Auctions - Chapter 9 PDF

Chapter 9: Auctions Jehle and Reny, Chapter 9 169 Ch 9.1: Introduction I In the first half of this class we have dealt with competitive markets for private goods. I We have shown that markets in general work very well. In particular, prices coord...

Chapter 9: Auctions Jehle and Reny, Chapter 9 169 Ch 9.1: Introduction I In the first half of this class we have dealt with competitive markets for private goods. I We have shown that markets in general work very well. In particular, prices coordinate demand and supply in an efficient manner. I But what happens if markets are “thin” and the valuation of goods in question depends on some private information? 170 Ch 9.1: Introduction I For example what if you are trying to sell a unique work of art? Or a uniquely used cell phone? I Or the sponsorship rights for the (Rai↵eisen) Super League? I How would you sell the right to radio spectrum used to transfer 5G mobile data? I Or, for post-communist countries, how would you sell a government-owned firm? 171 Ch 9.1: Introduction I Usually, the answer to these questions is to hold an auction. I But there are many auction formats — auctions are designed. I How should we optimally design an auction? I Auctions are not just used for selling, procurement auctions are used by firms and government in order to buy goods and services! I We will explore several common auction formats. The question of auction design falls more broadly into the field of mechanism design. 172 Ch 9.2: The four common auctions First Price Auction (FPA) I Each buyer (bidder) submits a sealed bid to the seller. I Once all bids have been submitted, the seller inspects the bids. I The bidder with the highest bid wins the object. I The price is equal to the highest bid. 173 Ch 9.2: The four common auctions Second Price Auction (SPA) I Each bidder submits a sealed bid to the seller. I Once all bids have been submitted, the seller inspects the bids. I The bidder with the highest bid wins the object. I The price is equal to the second highest bid. 174 Ch 9.2: The four common auctions Dutch Auction I The seller observes the bidders. I The seller announces a very high price. I If none of the bidders accept the price, the seller reduces the price. I The first bidder to accept a price receives the object and pays the accepted price. 175 Ch 9.2: The four common auctions English Auction I The seller observes the bidders. I The seller announces a very low price. I Bidders announce if they accept the price. Bidders who do not accept the price, leave the auction. I If more than one agent accepts the price, the seller increases the price and the previous step is repeated. I Once only one bidder remains, that bidder receives the object and pays the current price. 176 Ch 9.2: Classroom experiment I We will split into two groups: I a first price auction group and I a second price auction group. I I will collect your bids through Google Forms. I FPA group should put their bids at: igorletina.com/fpa I SPA group should put their bids at: igorletina.com/spa 177 Ch 9.3: Independent Private Values I There is a single, risk-neutral seller and a single, indivisible object for sale. I There are N risk-neutral buyers. I The seller values the object at 0. I Buyer i’s value of the object, denoted with vi , is a random variable. I vi is drawn from the interval [0, 1] according to the cdf Fi (vi ), with a density fi (vi ). I Assume that buyers are ex-ante symmetric, so that fi = fj for all i, j. 178 Ch 9.3: Independent Private Values I The buyer i knows her valuation vi. I The seller and the other buyers know only that vi is distributed according to f (vi ). I The utility of the agent who wins the object and pays p is given by vi p. The utility of the agents who do not win is 0. The utility of the seller is p. I Note that there are auctions where losers also pay, which is not the case in any of the four auctions we are considering. 