Auction Theory. Philip Selin. U.U.D.M. Project Report 2016:27. Department of Mathematics Uppsala University
|
|
- Alicia Hubbard
- 5 years ago
- Views:
Transcription
1 U.U.D.M. Project Report 2016:27 Auction Theory Philip Selin Examensarbete i matematik, 15 hp Handledare: Erik Ekström Examinator: Veronica Crispin Quinonez Juni 2016 Department of Mathematics Uppsala Uniersity
2
3 Auction Theory Philip Sehlin May 23, 2016 Abstract We study auction theory where bidders hae independent priate alues. We describe different auction types, and derie Nash equilibria for the symmetric case in which all bidders hae alues drawn from the same distribution. We also study a case with uncertainty about the number of bidders, and examples with asymmetric distributions. 1
4 Contents 1 Introduction Some Common Auction Forms English Auction Dutch Auction Sealed-Bid First Price Auction (FPA) Sealed-Bid Second Price Auction (SPA) Nash Equilibrium Indepentent Priate Values (IPV) Auctions with Symmetric Bidders First Price Auction Nash Equilibrium Second Price Auction Nash Equilibrium The Enelope Theorem Reenue Equialence First Price Auctions Second Price Auctions Expected Reenue from FPA and SPA Uncertain number of bidders First Price Auctions Second Price Auctions Asymmetric Auctions Strong and weak bidders Equilibrium for strong and weak bidders Reenue Equialence in Asymmetric Auctions Different length of the interal of alues, with uniform distributions Same length of interal of alues, but different distributions Uniform and Normal, both with interal [0,1] Uniform and Exponential, both with interal [0,1] Different settings with three players Uniform, Normal and Exponential, eeryone with interal [0,1] Two players uniform distributed oer [0,1] and one player uniform distributed oer [0,2] Uniform oer [0,1], Uniform oer [0,2] and Normal oer [0,1] References 19 2
5 1 Introduction We consider auctions with independent priate alues (IPV). We let the number of bidders be n, and the set of possible bids is [0, ). A generic bid by player i is denoted b i, and the alue of player i is denoted i. The distribution function of a player s alue is denoted by F. 1.1 Some Common Auction Forms English Auction The English auction is an open ascending price auction. This auction has a low start alue, and increases with small increments until there is only one interested bidder left. This will be the same as the Second Price Auction, since when the second highest bidder drops off the highest bidder wins and pays the second highest price. i max Bidders payoff function [1][2] : u i (b) = u i (b 1,..., b n ) = b j, if b i > max b j. j i j i 0, otherwise Dutch Auction The Dutch auction is the open descending price counterpart of the English auction. Here the auction has a start alue higher than eery interested bidder, and gradually lowered until there is a bidder that is interested at that price. This will be the same as the First Price Auction, since the highest bid wins and pays his own bid. Bidders payoff function [1][2] i b i, if b i = max : u i (b) = u i (b 1,..., b n ) = b i. i N 0, otherwise Sealed-Bid First Price Auction (FPA) In this form the bidders submit their bids in sealed enelopes. The bidder that has the highest bid when the auction has ended wins the auction and pays the price that he bids. Bidders payoff function [1][2] i b i, if b i = max : u i (b) = u i (b 1,..., b n ) = b i. i N 0, otherwise Sealed-Bid Second Price Auction (SPA) This auction is also known as a Vickrey Auction. Here also the bidders submit their bids in sealed enelopes, but in this case if you hae the highest bid, you will pay the second highest bid. i max Bidders payoff function [1][2] : u i (b) = u i (b 1,..., b n ) = b j, if b i > max b j. j i j i 0, otherwise. 3
6 1.2 Nash Equilibrium Nash Equilibrium is a stable state of a system inoling the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged. Definition Nash equilibrium of a strategic game [4] (N, (A i ) i N, (u i ) i N ) is a profile a A of actions with the property that for eery player i N, we hae where u i is the payoff function. u i (a i, a i) u i (a i, a i) for all a i A i a i is the strategy of player i, and a i if the strategies of all players except for player i. 4
7 2 Indepentent Priate Values (IPV) Auctions with Symmetric Bidders In the IPV case there are two important features that define an IPV auction, such as Bidder i s information is independent of j s information. Bidder i s alue is independent of j s information. 2.1 First Price Auction Nash Equilibrium Proposition In a First Price Auction the Nash Equilibrium is to bid [1] where is the lowest possible alue F (x) is the CDF of each player s alue ( ) n 1 F (x) β() = dx F () Proof. N = {1,..., n} : The set of bidders. β i () = [0, ) for each i N : The set of possible bids by player i. A generic bid by player i is denoted b i () β i. Assume that β i () β() since this is a symmetric equilibrium, and all bidders will use the bid function b = β(). i b i, if b i = max Bidders payoff function: u i (b) = u i (b 1,..., b n ) = b i. i N 0, otherwise. A player will bid b if i β 1 (b). ( n 1 A player with aluation i and who bids b i expects to earn ( i b i ) P r(b j i b i )). P r( i β 1 (b)) = F ( β 1 (b) ) and ( ) n 1 ( P r( j i β 1 (b)) = F ( β 1 (b) )) n 1. ( A player with aluation i and who bids b then expects to earn ( i b) F ( β 1 (b) )) n 1. To calculate the Nash Equilibrium we will take the deriatie with respect to b. 5
8 ( ( b) (F ( β 1 (b) )) ) n 1 = (F ( β 1 (b) )) n 1 ( b)(n 1) (F ( β 1 (b) )) n 2 F ( β 1 (b) ) + b β ( β 1 (b) ) = 0 Integraton by parts gies us: This gies us the Nash Equilibrium: ( ) n 2F ( ) n 1 β() (n 1) (F ()) () F () + β () ( ) n 1 ( n 2F β () F () + β()(n 1) F ()) () = ( n 2F = (n 1) F ()) () = 0 ( [ ( ) n 1 ] ( n 2F β() F () = x(n 1) F (x)) (x)dx ( ) n 1 [( n 1 ] dx β() F () = x F (x)) ( ) n 1 ( ) n 1 ( n 1dx β() F () = F () F (x)) β() = ( ) n 1 F (x) dx F () ) (1) 6
