Behavioral Implementation under Incomplete Information Mehmet Barlo - - PowerPoint PPT Presentation
Behavioral Implementation under Incomplete Information Mehmet Barlo 1 un Dalkran 2 Nuh Ayg October, 2020 Barlo and Dalkran Behavioral Implementation under Incomplete Information Behavioral Economics The premise of behavioral economics is
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The premise of behavioral economics is predictable irrationality.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
What shall a planner do if she were to implement a goal when the relevant information is distributed among “predictably irrational” individuals?
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We investigate the implementation problem under incomplete information when individuals’ choices need not be rational. Our results provide an important leap as incomplete information is inescapable in many economic settings. are complementary to “Behavioral Implementation” de Clippel (2014) [AER, 104(10): 2975-3002], which investigates the same problem under complete information.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
∗ Presidential Address to the European Meetings of the Econometric Society in Manchester, UK, 2019. ∗ We thank Professor Stephen Morris for allowing us to share this photo. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
∗ This slide is from Professor Stephen Morris’s Presidential Address to the European Meetings of the Econometric Society in Manchester, UK, August 27, 2019. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
People have cognitive limitations and are prone to behavioral biases:
the attraction effect the status-quo bias intransitive indifference limited attention the endowment effect temptation and self-control compromise and framing effects ...
Therefore, individual behavior may not be consistent with a rational preference relation. Behavioral economics and psychology offers insights on the systematic deviations from rationality.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The Behavioral Insights Team (BIT) (a.k.a the Nudge Unit) was established in 2010 in the U.K. ◮ “in order to improve the government policies using ideas drawn from the behavioral sciences” Thaler and Sunstein (2008) “Nudge: Improving Decisions about Health, Wealth, and Happiness ” Australia, Canada, Germany, India, Indonesia, Ireland, Jordan, Netherlands, Peru, Singapore, Turkey, and the US among others started applying behavioral insights to their policies and programs. International institutions such as the EU, OECD, UN, World Bank established behavioral insights units to support their programs.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Herne (1997) demonstrates how the presence of a decoy alternative causes attraction effect in a policy-making context: In 1993, Finland took the decision of building a new nuclear power plant to parliamentary vote. ◮ Opponents: Let’s go solar! ◮ Proponents: Let’s consider coal as well!
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Environment Reliability solar nuclear nuclear coal coal solar Nuclear dominates coal in both dimensions! But, solar dominates coal only in the environment dimension. Coal is a decoy! Attraction Effect: Decoy alternatives –alternatives that are known to be dominated by another alternative– can cause preference reversals when they are introduced in the consideration set!
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
It is expected by the proponents that nuclear is chosen from the grand set {coal, nuclear, solar} solar is chosen from the set {nuclear, solar}. This violates weak axiom of revealed preferences (WARP) Sen (1971): WARP ⇐ ⇒ Sen’s α ∧ Sen’s β Sen’s α (independence of irrelevant alternatives) fails! Sen’s β (expansion consistency)
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Alice and Bob are to decide what type of energy to employ or jointly invest in, be it coal energy, nuclear energy, or solar energy. Their individual choices may not be rational, i.e., they may violate weak axiom of revealed preferences (WARP). There is incomplete information between Ann and Bob regarding the true state of the world.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
There are four possible states of the world with the following predicted individual choices of Ann and Bob over X = {coal, nuclear, solar}:
State 1 State 2 State 3 State 4 Subsets of X Ann Bob Ann Bob Ann Bob Ann Bob {c, n, s} {n} {s} {n} {n} {n} {c} {c, s} {n, s} {c, n} {n} {n} {n} {n} {n} {c} {n} {c} {c, s} {c, s} {s} {s} {s} {c} {c} {c} {s} {n, s} {n} {s} {s} {s} {n, s} {n, s} {s} {s}
The set of states: Θ = {(ρA, ρB), (ρA, γB), (γA, ρB), (γA, γB)}
◮ Θ = ΘA × ΘB where Θi = {ρi, γi} for both i ∈ {A, B}.
There is distributed knowledge of the true state θ = (θA, θB):
◮ Ann learns only θA ∈ ΘA.
◮ Bob learns only θB ∈ ΘB.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The predicted choices of individuals at each state are:
C (ρA,ρB )
A
C (ρA,ρB )
B
C (ρA,γB )
A
C (ρA,γB )
B
C (γA,ρB )
A
C (γA,ρB )
B
C (γA,γB )
A
C (γA,γB )
B
{c, n, s} {n} {s} {n} {n} {n} {c} {c, s} {n, s} {c, n} {n} {n} {n} {n} {n} {c} {n} {c} {c, s} {c, s} {s} {s} {s} {c} {c} {c} {s} {n, s} {n} {s} {s} {s} {n, s} {n, s} {s} {s} n ≻A c ∼A s s ≻B n ≻B c Att “c” Att “c” S-quo “c” c ≻B s ∼B n Cycles Groups
At (ρA, ρB), both individuals’ choices can be rationalized: Ann is a proponent with n ≻A c ∼A s; Bob is an opponent with s ≻B n ≻B c. At (ρA, γB), both individuals display an attraction effect where c is the decoy! At (γA, ρB), Ann’s choices display a status-quo bias with the default option c whereas Bob’s choices can be rationalized according to c ≻B s ∼B n. At (γA, γB), both individuals can be thought of as a group where their choices are obtained via some sort of voting where Ann’s involves Condorcet cycles.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The social planner cares about the welfare of the individuals in terms of generalized Pareto optimality ` a la Bernheim and Rangel (2009), which we refer to as BR-optimality.
BR-optimality
We work with Social Choice Sets (SCSs) under incomplete information, see e.g., Jackson (1991), Bergemann and Morris (2008) among others. An SCS is a set of Social Choice Functions (SCFs) and an SCF is a state contingent allocation. In this example, we work with the SCS that is composed of “mutually exhaustive” selections from BR-optimal outcomes:
State: (ρA, ρB) (ρA, γB) (γA, ρB) (γA, ρB) BR-Optima {n, s} {n, s} {c, n} {c, s} f n n n s f ′ s s c c
The Social Choice Set: F = {f , f ′}.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
µ = (M, g) : MA = {U, M, D} and MB = {L, M, R} Bob Ann g L M R U n c n M c s c D n s s
Back to Simple Mechanisms Back to Example SM-1 Barlo and Dalkıran Behavioral Implementation under Incomplete Information
In case of a mechanism, a message sent by an individual restricts the alternatives the other agents can generate. De Clippel (2014): The opportunity set of agent i given m−i under the mechanism µ = (M, g) is given by Oµ
i (m−i) := {g(mi, m−i)|mi ∈ Mi} ⊆ X.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Bob Ann L M R U n c n M c s c D n s s E.g., Oµ
A(L) = {n, c} and Oµ B(D) = {n, s}.
i (Oµ i (m∗ −i)) ∀i.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Bob Ann L M R U n c n M c s c D n s s Choices C (γA,γB)
A
C (γA,γB)
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
Bob Ann L M R U
An Ac
n M c s
Ac
D
An
s
As
Choices C (γA,γB)
A
C (γA,γB)
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
Bob Ann L M R U n cB n M c sB c D n sB sB Choices C (γA,γB)
A
C (γA,γB)
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
Bob Ann L M R U
An AcB
n M c sB
Ac
D
An
sB
AsB
Nash Equilibrium message profiles at (γA, γB): (U, M); (D, R). Nash Equilibrium outcomes at (γA, γB): {c, s} {c, s} {c, s}.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
(ρA, ρB) (ρA, γB) L M R U n c n M c s c D n s s L M R U n c n M c s c D n s s N.Eq = {n, s} {n, s} {n, s} N.Eq = {n, s} {n, s} {n, s} (γA, ρB) (γA, γB) L M R U n c n M c s c D n s s L M R U n c n M c s c D n s s N.Eq = {c, n} {c, n} {c, n} N.Eq = {c, s} {c, s} {c, s}
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
State by state, we have BR-Optimal outcomes = Nash outcomes: (ρA, ρB) (ρA, γB) {n, s} {n, s} {n, s} {n, s} {n, s} {n, s} (γA, ρB) (γA, γB) {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
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., σi : Θi → Mi. 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 g(σ∗
i (θ′ i), σ∗ −i(θ−i)) ∈ C (θ′
i ,θ−i)
i
(Oµ
i (σ∗ −i(θ−i))), for all θ′ i ∈ Θi.