179 Ch 9.3: Independent Private Values I This model is called independent private values model. I Independent refers to the assumption that the realizations of vi and vj are statistically independent (i.e. not correlated). I Suppose we were trying to decide how much to bid on the right to exploit an oil field. Before bidding we conduct preliminary exploration study of the size of the oil field. This generates our private signal. But presumably other bidders would be conducting similar studies as well. Then, if we receive the signal that the oil field is large, it is likely that our competitors have received the same signal. 180 Ch 9.3: Independent Private Values I Private value refers to the assumption that the value of the object to the bidder i is fully captured by vi and does not depend on the private information of other bidders. I Consider again our oil field example. Presumably, the thing we care about is the true size of the oil field. But then private information of others would be valuable — if we were the only one with a study indicating that the oil field is large, while everyone else had a study indicating that it is small, we would be wise to consider the possibility that our study was flawed. 181 Ch 9.4: First Price Auction I Formally, specifying an auction format is specifying a game that the bidders will play. The game is one of incomplete information, and the solution concept is the Bayes-Nash equilibrium. I Each agent wants to win the object, so long as the price is lower than her valuation. I But conditional on winning, the bidder prefers to pay a lower price. I In choosing their bid, bidders trade o↵ the probability of winning with the price they will have to pay. I The strategy of each agent i is a bid made for each possible valuation. That is, a strategy is a bidding function bi : [0, 1] ! R+. 182 Ch 9.4: First Price Auction I Let us focus our attention by restricting the set of possible bidding strategies. We will later see that what we find is indeed the equilibrium bidding strategy. I First, since agents are ex-ante symmetric, it is natural to expect that the equilibrium bidding functions will be symmetric as well. I.e. bi = bk = b for all i, k. I Second, it seems natural to expect that higher vi should lead to a higher bid. Let us assume that b is increasing. 183 Ch 9.4: First Price Auction I To make sure that bidding according to b constitutes an equilibrium, we need to check if there are profitable deviations for a bidder with a type v. I Let us do this in a particular way. Suppose you still use the bidding function b, but calculate the bid according to some value r which is di↵erent from your true value v. I We want to find the expected payo↵ of this deviation, when all other bidders use the bidding function b truthfully. I The bidder i wins only when his bid is the highest, that is when b(r) > b(vj ) for all j 6= i. Since b is increasing, this occurs with probability F N 1 (r). I The value of winning is the di↵erence between the true valuation of the object and the bid, v b(r). 184 Ch 9.4: First Price Auction I Thus the expected value of this deviation is u(r, v) = F N 1 (r)(v b(r)). (3) I If b is an equilibrium strategy, then setting v = r must maximize the expected value of the above expression. I That is, the problem of the bidder is max u(r, v), r and the first order condition is du(r, v) ! = 0. dr 185 Ch 9.4: First Price Auction dF N 1 (r)(v b(r)) =0 dr (N 1)F N 2 (r)f (r)(v b(r)) FN 1 (r)b0 (r) = 0. Evaluating this expression at r = v gives (N 1)F N 2 (v)f (v)(v b(v)) FN 1 (v)b0 (v) = 0 and rearranging (N 1)F N 2 (v)f (v)b(v) + F N 1 (v)b0 (v) = (N 1)F N 2 (v)f (v)v dF N 1 (v)b(v) = (N 1)F N 2 (v)f (v)v. (4) dv 186 Ch 9.4: First Price Auction Since Equation (4) has to hold for every v then it must be the case that Z v F N 1 (v)b(v) = (N 1) xf (x)F N 2 (x)dx + const. 0 Since an agent with v = 0 has to bid zero, then the constant above also has to be zero. Hence, our candidate strategy is Z v N 1 b(v) = N 1 xf (x)F N 2 (x)dx. (5) F (v) 0 187 Ch 9.4: First Price Auction I It can be shown that b(v) is increasing. (You will see this in an exercise.) I Hence, we have provided a heuristic argument for the following result. Proposition 9.1 (First Price Auction Equilibrium) If N bidders have independent private values drawn from the common distribution F then bidding Z v N 1 b(v) = N 1 xf (x)F N 2 (x)dx. F (v) 0 constitutes a symmetric Nash equilibrium of a First Price Auction. Moreover, this is the only symmetric Nash equilibrium. 