierø (2013). Let F be a finite, nonempty set of feasible actions, and C be a finite, nonempty set of feasible consequences. Together these sets determine a conceivable state space, C F , whose elements depict the resolutions of uncertainty.

On this conceivable state space, we define what we refer to as conceivable acts. Formally,

where Δ ( C ) is the set of all lotteries over C . As is usually done, we abuse notation and use p to also denote the constant act that returns the lottery p in each state. We use both c and δ c to denote the lottery that returns consequence c with probability 1, depending on the context.

Discovery of new consequences expands the conceivable state space. Let C denote the initial set of consequences and suppose that a new consequence, c ¯ , is discovered. The set of consequences of which the decision maker is aware then expands to C ′ = C ∪ ​ { c ¯ } . As a result, the conceivable state space expands to ( C ′ ) F . The corresponding expanded set of conceivable acts is given by

We consider a decision maker whose choice behavior is characterized by a preference relation ≽ F ^ on the set of conceivable acts F ^ . We denote by ≻ F ^ and ∼ F ^ the asymmetric and symmetric parts of ≽ F ^ with the interpretations of strict preference and indifference, respectively. For any f ∈ F ^ , p ∈ Δ ( C ) , and E ⊆ C F , let p E f be the act in F ^ obtained from f by replacing its s –th coordinate with p for all s ∈ E . A state s ∈ C F is said to be null if p s f ∼ F ^ q s f for all p ,   q ∈ Δ ( C ) and for all f ∈ F ^ . A state is said to be nonnull if it is not null. Denote by E N the set of null states and let S ( F ,   C ) = C F − E N be the set of all nonnull states. Henceforth we refer to S ( F ,   C ) as the feasible state space.

When the state space expands so does the set of conceivable acts, which means that the preference relation must be redefined on the extended domain. Specifically, if F ^ * is the expanded set of conceivable acts in the wake of discoveries of new feasible consequences, then the corresponding preference relation is denoted by ≽ F ^ * . Let F be a family of sets of conceivable acts corresponding to increasing awareness of consequences.^[1]

In Karni and Vierø (2013) it is implicitly assumed that upon the expansion of the state space following the discovery of a new consequence, non-null states remain non-null. Formally, for all f ∈ F ^ and f ′ ∈ F ^ * ,   if p s f ≻ F ^ q s f then p s f ′ ≻ F ^ ∗ q s f ′ , for all s ∈ S ( F , C ) . Under monotonicity this is equivalent to assuming that S ( F ,   C ) ⊆ S ( F ,   C ′ ) . However, as we discussed in the introduction, there are important situations in which such inclusion is untenable.

For each F ^ ∈ F , f , g ∈ F ^ , and α ∈ [ 0 ,   1 ] define the convex combination α f + ( 1 − α ) g ∈ F ^ by: ( α f + ( 1 − α ) g ) ( s ) = α f ( s ) + ( 1 − α ) g ( s ) , for all s ∈ C F . Then, F ^ is a convex subset in a linear space.^[2] We assume that, for each F ^ ∈ F , ≽ F ^ abides by the axioms of Anscombe and Aumann (1963). Formally,

• (A.1) (Weak order ) For all F ^ ∈ F , the preference relation ≽ F ^ is transitive and complete.

• (A.2) (Archimedean ) For all F ^ ∈ F and f , g , h ∈ F ^ , if f ≻ F ^ g and g ≻ F ^ h then α f + ( 1 − α ) h ≻ F ^ g and g ≻ F ^ β f + ( 1 − β ) h , for some α , β ∈ ( 0,1 ) .

• (A.3) (Independence) For all F ^ ∈ F , f ,   g ,   h ∈ F ^ , and α ∈ ( 0 ,   1 ] , f ≽ F ^ g if and only if α f + ( 1 − α ) h ≽ F ^ α g + ( 1 − α ) h .

• (A.4) (Monotonicity) For all F ^ ∈ F , f ∈ F ^ , p ,   q ∈ Δ ( C ) and nonnull event E ⊆ C F , p E f ≽ F ^ q E f if and only if p ≽ F ^ q .

