1First-Price Sealed-Bid Auction{ Two bidders, one good{ Bidder i’s valuation for the good is vi, is known only by bidder i. Valuations are independently and uniformly distributed on [0,1].{ Each bidder i submits a nonnegative bid bi. The higher bidder wins and pays his bid. Other bidder pays and receives nothing.{ In case of a tie, the winner is determined by a coin flip1p{ Bidder i’s payoff, if wins and pays p, is vi-p{ Bidders are risk-neutral{ All of this information is common knowledgeFirst-Price Sealed-Bid Auction{ Action spacesA A [0)zA1= A2= [0,∞){ Type spacesz T1= T2= [0,1]{ Beliefsz p1(t2|t1)= p1(t2) z p2(t1| t2)= p2(t1)2{ Player i’s (expected) payoff function⎪⎩⎪⎨⎧<=>−−= ififif,0,2/)(,),;,(2121jijijiiiiiibbbbbbbvbvvvbbπ2First-Price Sealed-Bid Auction{ Strategy for player i: bi(vi){Strategies (b1(v1), b2(v2)) are a Bayesian Nash {Strategies (b1(v1), b2(v2)) are a Bayesian Nash equilibrium if for each viin [0,1], bi(vi) solvesmax (vi-bi)Prob{bi> bj(vj)}+ (vi-bi)Prob{bi= bj(vj)}/2{ What type of strategy might make sense? 3First-Price Sealed Bid Auction{ Let’s see if there is a linear equilibriumbi(vi) = ai+ civii=1,2Example strategiesbi(vi) ai+ civii1,2{ Assuming player j adopts the strategy bj(vj) = aj+cjvj, player i’s best response:bivipy pmax (vi-bi)Prob{bi> bj(vj)}= (vi-bi)Prob{bi> aj+cjvj}{ Player i knows:4Pdf of bjajaj+cjRecall for values in range that Uniform variables have pdf = 1/(b-a) and cdf =(x-a)/(b-a)3{ Given bj(vj) = aj+cjvj, player i should not bid below player j’s minimum or above player j’s maximum sobb First-Price Sealed-Bid Auctionzbi¸ aj ; bi· aj+ cj{ We expect a higher-value type to bid more than lower onesz cj¸ 0{ We know Prob{bi>aj+cjvj} = Prob {vj<(bi-aj)/cj}z = (bi-aj)/cj{ Player i’s objective:From cdf for vj= (x-0)/(1-0)5z max (vi-bi)Prob{bi> aj+cjvj} = (vibi-bi2+biaj-viaj)/cjz From FOC, bi= (vi+aj)/2 (and SOC okay){ Player i’s best response (thus far):z bi=ajif vi≤ aj(from constraint bi¸ aj), z bi= (vi+aj)/2 otherwiseFirst-Price Sealed-Bid Auction{ Player i’s best responsebi=ajif vi≤aj, bi= (vi+aj)/2 otherwise{ Can ajbe z Between 0 and 1?{ For some values, vi· ai, so not linearz Greater than or equal to 1?{ Since cj¸ 0, bj(vj) = aj+ cjvj≥ 1 { Then bj(vj) ¸ vj!z Less than or equal to zero?biviIf 0 < aj< 16q{ bi(vi)= (vi+aj)/2We have aj≤0, bi = ai+civi (from linear form)= aj/2+1/2(vi) (best response to j) → ai= aj/2 ci=1/24First-Price Sealed-Bid Auction{ Player i’s best responseaj≤0 ai+civi=aj/2+1/2(vi) →ai= aj/2 ci=1/2aj≤0, ai+civi=aj/2+1/2(vi) →ai= aj/2 ci=1/2{ Player j’s best responseai≤0, aj+cjvj=ai/2+1/2(vj) → aj= ai/2 cj=1/2We have a= a= 0 c= c=1/2 and b(v)= v/2 i=1 27ai= aj= 0, ci= cj=1/2 and bi(vi)= vi/2 i=1,2Each player bids half his/her valuation in a linear equilibrium.