188 Ch 9.5: Dutch Auction I In a Dutch Auction, each bidder has a single decision to make — when do I accept the price? I Imagine you are a manager sending a representative to a Dutch Auction. The only instruction you can give to the representative is which price to accept. I But this is the same as sending a sealed bid in a First Price Auction! I Thus, First Price Auction and a Dutch Auction are strategically equivalent! 189 Ch 9.5: Dutch Auction Proposition 9.2 (Dutch Auction Equilibrium) If N bidders have independent private values drawn from the common distribution F then accepting as soon as the price reaches Z v N 1 xf (x)F N 2 (x)dx. F N 1 (v) 0 constitutes a symmetric Nash equilibrium of a Dutch Auction. Moreover, this is the only symmetric Nash equilibrium. 190 Ch 9.6: Second Price Auction I Why even bother with the second-price auction? I After all, in the FPA the seller gets the highest bid, which is obviously better than getting the second-highest bid. I This argument is naive — it neglects that the bidders behave di↵erently in di↵erent auctions! I In the FPA the bidder pays her own bid, so she has an incentive to reduce the bid in order to pay less. In the SPA, the winner has no influence on the price! I Hence, we should expect the bidders to be more aggressive in the SPA! But, then it is not clear which auction format is better for the seller. 191 Ch 9.6: Second Price Auction I Luckily, figuring out the equilibrium in the SPA is easy. I Suppose that the bids of all bidders but i are fixed. Let the highest among those be bk. Suppose that the agent i knew what bk was. I What is the optimal bid by bidder i? I Bidding anything below bk results in i not winning. Bidding anything above bk results in i winning and paying exactly bk. I Thus i wants to win whenever bk < vi and wants to lose whenever bk > vi. Thus, no matter what the case, she can optimally bid vi. 192 Ch 9.6: Second Price Auction I But in the SPA the bidder i does not know the highest bid bk. I This does not matter! The above argument held for any bk. I Suppose bi > vi. This leads to losses whenever the highest bid by the competitors is in (vi , bi ) without any gains in any other case. I Similarly, bidding bi < vi leads to losing the auction when it would be profitable to win it by bidding vi , again without any gains. 193 Ch 9.6: Second Price Auction Proposition 9.3 (Second-Price Auction Equilibirum) If N bidders have independent private values, then bidding one’s value is the unique weakly dominant strategy for each bidder in a second-price auction. I Note that the above implies that bidding one’s value is a Nash equilibrium. 194 Ch 9.7: English Auction I What is the optimal behavior in the English Auction? I Suppose you drop out while the price is below your value, p < vi. You lose the chance of winning the object at price p0 , where p < p0 < vi which would be beneficial. I Similarly, staying in the auction after p > vi risks that you might win the object — but at a loss! I Hence, no matter how other agents are behaving, it is best to stay in the auction until the price reaches your value. 195 Ch 9.7: English Auction Proposition 9.4 (English Auction Equilibirum) If N bidders have independent private values, then dropping out when the price reaches one’s value is the unique weakly dominant strategy for each bidder in an English auction. I In this equilibrium the winner pays the price which is equal to the second-highest value, just like in the SPA. I Thus, SPA and English auction are equivalent (as long as the values are private). 196 Ch 9.8: Revenue comparison I We have seen that the FPA and the Dutch auction are equivalent. Hence, they will raise the same revenue. I Also, the SPA and the English auctions are equivalent and raise the same revenue. I Thus, we only need to compare the revenues from the FPA and the SPA. 197 Ch 9.8: Revenue comparison I In the FPA, if the buyer with value v wins, the seller receives b(v) in revenue. I Thus, if we denote the distribution of the highest value v as g(v) the expected revenue in the FPA is Z 1 RF P A = b(v)g(v)dv. (6) 0 I Let’s figure out what is the g(v). 