• (A.5) (Nontriviality) For all F ^ ∈ F , ≻ F ^ ≠ ∅ .

Karni and Vierø (2013) postulated the following awareness consistency axiom to characterize the decision maker's reaction to expansion in his awareness of consequences:

• (A.7) (Awareness consistency) For every given F , for all C ,   C ′ with C ⊂ C ′ , S ( F ,   C ) ⊆ S ( F ,   C ′ ) , f ,   g ∈ F ^ , and f ′ ,   g ′ ∈ F ^ * , such that f ′ = f and g ′ = g on S ( F ,   C ) and f ′ = g ′ on S ( F ,   C ′ ) − S ( F ,   C ) it holds that f ≽ F ^ g if and only if f ′ ≽ F ^ ∗ g ′ .

In words, axiom (A.7) posits that preferences conditional on events that are considered feasible before the expansion of awareness remain unchanged when awareness expands.

To allow for the possibility that some non-null states become null upon the discovery of new consequences, we modify the awareness consistency axiom (A.7) by restricting the events conditional on which preferences are required to remain unchanged when awareness expands to events that are considered feasible both before and after the awareness expansion.

• (A.7r) (Revised awareness consistency) For every given F , for all C ,   C ′ with C ⊂ C ′ , and for f ,   g ∈ F ^ , and f ′ ,   g ′ ∈ F ^ * , such that f ′ = f and g ′ = g on S ( F ,   C ) ∩ S ( F ,   C ′ ) , f = g on S ( F ,   C ) − [ S ( F ,   C ) ∩ S ( F ,   C ′ ) ] and f ′ = g ′ on S ( F ,   C ′ ) − [ S ( F ,   C ) ∩ S ( F ,   C ′ ) ] it holds that f ≽ F ^ g if and only if f ′ ≽ F ^ ∗ g ′ .

To grasp the nature of the extension of the “Reverse Bayesianism” result, consider the awareness consistency axiom (A.7). It is more restrictive than axiom (A.7r) in the sense that it requires that equalities f ′ = f and g ′ = g hold on the set S ( F ,   C ) while (A.7r) requires that these equalities only hold on the event S ( F ,   C ′ ) ∩ S ( F ,   C ) . This difference is significant because if, in (A.7r), we do not insist that f = g on S ( F ,   C ) − [ S ( F ,   C ) ∩ S ( F ,   C ′ ) ] , there could arise a preference reversal g ≻ F ^ f and f ′ ≽ F ^ ∗ g ′ . This reversal reflects the fact that the decision maker initially believes that all the states in S ( F ,   C ) are nonnull, and following the revision of her beliefs in the wake of the discovery of new consequences, some states in S ( F ,   C ) become null. Hence, it is possible that initially g ≻ F ^ f because the payoffs of g in the states that would later become null make it more attractive than f but once these states are nullified, f dominates g . The awareness consistency axiom (A.7) is compelling if none of the states in S ( F ,   C ) becomes null after the new consequences are discovered.

2.2 Representation theorem

Dominiak and Tserenjigmid (2018) have shown that the invariant risk preferences axiom (A.6) in Karni and Vierø (2013) is redundant. Therefore, in addition to invoking the revised awareness consistency axiom, we state and prove the theorem below without the invariant risk preferences axiom. Since the proof in Dominiak and Tserenjigmid (2018) was based on the awareness consistency axiom, the revised awareness consistency axiom requires a proof of the below theorem that is slightly different from the proofs in both Karni and Vierø (2013) and Dominiak and Tserenjigmid (2018).