9 Example If the alue of each player is uniformly distributed oer [0, 1] we hae We use P roposition that β() = And by soling this integral we get that ( F (x) F () F (x) = x ) n 1 dx = 0 β() = 1 (n 1) = n n So if we use 2 players for this equation we get that β() = 1 2 ( ) x n 1 dx Figure 1: Nash Eqiulibrium for p 1 and p 2, (where both is drawn from the same distribution Uni[0,1]) for any gien alue at [0, 1] 2.2 Second Price Auction Nash Equilibrium Proposition In a Second Price Auction the Nash Equilibrium [1] is to bid his true alue b i = i Proof. We want to show that the strategy (b 1,..., b n ) = ( 1,..., n ) i.e. a truthful bid. We aluate the bids as ( 1,..., n ) = (b 1,..., b n ). Then b 1 = 1 wins the auction, he will pay the second highest bid b 2 = 2. The payoff will then be u 1 = Likewise, for all other bidders i 1 i will need to change her payoff from 0 and will bid higher than 1, in which case the payoff will be u i = i 1 < 0 and will result a negatie payoff. Therefore no one will make a profit when deiate from the strategy. 7
10 2.3 The Enelope Theorem Suppose that b i = b( i ) is the symmetric equilibrium, then i s eqiulibrium payoff gien alue i is: since i is playing his best response we get U( i ) = ( i b( i ))F n 1 ( i ) U( i ) = max b i ( i b i )F n 1 (b 1 (b i )) And if we take the deriatie of this with respect of, we get: d d U() = F n 1 (b 1 (b( i ))) = F n 1 ( i ) =i and i U( i ) = U() + F n 1 ( s)d s and since a bidder with alue neer will win the auction gies us that U() = 0. And by combining the equations aboe we can sole for the equilibrium strategy [2][4] as 2.4 Reenue Equialence b() = i F n 1 (ṽ)dṽ F n 1 () Theorem Reenue Equialence Theorem (RET) [2][4] Suppose n bidders hae alues 1,..., n identically and independently distributed with cumulatie distribution function F ( ). Then any equilibrium of any auction game in which 1. The bidder with the highest alue wins the object. 2. The bidder with alue gets zero profits. With these settings the different auction types generate the same expected reenue, such that the seller will generate the same profit from any type of auction he choose. Proof. We consider a general auction where bidders will place there bids b 1,..., b n. The auction rule specifies for all i, such that x i : B 1... B n {0, 1} t i : B 1... B n R, (2) where x i is the probability that player i will win the object and t i is players i s payment as a function of (b i,..., b n ). Gien this rule, bidder i s expected payoff as a function is: E(U i ) = U i ( i, b i ) = i E b i [x i (b i, b i )] E b i [t i (b i, b i )]. b i, b i is the equilibrium of the auction, so bidder i s equilibrium payoff is therefore: U i ( i, b i ) = i F n 1 ( i ) E i [t i (b i ( i ), b i ( i )]. 8
11 and here we use condition (1) from T heorem where the highest alue wins the object, so we can write E i [x i (b i ( i ), b i ( i )] = F n 1 ( i ). d d U i() = E b i [x i (b i ( i ), b i ( i )] = F n 1 ( i ), =i and also that i i U i ( i ) = U i () + F n 1 (ṽ)dṽ = F n 1 (ṽ)dṽ where we can use (2) from T heorem to write U i () = 0 which uses the condition that the bidder with alue gets zero profits. If we combine our expressions for U i ( i ), we get bidder i s expected payment as: E i [t i (b i, b i )] = i F n 1 ( i ) i F n 1 (ṽ)dṽ = i ṽdf n 1 (ṽ), and since x i is not a part of this expression, bidder i s expected equilibrium payment gien his alue is the same under eery auction that satisfies both (1) and (2) of T heorem , then the expected payment is: E[V 1:n 1 V 1:n 1 i ] = E[V 2:n V 1:n = i ]. So therefore the expected reenue is: E[Reenue] = ne i [i s expected payment i ] = E[V 2:n ]. Just to show how the expected reenue is for the different auctions we show this for the example where there are two bidders with alues drawn from the same distribution U ni[0, 1]. Then the expected reenue is and the expected profit for bidder i with alue is First Price Auctions Bidder i has the alue i b i = (n 1) n i = i 2 The probability that b i wins the auction is P ( i ) = i The expected payoff if b i wins is then i 2 So therefore the expected profit is ( E[U i ( i )] = i i ) i = 2 i 2 2 9
12 2.4.2 Second Price Auctions Bidder i has the alue i b i = i The probability that b i wins the auction is P ( i ) = i The expected payoff if b i wins is then i 2 So therefore the expected profit is ( E[U i ( i )] = i i ) i = 2 i Expected Reenue from FPA and SPA Since the bidder with lowest alue has the expected profit E[U( i )] = 0, therefore: E[U( i )] = E[U( i )] + U (x)dx = 0 + xdx = And the expected profit for each bidder is: E[U 1 ()] = E[U 2 ()] = 1 0 U()d = This gies us the total expected profit for both bidders as: d = 1 6 E[Total bidder profit] = 1 3 And hence expected reenue is gien by: E[Reenue] = E[Surplus] E[Total bidder profit] where E[Surplus] = E[max{ i, j }] = 2 3 So this gies us that the expected reenue is E[Reenue] = =
13 3 Uncertain number of bidders 3.1 First Price Auctions Example If we let m be the number of possible bidders, and set the possibilities that player i will attend as (1 p i ) and p i else. Let m = 2 and p 1 (1) be the probability that player 1 beliees that there only will be 1 player (himself) in the auction. Since this is a symmetric auction with only two buyers we can write this as the following equation p 1 (1) = p and p 1 (2) = (1 p) F () = p( b) + (1 p)f (β 1 (b))( b) Maximize by taking the deriatie df () db If we use that =β 1 (b) we get = p (1 p)f (β 1 (b)) + (1 p)( b)f(β 1 1 (b)) β (β 1 (b)) = 0 df () (1 p)( β()f()) = p (1 p)f () + = 0 db β df () = (p + (1 p)f )β + (1 p)f()β() = (1 p)f() db df () ( (p ) ) = + (1 p)f β = (1 p)f() db So the equilibrium function is β = (1 p) 0 uf(u)du p + (1 p)f () (3) So to see the price difference between full information about the number of players who will attend we can take the example with two players in both situations with p 1 (1) = p 1 (2) = 0.5 for the equation with uncertain number of bidders. Uncertain number of bidders, FPA case gies us the Nash Equilibrium at b if P A = (1 p) 2 2(p + (1 p)) and then will gie us b if P A = (0.5) 2 2(0.5 + (0.5)) = 2 2(1 + ) 11
