Uncertainty in Mechanism Design Giuseppe Lopomo Luca Rigotti Chris - - PowerPoint PPT Presentation

Uncertainty in Mechanism Design

Giuseppe Lopomo

Duke

Luca Rigotti

Pitt (visiting Duke)

Chris Shannon

UC Berkeley

Notre Dame, 2017 March 29

Introduction

We impose a robustness requirement on standard mechanism design theory. For us robustness means mechanisms must allow misspecification of beliefs.

This is relatively simple to do.

Our approach differs from the one taken by the existing literature on robustness in mechanism design a-la Bergemann & Morris (2005).

They think of robustness as stemming from strategic considerations about the game agents play. Robustness allows players to make all possible conjectures about preferences and information (the type) of other players.

Although it can be related to this “higher order beliefs”, universal type space, vision of robustness, our approach, tools, and results are different. Our results help to shed light on some features of that literature.

Introduction II

Robustness The Principal chooses mechanisms that must provide incentives even if the agent beliefs are misspecified. This is done in a simple, almost trivial, way: Interim incentive compatibility must hold for sets of probability distributions

Results

Intuitively: as the size of these sets grows, robustness becomes more demanding; eventually, one gets ex post (dominant strategy) implementation. Surprisingly: robustness can be hard to achieve even when these sets are arbitrarily small; there is some discontinuity in the set of incentive compatible mechanisms. This simple notion of robustness yields results somewhat close to the more complicated version. More detailed discussion of the main result later, when we can all see precisely what it says. Now onto a description of the model.

Single Agent and State-Dependent Mechanism

The Setup in Words One agent with unobservable type. A (direct) mechanism determines something the agent cares about. The mechanism’s outcome depends on the agent’s report and the realization

• f an exogenous random variable.

The mechanism must be robust to possible errors in the specification of the agent’s beliefs about this random variable.

Main question

What does it mean to be incentive compatible in this environment?

Why this setup? Makes the driving forces as transparent as possible

Simple environment where robustness with respect to agents’ beliefs has a role. Single agent: robustness can be introduced without strategic interactions.

Easily generalizes to many agents.

We can talk about robustness without higher order beliefs.

Yet, connecting to the higher order beliefs literature is easy.

Monopolistic Screening

One object, one buyer whose valuation of the object is unknown to the seller The seller designs a mechanism to transfer the object for some payment. The buyer’s value for the object equals her type t ∈ T = [0, 1]. A direct mechanism is a function of the buyer’s reported type θ ∈ T, as well as an uncertain state of the world s ∈ S, that has two parts:

the allocation rule q : T × S → [0, 1] gives the probability the buyer gets the

• bject as a function of her report and the realized state.

the payment scheme p : T × S → R states how much the buyer pays as a function of her report and the realized state.

S is a compact metric space, and the realized s is publicly observable; typically, think of it as [0, 1]. When the buyer reports θ while her true type is t,

her utility in state s is: tq(θ, s) − p(θ, s) her expected utility, given some π ∈ ∆(S) (∆(·) denotes the simplex) is: Eπ[tq(θ, s) − p(θ, s)]

One should include mixed strategies (σ(t) ∈ ∆(T )), and then take expectations with respect to σ; I ignore that in the talk for simplicity.

This is (almost) a textbook setting: for many agents, set Si ⊃ ×j=iTj = T−i.

Interim Incentive Compatibility

Mechanism Design Refresher

Definition

A mechanism q(·), p(·) is interim incentive compatible if for each t, θ ∈ T Eπ(t)[tq(t, s) − p(t, s)]

• Expected utility of reporting true type

≥ Eπ(t)[tq(θ, s) − p(θ, s)]

• Expected utility of reporting another type

With many agents, this is standard Bayesian interim incentive compatibility. In this setting, beliefs are allowed to depend on the type of the agent (this is called correlation in the mechanism design literature).

The independence assumption rules that out by imposing the restriction π(t) = π for all t.

The relationship between π(·) and t is known to the designer: if he knew the agent’s type, he would also know her beliefs.

