Sales Talk Fr ed eric Koessler Vasiliki Skreta Paris School of - - PowerPoint PPT Presentation

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:

◮ tl likes only Lirac ◮ tr likes only Riesling ◮ ti 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...)

◮ sL house wine is Lirac ◮ sR 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

u(s, t) = tl tr ti sL 30 20 sR 32 20 (uniform priors) Optimal pooling posted price mechanism: sell at 15 euros to all types ρ = (p, x) = tl tr ti sL 1, 15 1, 15 1, 15 sR 1, 15 1, 15 1, 15 ⇒ interim revenue X(s) = 15 Optimal price with full info revelation: sell at 20 euros to 2/3 of types ρ = tl tr ti sL 1, 20 0, 0 1, 20 sR 0, 0 1, 20 1, 20 ⇒ X(s) = 20 × 2 3 ≃ 13.3 < 15 worse

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 ti?” if buyer says yes, seller asks a price of 20, buyer accepts if buyer says no, seller tells him whether he is sL or sR and asks a price of 30, tl accepts if seller said sL, tr accepts if seller said sR This protocol implements a feasible mechanism which is strictly better than posted prices: ρ = tl tr ti sL 1, 30 0, 0 1, 20 sR 0, 0 1, 30 1, 20 ⇒ X(sL) = X(sR) = 50/3 ≃ 16.7 > 15 Note: mechanism would extract all the surplus (and be optimal) if match function was u(sR, tr) = u(sL, tl) = 30

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 4 / 30

A better (ex-ante and interim) mechanism is: ρ = tl tr ti sL 1, 30 0, 0 1, 20 sR 0, 0 1, 32 1, 20 Buyer-IC and Buyer-PC are satisfied But not Seller-IC: X(sR) = 52/3 > X(sL) = 50/3 Modified feasible and equally profitable mechanism: ˜ ρ = tl tr ti sL 1, 15 0, 16 1, 20 sR 0, 15 1, 16 1, 20 ⇒ ˜ X(sL) = ˜ X(sR) = 51/3 > 50/3 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) = ET [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) = ES[U(s, t) | t]

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 10 / 30

Feasible Mechanisms

(Buyer-IC) U(t) ≥ ES

• p(s, t′)u(s, t) − x(s, t′)
• U(t′|t)

(Buyer-PC) U(t) ≥ 0 (Seller-IC), (Seller-PC) is equivalent to X(s) = X(s′) = ¯ X ≡ ES[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

• utcome (˜

p, ˜ x), such that for all s ∈ S: ET(x(s, t) | s) ≥ ET (˜ 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) = ES[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

Implication: seller ex-ante benefits from private information

Proposition

The optimal mechanism generates weakly higher ex-ante expected revenue compared to the full-information optimal mechanism.

Proof.

full-information optimal mech may not be S-IC when s private information can construct an equivalent S-IC mechanism as in the previous proposition an allocation with same ex-ante revenue for the seller is feasible

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 17 / 30

Other Implications

Access to Certification and Disclosure Rules does not benefit the seller ex-ante:

Proposition

The ability of the seller to certify his information or to commit to some information disclosure rule ex-ante does not lead to a higher ex-ante expected revenue. Any additional disclosure about the seller’s type would just make the IC and PC for the buyer harder to satisfy.

Corollary

If a mechanism satisfying (B-IC) and (B-PC) induces a positive ex-ante expected revenue, then there is a feasible mechanism that induces the same ex-ante expected revenue and interim payoff for the buyer.

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 18 / 30

Seller may interim regret having private information

Example

u(s, t) = t1 t2 s1 6 7 s2 1 With seller private information optimal mechanisms generate ex-ante revenue 6.5/2 ρ(s, t) = t1 t2 s1 1, 6 1, 1/2 s2 0, 0 1, 6.5 ˆ ρ(s, t) = t1 t2 s1 1, 3 1, 3.5 s2 0, 3 1, 3.5 Full-information optimum: s1 asks a price of 6 which is always accepted; s2 asks a price of 1 which is accepted with probability 1/2, yielding ex-ante revenue 6.5

2 .

with private info s1 is interim worse-off, while s2 is better-off

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 19 / 30

Properties of Optimal Mechanisms

Lemma

At an optimal mechanism U(t∗) = 0 for some t∗ ∈ T. B-PC binds at an endogenously determined type.