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
The following are (all) the ex-post equilibria of µ = (M, g): ◮ σ σ σ′∗: Ann → σ∗
A(ρA) = U
σ∗
A(γA) = D
Bob → σ∗
B(ρB) = L
σ∗
B(γB) = R
◮ σ σ σ′′∗: Ann → σ′∗
A (ρA) = D
σ′∗
A (γA) = U
Bob → σ′∗
B (ρB) = M
σ′∗
B (γB) = M
◮ σ σ σ′′′∗: Ann → σ′∗
A (ρA) = M
σ′∗
A (γA) = U
Bob → σ′∗
B (ρB) = M
σ′∗
B (γB) = M
◮ σ σ σ′′∗ and σ σ σ′′′∗ are outcome equivalent.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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: (ρA, ρB) State: (ρA, γB) State: (γA, ρB) State: (γA, γB) L M R U n n n c n M c s s s c D n s s s s L M R U n c n n n M c s s s c D n s s s s L M R U n c c c n M c s c D n n n s s L M R U n c c c n M c s c D n s s s s σ′∗ σ′∗ σ′∗(ρA, ρB) = (U, L) σ′∗ σ′∗ σ′∗(ρA, γB) = (U, R) σ′∗ σ′∗ σ′∗(γA, ρB) = (D, L) σ′∗ σ′∗ σ′∗(γA, γB) = (D, R) Outcome: n n n Outcome: n n n Outcome: n n n Outcome: s s s f (ρA, ρB) = n n n f (ρA, γB) = n n n f (γA, ρB) = n n n f (γA, γB) = s s s σ′′∗ σ′′∗ σ′′∗(ρA, ρB) = (D, M) σ′′∗ σ′′∗ σ′′∗(ρA, γB) = (D, M) σ′′∗ σ′′∗ σ′′∗(γA, ρB) = (U, M) σ′′∗ σ′′∗ σ′′∗(γA, γB) = (U, M) Outcome: s s s Outcome: s s s Outcome: c c c Outcome: c c c σ′′′∗ σ′′′∗ σ′′′∗(ρA, ρB) = (M, M) σ′′′∗ σ′′′∗ σ′′′∗(ρA, γB) = (M, M) σ′′′∗ σ′′′∗ σ′′′∗(γA, ρB) = (U, M) σ′′′∗ σ′′′∗ σ′′′∗(γA, γB) = (U, M) Outcome: s s s Outcome: s s s Outcome: c c c Outcome: c c c f ′(ρA, ρB) = s s s f ′(ρA, γB) = s s s f ′(γA, ρB) = c c c f ′(γA, γB) = c c c
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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 (ρA, ρB) (ρA, γB) (γA, ρB) (γA, γB) f n n n s σ′∗ n n n s The direct mechanism associated with the SCF f does not partially (ex-post) implement f : n n n is not chosen from {n, s} by Ann at (ρA, γB).
Bob Ann ρB γB ρA n n n n γA n s
The direct mechanism associated with the SCF f
C (ρA,γB)
A
C (ρA,γB)
B
{c, n, s} {n} {n} {c, n} {n} {n} {c, s} {s} {s} {n, s} {s} {s}
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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
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
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
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
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
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∈NΘi, denotes the set of all possible states of the world, ◮ θi 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
The Social Choice Set (SCS) is, F ⊂ {f |f : Θ → X}. µ = (M, g) denotes a mechanism where ◮ Mi = ∅ is the message space of agent i ∈ N and ◮ g : M → X is the outcome function where M := ×i∈NMi. σi : Θi → Mi 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 Oµ
i (m−i) := {g(mi, m−i) ∈ X : mi ∈ Mi}.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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 g(σ∗
i (θi), σ∗ −i(θ−i)) ∈ C (θi,θ−i) i
(Oµ
i (σ∗ −i(θ−i))).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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
αi : Θi → Θi denotes a possible deception by agent i ∈ N. ◮ αi(θi) can be interpreted as i’s reported type. α(θ) = (α1(θ1), α2(θ2), . . . , αn(θn)). denotes a profile of deceptions.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Definition A collection of sets S := {Si(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) for all i ∈ N, f (θ′
i, θ−i) ∈ C (θ′
i ,θ−i)
i
(Si(f , θ−i)) for each θ′
i ∈ Θi, and
(ii) for any deception profile α with f ◦ α / ∈ F, there exists θ∗ ∈ Θ and i∗ ∈ N such that f (α(θ∗)) / ∈ C θ∗
i∗ (Si∗(f , α−i∗(θ∗ −i∗))).
Barlo and Dalkıran Behavioral Implementation 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 Si(f , θ−i) at state (θ′ i , θ−i)
contains f (θ′
i , θ−i) for all θ′ i ∈ Θi; 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
Si∗(f , α−i∗(θ∗
−i∗)).
In Supplementary Materials, we provide
Python codes computing
consistent collections taking individuals’ choices and the SCS as inputs.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Theorem (1) If an SCS F is ex-post implementable, then there exists a collection of sets S := {Si(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
Suppose µ = (M, g) ex-post implements a given SCS F. Then,
◮ for any SCF f ∈ F, there is an EPE σf of µ such that f = g ◦ σf .
◮ for each θ ∈ Θ, g(σf (θ)) = f (θ) is in C θ
i (Oµ i (σf −i(θ−i))) for all
i ∈ N. Let S be such that Si(f , θ−i) := Oµ
i (σf −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
◮ σf ◦ α cannot be an EPE of µ.
Otherwise, by (ii) of ex-post implementability, there exists ˜ f ∈ F with ˜ f = g ◦ σf ◦ α. But, since f = g ◦ σf , we have ˜ f = f ◦ α ∈ F, a contradiction. ◮ Hence, there are θ∗ and i∗ who does not choose f (α(θ∗)) from Oµ
i∗(σf −i∗(α−i∗(θ−i∗))) = Si∗(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
Definition F is ex-post-choice monotonic if, for every f ∈ F and deception profile α with f ◦ α / ∈ F, there is θ∗ ∈ Θ, i∗ ∈ N, S∗ ∈ X such that (i) f (α(θ∗)) / ∈ C θ∗
i∗ (S∗),
(ii) f ((θ′
i∗, α−i∗(θ∗ −i∗))) ∈ C (θ′
i∗,α−i∗(θ∗ −i∗))
i∗
(S∗) for all θ′
i∗ ∈ Θi∗.
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
Ex-post Choice-Monotonicity requires that when there is an attempt of deception that would lead to a non-optimal outcome, there must exist
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
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 ∈ Θi},
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 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 ∈ Θi}).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition (3) If individual choices satisfy the IIA, then quasi-ex-post choice incentive compatibility implies the revelation principle.
choices satisfy the IIA!
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We show that under WARP quasi-ex-post choice incentive compatibility is equivalent to ex-post incentive compatibility of Bergemann and Morris (2008), and
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
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 µ2 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
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Definition S satisfies the choice incompatible pair property at θ if for each x ∈ S there exist i, j ∈ N such that x / ∈ C θ
i (S) and x /
∈ C θ
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
i (S) and x /
j (S).
