Bob’s “Best” Replies at ( γ A , γ B ) Bob L M R c B U n n Ann s B M c c s B s B D n C ( γ A ,γ B ) C ( γ A ,γ B ) Choices A B { c , n , s } { c , s } { n , s } { c , n } { n } { c } { c , s } { c } { s } { n , s } { s } { s } Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Nash Equilibrium at ( γ A , γ B ) Bob L M R A n A c B U n Ann s B A c M c A n s B A s B D Nash Equilibrium message profiles at ( γ A , γ B ): ( U , M ); ( D , R ). Nash Equilibrium outcomes at ( γ A , γ B ): { c , s } { c , s } { c , s } . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
State by State Nash Equilibrium Outcomes ( ρ A , ρ B ) ( ρ A , γ B ) L M R L M R U n c n U n c n M c s c M c s c D n s s D n s s N.Eq = { n , s } { n , s } { n , s } N.Eq = { n , s } { n , s } { n , s } ( γ A , ρ B ) ( γ A , γ B ) L M R L M R U n c n U n c n M c s c M c s c D n s s D n s s { c , n } { c , s } N.Eq = { c , n } { c , n } N.Eq = { c , s } { c , s } Barlo and Dalkıran Behavioral Implementation under Incomplete Information
BR-Optimal Outcomes = Nash Outcomes State by state, we have BR-Optimal outcomes = Nash outcomes : ( ρ A , ρ B ) ( ρ A , γ B ) { n , s } { n , s } { n , s } { n , s } { n , s } { n , s } ( γ A , ρ B ) ( γ A , γ B ) { c , n } { c , n } { c , n } { c , s } { c , s } { c , s } This is nice! But, how will they know the true state of the world ? Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Ex-Post Equilibrium Individuals observing only their own types when deciding their messages leads to interim considerations. A strategy of individual i is a function mapping i ’s types to her messages, i.e., σ i : Θ i → M i . An ex-post equilibrium of a mechanism µ = ( M , g ) is σ ∗ = ( σ ∗ i ) i ∈ N such that for all i ∈ N and all θ − i ∈ Θ − i − i ( θ − i )) ∈ C ( θ ′ i ,θ − i ) ( O µ g ( σ ∗ i ( θ ′ i ) , σ ∗ i ( σ ∗ − i ( θ − i ))) , for all θ ′ i ∈ Θ i . i An ex-post equilibrium consists of individuals’ strategies each measurable only with respect to that individual’s set of types and induces Nash equilibrium behavior at every state (type profile). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Motivating Example: Ex-post Equilibrium The following are (all) the ex-post equilibria of µ = ( M , g ): Ann → σ ∗ A ( ρ A ) = U σ ∗ A ( γ A ) = D ◮ σ σ σ ′∗ : Bob → σ ∗ B ( ρ B ) = L σ ∗ B ( γ B ) = R Ann → σ ′∗ A ( ρ A ) = D σ ′∗ A ( γ A ) = U ◮ σ σ σ ′′∗ : Bob → σ ′∗ B ( ρ B ) = M σ ′∗ B ( γ B ) = M Ann → σ ′∗ A ( ρ A ) = M σ ′∗ A ( γ A ) = U ◮ σ σ σ ′′′∗ : Bob → σ ′∗ B ( ρ B ) = M σ ′∗ B ( γ B ) = M σ ′′∗ and σ σ ′′′∗ are outcome equivalent . ◮ σ σ σ Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Ex-post Equilibrium Outcomes For all θ , g ( σ ′∗ σ ′∗ σ ′∗ ( θ )) = f ( θ ) and g ( σ ′′∗ σ ′′∗ σ ′′∗ ( θ )) = g ( σ ′′′∗ σ ′′′∗ σ ′′′∗ ( θ )) = f ′ ( θ ). This mechanism fully ex-post implements the SCS F . State: ( ρ A , ρ B ) State: ( ρ A , γ B ) State: ( γ A , ρ B ) State: ( γ A , γ B ) L M R L M R L M R L M R U n n n c n U n c n n n U n c c c n U n c c c n M c s s s c M c s s s c M c s c M c s c s s n s D n s s s D n s s s D n n s s D n s s s σ ′∗ σ ′∗ σ ′∗ ( ρ A , ρ B ) = ( U , L ) σ ′∗ ( ρ A , γ B ) = ( U , R ) σ ′∗ σ ′∗ σ ′∗ σ ′∗ σ ′∗ ( γ A , ρ B ) = ( D , L ) σ ′∗ σ ′∗ σ ′∗ ( γ A , γ B ) = ( D , R ) Outcome: n n n Outcome: n n n Outcome: n n n Outcome: s s s n n n s f ( ρ A , ρ B ) = n n f ( ρ A , γ B ) = n n f ( γ A , ρ B ) = n n f ( γ A , γ B ) = s s σ ′′∗ σ ′′∗ σ ′′∗ ( ρ A , ρ B ) = ( D , M ) σ ′′∗ σ ′′∗ ( ρ A , γ B ) = ( D , M ) σ ′′∗ σ ′′∗ σ ′′∗ σ ′′∗ ( γ A , ρ B ) = ( U , M ) σ ′′∗ σ ′′∗ σ ′′∗ ( γ A , γ B ) = ( U , M ) Outcome: s s s Outcome: s s s Outcome: c c c Outcome: c c c σ ′′′∗ σ ′′′∗ σ ′′′∗ σ ′′′∗ σ ′′′∗ σ ′′′∗ ( ρ A , ρ B ) = ( M , M ) σ ′′′∗ ( ρ A , γ B ) = ( M , M ) σ ′′′∗ σ ′′′∗ σ ′′′∗ ( γ A , ρ B ) = ( U , M ) σ ′′′∗ σ ′′′∗ ( γ A , γ B ) = ( U , M ) Outcome: s s s Outcome: s s s Outcome: c c c Outcome: c c c f ′ ( ρ A , ρ B ) = s s s f ′ ( ρ A , γ B ) = s s s f ′ ( γ A , ρ B ) = c c c f ′ ( γ A , γ B ) = c c c Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Revelation Principle Fails for Partial Implementation! The mechanism ( M , g ) partially implements the social choice function f in ex-post equilibrium as f ( θ ) = g ( σ ′∗ σ ′∗ σ ′∗ ( θ )) for all θ ∈ Θ because ( ρ A , ρ B ) ( ρ A , γ B ) ( γ A , ρ B ) ( γ A , γ B ) f n n n s σ ′∗ n n n s The direct mechanism associated with the SCF f does not partially n n is not chosen from { n , s } by Ann at ( ρ A , γ B ). (ex-post) implement f : n Bob C ( ρ A ,γ B ) C ( ρ A ,γ B ) ρ B γ B A B { c , n , s } { n } { n } n Ann ρ A n n n { c , n } { n } { n } γ A n s { c , s } { s } { s } The direct mechanism { n , s } { s } { s } associated with the SCF f Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Revelation Principle Fails for Partial Implementation! To our knowledge, the failure of the revelation principle due to behavioral aspects is first documented by Saran (2011): Models behavioral aspects via menu-dependent preferences over interim Anscombe-Aumann acts, and shows weak contraction consistency, a condition implied by the IIA (Sen’s α ), is sufficient for the revelation principle. We reaffirm that the revelation principle holds under the IIA in our setting. Thus, focusing on indirect mechanisms rather than direct mechanisms is crucial with behavioral aspects for full and partial implementation. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