(If players’ strategies are strictly increasing and differentiable, this is the unique symmetric equilibrium)A Double Auction Example{ A seller and a buyer have private valuations vsand vbz Assume drawn from independent uniform distributions on [0,1]{ Seller names asking price ps; buyer simultaneously names offer price pb{ If pb¸ ps, then trade occurs at price p = (pb+ps)/2; if pb< psthen no trade occurs{Utilities if trade occurs are pvand vp 8{Utilities if trade occurs are p-vsand vb–p (and 0 otherwise){ Find strategies that specify the price to offer (or demand) for each of the other party’s valuations5Double Auction Example{ A pair {pb(vb),ps(vs)} is a BNE if below true:{ For buyer, for each v_b in [0,1], p_b(v_b) solvesz (1) maxpb[vb–(pb+ EPS)/2]*Prob{pb¸ ps(vs){ where EPS=E[ps(vs)|pb¸ ps(vs) is the expected price the seller will demand, conditional on demand being less than the buyer’s offer of pb9ypb{ For seller, for each v_s in [0,1], p_s(v_s) solvesz (2) maxps[(ps+ EPB)/2 - vs]*Prob{pb(vb) ¸ ps){ where EPB = E[pb(vb)|pb(vb) ¸ ps] Double Auction{ Trades will never occur when a seller’s valuation is higher than the buyer’svaluation is higher than the buyer svb1Trade might occur in this region10vs1{ What are several simple strategies?6Double Auction{ Consider a 1-price equilibriumzFor any value x in [0 1] zFor any value x in [0,1], { buyer offers x if vb¸ x and 0 otherwise{ seller demands x if vs· x and 1 otherwisevbTRADE1Trade here would be “efficient” but does not occur11vsxx1Double Auction{ Is there a linear equilibrium?{If seller’s strategy is p(v) = a+cv{If seller s strategy is ps(vs) = as+csvsz vsis U[0,1] and psis U[as, as+cs]{ To determine buyer’s response we need1. Prob{pb¸ ps(vs)} which is Prob{pb¸ as+csvs} = Prob {(pb-as)/cs¸ vs}= CDF Î Prob = (pb-as)/cs.122. EPS=E[ps(vs)|pb¸ ps(vs)]= (as+pb)/2.Pdf of psasas+cspbps7Double Auction{ Then from buyer’s function (1):[{()/2}/2]*()/z maxpb[vb–{pb+(as+pb)/2}/2]*(pb-as)/csz FOC Æ pb= 2vb/3 + as/3 gives buyer’s response (which is linear)z And SOC okay13Double Auction{ If buyer’s strategy is pb(vb) = ab+cbvb,ygypb(b)bbb,z vbis U[0,1] and pbis U[ab, ab+cb]{ To find seller’s best response we need1. Prob{pb(vb) ¸ ps} which is Prob{ab+cbvb¸ ps} = Prob {vb¸ (ps-ab)/cb} FIX= CDF Î Prob = 1- (ps-ab)/cb= (cb- ps+ ab)/cb2EPB = E[p(v)|p(v) p] 2.EPB = E[pb(vb)|pb(vb) ¸ps] = (ps+ab+cb)/2.14Pdf of pbabab+cbpbps8Double Auction { Then from seller’s function (2):()z maxps[{ps+(ps+ab+cb)/2}/2-vb]*(ab+cb-ps)/cbz FOC Æ ps= 2vs/3 + 1/3(ab+cb) is seller’s best response (which is linear)z And SOC okay{ Now the best response functions arez pb= 2vb/3 + as/3z ps= 2vs/3 + 1/3(ab+cb)z So cb=2/3 and we can solve for a’s.15Double Auction{ 2 BR functions together give us:z Pb(vb) = 2/3 vb+ 1/12bbbz Ps(vs) = 2/3 vs+ ¼{ Trade occurs only if pb¸ psor if vb¸ vs+ 1/4169Double Auction{ Compare trades in 1-price and linear equilibrium{ Linear may “dominate” 1-price BNEz (for uniform valuations, gives higher expected gains than any other BNE)vbTRADE117vsxx1Revelation Principle{ There may be many different “hi” t hi l“mechanisms” to achieve a goalz Direct mechanism is a game where a player’s only action is to submit a (possibly dishonest) claim about his or her typez A direct mechanism in which truth-telling is a Bayesian Nash equilibrium is called “incentive-compatible”Players should be “willing to play”18{Players should be “willing to play”{ Myerson (1979):z Any Bayesian Nash equilibrium of any Bayesian game can be represented by an incentive-compatible direct