Lecture 3: Other Selling Mechanisms

Transcription

1 浙江大学经济学院系列讲座 : 博弈模型与机制设计 ( 三 ) Modeling Strategic Interactions Lecture Series 2008 Lecture 3: Other Selling Mechanisms by 王汝渠 Ruqu Wang 1

2 1. Introduction to Bargaining and Posted Prices Posted Prices & Take-It-or-Leave-It Offers -- Posted prices: usually the seller posts the prices -- Take-it-or-leave-it offers: can be made by the seller or the buyer e.g. A seller and a buyer discussing the price of transaction Vb=buyer s valuation, Vs=seller s valuation Vb ~F(v), Vs~G(v); both are private information Posted price: seller maximizes (p-vs)[1-f(p)] Take-it-or-leave-it offer made by the buyer: buyer maximizes (Vb-p)G(p) 2

3 Simultaneous Bargaining -- buyer and seller make offers simultaneously Pb=buyer s offer Ps=seller s offer -- determination of transaction price If Pb<Ps, no trade; If Pb>Ps, transaction at p=(pb+ps)/2 -- this is fully efficient if Pb=Vb, and Ps=Vs (truthful reporting of valuation) not an equilibrium without subsidies or surpluses -- Characterization of equilibrium strategy buyer P = BV ( ), seller P = SV ( ) b b s s 3

4 buyer maximizes V S s 1 ( Pb ) Pb S( Vs) V 2 + g( V ) dv b s s V b BV ( b) + Ps seller maximizes ( ) V B 1 ( Ps ) s f V b dv b 2 => 2 simultaneous differential equations => equilibrium not fully efficient => it is second best: the most efficient solution with budget balance -- If there is no uncertainty (i.e., Vs and Vb are known), then there are multiple equilibria with full efficiency. (There are also equilibria with no trade => not efficient) 4

5 2. Alternating-Offer Bargaining Models (perfect info) Rubinstein bargaining model -- 2 players sharing a pie -- Player 1 makes an offer, player 2 says Yes or No -- If Yes, negotiation ends -- If No, player 2 makes a counter-offer -- Player 1 can now say Yes or No -- Order of players making offers: 1,2,1,2,1,2, -- Game ends when an agreement is reached, or when pre-specified number of rounds (T) is reached -- If T=, negotiation could last indefinitely 5

6 3. Delays in Bargaining Models (private info) Delays or costly actions can be used to reveal private information e.g. Strikes, Protests A model of one-sided uncertainty -- seller s value = s [ ] buyer s value = b ~ F(b) s b -- continuous time -- identical discount rate -- once buyer s value is revealed, p=(s+b)/2 (splitting the surplus of b-s ) -- equilibrium strategy with delays seller makes an offer of p(t) at time t 6

7 a buyer of type b accepts the offer at time T(b) => the type of buyer accepting an offer at t must be T -- equilibrium conditions: s+ T 1 () t pt () = 2 t=t(b) maximizes 1 rt s+ T () t ( b p( t)) e = b e 2 rt 1 () t equivalently, if the buyer pretends to be type b*, then b*=b maximizes b s+ b* 2 rt ( b*) e 7

8 First order condition: r T'( b) = b s Let B(t) be the inverse function of T(b). Then B() t s B'( t) = r or B() t s ( Bt ( ) s)' = r This is a completely separating equilibrium. (There could be partially pooling equilibria in this game.) 8

9 A model of two-sided uncertainty -- seller s value = s ~ G(s) ----[ [ ] ] buyer s value = b ~ F(b) s b -- completely separating equilibrium strategy: b=b(t) reveals his type at t s=s(t) reveals his type at t -- First order condition: B() t S() t ( Bt ( ) St ( ))' = r S(t) B(t) 9

10 4. Auctions vs Posted Price Selling Model Setup -- Continuous time, infinite horizon, no discounting -- One seller, one object -- Buyers arrive according to Poisson process with rate of arrival λ -- A buyer s valuation v ~ F(v), i.i.d. -- The seller can use either auctions or posted prices -- If the object is not sold, the seller can sell it later -- Selling by posted prices => cost of display θ d -- Selling by auctions => cost of storage θ s + auction fee Θ a 10

11 Sell by Posted Prices -- Suppose the price is p -- An arriving buyer buys with probability 1-F(p) -- The seller s profit S d Π ( p) = p λ(1 F( p)) -- First order condition: d S θd ( ) Π ( p) = 1 f p = 0 2 dp λ(1 F( p)) -- Define virtual function: 1 Fv ( ) Jv () = v f () v S -- At the optimal price, Π ( p) = J( p) θ 11

12 Sell by Auctions -- Auctioning off the object T units of time apart -- Buyers arriving during [0,T] are notified of the auction to be held at T [T,2T] => auction held at 2T, T 2T 3T -- If k buyers arrive during the period of time, with reserve price p, the seller s profit becomes v k 2 (, ) ( 1)[1 ( )] ( ) ( ) p Π k p = vk k F v F v f vdv + pk F p F p + F p Π k 1 k A [1 ( )] ( ) ( ) 12

13 -- where A Π = Π( k, p) P ( T) θ T Θ k= 0 k s a P T is the prob. that k buyers arrived within T units of time -- ( ) k Conclusion: -- Auctions more profitable when the valuation distribution is more dispersed -- The dispersion concept is different from the variance of a distribution. -- It is related to the slope of the virtual function J(v) 13

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15 Plan for next lecture: Mechanism Design Reading materials 1. Optimal Auction Design, Roger B. Myerson, Mathematics of Operations Research, Vol. 6, No. 1 (Feb., 1981), pp A Note on Redistributive Fairness and Economic Reform, Anna Rubinchik and Ruqu Wang, Journal of Development Economics, Vol.86, No. 2 (June 2008), pp Market Design with Correlated Valuations, Yongmin Chen and Ruqu Wang, Economica, Vol.73, ,