aa r X i v : . [ ec on . T H ] N ov Constrained Information Design: Toolkit ∗ Laura Doval † Vasiliki Skreta ‡ November 9, 2018
Abstract
These notes show the tools in Le Treust and Tomala (2017) extend to thecase of multiple inequality and equality constraints. This showcases the powerof the results in that paper to analyze problems of information design subjectto constraints. In fact, we show in Doval and Skreta (2018) that they can beused to provide an upper bound on the number of posteriors a designer withlimited commitment uses in his optimal mechanism. ∗ The notes are currently in flux and will be updated soon to show how these tools can shed lighton some results in the literature and open the door to analyzing new and exciting problems. † California Institute of Technology, Pasadena, CA 91125. E-mail: [email protected] ‡ University of Texas at Austin, University College London, and CEPR. E-mail: [email protected] . Vasiliki Skreta acknowledges funding by the European Research Council(ERC) consolidator grant “Frontiers In Design.” Other acknowledgements to be added. Introduction
In Doval and Skreta (2018) we establish a canonical class of mechanisms for lim-ited commitment. Relying on this canonical class, we show how to characterize theprincipal’s optimal mechanism by solving a constrained information design problem.The results here provide tools to solve such general constrained information designproblems. These problems are becoming common: Since Kamenica and Gentzkow(2011) seminal paper on Bayesian persuasion, the literature on information designhas grown steadily. A bulk of new work analyzes constrained information designproblems, which can be classified in three groups1. The information designer faces constraints additional to the Bayes’ plausibil-ity constraint in Kamenica and Gentzkow (2011), like in Boleslavsky and Kim(2018) on persuasion and moral hazard, and Le Treust and Tomala (2017)) oninformation transmission with capacity constraints.2. The information designer is designing a mechanism that satisfies incentiveand participation constraints, like in Dworczak (2017) work on aftermarkets,Georgiadis and Szentes (2018) work on optimal monitoring in moral hazard,and in our own work Doval and Skreta (2018) canonical for limited commit-ment.3. Mechanism design problems that do not involve information design and stillcan be solved using information design tools, like Dworczak et al. (2018).The typical approach to tackle these constrained information design problems isto set up a Lagrangian to incorporate the constraints into the objective function,except for the Bayes’ plausibility constraint. If each constraint can be written asthe expectation over posteriors of some function, then the Lagrangian itself canbe written as an expectation over posteriors of some function given the Lagrangemultiplier. If there are N possible states of the world, one may be tempted to applyCarath´eodory’s theorem and conclude from this that the optimal information policyuses at most N posteriors. After all, the solution to the problem would correspond2o the concavification of the function whose expectation over posteriors determinesthe Lagrangian.In an inspiring contribution, Le Treust and Tomala (2017) show that the abovereasoning is flawed when the information designer faces one inequality constraint. Atthe heart of their result is the observation that the Lagrange multiplier is also partof the solution to the optimization problem. Indeed, they show that the solutioncorresponds to concavifying a function of N + 1 variables: the first N correspond toa belief and the last corresponds to the inequality constraint. It follows then thatthe optimal policy may involve N + 1 posteriors. The authors also show that theLagrangian approach is valid for their problem.In Doval and Skreta (2018), the designer designs both an allocation rule and aninformation structure; both have to satisfy the agent’s participation and incentivecompatibility constraints. To show the upper bound on the number of posteriorsthat the designer uses in an optimal policy, we apply the extension of the resultsLe Treust and Tomala (2017) which we present in these notes. It is our hope thatgiven the prevalence of constrained information design this simple extension is usefulto other researchers. Consider the following problem. Let Ω be a finite set of states. Let f, g , . . . , g r , g r +1 , . . . , g K :∆(Ω) R ∪ {−∞} be a tuple of functions defined on ∆(Ω). For µ ∈ ∆(Ω) and γ , . . . , γ K ∈ R , considercav g ,...,g K f ( µ, γ , . . . , γ K ) := sup X m λ m f ( µ m ) : P m λ m µ m = µ, P m λ m g l ( µ m ) ≥ γ l , l ∈ { , . . . , r } , P m λ m g l ( µ m ) = γ l , l ∈ { r + 1 , . . . , K } . (1) To make the comparison with Le Treust and Tomala (2017) simple, we follow their notationas much as possible. However, while they present their results for any convex set X , to make thepresentation closer to information design, we let X be the space of beliefs over the set of states Ω.