14 Example If we here instead of Example take three players instead of two where p 1 (1) = p, p 1 (2) = q and p 1 (3) = (1 p q) we will get F () = p( b) + qf (β 1 (b))( b) + (1 p q)f (β 1 (x)) 2 ( b) Maximize by taking the deriatie df () db If we use that =β 1 (b) we get = p qf (β 1 (b)) + q( b)f(β 1 1 (b)) β (β 1 (b)) (1 p q)f (β 1 (b)) ((1 p q)( b)f (β 1 (b))f(β() 1 1 β (β 1 ()) ) = 0 df () 1 = p qf () + q( β()f()) db β () (1 p q)f () (1 p q)( b)f ()f() β () = 0 df () = p qf () (1 p q)f () 2 q( b)f() + 2(1 p q)( b)(f ()f()) + db β () df () = (p + qf () + (1 p q)f () 2 )β (qf()β() + 2(1 p q)f ()f()β()) = db = (q + 2(1 p q)f ()f()) df () = ((p + qf () + (1 p q)f () 2 )β) = q + 2(1 p q)f ()f() db So the equilibrium function is β = q 0 udu + 2(1 p q) 0 uf (u)f(u)du p + qf () + (1 p q)f () 2 (4) And here we use as aboe that p 1 (1) = p 1 (2) = p 1 (3) = 1/3 for the equation with uncertain number of bidders. Uncertain number of bidders, FPA case gies us the Nash Equilibrium at b if P A = q (1 p q) 3 p + q + (1 p q) 12
15 3.2 Second Price Auctions For the Second Price Auction with uncertain number of bidders we still hae the same strategy and Nash equilibrium since it will not change with the uncertainty of players attendance. So the weakly dominant strategy is to bid his true alue b i = i. 4 Asymmetric Auctions Eery auction we hae studied so far has been symmetric in the way that all players hae the same distribution and the same interal of alues, so that all players draw eery alues equal likely. The scenario where it is an symmetric auction is not that common in practice, that is why I also want to study the asymmetric auctions where players are randomly drawn to hae different distributions and alues. We want to see how the Nash Equilibrium differ if we hae this conditions in different type of auctions. 4.1 Strong and weak bidders [3] When we talk about asymmetric auctions it is common to refer to strong and weak bidders, such that a strong bidder dominated the weak bidder with the alues, for example The strong bidder has a shifted distribution such that F S Uni[2, 3] and F W Uni[0, 1] where F S is the strong bidders distribution and F W is the weak bidders distribution. The strong bidder has a stretched distribution such that F S Uni[0, 2] and F W Uni[0, 1] Equilibrium for strong and weak bidders If we let b S and b W be the equilibrium bid functions, then we hae the problem For the strong bidder as which hae the first-order condition max F W (b 1 b W (b))( S b) f W (b 1 W (b)) F W (b 1 W (b))(b 1 W ) (b) 1 S b and if we set this equal to 0 at S = b 1 (b) we get For the weak bidder as which gies the first-order condition S f W (b 1 W (b)) F W (b 1 W (b))(b 1 W ) (b) 1 b 1 S (b) b max F S (b 1 b S (b))( W b) f S (b 1 S (b)) F S (b 1 S (b))(b 1 S ) (b) 1 W b 13
16 and if we set this equal to 0 at W = b 1 W (b) we get f S (b 1 S (b)) F S (b 1 S (b))(b 1 S ) (b) 1 b 1 W (b) b Theorem Suppose F S conditionally first-order stochastically dominates F W, such that F S (x) F W (x) [2]. Then comparing a first price auction and a second price auction, both who are uniform distributed, Eery type of strong bidder prefers the second price auction since the expected payoff is higher in the second price auction for the strong bidder. Eery type of weak bidder prefers the first price auction since the expected payoff is higher in the first price auction for the weak bidder. Proof. For this proof b S () and b W () hae the same range, so we define a matching function as m() b 1 W (b S()) as a weak bidder who bids equal as the strong bidder in the first price auction. Since b S () < b W () we know that m() <. We calculate the strong bidders expected payoff as E(U( i )) = P r(b W ( W ) < b)( b) and by taking the deriatie with respect to this gie us and by replacing b = b S () this gie us E (U( i )) = P r(b W ( W ) < b) E (U( i )) = P r(b W ( W ) < b S ()) = P r( W < m()) = F W (m()) since P r( < a) = F (a) when uniform distributed. By the enelope theorem we get VS F P () = F W (m())d For the second price auction, both bidders bid their true alue, so and so E (U( i )) = P r( W < ) = F W () VS SP () = F W ()d Since m() < and F W is strictly increasing, the strong bidder prefers the second price auction. By the exact same logic, the weak bidders expected payoff for the first price auction is and for the second price auction V F P W () = F S (m 1 ())ds VW SP () = F S ()d Since m 1 () > the expected payoff is higher with the first price auction for the weak bidder. 14
17 4.1.2 Reenue Equialence in Asymmetric Auctions Proposition With asymmetric bidders, the expected reenue in a first price auction may exceed that in an English auction [1][2]. 1 Example Suppose that the weak buyers aluation is distributed as b W Uni[0, 1+z ] and the 1 strong buyers aluation is distributed as b S Uni[0, 1 z ]. So the strong buyer has a wider interal than the weak buyer. If z = 0, both b W, b S Uni[0, 1] and b 1 i (b) = 2b is buyer i s equilibrium inerse bid function in the first price auction. A buyer with aluation 2b has a probability to win P r(win i = 2b) = 2b and the expected payment is therefore P r(win i = 2b)( i b i ) = 2b(2b b) = 2b 2 When z becomes positie, in the English auction the weak buyer with aluation 2b wins with probability 2b(1 z) and the expected payment is 2b 2 (1 z) In a high-bid auction buyers do not use b 1 1 i (b) = 2b, if they did the strong buyer would outbid the weaker buyer by 2(1 z) to 1 2(1+z), and so can reduce his bid and still win with probability 1. So for equilibrium the strong buyer must reduce his bid as a function of his aluation. A reduction would make the weak buyer to bid more aggressiely than with b 1 i (b) = 2b, since the strong buyers bids are distributed more densely than before. In equilibrium the weak and strong inerse bid functions are therefore b 1 W (b) = 2b 1 + z(2b) 2 and b 1 S (b) = 2b 1 z(2b) 2 The cumulatie distribution function for the winning bid of the first price auction is F F P A (b) = F S (b 1 S (b))f W (b 1 W (b)) = (1 z)b 1 S (b)(1 + z)b 1 W (b) = (1 z2 )(2b) 2 1 z 2 (2b) 4 For the English auction the second aluation is less than b iff it is not the case that both aluations are higher. So F EA (b) = 1 (1 F S (b))(1 F W (b)) = F S +F W F S F W = (1 z)b+(1+z)b (1 z 2 )b 2 = 2b (1 z)b 2 The cumulatie distribution function in the open auction is increasing in z. If z = 0 the two distributions yield the same expected reenue. When z > 0 the expected reenue is strictly greater for the first price auction than for the English auction. 15