◮ Yet, it is weaker as it is defined for a particular set.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
S∈S S ⊆ ¯
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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. In Supplementary Materials, we provide
Python codes computing
consistent collections S and ¯ X satisfying the choice incompatible pair property taking individuals’ choices and the SCS as inputs. In
Example SM-5 with three rational individuals, we show that even in the
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
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
Definition F satisfies the consistency-no-veto property whenever there exist (i) a collection of sets S := {Si(f , θ−i) : i ∈ N, f ∈ F, θ−i ∈ Θ−i} such that for all f ∈ F and for all i ∈ N, f (θ′
i , θ−i) ∈ C (θ′
i ,θ−i )
i
(Si(f , θ−i)) for each θ′
i ∈ Θi,
(ii) and a set of alternatives ¯ X ⊆ X with
S∈S S ⊆ ¯
X such that for any collection of product sets of states {¯ Θf }f ∈F with ¯ Θ =
f ∈F ¯
Θf ⊂ Θ, there exists f ∗ ∈ F such that (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 ∈ ¯ Θf , then there exists i∗ ∈ N and θ∗ ∈ ¯ Θf such that f (α(θ∗)) / ∈ C θ∗
i∗ (Si∗(f , α−i∗(θ∗ −i∗))). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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 Si(f , θ−i) at state (θ′ i , θ−i) contains f (θ′ i , θ−i)
for all θ′
i ∈ Θi; and
for any collection of product sets of states {¯ Θf }f ∈F with ¯ Θ =
f ∈F ¯
Θf ⊂ Θ, there is an SCF f ∗ in F such that
◮ if θ ∈ Θ \ ¯
Θ, then f ∗ obeys the choice no-veto-power property on ¯ 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 ∈ ¯ Θf , then there exists a whistle-blower i∗ ∈ N and an informant state θ∗ such that i∗ does not choose f (α(θ∗)) from Si∗(f , α−i∗(θ∗
−i∗)) at θ∗. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The
Python codes we provide in Supplementary Materials compute
◮ the collection S,
◮ the set of alternatives ¯ X, ◮ the collection of product sets {¯ Θf }f ∈F, ◮ SCF’s f ∗ ∈ F satisfying consistency-no-veto taking choices and the SCS as inputs. In
Example SM-4 of Supplementary Materials, we show that there are
collections S together with ¯ X, ¯ Θ, and f ∗ satisfying consistency-no-veto. Our codes identify S, ¯ X, {¯ Θf }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 (1) Let n ≥ 3. An SCF f : Θ → X is ex-post implementable if there exists a collection S := {Si(f , θ−i) : i ∈ N, θ−i ∈ Θ−i} s.t. f (θ′
i, θ−i) ∈ C (θ′
i ,θ−i)
i
(Si(f , θ−i)) for each θ′
i ∈ Θi and there exists ¯
X ⊆ X with
S∈S S ⊆ ¯
X s.t. for any product set ¯ 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 f (α(θ∗)) / ∈ C θ∗
i∗ (Si∗(f , α−i∗(θ∗ −i∗))).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
µ = (M, g): Mi = F × Θi × ¯ X × N and g : M → X is as follows:
Rule 1 : g(m) = f (θ) if mi = (f , θi, ·, ·) for all i ∈ N, Rule 2 : g(m) = xj if xj ∈ Sj(f , θ−j), ¯ x(j, f , θ−j)
if mi = (f , θi, ·, ·) for all i ∈ N \ {j} and mj = (˜ f , ˜ θj, xj, ·) with ˜ f = f , Rule 3 : g(m) = xj where j =
i ki (mod n)
where ¯ x(j, f , θ−j) is an arbitrary element from Sj(f , θ−j).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
de Clippel (2014) introduces the following notion of efficiency: Φeff(θ) :=
i (Yi) for all i ∈ N and X = ∪i∈NYi
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. Φeff : 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
The efficient SCS is F eff := {f : Θ → X | f (θ) ∈ Φeff(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
Let N = {A, B}, X = {x, y} and Θi = {θi, ωi} for i = A, B, and the following choices:
S C (θA,θB)
A
C (θA,θB)
B
C (θA,ωB)
A
C (θA,ωB)
B
C (ωA,θB)
A
C (ωA,θB)
B
C (ωA,ωB)
A
C (ωA,ωB)
B
{x, y} {x} {x} {x} {x} {y} {x} {y} {x}
Then, F eff = {f , f ′, f ′′, f ′′′} where
(θA, θB) (θA, ωB) (ωA, θB) (ωA, ωB) f x x x x f ′ x x x y f ′′ x x y x f ′′′ x x y y
SA, SB consistent with F eff under incomplete information implies x ∈ C (ωA,θB)
B
(SB(f ′, ωA)) and y ∈ C (ωA,ωB)
B
(SB(f ′, ωA)) so SB(f ′, ωA) = {x, y}. But, then y / ∈ C (ωA,ωB)
B
(SB(f ′, ωA)), a contradiction.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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
i
with for all θ, ∪i∈NY θ−i
i
= X, and (ii) e(˜ θi, θ−i) ∈ C (˜
θi,θ−i) i
(Y θ−i
i
) for all ˜ θi. 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
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
n objects are to be allocated among n individuals. ◮ H = {h1, . . . , hn} is the set of objects (e.g., houses or offices) ◮ H denotes the set of all non-empty subsets of H. ◮ X := {x ∈ Hn | xi = xj, for all i, j ∈ N with i = j} is the set
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
Individuals’ choices are independent and each individual cares only about her own object: ci(Z, θi) = ∅ – the chosen object(s) from Z ∈ H by i of type θi. Hi(S) := {xi ∈ H | x ∈ S} – the object(s) i gets in allocations in S. For any S ∈ X, individual choices on allocations are C θi
i (S) := {x ∈ S | xi ∈ ci(Hi(S), θi)}.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Constrained efficiency takes the following specific form: ˜ E c.eff
H
consists
e : N × Θ → H with ˜ ei(θ) = ˜ ej(θ) for all i = j and all θ ∈ Θ such that (i) for all i ∈ N and all θ−i ∈ Θ−i, there is Hi(θ−i) ∈ H with ˜ ei(˜ θi, θ−i) ∈ ci(Hi(θ−i), ˜ θi) for all ˜ θi ∈ Θi; and (ii) for all θ ∈ Θ, ∪i∈NHi(θ−i) = H. Corollary (2) Let n ≥ 3. ˜ E c.eff
H
is ex-post implementable on all domains satisfying the choice incompatibility on the set of all objects H.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We use the canonical model of choice with endowment effects (Masatlioglu & Ok, 2014): h∗
i ∈ H denotes i’s initial endowments;
θi = ♦i denotes i being of rational type; θi = h∗
i denotes i being of
behavioral type; Θi = {♦i, h∗
i } denotes i’s types.
i’s choices on objects are singleton-valued. Under reasonable assumptions, i’s choices are represented by Ui : H → R, i’s utility function on H, and Qi(h∗
i ) = {h ∈ H | h ∈ ci({h, h∗ i }, h∗ i )}, i’s
consideration set, such that for any Z ∈ H, ci(Z, θi) = arg maxh∈Z∩Qi(h∗
i ) Ui(h)
if θi = h∗
i and h∗ i ∈ Z,
arg maxh∈Z Ui(h) if θi = h∗
i but h∗ i /
∈ Z, arg maxh∈Z Ui(h) if θi = ♦i
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Then, i’s independent choices on allocations are: For any S ∈ X C θi
i (S) :=
{x ∈ S | xi = arg maxh∈Hi(S)∩Qi(h∗
i ) Ui(h)}
if θi = h∗
i and
h∗
i ∈ Hi(S),
{x ∈ S | xi = arg maxh∈Hi(S) Ui(h)} if θi = h∗
i and
h∗
i /
∈ Hi(S), {x ∈ S | xi = arg maxh∈Hi(S) Ui(h)} if θi = ♦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
Proposition (7) If an SCS F ∈ F is ex-post implementable, then there exists a collection of sets S := {Si(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 Hi(Si(f , θ−i)) ⊂ {h ∈ H | Ui(xi) ≥ Ui(h)}, (ii) if f (θ) = x and xi / ∈ Qi(h∗
i ), then h∗ i /
∈ Hi(Si(f , θ−i)), (iii) if f (θ) = x and xi = h∗
i , then Hi(Si(f , θ−i)) ∩ Qi(h∗ i ) = {h∗ i },
(iv) if f (θ) = x and xi = h∗
i , then either h∗ i /
∈ Hi(Si(f , θ−i)) or xi ∈ Qi(h∗
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
Proposition 7 tells that: (i) if f (θ) = x, then Hi(Si(f , θ−i)) ⊂ {h ∈ H | Ui(xi) ≥ Ui(h)},
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition 7 tells that: (i) if f (θ) = x, then objects offered to i for f and θ−i, Hi(Si(f , θ−i)), cannot provide strictly higher utilities than Ui(xi). Reason: Otherwise, xi / ∈ ci(Hi(Si(f , θ−i)), ♦i) when θi = ♦i; i.e., the rational type of i does not choose x from Si(f , θ−i).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition 7 tells that: (i) if f (θ) = x, then objects offered to i for f and θ−i, Hi(Si(f , θ−i)), cannot provide strictly higher utilities than Ui(xi). (ii) if f (θ) = x and xi / ∈ Qi(h∗
i ), then h∗ i /
∈ Hi(Si(f , θ−i)).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition 7 tells that: (i) if f (θ) = x, then objects offered to i for f and θ−i, Hi(Si(f , θ−i)), cannot provide strictly higher utilities than Ui(xi). (ii) if f (θ) = x and xi 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, xi / ∈ ci(Hi(Si(f , θ−i)), h∗
i ) when θi = h∗ i ; i.e.,
the behavioral type of i does not choose x from Si(f , θ−i).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition 7 tells that: (i) if f (θ) = x, then objects offered to i for f and θ−i, Hi(Si(f , θ−i)), cannot provide strictly higher utilities than Ui(xi). (ii) if f (θ) = x and xi 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 xi = h∗
i , then Hi(Si(f , θ−i)) ∩ Qi(h∗ i ) = {h∗ i },
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition 7 tells that: (i) if f (θ) = x, then objects offered to i for f and θ−i, Hi(Si(f , θ−i)), cannot provide strictly higher utilities than Ui(xi). (ii) if f (θ) = x and xi 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 xi = 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
i .
