Sales Talk Fr´ ed´ eric Koessler Vasiliki Skreta Paris School of Economics – CNRS University College London Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 1 / 30
Motivating Example: How to sell house wine? Buyer can be one of three types: ◮ t l likes only Lirac ◮ t r likes only Riesling ◮ t i is indifferent between Lirac and Riesling Tavern selling house wine: type of wine private information (buyer unable to tell by looking at the carafe/bottle...) ◮ s L house wine is Lirac ◮ s R house wine is Riesling Seller wants to maximize expected payment Buyer accepts the deal if the expected match is higher than the expected payment Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 2 / 30
Motivating Example: Match Function Example t l t r t i u ( s , t ) = s L 30 0 20 (uniform priors) s R 0 32 20 Optimal pooling posted price mechanism: sell at 15 euros to all types t l t r t i ρ = ( p , x ) = s L 1 , 15 1 , 15 1 , 15 ⇒ interim revenue X ( s ) = 15 s R 1 , 15 1 , 15 1 , 15 Optimal price with full info revelation: sell at 20 euros to 2/3 of types t l t r t i ⇒ X ( s ) = 20 × 2 ρ = s L 1 , 20 0 , 0 1 , 20 3 ≃ 13 . 3 < 15 worse s R 0 , 0 1 , 20 1 , 20 Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 3 / 30
Motivating Example: Bilateral Communication Consider the following scenario: seller asks the buyer: “are you t i ?” if buyer says yes, seller asks a price of 20, buyer accepts if buyer says no, seller tells him whether he is s L or s R and asks a price of 30, t l accepts if seller said s L , t r accepts if seller said s R This protocol implements a feasible mechanism which is strictly better than posted prices: t l t r t i ρ = s L 1 , 30 0 , 0 1 , 20 ⇒ X ( s L ) = X ( s R ) = 50 / 3 ≃ 16 . 7 > 15 s R 0 , 0 1 , 30 1 , 20 Note: mechanism would extract all the surplus (and be optimal) if match function was u ( s R , t r ) = u ( s L , t l ) = 30 Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 4 / 30
A better (ex-ante and interim) mechanism is: t l t r t i ρ = s L 1 , 30 0 , 0 1 , 20 s R 0 , 0 1 , 32 1 , 20 Buyer-IC and Buyer-PC are satisfied But not Seller-IC: X ( s R ) = 52 / 3 > X ( s L ) = 50 / 3 Modified feasible and equally profitable mechanism: t l t r t i ⇒ ˜ X ( s L ) = ˜ ρ = ˜ s L 1 , 15 0 , 16 1 , 20 X ( s R ) = 51 / 3 > 50 / 3 s R 0 , 15 1 , 16 1 , 20 The optimal mechanisms we derive are based on this idea Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 5 / 30
Remarks on the Example posted price is not optimal ◮ in contrast to Riley and Zeckhauser (1983), Myerson (1981) (seller has no private information) and Yilankaya (1999) (seller has private information–but buyer’s willingness to pay does not depend on it) bilateral cheap talk communication with partial information transmission followed by a conditional price is strictly better than posted price mediated selling mechanisms are even better the seller strictly benefits from having private information ◮ in contrast to Maskin and Tirole (1990) and Yilankaya (1999) Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 6 / 30
Model 1 seller, privately known type (product characteristic) s ∈ S 1 buyer, privately known type (taste) t ∈ T types are independently distributed match function (buyer’s valuation): u ( s , t ) ∈ R type-independent value for the seller (normalized to 0) information is soft selling procedure–the mechanism–is chosen by the seller Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 7 / 30
Questions methodology ◮ how do we formalize and solve the informed seller problem? characterization, implementation ◮ what are the characteristics of revenue-maximizing procedures? ◮ is the optimal mechanism an equilibrium of the mechanism selection game (and vice versa)? ◮ when are simple mechanisms, such as posted prices, optimal? ◮ can we implement the optimum with posted prices and unmediated communication between the seller and the buyer? ◮ is it possible for the seller to leverage his private information and to extract the entire surplus of the buyer? value of information ◮ does the seller benefit or regret from having private information about product characteristics? (and if yes, when) ◮ can the seller benefit from having access to a certifying technology? Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 8 / 30