18 4.2 Different length of the interal of alues, with uniform distributions In this case we compare uniform distributions with two players where p 1 Uni(0, 1) and p 2 Uni(0, 2) Figure 2: Nash Equilibrium for p 1 and p 2 for any gien alue at [0, 2] So here the Nash Equilibrium would be gien by b i = 2 3 for both p 1 and p 2 if p 1 alue is 1, and p 2 alue is Same length of interal of alues, but different distributions When using normal and exponential distribution I hae scaled these distributions such that the area under the interal is equal to 1. The normal distribution is scaled to the interal [0, 1], so C 2π e (x 0.5)2 2, C > 1 The exponential distribution is scaled to the interal [0, 1], so Ce x, C > Uniform and Normal, both with interal [0,1] In this first case we compare uniform distribution with normal distribution, both with the interal [0, 1]. Figure 3: Nash Equilibrium for p 1 Normal(0.5, 1) and p 2 Uni(0, 1) for any gien alue at [0, 1] 16
19 4.3.2 Uniform and Exponential, both with interal [0,1] In this second case we compare uniform distribution with exponential distribution, both with the interal [0, 1]. Figure 4: Nash Equilibrium for p 1 Exp(1) and p 2 Uni(0, 1) for any gien alue at [0, 1] 4.4 Different settings with three players We now just hae looked at asymmetric auctions with two players, but how will the equilibrium functions look when we instead uses three players? Here I also hae used the scaled distributions for normal and exponential, such that the normal distribution is scaled to the interal [0, 1], so C 2π e (x 0.5)2 2, C > 1 the exponential distribution is scaled to the interal [0, 1], so Ce x, C > Uniform, Normal and Exponential, eeryone with interal [0,1] Figure 5: Nash Equilibrium for p 1 Exp(1), p 2 Normal(0.5, 1) and p 3 Uni(0, 1) for any gien alue at [0, 1] 17
20 4.4.2 Two players uniform distributed oer [0,1] and one player uniform distributed oer [0,2] Figure 6: Nash Equilibrium for p 1 = p 2 Uni(0, 1) and p 3 Uni(0, 1) for any gien alue at [0, 2] Uniform oer [0,1], Uniform oer [0,2] and Normal oer [0,1] Figure 7: Nash Equilibrium for p 1 Normal(0.5, 1), p 2 Uni(0, 1) and p 3 Uni(0, 2) for any gien alue at [0, 2] 18
21 5 References [1] Krishna, Vijay. Auction Theory. San Diego: Academic, [2] Lein, J. Stanford. Auction Theory, [3] Maskin, E. and Riley, J. Harard Uniersity. Asymmetric Auctions. 1st ed, [4] Milgrom, Paul R. Putting Auction Theory to Work. Cambridge, UK: Cambridge UP,
Auction Theory Lecture Note, David McAdams, Fall Bilateral Trade
Auction Theory Lecture Note, Daid McAdams, Fall 2008 1 Bilateral Trade ** Reised 10-17-08: An error in the discussion after Theorem 4 has been corrected. We shall use the example of bilateral trade to
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012 The Revenue Equivalence Theorem Note: This is a only a draft
More informationAuction theory. Filip An. U.U.D.M. Project Report 2018:35. Department of Mathematics Uppsala University
U.U.D.M. Project Report 28:35 Auction theory Filip An Examensarbete i matematik, 5 hp Handledare: Erik Ekström Examinator: Veronica Crispin Quinonez Augusti 28 Department of Mathematics Uppsala University
More informationGames of Incomplete Information ( 資訊不全賽局 ) Games of Incomplete Information
1 Games of Incomplete Information ( 資訊不全賽局 ) Wang 2012/12/13 (Lecture 9, Micro Theory I) Simultaneous Move Games An Example One or more players know preferences only probabilistically (cf. Harsanyi, 1976-77)
More informationOptimal auctions with endogenous budgets
Optimal auctions with endogenous budgets Brian Baisa and Stanisla Rabinoich September 14, 2015 Abstract We study the benchmark independent priate alue auction setting when bidders hae endogenously determined
More informationAll-pay auctions with risk-averse players
Int J Game Theory 2006) 34:583 599 DOI 10.1007/s00182-006-0034-5 ORIGINAL ARTICLE All-pay auctions with risk-aerse players Gadi Fibich Arieh Gaious Aner Sela Accepted: 28 August 2006 / Published online:
More informationGame Theory Solutions to Problem Set 11
Game Theory Solutions to Problem Set. A seller owns an object that a buyer wants to buy. The alue of the object to the seller is c: The alue of the object to the buyer is priate information. The buyer
More informationAuctions: Types and Equilibriums
Auctions: Types and Equilibriums Emrah Cem and Samira Farhin University of Texas at Dallas emrah.cem@utdallas.edu samira.farhin@utdallas.edu April 25, 2013 Emrah Cem and Samira Farhin (UTD) Auctions April
More informationAll-Pay Auctions with Risk-Averse Players
All-Pay Auctions with Risk-Aerse Players Gadi Fibich Arieh Gaious Aner Sela December 17th, 2005 Abstract We study independent priate-alue all-pay auctions with risk-aerse players. We show that: 1) Players
More informationProblem Set 3: Suggested Solutions
Microeconomics: Pricing 3E00 Fall 06. True or false: Problem Set 3: Suggested Solutions (a) Since a durable goods monopolist prices at the monopoly price in her last period of operation, the prices must
More informationECON20710 Lecture Auction as a Bayesian Game
ECON7 Lecture Auction as a Bayesian Game Hanzhe Zhang Tuesday, November 3, Introduction Auction theory has been a particularly successful application of game theory ideas to the real world, with its uses
More informationInformative advertising under duopoly
Informatie adertising under duopoly Scott McCracken June 6, 2011 Abstract We consider a two-stage duopoly model of costless adertising: in the first stage each firm simultaneously chooses the accuracy
More informationRevenue Equivalence and Mechanism Design
Equivalence and Design Daniel R. 1 1 Department of Economics University of Maryland, College Park. September 2017 / Econ415 IPV, Total Surplus Background the mechanism designer The fact that there are
More information1 Auctions. 1.1 Notation (Symmetric IPV) Independent private values setting with symmetric riskneutral buyers, no budget constraints.