Interim Incentive Compatibility

Mechanism Design Refresher

Definition

A mechanism q(·), p(·) is interim incentive compatible if for each t, θ ∈ T Eπ(t)[tq(t, s) − p(t, s)]

• Expected utility of reporting true type

≥ Eπ(t)[tq(θ, s) − p(θ, s)]

• Expected utility of reporting another type

Standard result in mechanism design

Under independence (π(t) = π for all t), a mechanism is interim incentive compatible if and only if it is interim monotone and satisfies the interim envelope condition.

Definitions Reminder

A mechanism q(·), p(·) is interim monotone if Eπq(t, s) is increasing in t; satisfies the interim envelope condition if for each t, t ∈ T: Eπ

• tq(t, s) − p(t, s) − [tq(t, s) − p(t, s)] −

t

t q(τ, s)dτ

• = 0

Our Notion of Robust Incentive Compatibility

Definition

A mechanism q(·), p(·) is optimal incentive compatible if for each t, θ ∈ T Eπ[tq(t, s) − p(t, s)] ≥ Eπ[tq(θ, s) − p(θ, s)] for all π ∈ Π(t)

Optimal incentive compatibility and robustness

Incentives must be robust to (possibly small) errors in beliefs’ specification. The ‘size’ of Π(t) measures the difficulty of implementing truthful reporting:

when Π(t) is a singleton, this is standard interim incentive compatibility; when Π(t) is a small set, this is close to the standard case; as the ‘size’ of Π(t) grows incentive compatibility is harder to satisfy.

For lack of a better word, we call Π(t) the ‘beliefs’ of type t. As usual, the relationship between Π(·) and t is known to the designer, but she does not know the agent’s type.

We can have ‘independence’ by assuming Π(t) = Π for all t ∈ T . Decision theoretic justification for optimal incentive compatibility

Π(t) describes the agent’s preferences: Knightian uncertainty about S.

Optimal Incentive Compatibility: An Example

Example: The ε-contamination model

For a fixed ε ∈ (0, 1) and ¯ π ∈ ∆(S), define the set Πε as follows: Πε := {π ∈ ∆(S) : π = (1 − ε) ¯ π + ε ˜ π for some ˜ π ∈ ∆(S)}. Define the collection {Πε(t) : t ∈ T} as follows Πε(t) := {π ∈ ∆(S) : π = (1 − ε) ¯ π(t) + ε ˜ π for some ˜ π ∈ ∆(S)}

Πε(t) can be arbitrarily close to a singleton for each t. An example of optimal incentive compatibility

A mechanism q(·), p(·) is optimal incentive compatible if for each t, θ ∈ T (1 − ε)E ¯

π(t)[tq(t, s) − p(t, s)]

+ εE ˜

π[tq(t, s) − p(t, s)]

≥ (1 − ε)E ¯

π(t)[tq(θ, s) − p(θ, s)]

+ εE ˜

π[tq(θ, s) − p(θ, s)]

∀ ˜ π ∈ ∆(S)

Many Agents Notation and Definitions

Many buyers means a Bayesian game To allow for many agents one can, for example, set Si = ×j=iTj = T−i. The mechanism then specifies qi(·), pi(·) for all players in the game. The set of ‘beliefs’ is Π(ti) ⊂ ∆(T−i).

Definition

A mechanism qi(·), pi(·) is optimal incentive compatible if, for each player i, and for each ti, θ ∈ Ti Eπ[tiqi(ti, t−i) − pi(ti, t−i)]

• Expected utility of reporting true type

given truth telling from other players

≥ Eπ[tiqi(θ, t−i) − pi(θ, t−i)]

• Expected utility of reporting another type

given truth telling from other players for all

π ∈ Π(ti)

Optimal incentive compatibiity and equilibrium

Reporting truthfully must be a Bayesian Nash equilibrium for any π ∈ Π(ti). Equilibrium must be robust to (possibly small) misspecification of beliefs. From now on, back to a single agent.

Ex Post Incentive Compatibility

Mechanism Design Refresher II

Definition

A mechanism q(·), p(·) is ex post incentive compatible if for each t, θ ∈ T tq(t, s) − p(t, s) ≥ tq(θ, s) − p(θ, s) for each s Incentive compatibility must hold in each state. Deviations are not profitable even after the state is known. With many agents, this is dominant strategy incentive compatibility. Clearly, any ex post incentive compatibly mechanism is also interim incentive compatible and optimal incentive compatible.