Reason: Otherwise, h∗
i /
∈ ci(Hi(Si(f , θ−i)), h∗
i ) when θi = h∗ i ; i.e.,
the behavioral type of i does not choose x from Si(f , θ−i).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition 7 tells that: (i) if f (θ) = x, then objects offered to i for f and θ−i, Hi(Si(f , θ−i)), cannot provide strictly higher utilities than Ui(xi). (ii) if f (θ) = x and xi 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 xi = 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
i .
(iv) if f (θ) = x and xi = h∗
i , then either h∗ i /
∈ Hi(Si(f , θ−i)) or xi ∈ Qi(h∗
i ).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Proposition 7 tells that: (i) if f (θ) = x, then objects offered to i for f and θ−i, Hi(Si(f , θ−i)), cannot provide strictly higher utilities than Ui(xi). (ii) if f (θ) = x and xi 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 xi = 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
i .
(iv) if f (θ) = x and xi = h∗
i , then either xi 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. Reason: Otherwise, xi / ∈ ci(Hi(Si(f , θ−i)), h∗
i ) when θi = h∗ i ; i.e.,
the behavioral type of i does not choose x from Si(f , θ−i).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
For a practical display of Proposition 7, consider the following example:
Object U1 U2 U3 I 2 2 3 II 1 3 2 III 3 1 1 h∗
i
Qi(h∗
i )
II {I, II}
I {I, II, III}
III {III}
inducing the following choices
Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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: f (θ) =
if θ = (h∗
1, h∗ 2, h∗ 3)
(III, II, I)
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
For a practical display of Proposition 7, consider the following example:
Object U1 U2 U3 I 2 2 3 II 1 3 2 III 3 1 1 h∗
i
Qi(h∗
i )
II {I, II}
I {I, II, III}
III {III}
inducing the following choices
Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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
For a practical display of Proposition 7, consider the following example:
Object U1 U2 U3 I 2 2 3 II 1 3 2 III 3 1 1 h∗
i
Qi(h∗
i )
II {I, II}
I {I, II, III}
III {III}
inducing the following choices
Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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, H1(S1(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
The following indirect mechanism ex-post implements the SCF f in this example:
L R U (I, II, III) (III, II, I) 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):
3
h∗
2
♦2 h∗
1
(I, II, III) (III, II, I) ♦1 (III, II, I) (III, II, I)
h∗
2
♦2 h∗
1
(III, II, I) (III, II, I) ♦1 (III, II, I) (III, II, I)
At (h∗
1, h∗ 2, h∗ 3), Oµ 1 (h∗ −1) = {I, III} and the EPE σ∗ 1 (θ1) = ♦1 for all
θ1 ∈ {♦1, h∗
1}. This is an impasse due to f1(h∗ 1, h∗ 2, h∗ 3) = I. Back to Behavioral Aspects and Simplicity Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The behavioral-bias case (obtained by relabeling the above) involves choices with endowment effects where Θi = {♦i, h∗
i }, for all Z ∈ H
ci(Z, θi) =
i ) Ui(h)
if θi = h∗
i and h∗ i ∈ Z,
arg maxh∈Z Ui(h)
and
Object U1 U2 U3 I 2 2 3 II 1 3 2 III 3 1 1 h∗
i
Qi(h∗
i )
II {I, II}
I {I, II, III}
III {III}
This formulation induces the following choices:
Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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
Remark WARP holds for all choices but c1(·, h∗
1).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We construct a corresponding no-behavioral-bias case involving rational state-contingent choices where Θi = {♦i, h∗
i }, for all Z ∈ H
ci(Z, θi) = arg max
h∈Z Uθi i (h),
and the utilites and endowments are
Object U♦1
1
Uh∗
1
1
U♦2
2
Uh∗
2
2
U♦3
3
Uh∗
3
3
I 2 3 2 2 3 2 II 1 2 3 3 2 1 III 3 1 1 1 1 3 h∗
i
II
I
III
Thus, we obtain the the following choices:
Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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
Remark WARP holds for all choices.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The choices in the behavioral-bias and no-behavioral-bias cases are:
Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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 c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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 c1({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
Observations:
1
The difference between the behavioral-bias and no-behavioral-bias cases involves 1 of type h∗
1 facing {I, III}.
2
The only violation of WARP happens for 1 of type h∗
1 facing
{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
Observations:
1
The difference between the behavioral-bias and no-behavioral-bias cases involves 1 of type h∗
1 facing {I, III}.
2
The only violation of WARP happens for 1 of type h∗
1 facing
{I, II, III} and {I, III}.
3
In the behavioral-bias case, every consistent collection must be such that S1(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 ◮ H1(f (♦1, h∗
−1)) = III, H1(f (h∗)) = I, i.e., {I, III} are the
(h∗
2, h∗ 3), and
◮ c1({I, II, III}, ♦1) = I, c1({I, II, III}, h∗
1) = III, but
c1({I, III}, ♦1) = c1({I, III}, h∗
1) = III.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Observations:
1
The difference between the behavioral-bias and no-behavioral-bias cases involves 1 of type h∗
1 facing {I, III}.
2
The only violation of WARP happens for 1 of type h∗
1 facing
{I, II, III} and {I, III}.
3
In the behavioral-bias case, every consistent collection must be such that S1(f , h∗
−1) = {I, II, III} but not {I, III}: 1 must be offered her
initial endowment to make her choose “consistent” with f .
4
In the behavioral-bias case, the direct mechanism does not ex-post implement f while the indirect mechanism does: ◮ S1(f , h∗
−1) = {I, II, III} in the indirect mechanism, but
◮ S1(f , h∗
−1) = {I, III} in the direct mechanism.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Observations:
1
The difference between the behavioral-bias and no-behavioral-bias cases involves 1 of type h∗
1 facing {I, III}.
2
The only violation of WARP happens for 1 of type h∗
1 facing
{I, II, III} and {I, III}.
3
In the behavioral-bias case, every consistent collection must be such that S1(f , h∗
−1) = {I, II, III} but not {I, III}: 1 must be offered her
initial endowment to make her choose “consistent” with f .
4
In the behavioral-bias case, the direct mechanism does not ex-post implement f while the indirect mechanism does.
5
In the no-behavioral-bias case, both the direct and indirect mechanisms ex-post implement the SCF f .
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Conclusions:
1
Behavioral aspects enforce the planner’s use of individual 1’s initial endowment to ensure that 1’s choices are aligned with the social goal, while
2
in the absence of behavioral aspects, the planner may dispense with 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.
3
These display glimpses into the intricate nature of the distinction between direct mechanisms and indirect mechanisms in the context
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
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
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 (Θi, 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
Theorem 4–(i) says that direct ex-post implementability is equivalent to the consistency of the collection F := {f (Θi, 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 θi ∈ Θi with f (θ) = x.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We analyze the implications of independence: S = {Si(θ−i) | i ∈ N, θ−i ∈ Θ−i} is independent–consistent with the SCF f under incomplete information if (i) for all i and all θ′
i , f (θ′ i , θ−i) ∈ C θ′
i
i (Si(θ−i)) for all θ−i, and
(ii) for any deception α with f ◦ α = f , there are i∗ and θ∗ with f (α(θ∗)) / ∈ C
θ∗
i∗
i∗ (Si∗(α−i∗(θ∗ −i∗))).