Theorem. For each F ^ ∈ F , let ≽ F ^ be a binary relation on F ^ then, for all F ^ ,   F ^ ∗ ∈ F , the following two conditions are equivalent:

1. Each ≽ F ^ satisfies (A.1) – (A.5) and, jointly, ≽ F ^ and ≽ F ^ ∗ satisfy (A.7r).

2. There exist real-valued, non-constant, affine functions, U on Δ ( C ) and U * on Δ ( C ′ ) , and for any two F ^ ,   F ^ * ∈ F , there are probability measures, π F ^ on C F and π F ^ * on ( C ' ) F , such that for all f , g ∈ F ^ ,

and, for all f ′ ,   g ′ ∈ F ^ * ,

Moreover, U and U * are unique up to positive linear transformations, and there exists such transformations for which U ( p ) = U * ( p ) for all p ∈ Δ ( C ) . The probability distributions π F ^ and π F ^ * are unique, π F ^ ( S ( F ,   C ) ) = π F ^ * ( S ( F ,   C ′ ) ) = 1 , and

for all s ,   s ′ ∈ S ( F ,   C ) ∩ S ( F ,   C ′ ) .

Proof: See the appendix.

Note that if S ( F ,   C ) ⊆ S ( F ,   C ′ ) , then S ( F ,   C ) ∩ S ( F ,   C ′ ) = S ( F ,   C ) , and we have the result in Theorem 1 in Karni and Vierø (2013).

The generalization in the theorem allows for a wider range of applications of reverse Bayesianism than Karni and Vierø (2013). The generalization permits a particular type of belief revision on the prior feasible state space, namely extreme belief revision in which a state is nullified. For prior feasible states that are still considered feasible after the expansion in awareness, beliefs are updated according to reverse Bayesianism. This can be justified on the ground that it is possible to falsify a hypothesis, but one can only gain evidence that supports that something is true. Therefore, it is reasonable that one would nullify a state when presented with evidence that falsifies it but that with other types of evidence one maintains the relative beliefs.

The simultaneous expansion of the state space and nullification of prior feasible events may reflect an implicit belief of the DM in an underlying theory, or causal relation, under which the newly discovered consequences are inconsistent with particular actions resulting in some previously known consequences. Such theories are themselves subjective views, or interpretations, of the world that are manifested in the revision of beliefs. Because of their subjective nature these subjective interpretations are not formalized in this model. The examples in the introduction illustrates this point. The observations about the velocity of light in the Michelson-Morley experiment are inconsistent with Newtonian mechanics, therefore, once accepted as valid, it compelled nullifying states in which the experiment yield consequences implied by the aether theory. In the athlete example, there is an underlying cause, namely addiction, that simultaneously adds the new consequence of never competing again and nullifies the existing state in which the athlete can compete next month.

Another example is the theory of evolution and the subsequent discovery of genetics. These findings contradict Lamarck's hypothesis that an organism can pass on characteristics acquired through use or disuse during its lifetime to its offspring. In the 1930s, long after the discovery of genetics, Lysenko revived the ideas of Lamarck in the Soviet Union with Lysenkoism. Lysenkoism influenced Soviet agricultural policy and was later blamed for crop failures. This example illustrates that while the modern theory of genetics is interpreted by some as ruling out Lamarckism, others find it compatible with Lamarckism. Similarly, the evidence supporting evolution produced by research in molecular biology convinced some people that Darwin's theory is valid, while others continue to believe in intelligent design. These examples serve to illustrate the subjective nature of belief revision and nullification of states. Such beliefs can presumably be detected and quantified by the odds a decision maker would be willing to place on the outcomes of experiments testing the prediction of the underlying hypothesis.

Karni and Vierø (2013) imposed the assumption that S ( F ,   C ) ⊆ S ( F ,   C ′ ) . A strengthening of axiom (A.7r), in conjunction with monotonicity, will imply that nullification of prior feasible states will not occur. Axiom (A.7r') below gives a preference-based condition under which the assumption in Karni and Vierø (2013) is satisfied.

• (A.7r') For every given F , for all C ,   C ′ with C ⊂ C ′ , f ,   g ∈ F ^ and f ′ ,   g ′ ∈ F ^ * , such that f ′ = f and g ′ = g on S ( F ,   C ′ ) ∩ S ( F ,   C ) and f ′ = g ′ on S ( F ,   C ′ ) − [ S ( F ,   C ) ∩ S ( F ,   C ′ ) ] it holds that f ≽ F ^ g if and only if f ′ ≽ F ^ ∗ g ′ .