198 Ch 9.8: Revenue comparison I Consider bidder 1. She will have the valuation v with density f (v). I Her valuation will be higher than the valuation of all other agents with probability F N 1 (v). I Thus, bidder 1 will have the value v and it will be the highest value with density f (v)F N 1 (v). I But, any of the N agents could have the value v as the highest value. I Thus v will be the highest value with the density g(v) = N f (v)F N 1 (v). (7) 199 Ch 9.8: Revenue comparison I Now consider the SPA. If v is the second-highest value in the auction, the seller receives revenue equal to v. I If the distribution of the second-highest value is h(v), then the expected revenue in the SPA is Z 1 RSP A = vh(v)dv. (8) 0 I Again, let’s try to figure out h(v). 200 Ch 9.8: Revenue comparison I Consider bidder 1. She will have the valuation v with density f (v). I The probability that bidder 2 has a higher value than v is 1 F (v). The probability that bidder 3 has a higher value is the same, as is for all other agents. Thus the probability that any bidder has a higher value is (N 1)(1 F (v)). I The probability that all remaining bidders have a value lower than v is F N 2 (v). I Thus, bidder 1 will have the value v and it will be the second-highest value with density f (v)(N 1)(1 F (v))F N 2 (v). I But, any of the N agents could have the value v as the second-highest value, so h(v) is h(v) = N (N 1)f (v)(1 F (v))F N 2 (v). (9) 201 Ch 9.8: Revenue comparison I Combining Equations (6) and (7) gives us the expected revenue in the FPA Z 1 RF P A = N b(v)f (v)F N 1 (v)dv. (10) 0 I Combining Equations (8) and (9) gives us the expected revenue in the SPA Z 1 RSP A = N (N 1) vf (v)(1 F (v))F N 2 (v)dv. (11) 0 202 Ch 9.8: Revenue comparison I In an exercise, you will show that: RF P A = RSP A. I From this, the next result immediately follows. Proposition 9.5 (Revenue from the four auctions) If N bidders have independent private values, then the FPA, SPA, English and Dutch auction raise the same expected revenue for the seller. I This result holds more generally and is known as the Revenue Equivalence Theorem. 203 Ch 9.9: Extensions and qualifications I With correlated values we might have a “winner’s curse”. In general, the auctions are not equivalent and the English auction is better at revealing information. I With risk averse agents FPA will tend to raise more revenue than the SPA. I With budget-constrained bidders, FPA will also tend to raise more revenue than the SPA. I With less sophisticated bidders, the English auction is obviously strategy-proof. 204 Ch 9.10: General mechanisms I The four standard auctions are specific methods of selling an item. I Can the seller do any better? I Can we find the optimal selling mechanism, so that no matter what other mechanism the seller considers, she can do no better? 205 Ch 9.10: General mechanisms I Let us first define the most general (static) mechanism possible. I A general mechanism consists of: I A set of messages M ; I For each vector of reported messages (m1 ,... , mn ), a collection of n allocation functions p1 (m1 ,... , mn ),... , pn (m1 ,... , mn ) Pn such that i=1 pi (m1 ,... , mn )  1; I For each vector of reported messages (m1 ,... , mn ), a collection of n cost functions c1 (m1 ,... , mn ),... , cn (m1 ,... , mn ); 206 Ch 9.10: General mechanisms I Given a vector of messages m, a buyer i’s payo↵ is pi (m)vi ci (m) and the seller’s payo↵ is n X ci (m). i=1 I We want to maximize the expected payo↵ of the seller, given that buyers choose an equilibrium mi conditional on their realized vi. I But how do we do that? We do not even know what M looks like! I Luckily, we will be able to look at a simpler set of direct mechanisms without any loss! 207 Ch 9.11: Direct mechanisms I A direct mechanism is any mechanism where the set of messages is equal to the set of possible buyer valuations, i.e. M = [0, 1]. I That is, a direct mechanism consists of: I A set of messages [0, 1]; I For each vector of reported valuations (v1 ,... , vn ), a collection of n allocation functions p1 (v1 ,... , vn ),... , pn (v1 ,... , vn ) Pn such that i=1 pi (v1 ,... , vn )  1; I For each vector of reported valuations (v1 ,... , vn ),, a collection of n cost functions c1 (v1 ,... , vn ),... , cn (v1 ,... , vn ); 208 Ch 9.11: Direct mechanisms I Note that so far there is no requirement