Our necessity result (Theorem 1) implies: If an SCF f : Θ → X is ex-post implementable, then it is weak choice strategy-proof. An SCF f is weak choice strategy-proof if there is S∗ = {S∗
i (θ−i)}i,θ−i with
for all i and all θ−i, f (Θi, θ−i) ⊂ S∗
i (θ−i) and f (ˆ
θi, θ−i) ∈ C
ˆ θi i (S∗ i (θ−i)) for all ˆ
θi.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We analyze the implications of independence: S = {Si(θ−i) | i ∈ N, θ−i ∈ Θ−i} is independent–consistent with the SCF f under incomplete information if (i) for all i and all θ′
i , f (θ′ i , θ−i) ∈ C θ′
i
i (Si(θ−i)) for all θ−i, and
(ii) for any deception α with f ◦ α = f , there are i∗ and θ∗ with f (α(θ∗)) / ∈ C
θ∗
i∗
i∗ (Si∗(α−i∗(θ∗ −i∗))).
Our necessity result (Theorem 1) implies: If an SCF f : Θ → X is ex-post implementable, then it is weak choice strategy-proof. Under the IIA, weak choice strategy-proofness is equivalent to the following notion: An SCF f is choice strategy-proof holds if for all i and all θ−i, f (θ′
i , θ−i) ∈ C θ′
i
i (f (Θi, θ−i)) for all θ′ i . Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Given µ = (M, g), a strategy profile s∗ = (s∗
i )i with s∗ i : Θi → Mi is a
behavioral dominant equilibrium of µ if for all i and all θi g(s∗
i (θi), ˜
m−i) ∈ C θi
i (Oµ i ( ˜
m−i)) for all ˜ m−i ∈ M−i. We observe that with independent choices: (i) Any dominant equilibrium of µ is an EPE of µ; and (ii) if s∗ is an EPE of µ s.t. for all ˜ m−i there is ˜ θ−i with s∗
−i(˜
θ−i) = ˜ m−i, i.e., s∗ satisfies a full-range condition, then s∗ is a dominant equilibrium of µ. Remark Given an SCF f , the truthtelling strategy profile is a behavioral dominant equilibrium of f ’s direct mechanism µf if and only if it is an EPE of µf .
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
A strategy profile s∗ = (s∗
i )i with s∗ i : Θi → Mi is a behavioral dominant
equilibrium with interdependent choices of µ = (M, g) if for all i and all θ g(s∗
i (θi), ˜
m−i) ∈ C θ
i (Oµ i ( ˜
m−i)) for all ˜ m−i ∈ M−i. s∗
i is measurable only with respect to Θi for all i implies
s∗ being a behavioral dominant equilibrium with interdependent choices is equivalent to for all i and all θi, g(s∗
i (θi), ˜
m−i) ∈ C
(θi ,˜ θ−i ) i
(Oµ
i ( ˜
m−i)) for all ˜ m−i ∈ M−i and all ˜ θ−i ∈ Θ−i. That is, s∗
i is s.t. for any θi, s∗ i (θi) leads to a chosen alternative from
each opportunity set of i generated by any one of others’ messages ˜ m−i and any one type profile of others ˜ θ−i at the state (θi, ˜ θ−i). Remark Behavioral dominant equilibrium with interdependent choices imposes excessively stringent requirements and this reduces its appeal.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
In a two-individual mechanism, any message of an individual can be thought of as an opportunity set generated for the other. Therefore,
the mechanism of our motivating example can be rewritten as:
Bob Ann {c, n} {c, s} {c, n, s} {c, n} n c n {c, s} c s c {n, s} n s s
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Our necessity result implies that the
two-individual consistent collections should be
SA = {SA(f , ρB), {c, n, s}, {c, s}} SB = {{c, n}, {n, s}, SB(f ′, ρA), SB(f ′, γA)}. Therefore, Ann must have at least 3 strategies in any mechanism that ex-post implements F —to be able to generate {c, n, s}! The best we can hope for Bob is to have 2 strategies —to be able to let Ann generate {c, n, s} and {c, s}!
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Let us try a 3 × 2 mechanism: Bob Ann {c, n, s} {c, s} {c, n} x c {n, s} y s {t, z} z t Since {x, y, z} = {c, n, s}, we must have x = y = z. But also, we must have x = n and y = n —for Bob to be able to generate {c, n} and {n, s} respectively. A contradiction!
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Remark (1) Given the individual choices of Ann and Bob, any mechanism that ex-post implements the SCS F must have at least three messages for Ann and the total number of message profiles must be at least nine. Bob Ann L M R U n c n M c s c D n s s
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The following mechanism also works: Bob Ann {c, n} {c, s} {c, n, s} {c} c c c {c, n} n c n {n, s} n s s
Back to Example SM-1 Barlo and Dalkıran Behavioral Implementation under Incomplete Information
There has been a recent interest in simple mechanisms in the literature e.g., Li (2017), Borgers and Li (2018). Limited cognitive abilities and/or behavioral biases increases the relevance and importance of the simplicity of mechanisms. “The question as to what constitutes a “simple” mechanism is a difficult and controversial one” (Dutta, Sen, & Vohra (1995)). Our measure of simplicity of a mechanism is the total number of its message profiles.
◮ Our measure parallels the total size of message spaces used to
analyze communication complexity in Segal (2007, 2010).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
There could be different collections of sets consistent with the same SCS —as is the case for our motivating example.
◮ How many sets there are in a collection, and how small these sets
are, turn out to be important when designing simple mechanisms. Let {Sγ}γ∈Γ be the set of all the collections of sets that satisfy consistency (or two-individual consistency).
◮ Sγ = (Sγ
i )i∈N for γ ∈ Γ where Sγ i = {Sγ i (f , θ−i)|f ∈ F, θ−i ∈ Θ−i}. Barlo and Dalkıran Behavioral Implementation under Incomplete Information
(i) If Sγ is the consistent collection, i must have at least as many messages as the cardinality of the maximal set in Sγ
i .
(ii) For each set in Sγ
i , there must exist a particular message profile of the
individuals other than i that let i to generate this set. If the collection Sγ happens to be the collection of opportunity sets generated by the mechanism that ex-post implements F, then
◮ by (i) and (ii), the total number of message profiles must be at
least maxi∈N[#Sγ
i × maxS∈Sγ
i #S],
◮ by (i), the total number of message profiles must be also more than
i∈N maxS∈Sγ
i #S.
(iii) Combining, the total number of message profiles must exceed max
i × maxS∈Sγ
i #S], minγ∈Γ
i #S
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Theorem (5) In any mechanism that ex-post implements the social choice set F, (i) the minimum number of messages required for individual i is minγ∈Γ maxS∈Sγ
i #S
minγ∈Γ maxS∈Sγ
i #S
minγ∈Γ maxS∈Sγ
i #S,
(ii) the minimum number of message profiles required for the individuals
i
minγ∈Γ #Sγ
i
minγ∈Γ #Sγ
i , and
(iii) the minimum number of total messages required for all individuals is
max
i × maxS∈Sγ
i #S], minγ∈Γ
i #S
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Whether or not behavioral aspects imply simpler mechanisms is an interesting question and needs a structure for being well-defined. The example we used when analyzing allocation problems with endowment effects provides such a structure:
Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(Z, h∗
3)
{I, II, III} III I I I II II I III {I, II} I I II II I I {I, III} III III III III I I I III {II, III} III II II II II III The behavioral-bias case: WARP fails. Z c1(Z, ♦1) c1(Z, h∗
1)
c2(Z, ♦2) c2(Z, h∗
2)
c3(Z, ♦3) c3(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: WARP holds.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Observations: Recall that
1
The difference between the behavioral-bias and no-behavioral-bias cases involves 1 of type h∗
1 facing {I, III}. 2
The only violation of WARP happens for 1 of type h∗
1 facing {I, II, III}
and {I, III}.