If the inequality holds in every state it also holds in expectation.

Ex Post Incentive Compatibility

Mechanism Design Refresher II

Definition

A mechanism q(·), p(·) is ex post incentive compatible if for each t, θ ∈ T tq(t, s) − p(t, s) ≥ tq(θ, s) − p(θ, s) for each s

Standard result in mechanism design

A mechanism is ex post incentive compatible if and only if it is ex post monotone and satisfies the ex post envelope condition.

Reminder:

A mechanism q(·), p(·) is ex post monotone if q(t, s) is increasing in t for each s; satisfies the ex post envelope condition if for each t, t ∈ T: tq(t, s) − p(t, s) − [tq(t, s) − p(t, s)] =

t

t q(τ, s)dτ

for each s Here, the envelope condition reduces to revenue equivalence (ex post).

Optimal Incentive Compatibility

Back to our environment.

Definition

A mechanism q(·), p(·) is optimal incentive compatible if for each t, θ ∈ T Eπ[tq(t, s) − p(t, s)] ≥ Eπ[tq(θ, s) − p(θ, s)] for all π ∈ Π(t).

Question

What can one say about the set of optimal incentive compatible mechanisms?

Observations

An ex post incentive compatible mechanism is also interim and optimal. An optimal incentive compatible mechanism is also interim for any collection {π(t) : t ∈ T} that satisfies the condition π(t) ∈ Π(t) for each t. Optimal incentive compatibility falls somewhere in between interim and ex-post incentive compatibility.

Optimal Incentive Compatibility

Definition

A mechanism q(·), p(·) is optimal incentive compatible if for each t, θ ∈ T Eπ[tq(t, s) − p(t, s)] ≥ Eπ[tq(θ, s) − p(θ, s)] for all π ∈ Π(t).

Main Question

How does the set of optimal incentive compatible mechanisms depend on Π(t)?

Easy Results

If Π(t) is a singleton for each t ∈ T, optimal equals interim:

by definition.

If Π(t) is the simplex for each t ∈ T, optimal equals ex post:

For each state in S, there is a π ∈ ∆(S) that assigns probability 1 to that state.

This is (the modern version of) an old result (Ledyard, 1979).

What about cases in between? The answer depends on what Π(t) looks like.

Next: restrictions on Π(t) that yield our main result.

Aside: this framework lets one nicely go from interim to ex post.

Full Dimensionality

Definition

A set Π ⊂ ∆(S) has full dimension if, given any continuous function g : S → R,

• S g (s) dπ
• Eπ[g (s)]

= 0 ∀π ∈ Π ⇒ g = 0. Fully dimensional sets can be small.

Example: The ε-contamination model

For a fixed ε ∈ (0, 1) and ¯ π ∈ ∆(S), the set Πε := {π ∈ ∆(S) : π = (1 − ε) ¯ π + ε ˜ π for some ˜ π ∈ ∆(S)}. has full dimension for any ε ∈ (0, 1).

Full Dimensionality and Interior

A geometric characterization of full dimensionality

Definition

A set Π ⊂ ∆(S) has full dimension if, given any continuous function g : S → R,

• S g (s) dπ = 0

∀π ∈ Π ⇒ g = 0.

Characterization

A convex set Π ⊂ ∆(S) has full dimension whenever its algebraic interior {π ∈ Π : ∀ ˜ π ∈ ∆(S) there exists δ ∈ (0, 1] such that (1 − δ)π + δ ˜ π ∈ Π}. is non-empty. If Π has non-empty relative interior then it has full dimension.

Main Assumption: Fully Overlapping Beliefs

Our main assumption relates the set Π(t) with the sets of types close to t. This is done by placing restrictions on the entire collection of sets.

Definition

The collection {Π(t) : t ∈ T} is fully overlapping if for each t ∈ T there exists a neighborhood N(t) ⊂ T such that

• t∈N(t)

Π(t) has full dimension. Nearby types share a sufficiently rich (in the sense of full dimension) set of beliefs. Under independence, Π(t) = Π for all t, this simplifies to full dimension. With correlated types, Π(t) is not the same for all t, one can think of this as a local form of independence paired with full dimension.