1 Auctions 1.1 Notation (Symmetric IPV) Ancient market mechanisms. use. A lot of varieties. Widespread in Independent private values setting with symmetric riskneutral buyers, no budget constraints. Simple
More informationNotes on Auctions. Theorem 1 In a second price sealed bid auction bidding your valuation is always a weakly dominant strategy.
Notes on Auctions Second Price Sealed Bid Auctions These are the easiest auctions to analyze. Theorem In a second price sealed bid auction bidding your valuation is always a weakly dominant strategy. Proof
More informationRevenue Equivalence Theorem (RET)
Revenue Equivalence Theorem (RET) Definition Consider an auction mechanism in which, for n risk-neutral bidders, each has a privately know value drawn independently from a common, strictly increasing distribution.
More informationAuctions. Economics Auction Theory. Instructor: Songzi Du. Simon Fraser University. November 17, 2016
Auctions Economics 383 - Auction Theory Instructor: Songzi Du Simon Fraser University November 17, 2016 ECON 383 (SFU) Auctions November 17, 2016 1 / 28 Auctions Mechanisms of transaction: bargaining,
More informationRecap First-Price Revenue Equivalence Optimal Auctions. Auction Theory II. Lecture 19. Auction Theory II Lecture 19, Slide 1
Auction Theory II Lecture 19 Auction Theory II Lecture 19, Slide 1 Lecture Overview 1 Recap 2 First-Price Auctions 3 Revenue Equivalence 4 Optimal Auctions Auction Theory II Lecture 19, Slide 2 Motivation
More informationWe examine the impact of risk aversion on bidding behavior in first-price auctions.
Risk Aversion We examine the impact of risk aversion on bidding behavior in first-price auctions. Assume there is no entry fee or reserve. Note: Risk aversion does not affect bidding in SPA because there,
More informationSignaling in an English Auction: Ex ante versus Interim Analysis
Signaling in an English Auction: Ex ante versus Interim Analysis Peyman Khezr School of Economics University of Sydney and Abhijit Sengupta School of Economics University of Sydney Abstract This paper
More informationAuctions That Implement Efficient Investments
Auctions That Implement Efficient Investments Kentaro Tomoeda October 31, 215 Abstract This article analyzes the implementability of efficient investments for two commonly used mechanisms in single-item
More informationAuction Theory: Some Basics
Auction Theory: Some Basics Arunava Sen Indian Statistical Institute, New Delhi ICRIER Conference on Telecom, March 7, 2014 Outline Outline Single Good Problem Outline Single Good Problem First Price Auction
More informationStrategy -1- Strategic equilibrium in auctions
Strategy -- Strategic equilibrium in auctions A. Sealed high-bid auction 2 B. Sealed high-bid auction: a general approach 6 C. Other auctions: revenue equivalence theorem 27 D. Reserve price in the sealed
More informationLecture 6 Applications of Static Games of Incomplete Information
Lecture 6 Applications of Static Games of Incomplete Information Good to be sold at an auction. Which auction design should be used in order to maximize expected revenue for the seller, if the bidders
More informationAuctions. Microeconomics II. Auction Formats. Auction Formats. Many economic transactions are conducted through auctions treasury bills.
Auctions Microeconomics II Auctions Levent Koçkesen Koç University Many economic transactions are conducted through auctions treasury bills art work foreign exchange antiques publicly owned companies cars
More informationMicroeconomic Theory III Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 14.123 Microeconomic Theory III Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIT 14.123 (2009) by
More information1 Theory of Auctions. 1.1 Independent Private Value Auctions
1 Theory of Auctions 1.1 Independent Private Value Auctions for the moment consider an environment in which there is a single seller who wants to sell one indivisible unit of output to one of n buyers
More informationOnline Appendix for The E ect of Diversi cation on Price Informativeness and Governance
Online Appendix for The E ect of Diersi cation on Price Informatieness and Goernance B Goernance: Full Analysis B. Goernance Through Exit: Full Analysis This section analyzes the exit model of Section.
More informationBayesian Nash Equilibrium
Bayesian Nash Equilibrium We have already seen that a strategy for a player in a game of incomplete information is a function that specifies what action or actions to take in the game, for every possibletypeofthatplayer.