3
In the behavioral-bias case, every consistent collection must be such that S1(f , h∗
−1) = {I, II, III} but not {I, III}: 1 must be offered her initial
endowment to make her choose “consistent” with f .
4
In the behavioral-bias case,
the direct mechanism does not ex-post
implement f , but
the indirect mechanism does. 5
In the no-behavioral-bias case, both the direct and indirect mechanisms ex-post implement the SCF f .
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Observations: (continued)
6
In the behavioral-bias case, the simplest mechanism that ex-post implements f is
the indirect mechanism . 7
Individual 2’s behavior is independent of her type, so let Θ2 = {♦2}.
8
Then, the following direct mechanism of f is the simplest mechanism that ex-post implements f in the no-behavioral-bias case:
h∗
3
♦3 h∗
1
(I, II, III) (III, II, I) ♦1 (III, II, I) (III, II, I)
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Conclusions:
1
Whether or not the mechanism offers individual 1 her initial endowment as an option to ensure the consistency of her choices with f is crucial: She needs to switch her choices between her types ♦1 and h∗
1 (as called
for by f ) in the behavioral-bias case.
2
Thus, in the behavioral-bias case, she must have an additional message resulting in her initial endowment.
3
Therefore, the simplest mechanism in the behavioral-bias case is less simple than the simplest mechanism in the no-behavioral-bias case.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We investigate behavioral ex-post implementation under incomplete information with choices that need not satisfy rationality. We provide necessary as well as sufficient conditions. To display the applicability of our results in economically relevant domains, we analyze the behavioral ex-post implementation of efficiency and allocation problems with endowment effects. We present characterizations of situations where ex-post implementation is achievable only when it is attainable via direct mechanisms. We consider the size of the joint message space, the number of message profiles, of a mechanism as a measure of its simplicity and supply lower bounds on simplicity of mechanisms that ex-post implement an SCS.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Definition SA = {SA(f , θB)|f ∈ F, θB ∈ ΘB} and SB = {SB(f , θA)|f ∈ F, θA ∈ ΘA} are two-individual consistent with the SCS F under incomplete info if (i) for all f ∈ F, f (θ′
A, θB) ∈ C (θ′
A,θB)
A
(SA(f , θB)) for each θ′
A ∈ ΘA,
(ii) for all f ∈ F, f (θA, θ′
B) ∈ C (θA,θ′
B)
B
(SB(f , θA)) for each θ′
B ∈ ΘB,
(iii) for all f , f ′ ∈ F, SA(f , θB) ∩ SB(f ′, θA) = ∅ for each θA ∈ ΘA and θB ∈ ΘB, (iv) for all f ∈ F, if f ◦ α / ∈ F, then there exists θ∗ = (θ∗
A, θ∗ B) ∈ Θ s.t.
f (α(θ∗)) / ∈ C θ∗
A (SA(f , αB(θ∗ B))) or f (α(θ∗)) /
∈ C θ∗
B (SB(f , αA(θ∗ A))).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Theorem (B.1) Let n = 2. If an SCS F is ex-post implementable, then there exist collections of sets S1 and S2 that are two-individual consistent with F under incomplete information.
Necessity Sufficiency with Three or More Individuals Simple Mechanisms Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Let’s investigate the collections that satisfy two-individual consistency for the SCS F = f , f ′ in our motivating example: SA = {SA(f , ρB), SA(f , γB), SA(f ′, ρB), SA(f ′, γB)}, SB = {SB(f , ρA), SB(f , γA), SB(f ′, ρA), SB(f ′, γA)}.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
C (ρA,ρB )
A
C (ρA,ρB )
B
C (ρA,γB )
A
C (ρA,γB )
B
C (γA,ρB )
A
C (γA,ρB )
B
C (γA,γB )
A
C (γA,γB )
B
{c, n, s} {n} {s} {n} {n} {n} {c} {c, s} {n, s} {c, n} {n} {n} {n} {n} {n} {c} {n} {c} {c, s} {c, s} {s} {s} {s} {c} {c} {c} {s} {n, s} {n} {s} {s} {s} {n, s} {n, s} {s} {s}
State: (ρA, ρB) (ρA, γB) (γA, ρB) (γA, γB) SCR: BR-Optimal {n , s} {n , s} {c , n} {c , s} f n n n s f′ s s c c
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
(i) of two-individual consistency implies (for Ann): SA(f , ρB): f (ρA, ρB) = n and f (γA, ρB) = n imply n ∈ C (ρA,ρB )
A
(SA(f , ρB)) and n ∈ C (γA,ρB )
A
(SA(f , ρB)). There are four such sets: {c, n, s}, {c, n}, {n, s}, {n}. SA(f , γB): f (ρA, γB) = n and f (γA, γB) = s imply n ∈ C (ρA,γB )
A
(SA(f , γB)) and s ∈ C (γA,γB )
A
(SA(f , ρB)). There is only one such set: {c, n, s}. SA(f ′, ρB): f ′(ρA, ρB) = s and f ′(γA, ρB) = c imply s ∈ C (ρA,ρB )
A
(SA(f ′, ρB)) and c ∈ C (γA,ρB )
A
(SA(f ′, ρB)). There is only one such set as well: {c, s}. SA(f ′, γB): f ′(ρA, γB) = s and f ′(γA, γB) = c imply s ∈ C (ρA,γB )
A
(SA(f ′, ρB)) and c ∈ C (γA,γB )
A
(SA(f ′, ρB)). There is again only
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
(ii) of two-individual consistency implies (for Bob): SB(f , ρA): f (ρA, ρB) = n and f (ρA, γB) = n imply n ∈ C (ρA,ρB )
B
(SB(f , ρA)) and n ∈ C (ρA,γB )
B
(SB(f , ρA)). There are two such sets: {c, n} and {n}. SB(f , γA): f (γA, ρB) = n and f (γA, γB) = s imply n ∈ C (γA,ρB )
B
(SB(f , γA)) and s ∈ C (γA,γB )
B
(SB(f , γA)). There is only one such set: {n, s}. SB(f ′, ρA): f ′(ρA, ρB) = s and f ′(ρA, γB) = s imply s ∈ C (ρA,ρB )
B
(SB(f ′, ρA)) and s ∈ C (ρA,γB )
B
(SB(f ′, ρA)). There are three such sets {c, s} and {n, s} and {s}. SB(f ′, γA): f ′(γA, ρB) = c and f ′(γA, γB) = c imply c ∈ C (γA,ρB )
B
(SB(f ′, γA)) and c ∈ C (γA,γB )
B
(SB(f ′, γA)). There are two such sets {c, n} and {c}.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
(iii) of two-individual consistency: SA(f , θB) ∩ SB(f ′, θA) = ∅ and SA(f ′, θB) ∩ SB(f , θA) = ∅ for each θA ∈ {ρA, γA} and θB ∈ {ρB, γB}. In particular, SA(f ′, ρB) ∩ SB(f , ρA) = ∅ = ⇒ SB(f , ρA) = {c, n}. Two-individual consistency identifies 5 out of 8 sets: SA(f , γB) = {c, n, s}; SA(f ′, ρB) = {c, s}; SA(f ′, γB) = {c, s}, SB(f , γA) = {n, s}; SB(f , ρA) = {c, n}. Thus,
◮ SA = {SA(f , ρB), {c, n, s}, {c, s}}, and
◮ SB = {{c, n}, {n, s}, SB(f ′, ρA), SB(f ′, γA)}. In Supplementary Materials, we provide
Python codes computing two-individual
consistent collections taking individuals’ choices and the SCS as inputs. Using these codes in our motivating example, relabeled as
Example SM-1 , we
Necessity Sufficiency with Three or More Individuals Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Definition F involves choice incompatibility among a set of alternatives S ∈ X and collections of sets S1 := {S1(f , θ2)|f ∈ F, θ2 ∈ Θ2} ⊂ X and S2 := {S2(f , θ1)|f ∈ F, θ1 ∈ Θ1} ⊂ X at θ if (i) x ∈ C θ
i (S) implies x /
∈ C θ
j (S), i = j;
(ii) for any T ∈ Si, x ∈ C θ
i (T) implies x /
∈ C θ
j (S), i = 1, 2 and i = j;
(iii) for any deception profile α and any f , f ′ ∈ F with f = f ′, x ∈ C θ
i (Si(f , αj(θj))) implies x /
∈ C θ
j (Sj(f ′, αi(θi))), i = 1, 2, i = j.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
