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

Uncertainty in Mechanism Design Giuseppe Lopomo Luca Rigotti Chris Shannon UC Berkeley Duke Pitt (visiting Duke) 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 of 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 object 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: E π [ 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 S i ⊃ × j � = i T j = T − i .

Interim Incentive Compatibility Mechanism Design Refresher Definition A mechanism q ( · ) , p ( · ) is interim incentive compatible if for each t , θ ∈ T E π ( t ) [ tq ( t , s ) − p ( t , s )] ≥ E π ( t ) [ tq ( θ , s ) − p ( θ , s )] � �� � � �� � Expected utility of reporting true type 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 E π ( t ) [ tq ( t , s ) − p ( t , s )] ≥ E π ( t ) [ tq ( θ , s ) − p ( θ , s )] � �� � � �� � Expected utility of reporting true type 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 E π q ( t , s ) is increasing in t ; satisfies the interim envelope condition if for each t , t � ∈ T : � � � t � t � q ( t � , s ) − p ( t � , s ) − [ tq ( t , s ) − p ( t , s )] − t q ( τ , s ) d τ = 0 E π

Our Notion of Robust Incentive Compatibility Definition A mechanism q ( · ) , p ( · ) is optimal incentive compatible if for each t , θ ∈ T E π [ tq ( t , s ) − p ( t , s )] ≥ E π [ 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 Π ε as follows: Π ε : = { π ∈ ∆ ( S ) : π = ( 1 − ε ) ¯ π + ε ˜ π for some ˜ π ∈ ∆ ( S ) } . Define the collection { Π ε ( t ) : t ∈ T } as follows Π ε ( t ) : = { π ∈ ∆ ( S ) : π = ( 1 − ε ) ¯ π ( t ) + ε ˜ π for some ˜ π ∈ ∆ ( S ) } Π ε ( 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 )] ( 1 − ε ) E ¯ π ( t ) [ tq ( θ , s ) − p ( θ , s )] + ≥ + ∀ ˜ π ∈ ∆ ( S ) ε E ˜ π [ tq ( t , s ) − p ( t , s )] ε E ˜ π [ tq ( θ , s ) − p ( θ , s )]

Many Agents Notation and Definitions Many buyers means a Bayesian game To allow for many agents one can, for example, set S i = × j � = i T j = T − i . The mechanism then specifies q i ( · ) , p i ( · ) for all players in the game. The set of ‘beliefs’ is Π ( t i ) ⊂ ∆ ( T − i ) . Definition A mechanism q i ( · ) , p i ( · ) is optimal incentive compatible if, for each player i , and for each t i , θ ∈ T i for all E π [ t i q i ( t i , t − i ) − p i ( t i , t − i )] ≥ E π [ t i q i ( θ , t − i ) − p i ( θ , t − i )] π ∈ Π ( t i ) � �� � � �� � Expected utility of reporting true type Expected utility of reporting another type given truth telling from other players given truth telling from other players Optimal incentive compatibiity and equilibrium Reporting truthfully must be a Bayesian Nash equilibrium for any π ∈ Π ( t i ) . 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.