More informationIndependent Private Value Auctions
John Nachbar April 16, 214 ndependent Private Value Auctions The following notes are based on the treatment in Krishna (29); see also Milgrom (24). focus on only the simplest auction environments. Consider
More informationGame Theory Lecture #16
Game Theory Lecture #16 Outline: Auctions Mechanism Design Vickrey-Clarke-Groves Mechanism Optimizing Social Welfare Goal: Entice players to select outcome which optimizes social welfare Examples: Traffic
More informationGame Theory Problem Set 4 Solutions
Game Theory Problem Set 4 Solutions 1. Assuming that in the case of a tie, the object goes to person 1, the best response correspondences for a two person first price auction are: { }, < v1 undefined,
More informationRevenue Equivalence and Income Taxation
Journal of Economics and Finance Volume 24 Number 1 Spring 2000 Pages 56-63 Revenue Equivalence and Income Taxation Veronika Grimm and Ulrich Schmidt* Abstract This paper considers the classical independent
More informationAll Equilibrium Revenues in Buy Price Auctions
All Equilibrium Revenues in Buy Price Auctions Yusuke Inami Graduate School of Economics, Kyoto University This version: January 009 Abstract This note considers second-price, sealed-bid auctions with
More informationOptimal Auctions. Game Theory Course: Jackson, Leyton-Brown & Shoham
Game Theory Course: Jackson, Leyton-Brown & Shoham So far we have considered efficient auctions What about maximizing the seller s revenue? she may be willing to risk failing to sell the good she may be
More informationMultiunit Auctions: Package Bidding October 24, Multiunit Auctions: Package Bidding
Multiunit Auctions: Package Bidding 1 Examples of Multiunit Auctions Spectrum Licenses Bus Routes in London IBM procurements Treasury Bills Note: Heterogenous vs Homogenous Goods 2 Challenges in Multiunit
More informationAuction. Li Zhao, SJTU. Spring, Li Zhao Auction 1 / 35
Auction Li Zhao, SJTU Spring, 2017 Li Zhao Auction 1 / 35 Outline 1 A Simple Introduction to Auction Theory 2 Estimating English Auction 3 Estimating FPA Li Zhao Auction 2 / 35 Background Auctions have
More information(v 50) > v 75 for all v 100. (d) A bid of 0 gets a payoff of 0; a bid of 25 gets a payoff of at least 1 4
Econ 85 Fall 29 Problem Set Solutions Professor: Dan Quint. Discrete Auctions with Continuous Types (a) Revenue equivalence does not hold: since types are continuous but bids are discrete, the bidder with
More informationAuction is a commonly used way of allocating indivisible
Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 16. BIDDING STRATEGY AND AUCTION DESIGN Auction is a commonly used way of allocating indivisible goods among interested buyers. Used cameras, Salvator Mundi, and
More informationUp till now, we ve mostly been analyzing auctions under the following assumptions:
Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 7 Sept 27 2007 Tuesday: Amit Gandhi on empirical auction stuff p till now, we ve mostly been analyzing auctions under the following assumptions:
More informationAuctions. Episode 8. Baochun Li Professor Department of Electrical and Computer Engineering University of Toronto
Auctions Episode 8 Baochun Li Professor Department of Electrical and Computer Engineering University of Toronto Paying Per Click 3 Paying Per Click Ads in Google s sponsored links are based on a cost-per-click
More informationAuctions. Agenda. Definition. Syllabus: Mansfield, chapter 15 Jehle, chapter 9
Auctions Syllabus: Mansfield, chapter 15 Jehle, chapter 9 1 Agenda Types of auctions Bidding behavior Buyer s maximization problem Seller s maximization problem Introducing risk aversion Winner s curse
More informationDay 3. Myerson: What s Optimal
Day 3. Myerson: What s Optimal 1 Recap Last time, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] and density f i residual valuation
More informationOctober An Equilibrium of the First Price Sealed Bid Auction for an Arbitrary Distribution.
October 13..18.4 An Equilibrium of the First Price Sealed Bid Auction for an Arbitrary Distribution. We now assume that the reservation values of the bidders are independently and identically distributed
More informationGames with Private Information 資訊不透明賽局
Games with Private Information 資訊不透明賽局 Joseph Tao-yi Wang 00/0/5 (Lecture 9, Micro Theory I-) Market Entry Game with Private Information (-,4) (-,) BE when p < /: (,, ) (-,4) (-,) BE when p < /: (,, )
More informationCS 573: Algorithmic Game Theory Lecture date: March 26th, 2008
CS 573: Algorithmic Game Theory Lecture date: March 26th, 28 Instructor: Chandra Chekuri Scribe: Qi Li Contents Overview: Auctions in the Bayesian setting 1 1 Single item auction 1 1.1 Setting............................................
More informationAuctions. Michal Jakob Agent Technology Center, Dept. of Computer Science and Engineering, FEE, Czech Technical University
Auctions Michal Jakob Agent Technology Center, Dept. of Computer Science and Engineering, FEE, Czech Technical University AE4M36MAS Autumn 2015 - Lecture 12 Where are We? Agent architectures (inc. BDI
More informationSocial Network Analysis
Lecture IV Auctions Kyumars Sheykh Esmaili Where Are Auctions Appropriate? Where sellers do not have a good estimate of the buyers true values for an item, and where buyers do not know each other s values
More informationToday. Applications of NE and SPNE Auctions English Auction Second-Price Sealed-Bid Auction First-Price Sealed-Bid Auction
Today Applications of NE and SPNE Auctions English Auction Second-Price Sealed-Bid Auction First-Price Sealed-Bid Auction 2 / 26 Auctions Used to allocate: Art Government bonds Radio spectrum Forms: Sequential
More informationKIER DISCUSSION PAPER SERIES
KIER DISCUSSION PAPER SERIES KYOTO INSTITUTE OF ECONOMIC RESEARCH http://www.kier.kyoto-u.ac.jp/index.html Discussion Paper No. 657 The Buy Price in Auctions with Discrete Type Distributions Yusuke Inami
More informationAuctioning one item. Tuomas Sandholm Computer Science Department Carnegie Mellon University
Auctioning one item Tuomas Sandholm Computer Science Department Carnegie Mellon University Auctions Methods for allocating goods, tasks, resources... Participants: auctioneer, bidders Enforced agreement
More informationUCLA Department of Economics Ph.D. Preliminary Exam Industrial Organization Field Exam (Spring 2010) Use SEPARATE booklets to answer each question
Wednesday, June 23 2010 Instructions: UCLA Department of Economics Ph.D. Preliminary Exam Industrial Organization Field Exam (Spring 2010) You have 4 hours for the exam. Answer any 5 out 6 questions. All
More informationMarch 30, Why do economists (and increasingly, engineers and computer scientists) study auctions?
March 3, 215 Steven A. Matthews, A Technical Primer on Auction Theory I: Independent Private Values, Northwestern University CMSEMS Discussion Paper No. 196, May, 1995. This paper is posted on the course
More informationECO 426 (Market Design) - Lecture 8
ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Revenue equivalence Model: N bidders Bidder i has valuation v i Each v i is drawn independently from the same distribution F (e.g. U[0,
More informationECON Microeconomics II IRYNA DUDNYK. Auctions.
Auctions. What is an auction? When and whhy do we need auctions? Auction is a mechanism of allocating a particular object at a certain price. Allocating part concerns who will get the object and the price
More informationAd Auctions October 8, Ad Auctions October 8, 2010
Ad Auctions October 8, 2010 1 Ad Auction Theory: Literature Old: Shapley-Shubik (1972) Leonard (1983) Demange-Gale (1985) Demange-Gale-Sotomayor (1986) New: Varian (2006) Edelman-Ostrovsky-Schwarz (2007)
More informationAuctions 1: Common auctions & Revenue equivalence & Optimal mechanisms. 1 Notable features of auctions. use. A lot of varieties.