Main Contributions and Results An optimal (revenue maximizing) mechanism is well defined and is an equilibrium of the mechanism selection game, but other (sub-optimal) equilibria exist Incentives constraints of the seller are irrelevant for feasible ex-ante revenues ◮ Information certification or commitment to disclosure rules cannot be ex ante valuable for the seller ◮ The seller (in most cases strictly) benefits from being privately informed Characterization of optimal mechanisms for convex match functions ◮ Application to a continuous version of the house wine example ◮ Sufficient condition on the match function and type distribution for the ex-ante optimal revenue to be equal to the full-information revenue In progress: hard (certifiable) information about product characteristics Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 9 / 30
Mechanisms Mechanism (direct): ρ = ( p , x ) : S × T → [0 , 1] × R ◮ p ( s , t ): probability of sale ◮ x ( s , t ): expected transfer (price) Seller’s payoff: x ( s , t ) seller’s interim payoff: X ( s ) = E T [ x ( s , t ) | s ] Buyer’s payoff: U ( s , t ) = p ( s , t ) u ( s , t ) − x ( s , t ) buyer’s interim payoff: U ( t ) = E S [ U ( s , t ) | t ] Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 10 / 30
Feasible Mechanisms (Buyer-IC) � � p ( s , t ′ ) u ( s , t ) − x ( s , t ′ ) U ( t ) ≥ E S � �� � U ( t ′ | t ) (Buyer-PC) U ( t ) ≥ 0 (Seller-IC), (Seller-PC) is equivalent to X ( s ) = X ( s ′ ) = ¯ X ≡ E S [ X ( s )] ≥ 0 Proposition At every feasible mechanism the seller’s interim revenue is constant across types and equal to his ex-ante revenue. Hence, under feasibility, maximizing revenue ex-ante is the same as maximizing interim. Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 11 / 30
Mechanism Selection Game Is an optimal mechanism, that maximizes the seller’s ex-ante revenue under (B-IC), (B-PC), (S-IC) and (S-PC), an equilibrium of the mechanism selection game? Myerson (1983): Inscrutability Principle: w.l.o.g., all seller types propose the same mechanism ρ Revelation Principle: w.l.o.g, direct mechanisms are used along the equilibrium path Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 12 / 30
Mechanism Selection Game: Solution Concepts Core mechanism. ρ is a core mechanism if it is feasible, and if ν is preferred by some s ∈ S , ν not feasible for at least some some ¯ S such that S ∗ ⊆ ¯ S where S ∗ contains all s that strictly prefer ν to ρ . We show that the set of ex-ante optimal mechanisms coincide with the set of core mechanisms Expectational Equilibrium. ρ = ( p , x ) is an expectational equilibrium iff it is feasible, and for every generalized mechanism M , there exists a belief µ for the buyer, reporting and participation strategy profiles that form a Nash equilibrium given M and µ , with outcome (˜ p , ˜ x ), such that for all s ∈ S : E T ( x ( s , t ) | s ) ≥ E T (˜ x ( s , t ) | s ) Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 13 / 30
The Informed Principal Problem is Not a Problem Proposition An optimal mechanism is an expectational equilibrium. Proof. let ( p , x ) be an optimal mechanism the seller proposes ( p , x ) along the equilibrium path deviation to a mechanism (not necessarily direct) inducing (˜ p , ˜ x ) in equilibrium, the interim revenue ˜ X ( s ) of the seller should be the same ˜ X for all s passive beliefs if the deviation is profitable for s then ˜ X ( s ) > X ( s ), and thus ˜ X > X , which contradicts the optimality of ( p , x ) ⇒ the informed principal problem is a constrained optimization problem Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 14 / 30
What other mechanisms are expectational equilibria? Proposition Every feasible mechanism in which the interim revenue is higher than the full-information interim revenue (optimal revenue when the seller’s type is known) is an expectational equilibrium. Consider wine example. The following mechanisms are expectational equilibria full-information mechanism (IC for seller): 40 / 3 ≥ 40 / 3 pooling posted price mechanism: 15 ≥ 40 / 3 bilateral cheap talk and contingent prices: 50 / 3 ≥ 40 / 3 ◮ however, none of these mechanisms is a core mechanism ex-ante optimal yields 51 / 3 ≥ 40 / 3 Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 15 / 30
Seller-IC does NOT restrict implementable ex ante revenues Lemma Take a direct mechanism ( p , x ) that gives the buyer interim payoff U ( t ′ | t ) , t , t ′ ∈ T. There exists a mechanism (˜ p , ˜ x ) that satisfies the seller’s incentive constraint, generates the same ex-ante revenue for the seller, and gives the buyer the same interim payoff, that is U ( t ′ | t ) = U ( t ′ | t ) , for all t , t ′ ∈ T. ˜ Proof. Fix a mechanism ρ = ( p , x ), and let x ( s , t ) = E S [ x ( s , t )] and ˜ ˜ p ( s , t ) = p ( s , t ) for all s , t ∈ S × T (recall house wine example) Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 16 / 30
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