A set that has full dimension cannot be empty.

Fully Overlapping Beliefs

Definition

The collection {Π(t) : t ∈ T} is fully overlapping if for each t ∈ T there exists a neighborhood N(t) ⊂ T such that

• t∈N(t)

Π(t) has full dimension. Another way to think about this restriction is in terms of the correspondence that assigns a set of probability distributions to each type.

Theorem

The correspondence Π : T → 2∆(S) that maps types to sets of probability distributions is fully overlapping if it is lower hemi-continuous and Π(t) has non-empty relative interior for each t ∈ T.

Example

Suppose the mapping t → ( ¯ π(t), ε(t)) is continuous and let Πε(t)(t) := {π ∈ ∆(S) : π = (1 − ε(t)) ¯ π(t) + ε(t) ˜ π for some ˜ π ∈ ∆(S)} Then {Πε(t)(t) : t ∈ T} is fully overlapping. Π(t) can be arbitrarily close to a singleton for each t, yet fully overlapping.

Main Result

Robustness can be restrictive

Theorem

Assume {Π(t) : t ∈ T} is fully overlapping. Then, any optimal incentive compatible mechanism satisfies the ex post envelope condition. The proof has two main ingredients: the envelope theorem (so that one replaces the incentive compatibility inequalities with their ‘first order conditions’ equalities) and fully overlapping beliefs. Continuum of types (for the envelope theorem) and fully overlapping beliefs are crucial. There is a hidden assumption that is trivially satisfied in this simple example (smoothness of agent’s utility function). With that assumption, the result generalizes to any mechanism design setting in which the envelope theorem applies.

Proof of Main Result (Sketch)

Proof.

Optimal incentive compatibility can be written as:

• S [tq(t, s) − p(t, s) − tq(t, s) + p(t, s)]dπ
• Eπ[tq(t,s)−p(t,s)−[tq(t,s)−p(t,s)]]

≥ 0 ∀π ∈ Π (t)

1

By the envelope theorem: for any t ∈ N (t)

• S

  tq(t, s) − p(t, s) − [tq(t, s) − p(t, s)] −

t

t q(τ, s)dτ

  dπ = 0 ∀π ∈

• t∈N(t)

Π(t) This is the interim envelope condition, for each π ∈

t∈N(t) Π(t).

2

Π(t) is fully overlapping, hence

• t∈N(t)

Π(t) has full dimension: thus tq(t, s) − p(t, s) − [tq(t, s) − p(t, s)] =

t

t q(τ, s)dτ

which is the ex post envelope condition.

Main Result: Discussion

Theorem

Assume {Π(t) : t ∈ T} is fully overlapping. Then, any optimal incentive compatible mechanism satisfies the ex post envelope condition. There are (almost) no restriction on the ‘size’ of Π(t). It does not need to be large; on the contrary, it can be arbitrarily small.

Suppose ¯ π(t) depends continuously on t and let Πε(t) := {π ∈ ∆(S) : π = (1 − ε) ¯ π(t) + ε ˜ π for some ˜ π ∈ ∆(S)}. Then {Πε(t) : t ∈ T } is fully overlapping. How to think about this?

If each Π(t) is a singleton, interim incentive compatible mechanisms must satisfy the interim envelope condition. In our case, any optimal incentive compatible mechanism must satisfy the ex post envelope condition even when each Π(t) is arbitrarily small. The latter is always more restrictive. If one thinks about the limit as each Π(t) converges to some π(t), there is a ‘discontinuity’ in the set of feasible mechanisms.

Main Result: Implications I

Theorem

Assume {Π(t) : t ∈ T} is fully overlapping. Then, any optimal incentive compatible mechanism satisfies the ex post envelope condition.

What does this imply for mechanism design?

In many settings, ex-post envelope condition means ex-post revenue equivalence. In those settings, our main result says that any two optimal incentive compatible mechanisms that yield the same allocation rule must be revenue equivalent ex-post. This can restrict significantly the set of feasible mechanisms.