In This Paper We provide necessary as well as sufficient conditions for behavioral implementation under incomplete information. In doing so, we restrict our attention to ◮ full implementation the set of equilibrium outcomes fully coincide with a predetermined social goal, ◮ ex-post equilibrium measurable strategies that induce a Nash equilibrium at every state of the world. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Why Full Implementation? The power of partial implementation relies heavily on the revelation principle . [ The direct revelation partial implementation ] does assure that the resulting outcome will be an equilibrium of some game; however, there may be others as well. This problem is sometimes dismissed with an argument that as long as truthful revelation is an equilibrium, it will somehow be the salient equilibrium even if there are other equilibria. (Postlewaite and Schmeidler, 1986). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Why Full Implementation? In our setup, the revelation principle does not hold! ◮ The salience of a truth-telling equilibrium is not reasonable. We cannot restrict attention to direct revelation mechanisms without loss of generality if individual choices are not rational. Identifying mechanisms for full implementation are also useful as full implementation implies partial implementation. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Why Ex-Post Equilibrium? Ex-post equilibrium (EPE) is plausible in our setup since: The EPE makes no use of any probabilistic information, it is belief-free, it involves no belief updating or expectation considerations, and no common prior assumption. ◮ Expected utility hypothesis fails due to lack of rationality. ◮ Bayesian Nash equilibrium is impractical in our setup. The EPE induces robust behavior on account of the ex-post no-regret property: No individual would seek to change her message even if she were to know others’ type profile. The EPE provides a plausible extension of dominant equilibrium to the case of interdependence: ◮ Under independence, the EPE is equivalent to (behavioral) dominant equilibrium under some full-range conditions that hold in direct mechanisms, while ◮ dominant equilibrium with interdependent choices imposes excessively stringent requirements reducing its appeal. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Preliminaries I The environment is � N , X , Θ , { C θ i } i ∈ N ,θ ∈ Θ � . N = { 1 , ..., n } set of players , X denotes the set of all possible alternatives ◮ X denotes the set of all subsets of X , Θ = × i ∈ N Θ i , denotes the set of all possible states of the world, ◮ θ i denotes the private information of i . C θ i : X → X describes the choice behavior of agent i at θ . ◮ C θ i ( S ) ⊆ S , for all S ∈ X . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Preliminaries II The Social Choice Set (SCS) is, F ⊂ { f | f : Θ → X } . µ = ( M , g ) denotes a mechanism where ◮ M i � = ∅ is the message space of agent i ∈ N and ◮ g : M → X is the outcome function where M := × i ∈ N M i . σ i : Θ i → M i denotes a strategy of agent i in the mechanism µ . The opportunity set of agent i under µ for each m − i is given by O µ i ( m − i ) := { g ( m i , m − i ) ∈ X : m i ∈ M i } . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Ex-Post Equilibrium of a Mechanism Definition A strategy profile σ ∗ : Θ → M is an ex-post equilibrium of µ = ( M , g ) if for all θ ∈ Θ and all i ∈ N − i ( θ − i )) ∈ C ( θ i ,θ − i ) ( O µ g ( σ ∗ i ( θ i ) , σ ∗ i ( σ ∗ − i ( θ − i ))) . i Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Ex-Post Implementation Definition A social choice set F is is said to be ex-post implementable if there exists a mechanism µ = ( M , g ) such that: (i) For every f ∈ F , there exists an ex-post equilibrium σ ∗ of µ = ( M , g ) that satisfies [ f = g ◦ σ ∗ ] , i.e., f ( θ ) = g ( σ ∗ ( θ )) for all θ ∈ Θ; (ii) For every ex post equilibrium σ ∗ of µ = ( M , g ), there exists f ∈ F such that: [ g ◦ σ ∗ = f ] , i.e., g ( σ ∗ ( θ )) = f ( θ ) for all θ ∈ Θ . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Deception and Deception Profile α i : Θ i → Θ i denotes a possible deception by agent i ∈ N . ◮ α i ( θ i ) can be interpreted as i ’s reported type. α ( θ ) = ( α 1 ( θ 1 ) , α 2 ( θ 2 ) , . . . , α n ( θ n )). denotes a profile of deceptions . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Necessity Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistent Collections of Sets under Incomplete Info Definition A collection of sets S := { S i ( f , θ − i ) | i ∈ N , f ∈ F , θ − i ∈ Θ − i } ⊂ X is consistent with the SCS F ∈ F under incomplete information if for every SCF f ∈ F , we have i , θ − i ) ∈ C ( θ ′ i ,θ − i ) ( i ) for all i ∈ N , f ( θ ′ ( S i ( f , θ − i )) for each θ ′ i ∈ Θ i , and i ∈ F , there exists θ ∗ ∈ Θ and ( ii ) for any deception profile α with f ◦ α / i ∗ ∈ N such that f ( α ( θ ∗ )) / ∈ C θ ∗ i ∗ ( S i ∗ ( f , α − i ∗ ( θ ∗ − i ∗ ))). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency under Incomplete Information A collection of sets S is consistent with an SCS F under incomplete information if ( i ) Given any i ∈ N , any f ∈ F , and any θ − i ∈ Θ − i , it must be that when i ’s type is θ ′ i , his choices from S i ( f , θ − i ) at state ( θ ′ i , θ − i ) contains f ( θ ′ i , θ − i ) for all θ ′ i ∈ Θ i ; and ( ii ) given any f ∈ F , whenever there is a deception profile α that leads to an outcome not compatible with the SCS F, ( f ◦ α / ∈ F ), there exists an informant state θ ∗ and an informant individual i ∗ such that - i ∗ does not choose at state θ ∗ the alternative f ( α ( θ ∗ )) from S i ∗ ( f , α − i ∗ ( θ ∗ − i ∗ )). Python codes computing In Supplementary Materials, we provide consistent collections taking individuals’ choices and the SCS as inputs. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Necessity Result Theorem (1) If an SCS F is ex-post implementable, then there exists a collection of sets S := { S i ( f , θ − i ) | i ∈ N , f ∈ F , θ − i ∈ Θ − i } consistent with F under incomplete information. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proof of the Necessity Result Suppose µ = ( M , g ) ex-post implements a given SCS F . Then, ◮ for any SCF f ∈ F , there is an EPE σ f of µ such that f = g ◦ σ f . ◮ for each θ ∈ Θ, g ( σ f ( θ )) = f ( θ ) is in C θ i ( O µ i ( σ f − i ( θ − i ))) for all i ∈ N . Let S be such that S i ( f , θ − i ) := O µ i ( σ f − i ( θ − i )), i.e., the collection sustained by the opportunity sets associated with the EPE of µ , ◮ ( i ) of consistency of S with F holds. If a deception profile α is such that f ◦ α / ∈ F , then ◮ σ f ◦ α cannot be an EPE of µ . Otherwise, by ( ii ) of ex-post implementability, there exists f = g ◦ σ f ◦ α . But, since f = g ◦ σ f , we have f ∈ F with ˜ ˜ ˜ f = f ◦ α ∈ F , a contradiction. ◮ Hence, there are θ ∗ and i ∗ who does not choose f ( α ( θ ∗ )) from O µ i ∗ ( σ f − i ∗ ( α − i ∗ ( θ − i ∗ ))) = S i ∗ ( f , α − i ∗ ( θ − i ∗ )) at θ ∗ . That is, ( ii ) of consistency of S with F holds as well. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Ex-Post Choice Monotonicity Definition F is ex-post-choice monotonic if, for every f ∈ F and deception profile ∈ F , there is θ ∗ ∈ Θ , i ∗ ∈ N , S ∗ ∈ X such that α with f ◦ α / ∈ C θ ∗ (i) f ( α ( θ ∗ )) / i ∗ ( S ∗ ) , ( θ ′ i ∗ ,α − i ∗ ( θ ∗ − i ∗ )) f (( θ ′ i ∗ , α − i ∗ ( θ ∗ ( S ∗ ) for all θ ′ (ii) − i ∗ ))) ∈ C i ∗ ∈ Θ i ∗ . i ∗ Consistency under Incomplete Information implies Ex-post Choice Monotonicity. Proposition (1) If there is a collection of sets consistent with an SCS F under incomplete information, then F is ex-post choice monotonic. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
What is Ex-Post Choice Monotonicity? Ex-post Choice-Monotonicity requires that when there is an attempt of deception that would lead to a non-optimal outcome, there must exist - an informant state, - an informant/whistle-blower agent for this state, and - an informant/reward set for this whistle-blower, such that (i) the whistle-blower would not choose the outcome arising due to the undesirable deception from the reward set in the informant state; (ii) the whistle-blower does not have an incentive to falsely alert the designer when the outcome is optimal, i.e., compatible with the social choice set in question. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Quasi-Ex-Post Choice Incentive Compatibility Definition F is quasi-ex-post choice incentive compatible if, for every f ∈ F , θ ∈ Θ, i ∈ N there exists S ∈ X such that i) S ⊇ { f ( θ ′ i , θ − i ) | θ ′ i ∈ Θ i } , ii) f ( θ ) ∈ C θ i ( S ). Consistency under Incomplete Information implies Quasi-Ex-post Choice Incentive Compatibility. Proposition (2) If there is a collection of sets consistent with an SCS F under incomplete information, then F is quasi-ex-post choice incentive compatible. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Quasi-Ex-Post Choice Incentive Compatibility Quasi-ex-post choice incentive compatibility condition describes a necessary condition for partial ex-post implementation of any f ∈ F . A necessary and sufficient condition for revelation principle (direct revelation partial ex-post implementation) of any f ∈ F is ◮ For every θ ∈ Θ, i ∈ N , f ( θ ) ∈ C θ i ( { f ( θ ′ i , θ − i ) | θ ′ i ∈ Θ i } ). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
IIA is sufficient for Revelation Principle Proposition (3) If individual choices satisfy the IIA, then quasi-ex-post choice incentive compatibility implies the revelation principle. - Put differently, the revelation principle holds whenever individuals’ choices satisfy the IIA ! Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Necessity under WARP We show that under WARP quasi-ex-post choice incentive compatibility is equivalent to ex-post incentive compatibility of Bergemann and Morris (2008), and our ex-post-choice monotonicity implies ex-post monotonicity of Bergemann and Morris (2008), while ex-post monotonicity coupled with ex-post incentive compatibility implies ex-post-choice monotonicity. Remark Under WARP, ex-post-choice monotonicity and quasi-ex-post choice incentive compatibility hold if and only if ex-post monotonicity and ex-post incentive compatibility of Bergemann and Morris (2008) hold. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Necessity with Two Individuals The case of two individuals offers sharper descriptions of the consistent collections: Any one of the choice sets of the first individual must have a non-empty intersection with any one of the choice sets of the second individual. This parallels the restrictions featured in condition µ 2 of Moore and Repullo (1990) and condition β of Dutta and Sen (1990). For reasons of exposition, we present the case of two individuals separately. Two Individuals Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Sufficiency with three or more individuals Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The Choice Incompatible Pair Property Definition S satisfies the choice incompatible pair property at θ if for each x ∈ S ∈ C θ ∈ C θ there exist i , j ∈ N such that x / i ( S ) and x / j ( S ). A set satisfies the choice incompatible pair property at a particular state if for each alternative in this set, there exists a pair of individuals who do not choose this alternative from this set at the particular state. This guarantess any alternative in this set can be chosen by at most n − 2 individuals at the particular state. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The Choice Incompatible Pair Property Definition We say that S satisfies the choice incompatible pair property at θ if ∈ C θ ∈ C θ for each x ∈ S there exist i , j ∈ N s.t. x / i ( S ) and x / j ( S ). The choice incompatible pair property is similar to the economic environment assumption of the rational domain. ◮ Yet, it is weaker as it is defined for a particular set. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Sufficiency with the Choice Incompatible Pair Property Theorem (2) Let n ≥ 3 . If F is an SCS for which there exist ( i ) a collection of sets S := { S i ( f , θ − i ) : i ∈ N , f ∈ F , θ − i ∈ Θ − i } consistent with F under incomplete information, and ( ii ) a set of alternatives ¯ S ∈ S S ⊆ ¯ X ⊆ X with � X which satisfies the choice incompatible pair property at every state θ ∈ Θ , then F is ex-post implementable . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Sufficiency with the Choice Incompatible Pair Property When there are three or more individuals an SCS F is ex-post implementable whenever ( i ) there exists a collection of sets S consistent with F under incomplete information, and ( ii ) there exists a set of alternatives ¯ X which contains every alternative in S and satisfies the choice incompatible pair property at every state of the world. Python codes computing In Supplementary Materials, we provide consistent collections S and ¯ X satisfying the choice incompatible pair property taking individuals’ choices and the SCS as inputs. Example SM-5 with three rational individuals, we show that even in the In rational domain our Theorem 2 extends Theorem 2 of Bergemann and Morris (2008), which uses the economic environment assumption. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The Choice No-Veto-Power Property Definition A social choice function f satisfies choice no-veto-power property on S at θ if x ∈ C θ i ( S ) for all i ∈ N \ { j } implies f ( θ ) = x . The choice-no-veto power property of f on S at θ requires that if a particular alternative is chosen from S by at least n − 1 individuals at θ , then this particular alternative must be f -optimal at θ . ◮ We note that it is weaker than its analog in the rational domain since it is defined on a particular set. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The Consistency-No-Veto Property Definition F satisfies the consistency-no-veto property whenever there exist ( i ) a collection of sets S := { S i ( f , θ − i ) : i ∈ N , f ∈ F , θ − i ∈ Θ − i } such that ( θ ′ i ,θ − i ) for all f ∈ F and for all i ∈ N , f ( θ ′ i , θ − i ) ∈ C ( S i ( f , θ − i )) for each i θ ′ i ∈ Θ i , ( ii ) and a set of alternatives ¯ S ∈ S S ⊆ ¯ X ⊆ X with � X such that for any collection of product sets of states { ¯ Θ f } f ∈ F with Θ f ⊂ Θ, there exists f ∗ ∈ F such that ¯ f ∈ F ¯ Θ = � ( iii ) f ∗ satisfies choice no-veto-power property on ¯ X at every θ ∈ Θ \ ¯ Θ, and ( iv ) if for any f ∈ F and any deception profile α , f ( α ( θ )) � = f ∗ ( θ ) for some Θ f , then there exists i ∗ ∈ N and θ ∗ ∈ ¯ θ ∈ ¯ Θ f such that ∈ C θ ∗ f ( α ( θ ∗ )) / i ∗ ( S i ∗ ( f , α − i ∗ ( θ ∗ − i ∗ ))). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Sufficiency with the Consistency-No-Veto Property Theorem (3) Let n ≥ 3 . If an SCS F satisfies the consistency-no-veto property, then F is ex-post implementable. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The Consistency-No-Veto Property Given an SCS F , the consistency-no-veto property requires the existence of a collection of sets S and a set of alternatives ¯ X which contains every alternative that appears in S such that: Given any i ∈ N , any f ∈ F , and any θ − i ∈ Θ − i , it must be that when i ’s type is θ ′ i , his choices from S i ( f , θ − i ) at state ( θ ′ i , θ − i ) contains f ( θ ′ i , θ − i ) for all θ ′ i ∈ Θ i ; and for any collection of product sets of states { ¯ Θ f } f ∈ F with Θ f ⊂ Θ, there is an SCF f ∗ in F such that ¯ f ∈ F ¯ Θ = � Θ, then f ∗ obeys the choice no-veto-power property on ¯ ◮ if θ ∈ Θ \ ¯ X at θ , and ◮ if a deception profile α and an SCF f ∈ F lead to an outcome different than f ∗ ( θ ) for some θ ∈ ¯ Θ f , then there exists a whistle-blower i ∗ ∈ N and an informant state θ ∗ such that i ∗ does not choose f ( α ( θ ∗ )) from S i ∗ ( f , α − i ∗ ( θ ∗ − i ∗ )) at θ ∗ . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Checking the Consistency-No-Veto Property Python codes we provide in Supplementary Materials compute The ◮ the collection S , ◮ the set of alternatives ¯ X , ◮ the collection of product sets { ¯ Θ f } f ∈ F , ◮ SCF’s f ∗ ∈ F satisfying consistency-no-veto taking choices and the SCS as inputs. Example SM-4 of Supplementary Materials, we show that there are In Θ, and f ∗ satisfying consistency-no-veto. collections S together with ¯ X , ¯ Our codes identify S , ¯ X , { ¯ Θ f } f , and f ∗ ’s associated with all of the collections satisfying consistency-no-veto. Therefore, our codes induce better understanding and application capabilities by mitigating the effects of complications due to conditions such as monotonicity-no-veto and ex-post-monotonicity-no-veto. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Corollary for a Social Choice Function Corollary (1) Let n ≥ 3 . An SCF f : Θ → X is ex-post implementable if there exists a collection S := { S i ( f , θ − i ) : i ∈ N , θ − i ∈ Θ − i } s.t. i , θ − i ) ∈ C ( θ ′ i ,θ − i ) i ∈ Θ i and there exists ¯ f ( θ ′ ( S i ( f , θ − i )) for each θ ′ X ⊆ X i S ∈ S S ⊆ ¯ X s.t. for any product set ¯ with � Θ ⊂ Θ , ( i ) f satisfies choice no-veto-power property on ¯ X at every θ ∈ Θ \ ¯ Θ , and ( ii ) for any deception profile α with f ( α ( θ )) � = f ( θ ) for some θ ∈ ¯ Θ , there exists i ∗ ∈ N and θ ∗ ∈ ¯ Θ such that ∈ C θ ∗ f ( α ( θ ∗ )) / i ∗ ( S i ∗ ( f , α − i ∗ ( θ ∗ − i ∗ ))) . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The Mechanism with Three or More Individuals µ = ( M , g ): M i = F × Θ i × ¯ X × N and g : M → X is as follows: Rule 1 : g ( m ) = f ( θ ) if m i = ( f , θ i , · , · ) for all i ∈ N , � x j if m i = ( f , θ i , · , · ) for all i ∈ N \ { j } if x j ∈ S j ( f , θ − j ) , Rule 2 : g ( m ) = and m j = (˜ f , ˜ θ j , x j , · ) with ˜ ¯ x ( j , f , θ − j ) otherwise. f � = f , Rule 3 : g ( m ) = x j where j = � i k i (mod n) otherwise. where ¯ x ( j , f , θ − j ) is an arbitrary element from S j ( f , θ − j ). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Sufficiency with Two Individuals With two individuals , the above mechanism is not well-defined and hence another mechanism is needed. We provide two methods to strengthen (two-individual) consistency to deliver sufficiency as in the case of three or more individuals. For reasons of exposition, we present the case of two individuals separately. Two Individuals Sufficiency with Two Individuals Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Efficiency Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Efficiency Notion of de Clippel (2014) de Clippel (2014) introduces the following notion of efficiency : Φ eff ( θ ) := x | ∃ ( Y i ) i ∈ N with x ∈ C θ � � i ( Y i ) for all i ∈ N and X = ∪ i ∈ N Y i . An alternative x is efficient at θ if ◮ each individual has an implicit opportunity set such that she chooses x from this set at θ and ◮ each alternative is in at least one of the implicit opportunity sets of an individual. Φ eff : Θ → X is a social choice correspondence (SCC) that we refer to as the de Clippel efficient SCC . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Efficient SCS is not ex-post implementable The efficient SCS is F eff := { f : Θ → X | f ( θ ) ∈ Φ eff ( θ ) for all θ ∈ Θ } . F eff consists of all SCFs that are selections from the de Clippel efficient SCC. Proposition (4) F eff is not ex-post implementable. Reason: F eff may contain an SCF that violates quasi-ex-post choice incentive compatibility. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
F eff is not ex-post implementable Let N = { A , B } , X = { x , y } and Θ i = { θ i , ω i } for i = A , B , and the following choices: C ( θ A ,θ B ) C ( θ A ,θ B ) C ( θ A ,ω B ) C ( θ A ,ω B ) C ( ω A ,θ B ) C ( ω A ,θ B ) C ( ω A ,ω B ) C ( ω A ,ω B ) S A B A B A B A B { x , y } { x } { x } { x } { x } { y } { x } { y } { x } Then, F eff = { f , f ′ , f ′′ , f ′′′ } where ( θ A , θ B ) ( θ A , ω B ) ( ω A , θ B ) ( ω A , ω B ) f x x x x f ′ x x x y f ′′ x x y x f ′′′ x x y y S A , S B consistent with F eff under incomplete information implies x ∈ C ( ω A ,θ B ) ( S B ( f ′ , ω A )) and y ∈ C ( ω A ,ω B ) ( S B ( f ′ , ω A )) so B B S B ( f ′ , ω A ) = { x , y } . ∈ C ( ω A ,ω B ) But, then y / ( S B ( f ′ , ω A )), a contradiction. B Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Constrained Efficiency Our constrained efficiency combines efficiency and quasi-ex-post choice incentive compatibility: E c.eff consists of e : Θ → X such that ( i ) for all i and all θ − i , there is Y θ − i with for all θ , ∪ i ∈ N Y θ − i = X , and i i θ i , θ − i ) ∈ C (˜ θ i ,θ − i ) ( Y θ − i ( ii ) e (˜ ) for all ˜ θ i . i i A state-contingent allocation e is constrained efficient if for any individual and for any type profile of the others’, there exists an implicit opportunity set such that ◮ her choices from this set for each of her types is aligned with e and ◮ at every state each alternative is in at least one of the implicit opportunity sets of an individual. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Constrained Efficiency is ex-post implementable Proposition (5) E c.eff has a consistent collection of sets under incomplete information. Reason : The implicit opportunity sets associated with constrained efficiency constitute a collection of sets consistent with constrained efficiency under incomplete information. Then, Theorem 2 and Proposition 5 deliver the following result: Proposition (6) Let n ≥ 3 . E c.eff is ex-post implementable on all domains with X satisfying the choice incompatible pair property at every state θ ∈ Θ . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Allocation Problems Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Allocation Problems with Endowment Effects n objects are to be allocated among n individuals. ◮ H = { h 1 , . . . , h n } is the set of objects (e.g., houses or offices) ◮ H denotes the set of all non-empty subsets of H . ◮ X := { x ∈ H n | x i � = x j , for all i , j ∈ N with i � = j } is the set of allocations such that each individual gets only one object. There are two types of individual choice behavior: ◮ choices on objects and ◮ choices on allocations of objects . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Allocation Problems with Endowment Effects Individuals’ choices are independent and each individual cares only about her own object: c i ( Z , θ i ) � = ∅ – the chosen object(s) from Z ∈ H by i of type θ i . H i ( S ) := { x i ∈ H | x ∈ S } – the object(s) i gets in allocations in S . For any S ∈ X , individual choices on allocations are C θ i i ( S ) := { x ∈ S | x i ∈ c i ( H i ( S ) , θ i ) } . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Allocation Problems with Endowment Effects Constrained efficiency takes the following specific form: ˜ E c.eff consists H of ˜ e : N × Θ → H with ˜ e i ( θ ) � = ˜ e j ( θ ) for all i � = j and all θ ∈ Θ such that ( i ) for all i ∈ N and all θ − i ∈ Θ − i , there is H i ( θ − i ) ∈ H with e i (˜ θ i , θ − i ) ∈ c i ( H i ( θ − i ) , ˜ θ i ) for all ˜ ˜ θ i ∈ Θ i ; and ( ii ) for all θ ∈ Θ, ∪ i ∈ N H i ( θ − i ) = H . Corollary (2) Let n ≥ 3 . ˜ E c.eff is ex-post implementable on all domains satisfying the H choice incompatibility on the set of all objects H. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Endowment Effects ` a la Masatlioglu and Ok (2014) We use the canonical model of choice with endowment effects (Masatlioglu & Ok, 2014): h ∗ i ∈ H denotes i ’s initial endowments; θ i = ♦ i denotes i being of rational type; θ i = h ∗ i denotes i being of behavioral type; Θ i = { ♦ i , h ∗ i } denotes i ’s types. i ’s choices on objects are singleton-valued. Under reasonable assumptions, i ’s choices are represented by U i : H → R , i ’s utility function on H , and Q i ( h ∗ i ) = { h ∈ H | h ∈ c i ( { h , h ∗ i } , h ∗ i ) } , i ’s consideration set , such that for any Z ∈ H , if θ i = h ∗ i and h ∗ i ∈ Z , arg max h ∈ Z ∩ Q i ( h ∗ i ) U i ( h ) c i ( Z , θ i ) = if θ i = h ∗ i but h ∗ ∈ Z , arg max h ∈ Z U i ( h ) i / arg max h ∈ Z U i ( h ) if θ i = ♦ i Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Endowment Effects ` a la Masatlioglu and Ok (2014) Then, i ’s independent choices on allocations are: For any S ∈ X { x ∈ S | x i = arg max h ∈ H i ( S ) ∩ Q i ( h ∗ i ) U i ( h ) } if θ i = h ∗ i and h ∗ i ∈ H i ( S ) , C θ i i ( S ) := { x ∈ S | x i = arg max h ∈ H i ( S ) U i ( h ) } if θ i = h ∗ i and h ∗ i / ∈ H i ( S ) , { x ∈ S | x i = arg max h ∈ H i ( S ) U i ( h ) } if θ i = ♦ i . If a mechanism does not offer i her initial endowment, h ∗ i , then she makes her choices as if she is rational. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition (7) If an SCS F ∈ F is ex-post implementable, then there exists a collection of sets S := { S i ( f , θ − i ) | i ∈ N , f ∈ F , θ − i ∈ Θ − i } consistent with F under incomplete information such that ( i ) if f ( θ ) = x, then H i ( S i ( f , θ − i )) ⊂ { h ∈ H | U i ( x i ) ≥ U i ( h ) } , ( ii ) if f ( θ ) = x and x i / ∈ Q i ( h ∗ i ) , then h ∗ i / ∈ H i ( S i ( f , θ − i )) , ( iii ) if f ( θ ) = x and x i = h ∗ i , then H i ( S i ( f , θ − i )) ∩ Q i ( h ∗ i ) = { h ∗ i } , ( iv ) if f ( θ ) = x and x i � = h ∗ i , then either h ∗ ∈ H i ( S i ( f , θ − i )) or x i ∈ Q i ( h ∗ i / i ) . Remark Proposition 7 demonstrates the critical use of initial endowments and how they enable the planner to induce behavioral individuals make choices aligned with the social goal. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then H i ( S i ( f , θ − i )) ⊂ { h ∈ H | U i ( x i ) ≥ U i ( h ) } , Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then objects offered to i for f and θ − i , H i ( S i ( f , θ − i )), cannot provide strictly higher utilities than U i ( x i ). Reason: Otherwise, x i / ∈ c i ( H i ( S i ( f , θ − i )) , ♦ i ) when θ i = ♦ i ; i.e., the rational type of i does not choose x from S i ( f , θ − i ). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then objects offered to i for f and θ − i , H i ( S i ( f , θ − i )), cannot provide strictly higher utilities than U i ( x i ). ( ii ) if f ( θ ) = x and x i / ∈ Q i ( h ∗ i ), then h ∗ i / ∈ H i ( S i ( f , θ − i )). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then objects offered to i for f and θ − i , H i ( S i ( f , θ − i )), cannot provide strictly higher utilities than U i ( x i ). ( ii ) if f ( θ ) = x and x i is not in i ’s consideration set, then in any consistent collection of sets objects offered to i for f and θ − i cannot involve i ’s initial endowment h ∗ i . Reason: Otherwise, x i / ∈ c i ( H i ( S i ( f , θ − i )) , h ∗ i ) when θ i = h ∗ i ; i.e., the behavioral type of i does not choose x from S i ( f , θ − i ). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then objects offered to i for f and θ − i , H i ( S i ( f , θ − i )), cannot provide strictly higher utilities than U i ( x i ). ( ii ) if f ( θ ) = x and x i is not in i ’s consideration set, then in any consistent collection of sets objects offered to i for f and θ − i cannot involve i ’s initial endowment h ∗ i . ( iii ) if f ( θ ) = x and x i = h ∗ i , then H i ( S i ( f , θ − i )) ∩ Q i ( h ∗ i ) = { h ∗ i } , Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then objects offered to i for f and θ − i , H i ( S i ( f , θ − i )), cannot provide strictly higher utilities than U i ( x i ). ( ii ) if f ( θ ) = x and x i is not in i ’s consideration set, then in any consistent collection of sets objects offered to i for f and θ − i cannot involve i ’s initial endowment h ∗ i . ( iii ) if f ( θ ) = x and x i = h ∗ i , then the only object in the consideration set of i that is offered to i for f and θ − i in any consistent collection of sets must be i ’s initial endowment h ∗ i . Reason: Otherwise, h ∗ i / ∈ c i ( H i ( S i ( f , θ − i )) , h ∗ i ) when θ i = h ∗ i ; i.e., the behavioral type of i does not choose x from S i ( f , θ − i ). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then objects offered to i for f and θ − i , H i ( S i ( f , θ − i )), cannot provide strictly higher utilities than U i ( x i ). ( ii ) if f ( θ ) = x and x i is not in i ’s consideration set, then in any consistent collection of sets objects offered to i for f and θ − i cannot involve i ’s initial endowment h ∗ i . ( iii ) if f ( θ ) = x and x i = h ∗ i , then the only object in the consideration set of i that is offered to i for f and θ − i in any consistent collection of sets must be i ’s initial endowment h ∗ i . ( iv ) if f ( θ ) = x and x i � = h ∗ i , then either h ∗ i / ∈ H i ( S i ( f , θ − i )) or x i ∈ Q i ( h ∗ i ). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects Proposition 7 tells that: ( i ) if f ( θ ) = x , then objects offered to i for f and θ − i , H i ( S i ( f , θ − i )), cannot provide strictly higher utilities than U i ( x i ). ( ii ) if f ( θ ) = x and x i is not in i ’s consideration set, then in any consistent collection of sets objects offered to i for f and θ − i cannot involve i ’s initial endowment h ∗ i . ( iii ) if f ( θ ) = x and x i = h ∗ i , then the only object in the consideration set of i that is offered to i for f and θ − i in any consistent collection of sets must be i ’s initial endowment h ∗ i . ( iv ) if f ( θ ) = x and x i � = h ∗ i , then either x i is in i ’s consideration set or i ’s initial endowment is not offered to i for f and θ − i in any consistent collection of sets. ∈ c i ( H i ( S i ( f , θ − i )) , h ∗ i ) when θ i = h ∗ Reason: Otherwise, x i / i ; i.e., the behavioral type of i does not choose x from S i ( f , θ − i ). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects For a practical display of Proposition 7, consider the following example: h ∗ Q i ( h ∗ Object U 1 U 2 U 3 i ) i { I , II } I 2 2 3 Indv. 1 II II 1 3 2 Indv. 2 I { I , II , III } III 3 1 1 Indv. 3 III { III } inducing the following choices Z c 1 ( Z , ♦ 1 ) c 1 ( Z , h ∗ 1 ) c 2 ( Z , ♦ 2 ) c 2 ( Z , h ∗ 2 ) c 3 ( Z , ♦ 3 ) c 3 ( Z , h ∗ 3 ) { I , II , III } III I II II I III { I , II } I I II II I I { I , III } III III I I I III { II , III } III II II II II III The planner aims to implement the following social choice function: � ( I , II , III ) if θ = ( h ∗ 1 , h ∗ 2 , h ∗ 3 ) f ( θ ) = ( III , II , I ) otherwise. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects For a practical display of Proposition 7, consider the following example: h ∗ Q i ( h ∗ Object U 1 U 2 U 3 i ) i { I , II } I 2 2 3 Indv. 1 II II 1 3 2 Indv. 2 I { I , II , III } III 3 1 1 Indv. 3 III { III } inducing the following choices Z c 1 ( Z , ♦ 1 ) c 1 ( Z , h ∗ 1 ) c 2 ( Z , ♦ 2 ) c 2 ( Z , h ∗ 2 ) c 3 ( Z , ♦ 3 ) c 3 ( Z , h ∗ 3 ) { I , II , III } III I II II I III { I , II } I I II II I I { I , III } III III I I I III { II , III } III II II II II III The SCF f assigns each individual her unconstrained utility maximizing object unless all agents are of behavioral type, and in that case f allocates each agent her constrained utility maximizer. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Consistency with Endowment Effects For a practical display of Proposition 7, consider the following example: h ∗ Q i ( h ∗ Object U 1 U 2 U 3 i ) i { I , II } I 2 2 3 Indv. 1 II II 1 3 2 Indv. 2 I { I , II , III } III 3 1 1 Indv. 3 III { III } inducing the following choices Z c 1 ( Z , ♦ 1 ) c 1 ( Z , h ∗ 1 ) c 2 ( Z , ♦ 2 ) c 2 ( Z , h ∗ 2 ) c 3 ( Z , ♦ 3 ) c 3 ( Z , h ∗ 3 ) { I , II , III } III I II II I III { I , II } I I II II I I { I , III } III III I I I III { II , III } III II II II II III In every consistent collection, H 1 ( S 1 ( f , h ∗ − 1 )) = { I , II , III } . So, implementation demands offering 1 his initial endowment, II , even though in no socially optimal outcome she gets II . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Ex-Post Implementation with Endowment Effects The following indirect mechanism ex-post implements the SCF f in this example: Ind. 2 chooses M Ind. 3 L R U ( I , II , III ) ( III , II , I ) Ind. 1 C ( II , I , III ) ( III , II , I ) D ( III , II , I ) ( III , II , I ) However, f ’s direct mechanism does not ex-post implement the SCF f (and hence the revelation principle fails): Ind. 3 chooses h ∗ Ind. 3 chooses ♦ 3 3 Ind. 2 Ind. 2 h ∗ h ∗ ♦ 2 ♦ 2 2 2 h ∗ h ∗ Ind. 1 ( I , II , III ) ( III , II , I ) Ind. 1 ( III , II , I ) ( III , II , I ) 1 1 ♦ 1 ( III , II , I ) ( III , II , I ) ♦ 1 ( III , II , I ) ( III , II , I ) 3 ), O µ At ( h ∗ 1 , h ∗ 2 , h ∗ 1 ( h ∗ − 1 ) = { I , III } and the EPE σ ∗ 1 ( θ 1 ) = ♦ 1 for all θ 1 ∈ { ♦ 1 , h ∗ 1 } . This is an impasse due to f 1 ( h ∗ 1 , h ∗ 2 , h ∗ 3 ) = I . Back to Behavioral Aspects and Simplicity Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects The behavioral-bias case (obtained by relabeling the above) involves choices with endowment effects where Θ i = { ♦ i , h ∗ i } , for all Z ∈ H � if θ i = h ∗ i and h ∗ i ∈ Z , arg max h ∈ Z ∩ Q i ( h ∗ i ) U i ( h ) c i ( Z , θ i ) = arg max h ∈ Z U i ( h ) otherwise, and h ∗ Q i ( h ∗ Object U 1 U 2 U 3 i ) i I 2 2 3 Indv. 1 II { I , II } II 1 3 2 Indv. 2 I { I , II , III } { III } III 3 1 1 Indv. 3 III This formulation induces the following choices: c 1 ( Z , h ∗ c 2 ( Z , h ∗ c 3 ( Z , h ∗ Z c 1 ( Z , ♦ 1 ) 1 ) c 2 ( Z , ♦ 2 ) 2 ) c 3 ( Z , ♦ 3 ) 3 ) { I , II , III } III I II II I III { I , II } I I II II I I { I , III } III III I I I III { II , III } III II II II II III Remark WARP holds for all choices but c 1 ( · , h ∗ 1 ) . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects We construct a corresponding no-behavioral-bias case involving rational state-contingent choices where Θ i = { ♦ i , h ∗ i } , for all Z ∈ H h ∈ Z U θ i c i ( Z , θ i ) = arg max i ( h ) , and the utilites and endowments are U h ∗ U h ∗ U h ∗ U ♦ 1 U ♦ 2 U ♦ 3 h ∗ Object 1 2 3 i 1 1 2 2 3 3 Indv. 1 II I 2 3 2 2 3 2 Indv. 2 I II 1 2 3 3 2 1 Indv. 3 III III 3 1 1 1 1 3 Thus, we obtain the the following choices: c 1 ( Z , h ∗ c 2 ( Z , h ∗ c 3 ( Z , h ∗ Z c 1 ( Z , ♦ 1 ) 1 ) c 2 ( Z , ♦ 2 ) 2 ) c 3 ( Z , ♦ 3 ) 3 ) { I , II , III } III I II II I III { I , II } I I II II I I { I , III } III I I I I III { II , III } III II II II II III Remark WARP holds for all choices. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects The choices in the behavioral-bias and no-behavioral-bias cases are: Z c 1 ( Z , ♦ 1 ) c 1 ( Z , h ∗ 1 ) c 2 ( Z , ♦ 2 ) c 2 ( Z , h ∗ 2 ) c 3 ( Z , ♦ 3 ) c 3 ( Z , h ∗ 3 ) { I , II , III } III I II II I III { I , II } I I II II I I { I , III } III III I I I III { II , III } III II II II II III The behavioral-bias case Z c 1 ( Z , ♦ 1 ) c 1 ( Z , h ∗ 1 ) c 2 ( Z , ♦ 2 ) c 2 ( Z , h ∗ 2 ) c 3 ( Z , ♦ 3 ) c 3 ( Z , h ∗ 3 ) { I , II , III } III I II II I III { I , II } I I II II I I { I , III } III I I I I III { II , III } III II II II II III The no-behavioral-bias case Remark Choices coincide except c 1 ( { I , III } , h ∗ 1 ) . That is, choices in two cases coincide unless individual 1 is of type h ∗ 1 and faces { I , III } . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects Observations: The difference between the behavioral-bias and no-behavioral-bias 1 cases involves 1 of type h ∗ 1 facing { I , III } . The only violation of WARP happens for 1 of type h ∗ 1 facing 2 { I , II , III } and { I , III } : ◮ Individual 1 chooses I from { I , II , III } in both cases; ◮ Her choice III from { I , III } violates the IIA in the behavioral-bias case while her choices satisfy the IIA in the no-behavioral-bias case as she chooses I from { I , III } . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects Observations: The difference between the behavioral-bias and no-behavioral-bias 1 cases involves 1 of type h ∗ 1 facing { I , III } . The only violation of WARP happens for 1 of type h ∗ 1 facing 2 { I , II , III } and { I , III } . In the behavioral-bias case , every consistent collection must be such 3 that S 1 ( f , h ∗ − 1 ) = { I , II , III } but not { I , III } : 1 must be offered her initial endowment to make her choose “consistent” with f as ◮ H 1 ( f ( ♦ 1 , h ∗ − 1 )) = III , H 1 ( f ( h ∗ )) = I , i.e., { I , III } are the objects f assigns to 1 depending on her types when others’ are ( h ∗ 2 , h ∗ 3 ), and ◮ c 1 ( { I , II , III } , ♦ 1 ) = I , c 1 ( { I , II , III } , h ∗ 1 ) = III , but c 1 ( { I , III } , ♦ 1 ) = c 1 ( { I , III } , h ∗ 1 ) = III . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects Observations: The difference between the behavioral-bias and no-behavioral-bias 1 cases involves 1 of type h ∗ 1 facing { I , III } . The only violation of WARP happens for 1 of type h ∗ 1 facing 2 { I , II , III } and { I , III } . In the behavioral-bias case , every consistent collection must be such 3 that S 1 ( f , h ∗ − 1 ) = { I , II , III } but not { I , III } : 1 must be offered her initial endowment to make her choose “consistent” with f . In the behavioral-bias case, the direct mechanism does not ex-post 4 implement f while the indirect mechanism does: ◮ S 1 ( f , h ∗ − 1 ) = { I , II , III } in the indirect mechanism, but ◮ S 1 ( f , h ∗ − 1 ) = { I , III } in the direct mechanism. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects Observations: The difference between the behavioral-bias and no-behavioral-bias 1 cases involves 1 of type h ∗ 1 facing { I , III } . The only violation of WARP happens for 1 of type h ∗ 1 facing 2 { I , II , III } and { I , III } . In the behavioral-bias case , every consistent collection must be such 3 that S 1 ( f , h ∗ − 1 ) = { I , II , III } but not { I , III } : 1 must be offered her initial endowment to make her choose “consistent” with f . In the behavioral-bias case, the direct mechanism does not ex-post 4 implement f while the indirect mechanism does. In the no-behavioral-bias case, both the direct and indirect 5 mechanisms ex-post implement the SCF f . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Comparative Statics with Endowment Effects Conclusions: Behavioral aspects enforce the planner’s use of individual 1’s initial 1 endowment to ensure that 1’s choices are aligned with the social goal, while in the absence of behavioral aspects, the planner may dispense with 2 the assignment of individual 1 to her initial endowment as it does not appear in the social choice function at any state of the world. These display glimpses into the intricate nature of the distinction 3 between direct mechanisms and indirect mechanisms in the context of behavioral ex-post implementation under incomplete information. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Direct Mechanisms Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Ex-Post Implementation with Direct Mechanisms We analyze the significance of direct mechanisms pertinent to ex-post implementation in general environments. The intuitive nature of direct mechanisms turns out to be helpful in behavioral implementation under incomplete information: We provide two characterizations of the scope of situations in which ex-post implementation is possible only when direct ex-post implementation is achievable. In what follows, we focus on SCFs instead of SCSs since direct mechanisms cannot coordinate selections of SCFs from an SCS. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Extent of Direct Behavioral Ex-Post Implementation Theorem (4) Let f : Θ → X be an SCF. ( i ) f is (fully) ex-post implementable by its associated direct mechanism possessing a truthful EPE if and only if the collection F := { f (Θ i , θ − i ) : i ∈ N , θ − i ∈ Θ − i } is consistent with f under incomplete information. ( ii ) If f is full-range, then f is ex-post implementable if and only if it is (fully) ex-post implementable via its direct mechanism. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Extent of Direct Behavioral Ex-Post Implementation Theorem 4–( i ) says that direct ex-post implementability is equivalent to the consistency of the collection F := { f (Θ i , θ − i ) : i ∈ N , θ − i ∈ Θ − i } . Theorem 4–( ii ) provides another characterization of direct ex-post implementability involving a full-range condition for the SCF f that we borrow from Bergemann and Morris (2008): An SCF f is full-range if for all x ∈ X , all i ∈ N , and all θ − i ∈ Θ − i , there is θ i ∈ Θ i with f ( θ ) = x . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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