1 Notable features of auctions Ancient market mechanisms. use. A lot of varieties. Widespread in Auctions 1: Common auctions & Revenue equivalence & Optimal mechanisms Simple and transparent games (mechanisms).
More informationCSV 886 Social Economic and Information Networks. Lecture 4: Auctions, Matching Markets. R Ravi
CSV 886 Social Economic and Information Networks Lecture 4: Auctions, Matching Markets R Ravi ravi+iitd@andrew.cmu.edu Schedule 2 Auctions 3 Simple Models of Trade Decentralized Buyers and sellers have
More informationBayesian games and their use in auctions. Vincent Conitzer
Bayesian games and their use in auctions Vincent Conitzer conitzer@cs.duke.edu What is mechanism design? In mechanism design, we get to design the game (or mechanism) e.g. the rules of the auction, marketplace,
More informationAlgorithmic Game Theory
Algorithmic Game Theory Lecture 10 06/15/10 1 A combinatorial auction is defined by a set of goods G, G = m, n bidders with valuation functions v i :2 G R + 0. $5 Got $6! More? Example: A single item for
More informationISSN BWPEF Uninformative Equilibrium in Uniform Price Auctions. Arup Daripa Birkbeck, University of London.
ISSN 1745-8587 Birkbeck Working Papers in Economics & Finance School of Economics, Mathematics and Statistics BWPEF 0701 Uninformative Equilibrium in Uniform Price Auctions Arup Daripa Birkbeck, University
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2017
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2017 These notes have been used and commented on before. If you can still spot any errors or have any suggestions for improvement, please
More informationAuctions. Michal Jakob Agent Technology Center, Dept. of Computer Science and Engineering, FEE, Czech Technical University
Auctions Michal Jakob Agent Technology Center, Dept. of Computer Science and Engineering, FEE, Czech Technical University AE4M36MAS Autumn 2014 - Lecture 12 Where are We? Agent architectures (inc. BDI
More informationSubjects: What is an auction? Auction formats. True values & known values. Relationships between auction formats
Auctions Subjects: What is an auction? Auction formats True values & known values Relationships between auction formats Auctions as a game and strategies to win. All-pay auctions What is an auction? An
More informationOn Existence of Equilibria. Bayesian Allocation-Mechanisms
On Existence of Equilibria in Bayesian Allocation Mechanisms Northwestern University April 23, 2014 Bayesian Allocation Mechanisms In allocation mechanisms, agents choose messages. The messages determine
More informationSecond-chance offers
Second-chance offers By Rodney J. Garratt and Thomas Tröger February 20, 2013 Abstract We study the second-price offer feature of ebay auctions in which the seller has multiple units. Perhaps surprisingly,
More informationConsider the following (true) preference orderings of 4 agents on 4 candidates.
Part 1: Voting Systems Consider the following (true) preference orderings of 4 agents on 4 candidates. Agent #1: A > B > C > D Agent #2: B > C > D > A Agent #3: C > B > D > A Agent #4: D > C > A > B Assume
More informationAuction Theory - An Introduction
Auction Theory - An Introduction Felix Munoz-Garcia School of Economic Sciences Washington State University February 20, 2015 Introduction Auctions are a large part of the economic landscape: Since Babylon
More informationNBER WORKING PAPER SERIES BIDDING WITH SECURITIES: AUCTIONS AND SECURITY DESIGN. Peter M. DeMarzo Ilan Kremer Andrzej Skrzypacz
NBER WORKING PAPER SERIES BIDDING WITH SECURITIES: AUCTIONS AND SECURITY DESIGN Peter M. DeMarzo Ilan Kremer Andrzej Skrzypacz Working Paper 10891 http://www.nber.org/papers/w10891 NATIONAL BUREAU OF ECONOMIC
More informationParkes Auction Theory 1. Auction Theory. Jacomo Corbo. School of Engineering and Applied Science, Harvard University
Parkes Auction Theory 1 Auction Theory Jacomo Corbo School of Engineering and Applied Science, Harvard University CS 286r Spring 2007 Parkes Auction Theory 2 Auctions: A Special Case of Mech. Design Allocation
More informationGame Theory I 1 / 38
Game Theory I 1 / 38 A Strategic Situation (due to Ben Polak) Player 2 α β Player 1 α B-, B- A, C β C, A A-, A- 2 / 38 Selfish Students Selfish 2 α β Selfish 1 α 1, 1 3, 0 β 0, 3 2, 2 3 / 38 Selfish Students
More informationAnswers to Problem Set 4
Answers to Problem Set 4 Economics 703 Spring 016 1. a) The monopolist facing no threat of entry will pick the first cost function. To see this, calculate profits with each one. With the first cost function,
More informationGame Theory I 1 / 38
Game Theory I 1 / 38 A Strategic Situation (due to Ben Polak) Player 2 α β Player 1 α B-, B- A, C β C, A A-, A- 2 / 38 Selfish Students Selfish 2 α β Selfish 1 α 1, 1 3, 0 β 0, 3 2, 2 No matter what Selfish
More informationRecalling that private values are a special case of the Milgrom-Weber setup, we ve now found that
Econ 85 Advanced Micro Theory I Dan Quint Fall 27 Lecture 12 Oct 16 27 Last week, we relaxed both private values and independence of types, using the Milgrom- Weber setting of affiliated signals. We found
More informationChapter 3. Dynamic discrete games and auctions: an introduction
Chapter 3. Dynamic discrete games and auctions: an introduction Joan Llull Structural Micro. IDEA PhD Program I. Dynamic Discrete Games with Imperfect Information A. Motivating example: firm entry and
More informationThe FedEx Problem (Working Paper)
The FedEx Problem (Working Paper) Amos Fiat Kira Goldner Anna R. Karlin Elias Koutsoupias June 6, 216 Remember that Time is Money Abstract Benjamin Franklin in Adice to a Young Tradesman (1748) Consider
More informationMeans of Payment and Timing of Mergers and Acquisitions in a Dynamic Economy
Means of Payment and Timing of Mergers and Acquisitions in a Dynamic Economy Alexander S. Gorbenko London Business School Andrey Malenko MIT Sloan School of Management This ersion: January 2014 We are
More informationBlind Portfolio Auctions via Intermediaries
Blind Portfolio Auctions via Intermediaries Michael Padilla Stanford University (joint work with Benjamin Van Roy) April 12, 2011 Computer Forum 2011 Michael Padilla (Stanford University) Blind Portfolio