3e Treust and Tomala (2017) consider the above problem for r = 1 and no equalityconstraints. Proposition 3.1.
Define f g ,...,g K : ∆(Ω) × R K R ∪ {−∞} f g ,...,g K ( µ, γ , . . . , γ K ) = ( f ( µ ) if γ i ≤ g i ( µ ) , i ∈ { , . . . , r } ∧ γ i = g i ( µ ) , i ∈ { r + 1 , . . . , K }−∞ otherwise . Then, for each ( µ, γ , . . . , γ K ) ∈ ∆(Ω) × R K ,cav g ,...,g K f ( µ, γ , . . . , γ K ) = cav f g ,...,g K ( µ, γ , . . . , γ K ) . Proof.
The function cav f g ,...,g K ( µ, γ , . . . , γ K ) is given by the following program:sup X m λ m f g ,...,g K ( µ m , γ ,m , . . . , γ K,m )s.t. ( P m λ m µ m = µ P m λ m γ l,m = γ l , l ∈ { , . . . , K } . Take a family ( λ m , µ m , γ ,m , . . . , γ K,m ) that is feasible for this program. Then, for l ≤ r , we have P λ m g l ( µ m ) ≥ P m λ m γ l,m = γ m , and for l ∈ { r + 1 , . . . , K } , wehave P λ m g l ( µ m ) P m λ m γ l,m = γ m . Thus, ( λ m , µ m , γ ,m , . . . , γ K,m ) is feasible forcav g ,...,g K f ( µ, γ , . . . , γ K ). Thus, cav g ,...,g K f ( µ, γ , . . . , γ K ) ≥ cav f g ,...,g K ( µ, γ , . . . , γ K ).On the other hand, let ( λ m , µ m ) such that P m λ m µ m = µ and P λ m g l ( µ m ) ≥ γ l , l ≤ r and P m λ m g l ( µ m ) = γ l , l ∈ { r +1 , . . . , K } . For each l , let γ l = P m λ m g l ( x m )and for each m , let γ l,m = g l ( µ m ) + γ l − γ l . Then, P m λ m γ l,m = γ l . Because γ l ≥ γ l (with equality for l ∈ { r + 1 , . . . , K } ), g l ( µ m ) ≥ γ l,m for l ∈ { , . . . , r } and g l ( x m ) = γ l,m for l ∈ { r + 1 , . . . , K } . Thus, ( λ m , µ m , γ ,m , . . . , γ K,m ) is feasible forcav f g ,...,g K . Hence, cav g ,...,g K f ( µ, γ , . . . , γ K ) ≤ cav f g ,...,g K ( µ, γ , . . . , γ K ). Corollary 3.1.
The solution to problem (1) uses at most N + K posteriors. roof. The result follows from Proposition 3.1 and an application of Carathe´odory’stheorem (see, e.g., Rockafellar (1970)), because f g ,...,g K has domain ∆(Ω) × R K . Corollary 3.2.
Suppose that in problem (1) , only
M < r inequality constraints bind.Then the solution to problem (1) uses at most N + M + K − r posteriors.Proof. Suppose that in the solution to program cav g ,...,g K f ( µ, γ , . . . , γ K ), M ≤ r constraints bind and r − M are slack. Thencav g ,...,g K f ( µ, γ , . . . , γ K ) = cav g M ,g r +1 ,...,g K f ( µ, γ M , γ r +1 , . . . , γ K ) , where γ M is the projection of vector ( γ , . . . , γ r ) on the binding constraints and g M isthe projection of vector ( g , . . . , g r ) on the set M . It follows from Proposition 3.1 thatcav g M ,g r +1 ,...,g K f ( µ, γ M , γ r +1 , . . . , γ K ) = cav f g M ,g r +1 ,...,g K ( µ, γ M , . . . , γ r +1 , . . . , γ K ). Thusthe solution to (1) uses at most N + M + K − r beliefs.5 eferencesBoleslavsky, R. and K. Kim (2018): “Bayesian persuasion and moral hazard,”Tech. rep. Doval, L. and V. Skreta (2018): “Mechanism design with limited commitment,”Tech. rep.
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