Optimal and Ex Post Incentive Compatibility

Theorem

Suppose {Π(t) : t ∈ T} is fully overlapping. Then, any optimal incentive compatible mechanism that is also ex-post monotone must be ex-post incentive compatible. Let qepmon(·) denote an ex post monotone allocation rule; if qepmon(·), p(·) is an optimal incentive compatible mechanism, then it must be ex-post incentive compatible. This follows from our main result and the standard mechanism design equivalence of ex-post incentive compatibility with ex-post monotonicity and ex-post envelope condition.

Remark

For a given qepmon, the importance of this result is not to say that it is possible to implement this allocation rule in an ex-post incentive compatible way.

This is well-known: use a payment rule satisfying the ex-post envelope condition.

Instead, the result says that only payment schemes that with qepmon form an ex post incentive compatible mechanism can be implemented robustly.

Main Result: Implications II

Theorem

Assume {Π(t) : t ∈ T} is fully overlapping. Then, any optimal incentive compatible mechanism satisfies the ex post envelope condition. Let qepmon denote an ex post monotone allocation rule.

What can the seller achieve if she is constrained to use qepmon as part of a Bayesian incentive compatible mechanism with correlated types?

A lot, since full extraction mechanisms are possible for ex post monotone allocation rules.

What can the seller achieve if she is constrained to use qepmon as part of an optimal incentive compatible mechanism with correlated types?

Much less, since she must pair that allocation rule with a transfer scheme that satisfies the ex-post envelope condition.

Optimal incentive compatibility is restrictive

Even allowing for correlation, the designer cannot robustly do as well.

Relationship With Other Ideas of Robustness

Bergemann and Morris (2005): Robustness and Higher Order Beliefs Robust (in our sense) mechanisms must satisfy the ex post envelope condition.

(A rough version of) Bergemann and Morris (2005)

They require mechanisms to be interim incentive compatible in the universal type space (for all possible hierarchies of higher order beliefs). They show this is (sometimes) equivalent to ex-post implementation.

How is that related to our result?

Think about the version of our model with many players: our setup corresponds to a subset of the universal type space. In their language, we implement in type spaces where only the ‘first order’ beliefs change for a given ‘payoff type’.

One can consider all such type spaces by assuming Π (t) = ∆(S), in which case ex-post implementation obtains.

We show that one may get close to ex-post implementation in a smaller set of type spaces. Their notion of robustness appears unnecessarily demanding.

Monotonicity of the Allocation Rule

The main result is about the envelope condition. Is there anything one can say about the monotonicity conditions that sometimes play a central role in mechanism design?

Lemma

If a mechanism q(·), p(·) is optimal incentive compatible then whenever t > t, Eπ[q(t, s)] ≥ Eπ[q(t, s)] for all π ∈ Π(t) ∩ Π(t) Optimal incentive compatibility requires that the expected value of the allocation rule taken with respect to any common belief must be larger for the higher type.

Monotonicity of the Allocation Rule

Lemma

If q(·), p(·) is optimal incentive compatible then whenever t > t, Eπ[q(t, s)] ≥ Eπ[q(t, s)] for all π ∈ Π(t) ∩ Π(t)

Proof.

By optimal incentive compatibility, for any t, t ∈ T Eπ[tq(t, s) − p(t, s)] ≥ Eπ

• tq(t, s) − p(t, s)
• ∀π ∈ Π(t)

Eπ[tq(t, s) − p(t, s)] ≥ Eπ

• tq(t, s) − p(t, s)
• ∀π ∈ Π(t)

For any π ∈ Π(t) ∩ Π(t) this reduces to Eπ[tq(t, s) − tq(t, s)] ≥ Eπ

• tq(t, s) − tq(t, s)
• r

(t − t)Eπ[q(t, s)] ≥ (t − t)Eπ

• q(t, s)
• If t > t, this implies Eπ[q(t, s)] ≥ Eπ[q(t, s)] for all π ∈ Π(t) ∩ Π(t).