F involves choice incompatibility among a non-empty set of alternatives S and non-empty collections of sets S1 and S2 at state θ means that the individual choices at θ are not aligned when
with i = j; and
with a particular SCF f and the other individual, j, makes a choice from a set in Sj which is associated with a different SCF f ′ = f while f , f ′ ∈ F and i, j = 1, 2 with i = j. That is, choice incompatibility conditions require that there is sufficient disagreement between the two individuals at a given state of the world.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Theorem (B.2) Let n = 2. If F is an SCS for which there exist (i) S1 := {S1(f , θ2)|f ∈ F, θ2 ∈ Θ2}, S2 := {S2(f , θ1)|f ∈ F, θ1 ∈ Θ1} that are two-individual consistent with F under incomplete info, (ii) a set of alternatives ¯ X ⊆ X with
S∈ S1∪ S2 S ⊆ ¯
X s.t F involves choice incompatibility among ¯ X and S1 and S2 at every θ ∈ Θ, then F is ex-post implementable.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
When there are two individuals, F is ex-post implementable whenever
(i) there exist collections of sets S1 and S2 that are two-individual
consistent with F under incomplete information, and (ii) there exists a set of alternatives ¯ X that contains every alternative in S1 and S2; and choice incompatibility among ¯ X and S1 and S2 hold at every state of the world. In Supplementary Materials, we provide
Python codes computing
two-individual consistent collections S and ¯ X satisfying choice incompatibility taking individuals’ choices and the SCS as inputs. In
Example SM-3 , we show that there are collections S and ¯
X satisfying choice incompatibility.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Definition F respects choice unanimity on a non-empty set of alternatives S ∈ X and non-empty collections of sets S1 := {S1(f , θ2)|f ∈ F, θ2 ∈ Θ2} ⊂ X and S2 := {S2(f , θ1)|f ∈ F, θ1 ∈ Θ1} ⊂ X at θ if there exists f ∗ ∈ F s.t. (i) x ∈ C θ
1 (S) ∩ C θ 2 (S) implies f ∗(θ) = x; and
(ii) for any T ∈ Si, x ∈ C θ
i (T) ∩ C θ j (S) implies f ∗(θ) = x, for i, j = 1, 2
with i = j; and (iii) for any deception profile α and f , f ′ ∈ F with f = f ′, x ∈ C θ
i (Si(f , αj(θj))) ∩ C θ j (Sj(f ′, αi(θi))) implies f ∗(θ) = x.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Definition Let n = 2. F satisfies the consistency-unanimity whenever there exist S1 := {S1(f , θ2)|f ∈ F, θ2 ∈ Θ2}, S2 := {S2(f , θ1)|f ∈ F, θ1 ∈ Θ1} s.t. (i/ii) for all f ∈ F, f (θ′
i , θj) ∈ C (θ′
i ,θj )
i
(Si(f , θj)) for each θ′
i ∈ Θi,
(iii) for all f , f ′ ∈ F, S1(f , θ2) ∩ S2(f ′, θ1) = ∅ for each θ1 ∈ Θ1 and θ2 ∈ Θ2, and there is a set of alternatives ¯ X ⊆ X with
S∈S1∪ S2 S ⊆ ¯
X such that for any collection of product sets {¯ Θf }f ∈F with ¯ Θ =
f ∈F ¯
Θf ⊂ Θ, (iv) F respects choice unanimity on ¯ X and S1 and S2 at every θ ∈ Θ \ ¯ Θ, and (v) for all f ∈ F and deception profile α, if f (α(θ)) = f ∗(θ) for some θ ∈ ¯ Θf where f ∗ is the SCF that satisfies (i)–(iii) of choice unanimity, then there exists θ∗ ∈ ¯ Θf such that either f (α(θ∗)) / ∈ C θ∗
1 (S1(f , α2(θ∗ 2 ))) or
f (α(θ∗)) / ∈ C θ∗
2 (S2(f , α1(θ∗ 1 ))). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Theorem (B.3) Let n = 2. If an SCS F satisfies the consistency-unanimity property, then F is ex-post implementable.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Given a non-empty SCS F, there exist collections of sets Si (i = 1, 2) and a set of alternatives ¯ X which contains every alternative in S1 ∪ S2 such that the following hold: ◮ when i’s type is θ′
i his choice from Si(f , θj) at state (θ′ i, θj)
contains f (θ′
i, θj) for all θ′ i ∈ Θi, with j = 1, 2 and i = j;
◮ any set in S1 must have a common element with any set in S2; ◮ for any collection of product sets of states {¯ Θf }f ∈F with ¯ Θ =
f ∈F ¯
Θf ⊂ Θ, there is an SCF f ∗ in F s.t.
F respects choice unanimity on ¯ X and S1 and S2 whenever θ ∈ Θ \ ¯ Θ; and for any deception profile α and SCF f ∈ F that lead to an
Θf , there exists a whistle-blower i∗ ∈ {1, 2} and an informant state θ∗ ∈ ¯ Θf such that i∗ does not choose f (α(θ∗)) from Si∗(f , αj(θ∗
j )) at
θ∗ where j ∈ {1, 2} and j = i∗.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
With two individuals, the Python codes we provide in Supplementary Materials compute
◮ the collection S,
◮ the set of alternatives ¯ X, ◮ the collection of product sets {¯ Θf }f ∈F, ◮ SCF’s f ∗ ∈ F satisfying consistency-unanimity taking choices and the SCS as inputs. In
Example SM-3 of Supplementary Materials, we show that there are
collections S together with ¯ X, ¯ Θ, and f ∗ satisfying consistency-unanimity. Our codes identify S, ¯ X, {¯ Θf }f , and f ∗’s associated with all of the collections satisfying consistency-no-veto.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
µ = (M, g): Mi = {0, 1} × F × Θi × ¯ Si × {0, 1} and g : M → X is:
Rule 1 : g(m) = f (θ) if mi = (0, f , θi, ·, ·) for both i ∈ {1, 2}, Rule 2.1 : g(m) =
if x1 ∈ S1(f2, θ2) ¯ x(1, f2, θ2)
if m1 = (1, f1, θ1, x1, ·) and m2 = (0, f2, θ2, x2, ·), Rule 2.2 : g(m) =
if x2 ∈ S2(f1, θ1) ¯ x(2, f1, θ1)
if m1 = (0, f1, θ1, x1, ·) and m2 = (1, f2, θ2, x2, ·), Rule 3 : g(m) = ¯ x(f1, f2, θ1, θ2) if m1 = (0, f1, θ1, x1, k1) and m2 = (0, f2, θ2, x2, k2) with f1 = f2, Rule 4 : g(m) = xj if m1 = (1, f1, θ1, x1, k1) and m2 = (1, f2, θ2, x2, k2) with j =
if k1 + k2 is odd, 2 if k1 + k2 is even.
where ¯ x(j, f , θ−j) ∈ Sj(f , θ−j) and ¯ x(f , f ′, θ1, θ2) ∈ S1(f , θ2) ∩ S2(f ′, θ1).
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Corollary (B.1) Let n = 2. An SCF f : Θ → X is ex-post implementable whenever there are collections Si := {Si(f , θj)|θj ∈ Θj} ⊂ X with i, j = 1, 2, i = j s.t. (i) f (θ′
1, θ2) ∈ C (θ′
1,θ2)
1
(S1(f , θ2)) for each θ′
1 ∈ Θ1, and
f (θ1, θ′
2) ∈ C (θ1,θ′
2)
2
(S2(f , θ1)) for each θ′
2 ∈ Θ2, and
S1(f , θ2) ∩ S2(f , θ1) = ∅ for each θ1 ∈ Θ1 and θ2 ∈ Θ2, and there is a set of alternatives ¯ X ⊆ X with
S∈S1∪ S2 S ⊆ ¯
X such that for any product set ¯ Θ ⊆ Θ, (ii) x ∈ C θ
1 ( ¯
X) ∩ C θ
2 ( ¯
X) implies f (θ) = x, x ∈ C θ
1 (T) ∩ C θ 2 ( ¯
X) with T ∈ S1 implies f (θ) = x, and x ∈ C θ
1 ( ¯
X) ∩ C θ
2 (T ′) with T ′ ∈ S2 implies
f (θ) = x, for each θ ∈ Θ \ ¯ Θ, and (iii) for any deception profile α, if f (α(θ)) = f (θ) for some θ ∈ ¯ Θ, then there exists θ∗ ∈ ¯ Θ such that either f (α(θ∗)) / ∈ C θ∗
1 (S1(f , α2(θ∗ 2 ))) or
f (α(θ∗)) / ∈ C θ∗
2 (S2(f , α1(θ∗ 1 ))). Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The following example is inspired from Masatlioglu and Ok (2014), which presents a “model of individual decision making when the endowment of an agent provides a reference point that may influence her choices” Θ = {(♦, ♦), (♦, c), (c, ♦), (c, c)}.