More information2 Comparison Between Truthful and Nash Auction Games
CS 684 Algorithmic Game Theory December 5, 2005 Instructor: Éva Tardos Scribe: Sameer Pai 1 Current Class Events Problem Set 3 solutions are available on CMS as of today. The class is almost completely
More informationby open ascending bid ("English") auction Auctioneer raises asking price until all but one bidder drops out
Auctions. Auction off a single item (a) () (c) (d) y open ascending id ("English") auction Auctioneer raises asking price until all ut one idder drops out y Dutch auction (descending asking price) Auctioneer
More informationThe Impact of a Right of First Refusal Clause in a First-Price Auction with Unknown Heterogeneous Risk-Aversion
The Impact of a Right of First Refusal Clause in a First-Price Auction with Unknown Heterogeneous Risk-Aversion Karine Brisset, François Cochard and François Maréchal January 2017 Abstract We consider
More informationCUR 412: Game Theory and its Applications, Lecture 4
CUR 412: Game Theory and its Applications, Lecture 4 Prof. Ronaldo CARPIO March 22, 2015 Homework #1 Homework #1 will be due at the end of class today. Please check the website later today for the solutions
More informationCS711: Introduction to Game Theory and Mechanism Design
CS711: Introduction to Game Theory and Mechanism Design Teacher: Swaprava Nath Domination, Elimination of Dominated Strategies, Nash Equilibrium Domination Normal form game N, (S i ) i N, (u i ) i N Definition
More informationInefficiency of Collusion at English Auctions
Inefficiency of Collusion at English Auctions Giuseppe Lopomo Duke University Robert C. Marshall Penn State University June 17, 2005 Leslie M. Marx Duke University Abstract In its attempts to deter and
More informationLECTURE 7: SINGLE OBJECT AUCTIONS. 9/11/2010 EC3322 Autumn
LECTURE 7: SINGLE OBJECT AUCTIONS 9/11/2010 EC3322 Autumn 2010 1 Reading Kagel, John H. (1995) Auctions: A survey of experimental results. In: Kagel, John H., Roth, Alvin (Eds.), The Handbook of Experimental
More informationHW Consider the following game:
HW 1 1. Consider the following game: 2. HW 2 Suppose a parent and child play the following game, first analyzed by Becker (1974). First child takes the action, A 0, that produces income for the child,
More informationZero Intelligence Plus and Gjerstad-Dickhaut Agents for Sealed Bid Auctions
Zero Intelligence Plus and Gjerstad-Dickhaut Agents for Sealed Bid Auctions A. J. Bagnall and I. E. Toft School of Computing Sciences University of East Anglia Norwich England NR4 7TJ {ajb,it}@cmp.uea.ac.uk
More informationSYMMETRIC AUCTIONS: FINDING NUMERICAL SOLUTIONS BY USING AUCTION SOLVER
UDK 35.073.53.01 SYMMETRIC AUCTIONS: FINDING NUMERICAL SOLUTIONS BY USING AUCTION SOLVER Dushko Josheski 30 Abstract This essay theoretically investigates symmetric types of auctions and provides simulation
More informationSimon Fraser University Spring 2014
Simon Fraser University Spring 2014 Econ 302 D200 Final Exam Solution This brief solution guide does not have the explanations necessary for full marks. NE = Nash equilibrium, SPE = subgame perfect equilibrium,
More informationA Systematic Presentation of Equilibrium Bidding Strategies to Undergradudate Students
A Systematic Presentation of Equilibrium Bidding Strategies to Undergradudate Students Felix Munoz-Garcia School of Economic Sciences Washington State University April 8, 2014 Introduction Auctions are
More informationAuctions in the wild: Bidding with securities. Abhay Aneja & Laura Boudreau PHDBA 279B 1/30/14
Auctions in the wild: Bidding with securities Abhay Aneja & Laura Boudreau PHDBA 279B 1/30/14 Structure of presentation Brief introduction to auction theory First- and second-price auctions Revenue Equivalence
More informationECO 426 (Market Design) - Lecture 9
ECO 426 (Market Design) - Lecture 9 Ettore Damiano November 30, 2015 Common Value Auction In a private value auction: the valuation of bidder i, v i, is independent of the other bidders value In a common
More informationLecture #6: Auctions: Theory and Applications. Prof. Dr. Sven Seuken
Lecture #6: Auctions: Theory and Applications Prof. Dr. Sven Seuken 15.3.2012 Housekeeping Questions? Concerns? BitTorrent homework assignment? Posting on NB: do not copy/paste from PDFs Game Theory Homework:
More informationAnalyses of an Internet Auction Market Focusing on the Fixed-Price Selling at a Buyout Price
Master Thesis Analyses of an Internet Auction Market Focusing on the Fixed-Price Selling at a Buyout Price Supervisor Associate Professor Shigeo Matsubara Department of Social Informatics Graduate School
More informationCUR 412: Game Theory and its Applications, Lecture 4
CUR 412: Game Theory and its Applications, Lecture 4 Prof. Ronaldo CARPIO March 27, 2015 Homework #1 Homework #1 will be due at the end of class today. Please check the website later today for the solutions
More information1 Intro to game theory
These notes essentially correspond to chapter 14 of the text. There is a little more detail in some places. 1 Intro to game theory Although it is called game theory, and most of the early work was an attempt
More informationStrategy -1- Strategy
Strategy -- Strategy A Duopoly, Cournot equilibrium 2 B Mixed strategies: Rock, Scissors, Paper, Nash equilibrium 5 C Games with private information 8 D Additional exercises 24 25 pages Strategy -2- A
More informationLast-Call Auctions with Asymmetric Bidders
Last-Call Auctions with Asymmetric Bidders Marie-Christin Haufe a, Matej Belica a a Karlsruhe nstitute of Technology (KT), Germany Abstract Favoring a bidder through a Right of First Refusal (ROFR) in
More informationOptimal selling rules for repeated transactions.
Optimal selling rules for repeated transactions. Ilan Kremer and Andrzej Skrzypacz March 21, 2002 1 Introduction In many papers considering the sale of many objects in a sequence of auctions the seller
More informationEC476 Contracts and Organizations, Part III: Lecture 3
EC476 Contracts and Organizations, Part III: Lecture 3 Leonardo Felli 32L.G.06 26 January 2015 Failure of the Coase Theorem Recall that the Coase Theorem implies that two parties, when faced with a potential
More information