Monotonicity of the Allocation Rule

Lemma

If a mechanism q(·), p(·) is optimal incentive compatible then whenever t > t, Eπ[q(t, s)] ≥ Eπ[q(t, s)] for all π ∈ Π(t) ∩ Π(t). Optimal incentive compatibility requires that the expected value of the allocation rule taken with respect to any common belief must be larger for the higher type.

Contrast with Standard Model

Paired with independence, this lemma implies that interim monotonicity is necessary for interim incentive compatibility:

for each type t, Π(t) = {π(t)} for some π(t) ∈ ∆(S), and independence gives π(t) = π(t) = π ∈ ∆(S) for any pair of types t, t.

In contrast, if beliefs depend on types, interim monotonicity is no longer necessary. In our setup, types can share common beliefs even without assuming that those beliefs are independent of types. Hence, there are different ways in which monotonicity can arise.

Monotonicity

Beliefs can have different degrees of commonality.

Definitions

Given a collection {Π(t) : t ∈ T} we say that: types share common beliefs when ∩t∈T Π(t) = ∅; beliefs are independent when Π(t) = Π ⊂ ∆(S) for all t; beliefs are overlapping when for every t there is a neighborhood N(t) such that ∩t∈N(t)Π(t) = ∅.

Remark

Using these definitions, we derive different versions of monotonicity.

Monotonicity

Theorem

Let q(·), p(·) be an optimal incentive compatible mechanism for {Π(t) : t ∈ T}. If types share common beliefs, then Eπ[q(·, s)] must be increasing in t for all π ∈ ∩t∈T Π(t). If beliefs are independent, then Eπ[q(·, s)] must be increasing in t for all π ∈ Π. If beliefs are overlapping, then q must be locally monotone (∀¯ t there exists a non-empty set ¯ Π ⊂ ∆(S) and a neighborhood N(¯ t) such that Eπ[q(t, s)] is an increasing function of t on N(¯ t) for all π ∈ ¯ Π). If Π(t) = ∆(S) for all t, then q(·) must be ex-post monotone. If q(·) is not ex-post monotone, then there exists Π = ∆(S) sufficiently large such that q(·), p(·) is not optimal incentive compatible for any payment function p(·) and any {Π(t) : t ∈ T} with Π ⊂ ∩t∈T Π(t).

Remark

In our setting, one can think about local as well as global monotonicity.

General Setting

O is the set of outcomes, and a direct mechanism φ : T × S → O maps reports and states into outcomes. The agent cares about outcome, type, and the state of the world. If the announced type is θ and the true type is t, the agent’s utility in state s is: u (φ (θ, s) , t, s) where u : O × T × S → R. The main result goes through, but we need an additional assumption (that was hidden in the simple setup).

Assumption (A)

The payoff function u : O × T × S → R is differentiable with respect to t, and ∂u

∂t

is non-negative and bounded. This is a standard assumption, necessary to use the envelope theorem.

General Result

Theorem

Suppose assumption A holds and {Π(t) : t ∈ T} is fully overlapping. Then, any

• ptimal incentive compatible mechanism satisfies the ex post envelope condition.

The proof is the same:

by the envelope theorem one can replace the incentive compatibility inequalities with the ‘first order conditions’ equalities by fully overlapping beliefs these equalities must hold ex-post.

Continuum of types, smoothness of the agent’s utility function, and fully

• verlapping beliefs are crucial.

Definition

A mechanism φ : T × S → O satisfies the ex-post envelope condition if for each s ∈ S u(φ(t, s), t, s) − u(φ(t, s), t, s) =

t

t

u2(φ(τ, s), τ, s)dτ ∀t, t ∈ T

Optimal and Ex Post Incentive Compatibility

Ex-post envelope is not enough for ex-post incentive compatibility.

Definition

A mechanism φ : T × S → O satisfies ex-post monotone type sensitivity if for each s ∈ S, t > t ⇒ u2(φ(t, s), t, s) ≥ u2(φ(t, s), t, s). This condition is weaker than Rochet’s cyclical monotonicity and therefore it is not enough for ex-post incentive compatibility. On the other hand, fully overlapping beliefs and monotone type sensitivity imply that an optimal incentive compatible mechanism must be ex-post incentive compatible.