Lemma

The buyer’s interim expected utility is minimized at the same type tmin ∈ T at all B-IC mechanisms if and only if for every s tmin(s) ∈ arg mint u(s, t) is independent of s.

Corollary

Assume that tmin ∈ arg mint u(s, t) for every s. At an optimal mechanism, B-PC binds at tmin, i.e., U(tmin) = 0.

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 20 / 30

Reformulating the Seller’s Problem

Convex Environments

Assumption (C)

For every s ∈ S, u(s, t) is convex in t. For any t∗ ∈ T ≡ [t,¯ t], let J(s, t; t∗) =

• JL(s, t) = u(s, t) + Ft(t)

ft(t) ∂u(s,t) ∂t

if t < t∗ JR(s, t) = u(s, t) − 1−Ft(t)

ft(t) ∂u(s,t) ∂t

if t > t∗ For any t∗, ex-ante expected revenue: ¯ X = ES ¯

t t

p(s, t)J(s, t; t∗)f (t)dt

• − U(t∗)

At an optimal mechanism U(t∗) = 0 for some t∗.

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 21 / 30

Solving the Seller’s Problem

for every t∗ ∈ T, set U(t∗) = 0, and choose the assignment rule that maximises ¯ X subject to P(t) ≡ ES

• p(s, t)∂u(s,t)

∂t

• is increasing and

P(t) ≤ 0 for t < t∗ and P(t) ≥ 0 for t > t∗. denote the maximal revenue R(t∗). Then repeat the process for all t∗ ∈ T. the optimal mechanism is obtained for the t∗ that maximises R(t∗). we illustrate the process in examples

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 22 / 30

On the Irrelevance of the Seller’s Information

Seller benefits from having private information because uncertainty about the seller’s type relaxes (B-IC) or (B-PC) or both

Definition

(Single Crossing) A function f : X → R is single crossing if for every x ≤ x′, f (x) ≥ (>)0 implies that f (x′) ≥ (>)0.

Assumption (Single Crossing)

Both JR and −JL satisfy single-crossing in t for every s.

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 23 / 30

Proposition

Suppose that Assumptions C and Single-Crossing hold. Furthermore, suppose that tmin(s) is the same for all s. The optimal mechanism (p∗, x∗) is given by p∗(s, t) ≡

• 1

if J(s, t; tmin) > 0 if J(s, t; tmin) ≤ 0, x∗(s, t) = ES

• p(s, t)u(s, t) −

t

• tmin

p(s, τ)∂u(s,τ)

∂τ

dτ

• Corollary (Irrelevance of the Seller’s Information)

Under the above conditions the optimal revenue is equal to the full-information ex-ante revenue.

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 24 / 30

Illustrative Example

Seller benefits from private information even if u is strictly increasing in both arguments

Example

s ∈ {s1, s2}, prior prob of s1 is σ ∈ (0, 1); t ∈ [0, 1] uniform. Match function: u(s1, t) = t2 + 1/4, u(s2, t) = t + 2/3, t∗ = 0 so uncertainty about s does not relax (B-PC) match function strictly increasing in s and in t Virtual valuations are given by J(s1, t) = 3t2 − 2t + 1/4 and J(s2, t) = 2t − 1/3.

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 25 / 30

0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5

Figure: Left graph the match functions; right the virtual valuations.

Pointwise optimisation yields: p(s1, t) =

• if t ∈ (1/6, 1/2)

1

• therwise,

p(s2, t) =

• if t < 1/6

1 if t > 1/6. Mechanism not B-IC when seller’s type known: p(s1, t)∂u(s1,t)

∂t

is not monotonic in t, but P(t) = ES

• p(s, t)∂u(s,t)

∂t

• is for σ < 3

4!

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 26 / 30

Full-information optimal mechanism: price 1/4 or 1/2 when s1, price 5/6 when s2. Ex-ante: ¯ X = 25 − 16σ 36 . When seller has private info: Ex-ante: ¯ X = 29 108σ + 25 36(1 − σ) > 25 − 16σ 36 , when σ < 3/4. In this example uncertainty relaxes the buyer’s incentive constraint.