that the buyers report their type truthfully. They could say that their type is ri when in reality it is vi. I We will focus on incentive-compatible direct mechanisms, where it is an equilibrium for all agents to report their type truthfully. I The expected payo↵ of a buyer i with a valuation vi who reports the type ri , given that all other buyers report truthfully Z 1 Z 1 ui (ri , vi ) =... [pi (ri , v i )vi ci (ri , v i )] f i (v i )dv i 0 0 209 Ch 9.11: Direct mechanisms I We can simplify the notation by denoting the probability of winning p̄ and the expected payment with c̄ as follows Z 1 Z 1 p̄i (ri ) =... pi (ri , v i )f i (v i )dv i 0 0 Z 1 Z 1 c̄i (ri ) =... ci (ri , v i )f i (v i )dv i 0 0 I Then, the expected payo↵ of the buyer is ui (ri , vi ) = p̄i (ri )vi c̄i (ri ). 210 Ch 9.11: Direct mechanisms Definition 9.6 (IC direct mechanism) We say that a direct mechanism is IC if for every i and for every vi ui (vi , vi ) ui (ri , vi ). I We can think of the SP auction as an IC direct mechanism: report your valuation, highest valuation wins, and pays the second-highest valuation. I An FP auction is not an IC direct mechanism: equilibrium bids are b(vi ) < vi. 211 Ch 9.11: Direct mechanisms Proposition 9.7 (Revelation Principle) For any general mechanism and any equilibrium of that mechanism, there exists an IC direct mechanism such that the outcomes are the same in the equilibrium of the general mechanism and in the IC direct mechanism. I Proof omitted, we will provide intuition. I Importance of this result is immense – now we know that it is sufficient to look for the optimal mechanism in the set of IC direct mechanisms, which is a much easier task than optimizing over the set of general mechanisms. 212 Ch 9.11: Direct mechanisms Messages ✓ @ @ m(v) @ p(m), c(m) @ @ @ @ R @ Values Outcomes 213 Ch 9.11: Direct mechanisms Messages ✓ @ @ m(v) @ p(m), c(m) @ @ @ @ R @ Values - Outcomes p(m(v)), c(m(v)) 213 Ch 9.11: Direct mechanisms Proposition 9.8 (Characterization of IC direct mechanisms) A direct selling mechanism (pi (·), ci (·))ni=1 is incentive compatible if and only if for every buyer i (i) p̄i (vi ) is non-decreasing in vi and R vi (ii) c̄i (vi ) = c̄i (0) + p̄i (vi )vi 0 p̄i (x)dx, 8vi 2 [0, 1]. I Proof omitted. I Part (ii) is important – it says that in any IC direct mechanism, the payment of any buyer is completely determined by the payment of the lowest type and the allocation functions p. 214 Ch 9.11: Direct mechanisms Proposition 9.9 (Revenue equivalence) If two IC direct mechanisms have the same (i) assignment functions p and (ii) every buyer with value zero is indi↵erent between the two mechanisms, then the two mechanisms generate the same expected revenue for the seller. I Proof omitted. I This explains why our four standard auctions generated the same expected revenues! 215 Ch 9.12: Optimal mechanisms I We cannot force anyone to participate in our mechanism. I it is individually rational (IR) for all buyers to participate if ui (vi , vi ) 0 for all agents and all valuations. I By Proposition 9.8(ii), we have Z vi c̄i (vi ) = c̄i (0) + p̄i (vi )vi p̄i (x)dx 0 Z vi p̄i (vi )vi c̄i (vi ) = c̄i (0) + p̄i (x)dx Z 0 vi ui (vi , vi ) = c̄i (0) + p̄i (x)dx 0 R vi I Since 0 p̄i (x)dx is increasing in vi , if ui (0, 0) 0 then the IR constraint is satisfied for all types. I Thus, IR constraint is satisfied i↵ c̄i (0)  0. 216 Ch 9.12: Optimal mechanisms I We are now in position to analyze the optimal design problem: 1. By the Revelation Principle we know we look for optimal mechanisms in the class of IC direct mechanisms. 2. By Proposition 9.8 we know how these mechanisms look like, and 3. we know it is sufficient to choose ci (0) and pi functions. I Our objective is maximization of expected revenues for the principal, that is n Z X 1 max c̄i (vi )fi (vi )dvi ci (0),pi 0 i=1 s.t. IC and IR hold 217 Ch 9.12: Optimal mechanisms I Using Proposition 9.8 we can rewrite the maximization problem as n Z X 1 Z vi n X max p̄i (vi )vi p̄i (x)dx fi (vi )dvi + c̄i (0) ci (0),pi 0 0 i=1 i=1 subject to: p̄i (vi ) is non-decreasing in vi c̄i (0)  0 I Clearly, the objective function is maximized for c̄i (0) = 0, so the remaining problem is... 218 Ch 9.12: Optimal mechanisms n Z X 1 Z vi max p̄i (vi )vi p̄i (x)dx fi (vi )dvi pi 0 0 i=1 subject to: p̄i (vi ) is non-decreasing in vi I Let us focus on the objective function for the moment. 