◮ ♦ stands for not having a status-quo,
◮ c stands for status-quo being coal. Individual choices of Ann and Bob from (the subsets) of X = {c, n, s} are: S C (♦,♦)
A
C (♦,♦)
B
C (♦,c)
A
C (♦,c)
B
C (c,♦)
A
C (c,♦)
B
C (c,c)
A
C (c,c)
B
{c, n, s} {s} {s} {s} {n} {n} {s} {n} {n} {c, n} {n} {n} {n} {n} {n} {n} {n} {n} {c, s} {s} {s} {s} {c} {c} {s} {c} {c} {n, s} {s} {s} {s} {s} {s} {s} {n} {n}
Back to Python Codes Barlo and Dalkıran Behavioral Implementation under Incomplete Information
A social planner wants to ex-post implement the SCF f , a particular selection from the BR-optimal outcomes, described below: State (♦, ♦) (♦, c) (c, ♦) (c, c) BR-optimal {s} {n, s} {n, s} {n} f f f s s s n The social planner breaks the tie in favor of s whenever n and s are both BR-optimal.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The SCF f satisfies the consistency-unanimity property since the collections and SA := {SA(f , ♦), SA(f , c)} and SB := {SB(f , ♦), SB(f , c)} such that ◮ SA(f , ♦) = {n, s} and SA(f , c) = {c, n, s}, ◮ SB(f , ♦) = {n, s} and SB(f , c) = {c, n, s}, and ¯ X = {c, n, s} satisfy conditions (i), (ii), and (iii) of the Corollary. ∴ f is ex-post implementable.
Necessity Sufficiency with Three or More Individuals Efficiency Back to Example SM-2 Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We provide Python codes associated with our necessity and sufficiency results: Using the SCS and individuals’ choices as inputs, these codes deliver consistent collections; and consistent collections satisfying the choice incompatible pair property and hence the hypothesis of our sufficiency result, Theorem 3; and collections of sets satisfying consistency-no-veto and hence the hypothesis
We supply Python codes for the case of two-individuals that compute
◮ two-individual consistent collections,
◮ two-individual consistent collections satisfying choice incompatibility (Theorem B.2), and ◮ collections satisfying consistency-unanimity (Theorem B.3).
Back to Consistency Back to Theorem 2 Back to Theorem 3 Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We provide examples to display computations with our Python codes. Example SM-1 is our motivating example. Example SM-2 is
the two-individual application of Corollary B.1 , which has a
consistent collection of sets satisfying consistency-unanimity but not choice incompatibility. Example SM-3 is a two-individual example, in which there are consistent collections satisfying choice incompatibility and consistency-unanimity. Example SM-4 has three individuals and there are consistent collections satisfying both choice incompatible pair property and consistency-no-veto. Example SM-5 is an example with three rational individuals and shows that the choice incompatible pair property imposes a weaker sufficiency requirement than the one the economic environment assumption imposes.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Please refer to “Example-SM1-The-Motivating-Example-summary.xlsx” provided in “SM-BHIMP.zip”.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
There are 14 two-individual consistent collections with F. The collection # 9 is the two-individual consistent collection associated with
the mechanism of our motivating example in our paper.
The collection # 10 is the two-individual consistent collection associated with
the other simplest mechanism of our motivating example .
None of the 14 two-individual consistent collections satisfies choice incompatibility, and there is no collection satisfying choice-unanimity. This shows that our sufficiency conditions are not necessary in general.
Back to Implications of Two-Individual Consistency Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Please refer to “Example-SM2(two-individuals)-summary.xlsx” provided in “SM-BHIMP.zip”.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
There are 12 two-individual consistent collections with f . The collection # 7 is the two-individual consistent collection analyzed
in conjunction with the application of Corollary B.1 .
All of the 12 two-individual consistent collections (together with the appropriate ¯ X) satisfy consistency-unanimity. Therefore, the hypothesis of our two-individual sufficiency result involving consistency-unanimity, Theorem B.3, is satisfied in this example. None of the 12 two-individual consistent collections satisfies choice incompatibility.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Please refer to “Example-SM3-summary.xlsx” provided in “SM-BHIMP.zip”.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
There are 12 two-individual consistent collections with f . The collections # 4 and # 12 are two-individual consistent collections satisfying both choice incompatibility and consistency-unanimity. In this example, the hypotheses of both our two-individual sufficiency results, Theorems B.2 and B.3, are satisfied. The other two-individual consistent collections do not satisfy choice incompatibility. Therefore, using collections # 4 or # 12 when defining our two-individual mechanism results in EPE only under Rule 1.
Back to Choice Incompatibility Back to Consistency-Unanimity Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Please refer to “Example-SM4-summary.xlsx” provided in “SM-BHIMP.zip”.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
This is an example with four alternatives, x, y, z, t, and three individuals and there are 4560 consistent collections with f . The consistent collection # 1441 satisfies consistency-no-veto but not choice incompatible pair property with ¯ X = {x, y}. But it satisfies both with ¯ X = {x, y, z}. As a result, in this example, the hypotheses of both our sufficiency results, Theorems 2 and 3, are satisfied. We also provide ¯ X, the corresponding product sets of states, ¯ Θ, 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.
Back to Consistency-No-Veto Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Please refer to “Example-SM5-summary.xlsx” provided in “SM-BHIMP.zip”.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
This is an example with four alternatives x, y, z, t, and three rational individuals and there are 16 consistent collections with f . In this example, the environment is not economic as alternative t is the top-ranked alternative for every individual at every state. All the 16 consistent collections along with ¯ X = {x, y, z} satisfy the choice incompatible pair property at every state. The choice incompatible pair property imposes a weaker sufficiency requirement than the one the economic environment assumption imposes. Remark Our Theorem 2 extends Theorem 2 of Bergemann and Morris (2008), their sufficiency result with the economic environment assumption, even in the rational domain.
Back to Theorem 2 Barlo and Dalkıran Behavioral Implementation under Incomplete Information
The welfare of individuals (whose choices are not necessarily derived from maximization of preferences) ` a la Bernheim and Rangel (2009) is: “An alternative x is strictly unambiguously chosen over another alternative z iff z is never chosen whenever x is available.”
“An alternative x is weakly unambiguously chosen over another alternative z iff whenever they are both available, z is never chosen unless x is chosen as well.”
both available.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information
Generalized Pareto Optimality ` a la Bernheim and Rangel (2009): An alternative x is strictly generalized Pareto-optimal if there does not exists y such that y is weakly unambiguously chosen over x for every agent and y is strictly unambiguously chosen over x for some agent. We refer to strictly generalized Pareto-optimal alternatives as BR-optimal alternatives.
Back to the Motivating Example Barlo and Dalkıran Behavioral Implementation under Incomplete Information
We investigate behavioral ex-post implementation under incomplete information with choices that need not satisfy rationality. We provide necessary as well as sufficient conditions. To display the applicability of our results in economically relevant domains, we analyze the behavioral ex-post implementation of efficiency and allocation problems with endowment effects. We present characterizations of situations where ex-post implementation is achievable only when it is attainable via direct mechanisms. We consider the size of the joint message space, the number of message profiles, of a mechanism as a measure of its simplicity and supply lower bounds on simplicity of mechanisms that ex-post implement an SCS.
Barlo and Dalkıran Behavioral Implementation under Incomplete Information