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 27 / 30

Background–informed principal

Myerson (1983)

◮ inscrutability principle introduces core, of undominated, safe mechanisms, expectational equilibrium

Maskin and Tirole (1990) and Yilankaya (1999)

◮ private values: informed principal neither gains nor loses from his private information

Maskin and Tirole (1992), Tisljar (2002, 2003):

◮ common values but agent no private information

Yilankaya (1999)

◮ private values, bilateral trade problem not covered by Maskin and Tirole (1990)

Mylovanov and Troeger (2013a,b)

◮ private values: neologism-proof allocations are equilibria, conditions for irrelevance of seller’s information.

Balestrieri and Izmalkov (2012)

◮ horizontal differentiation problem: particular case of our framework–generalisation to continuous types of an example that appeared in Koessler and Renault (2012) Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 28 / 30

Background–other related literatures

signalling through choice of reserve price

◮ Cai et al. (2007) and Kremer and Skrzypacz (2004)

information disclosure

◮ Es˝

• and Szentes (2007), Ottaviani and Prat (2001), Rayo and Segal

(2010)

advertising

◮ Anderson and Renault (2006), Johnson and Myatt (2006), Sun

(2011), Koessler and Renault (2012)

Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 29 / 30

References

Anderson, S. and R. Renault (2006): “Advertising Content,” American Economic Review, 96, 93–113. Balestrieri, F. and S. Izmalkov (2012): “Informed seller in a Hotelling market,” mimeo. Cai, H., J. Riley, and L. Ye (2007): “Reserve price signaling,” Journal of Economic Theory, 135, 253–268. Es˝

• , P. and B. Szentes (2007): “Optimal information disclosure in auctions and the handicap auction,” The Review of

Economic Studies, 74, 705. Gershkov, A., J. K. Goeree, A. Kushnir, B. Moldovanu, and X. Shi (2013): “On the equivalence of Bayesian and dominant strategy implementation,” Econometrica, 81, 197–220. Johnson, J. P. and D. P. Myatt (2006): “On the Simple Economics of Advertising, Marketing, and Product Design,” American Economic Review, 96, 756–784. Koessler, F. and R. Renault (2012): “When Does a Firm Disclose Product Information?” Rand Journal of Economics, forthcoming. Kremer, I. and A. Skrzypacz (2004): “Auction selection by an informed seller,” Unpublished Manuscript. Maskin, E. and J. Tirole (1990): “The principal-agent relationship with an informed principal: The case of private values,” Econometrica: Journal of the Econometric Society, 379–409. ——— (1992): “The principal-agent relationship with an informed principal, II: Common values,” Econometrica: Journal of the Econometric Society, 1–42. Myerson, R. (1981): “Optimal Auction Design,” Mathematics of Operations Research, 6, 58. ——— (1983): “Mechanism design by an informed principal,” Econometrica: Journal of the Econometric Society, 1767–1797. Mylovanov, T. and T. Troeger (2013a): “Informed principal problems in generalized private values environments,” Theoretical Economics, 7, 465–488. ——— (2013b): “Mechanism design by an informed principal: the quasi-linear private-values case,” mimeo. Ottaviani, M. and A. Prat (2001): “The value of public information in monopoly,” Econometrica, 69, 1673–1683. Rayo, L. and I. Segal (2010): “Optimal information disclosure,” Journal of Political Economy, 118, 949–987. Riley, J. and R. Zeckhauser (1983): “Optimal selling strategies: When to haggle, when to hold firm,” The Quarterly Journal of Economics, 98, 267–289. Sun, M. J. (2011): “Disclosing Multiple Product Attributes,” Journal of Economics and Management Strategy, 20, 195–224. Tisljar, R. (2002): “Mechanism Design by an Informed Principal: Pure-Strategy Equilibria for a Common Value Model,” Bonn econ discussion papers. ——— (2003): “Optimal trading mechanisms for an informed seller,” Economics Letters, 81, 1–8. Yilankaya, O. (1999): “A note on the seller’s optimal mechanism in bilateral trade with two-sided incomplete information,” Journal of Economic Theory, 87, 267–271. Koessler – Skreta (PSE – UCL) Sales Talk December 15, 2014 30 / 30