Theorem

Suppose Assumption A holds and that {Π(t) : t ∈ T} is fully overlapping. Then any optimal incentive compatible mechanism that satisfies ex-post monotone type sensitivity is ex-post incentive compatible. This follows from the general result and the fact ex-post envelope and ex-post monotone type sensitivity imply ex-post incentive compatibility (this is a familiar result in an unfamiliar setting).

Decision Theory Slide

Definition

A mechanism φ(·) is optimal incentive compatible if for each t, θ ∈ T Eπ[u (φ(t, s), t, s)] ≥ Eπ[u (φ(θ, s), t, s)] for all π ∈ Π(t). Each type t is associated with a utility function u (·) and beliefs Π(t).

Those two items describe a type’s preferences.

A theory of decision making where beliefs are described by sets of probabilities can yield the primitives used in the definition above. Sets of probabilities obtain from Knightian uncertainty (ambiguity). But only one of the many existing ways to model Knightian uncertainty works, however, since all probability distributions must matter.

Expected Utility without Completeness: Bewley (1986)

Bewley generalizes expected utility by dropping the completeness axiom. The corresponding representation yields expected utility with sets: x y ⇔ Eπ[u (x)] > Eπ[u (y)] for all π ∈ Π(t) In this setting, optimal incentive compatibility is a natural notion of incentive compatibility (but not the only one).

Conclusions

We introduce a natural notion of robustness to errors in the specification of the agent’s beliefs (sets instead of singletons) in mechanism design.

This notion is consistent with Bewley’s theory of decision making under Knightian uncertainty. Our model gives an easy way to go from Bayesian interim incentive compatibility to dominant strategy implementation.

The consequences of imposing this type of robustness on mechanisms can be dramatic:

• nly mechanisms that satisfy the ex-post envelope conditions are feasible if

beliefs are fully overlapping.

Robustness (even to arbitrarily small sets) can significantly affect equilibrium

• utcomes.

Application: auctions:

in the standard auction environment, all robust auctions must satisfy ex-post revenue equivalence.

Since our framework moves from ex post implementation to bayesian implementation, it could highlight connections between theoretical results in the two cases.

And Now... The Second Half of the Paper

Robustness and Information Rents The asymmetric information literature (beginning with Akerlof’s lemons model) shows that private information can generate information rents. More recently, however, a sequence of papers has shown how these rents can be appropriated by a sufficiently clever designer in a wide variety of settings. These are the so-called ‘full extraction theorems’. When full extraction is possible, private information has no welfare consequences. Many have argued that these results illustrate one of the unrealistic features of standard mechanism design theory, and motivate the need for robust mechanisms. In the second half of the paer, we tackle the question of full extraction when mechanisms have to satisfy our idea of robustness.

Full Extraction and Robustness

A few authors (Crémer & McLean [1985], [1988], and McAfee & Reny [1992]) have found conditions on beliefs under which information rents disappear.

These conditions hold unless beliefs are linearly dependent across types. Therefore, generically, one expects rent extraction to be possible.

Private information has no welfare consequences.

Optimal incentive compatibility and full extraction

What happens if incentive compatibility has to hold for sets of probability distributions? Full rent extraction is harder... I will spare you all the details and just give the result (apologies if this makes no sense).

Optimal Full Extraction Condition

Theorem

The designer can achieve full rent extraction with an optimal incentive compatible mechanism if ∀t ∈ T : Π(t) ∩ co ∪t=tΠ(t) = ∅ The condition is a set version of the standard condition, and requires that beliefs be sufficiently different across types.

Π(tM) The optimal full extraction condition is satisfied Π(tH) Π(tL)

Implications for Auction Theory

Theorem

Assume {Π(t) : t ∈ T} is fully overlapping. Then, any optimal incentive compatible mechanism satisfies the ex post envelope condition.

What does this imply for auctions?

Under independence, we know that all auction types that allocate the object to the agent with the highest valuation are revenue equivalent in expectation as a consequence of the interim envelope condition. Therefore, first and second price auctions are interim revenue equivalent. However, the first price auction does not satisfy the ex post envelope condition, only the second price auction does. Therefore, only auction formats that generate the same ex post revenue as the second price auction can be robust.