219 Ch 9.12: Optimal mechanisms n Z X 1 Z vi p̄i (vi )vi p̄i (x)dx fi (vi )dvi = i=1 0 0 n Z X 1 Z 1 Z vi p̄i (vi )vi fi (vi )dvi p̄i (x)fi (vi )dxdvi = i=1 0 0 0 Xn Z 1 Z 1Z 1 p̄i (vi )vi fi (vi )dvi p̄i (x)fi (vi )dvi dx = i=1 0 0 x Xn Z 1 Z 1 p̄i (vi )vi fi (vi )dvi p̄i (x)(1 Fi (x))dx = i=1 0 0... 220 Ch 9.12: Optimal mechanisms... n Z X 1 Z 1 p̄i (vi )vi fi (vi )dvi p̄i (vi )(1 Fi (vi ))dvi = i=1 0 0 n Z X 1  1 Fi (vi ) p̄i (vi ) vi fi (vi )dvi = 0 fi (vi ) i=1 n Z 1 X Z 1  1 Fi (vi ) · · · pi (v1 ,... , vn ) vi f1 (v1 ) · · · fn (vn )dv1 · · · dvn = fi (vi ) i=1 0 0 Z 1 Z 1 (X n  ) 1 Fi (vi ) ··· pi (v1 ,... , vn ) vi f (v)dv 0 0 fi (vi ) i=1 221 Ch 9.12: Optimal mechanisms I Thus our problem is Z 1 Z 1 (X n  ) 1 Fi (vi ) max ··· pi (v1 ,... , vn ) vi f (v)dv pi 0 0 fi (vi ) i=1 subject to: p̄i (vi ) is non-decreasing in vi I Maximizing integrals is difficult... but, if we can manage to maximize the value of the integrand for each v, then we have also maximized the integral. 222 Ch 9.12: Optimal mechanisms I So let’s think about this problem (ignore the constraint) ( n  ) X 1 Fi (vi ) max pi (v1 ,... , vn ) vi pi fi (vi ) i=1 I Since  pi 0, then for any vi for which 1 Fi (vi ) vi < 0 we should set pi = 0. fi (vi ) P I Remember that ni=1 pi (v1 ,... , vn )  1. I If at least one of the elements in the square  brackets is 1 Fi (vi ) positive, then put pi = 1 for the highest vi. fi (vi ) 223 Ch 9.12: Optimal mechanisms I If we construct pi ’s as on the previous slide, and it turns out that p̄i (vi ) is non-decreasing in vi , then we have our optimal mechanism. I This will be the case if 1 Fi (vi ) vi is strictly increasing in vi. (12) fi (vi ) I If the above is not satisfied, we need to adjust the mechanism slightly (called ironing) but our main insights are still valid. 224 Ch 9.12: Optimal mechanisms Proposition 9.10 (Optimal Mechanism) If n bidders have independent private values with fi satisfying (12) for all i, then the optimal mechanism is characterized by 8 ⇢ max j6=i 0, vj pi (v) = fi (vi ) fj (vj ) : 0, otherwise and c̄i (0) = 0. I Recall that c̄i is characterized by Proposition 9.8(ii). 225 Ch 9.12: Optimal mechanisms I Note that the optimal mechanism is inefficient. I It sometimes does not sell the object even though each buyer values it strictly more than the seller. I This should not be surprising – monopolists tend to reduce output in order to increase prices. I The four auctions we studied in the last class are efficient but they are suboptimal for the seller. I Can they be made optimal? 226 Ch 9.12: Optimal mechanisms Proposition 9.11 (Optimal Auction under Symmetry) If n bidders have independent private values, each drawn from the same density f satisfying (12), then the second price auction with a reserve price ⇢⇤ is optimal, where 1 F (⇢⇤ ) ⇢⇤ = 0. f (⇢⇤ ) 227 Ch 9.13: Conclusion I Why should we care? 228 Ch 9.13: Conclusion I Auctions are everywhere. 229 Ch 9.13: Conclusion 230 Ch 9.13: Conclusion 231 Ch 9.13: Conclusion 232 Ch 9.13: Conclusion 233 Ch 9.13: Conclusion 234 Ch 9.13: Conclusion 235 Ch 9.13: Conclusion 236 Ch 9.13: Conclusion 237 Ch 9.13: Conclusion I Bad design can be costly. 238 Ch 9.13: Conclusion 239 Ch 9.13: Conclusion 240 Ch 9.13: Conclusion Source: Wolfstetter, Elmar (2001), “The Swiss UMTS spectrum auction flop: Bad luck or bad design?”, SFB 373 Discussion Paper, No. 2001/50, Humboldt University of Berlin. 241 Ch 9.13: Conclusion New Zealand’s spectrum auctions: Source: Mueller, Milton (1993) “New Zealand’s revolution in spectrum management”, Information Economics and Policy 5.2: 159-177. 242 Ch 9.13: Conclusion I Auctions are just one type of mechanisms that economists are helping design. 243 Ch 9.13: Conclusion 244 Ch 9.13: Conclusion 245 Ch 9.13: Conclusion 246 Chapter 10: Final words 247 I We started by studying idealized markets in a general equilibrium setting. I We have shown that such ideal markets have many desirable properties. I We were able to study market imperfections and some ways to correct them. I Finally, we discussed how economic mechanisms could be designed. 248