Auctions - Revision history

Pilar: /* Auctions with imperfect information and common value component */

2013-01-25T17:58:24Z

‎Auctions with imperfect information and common value component

Pilar: /* Things to remember */

2013-01-25T17:03:36Z

‎Things to remember

Pilar: /* Auctions with imperfect information and common value component */

2013-01-25T17:02:16Z

‎Auctions with imperfect information and common value component

Pilar: /* First-price sealed-bid auction */

2013-01-25T16:52:19Z

‎First-price sealed-bid auction

Pilar: /* Second-price sealed-bid auction */

2013-01-25T16:48:55Z

‎Second-price sealed-bid auction

Pilar: /* Auctions with imperfect information */

2013-01-25T16:46:35Z

‎Auctions with imperfect information

Pilar: /* Revenue equivalency of first-price and second-price auctions */

2013-01-25T16:45:08Z

‎Revenue equivalency of first-price and second-price auctions

Pilar: /* First-price sealed-bid auction */

2013-01-25T16:43:37Z

‎First-price sealed-bid auction

Pilar: /* Exercise 1 */

2013-01-25T16:38:35Z

‎Exercise 1

Pilar: /* Second-price sealed-bid auction */

2013-01-25T16:36:54Z

‎Second-price sealed-bid auction

@@ Line 128: / Line 128: @@
 == Auctions with imperfect information and common value component ==
-Auctions with imperfect information and common value component are more realistic but more complicated to analyze as well. The main difference is that valuation of the auctioned item for player ''i'' is based not only on his estimate but also on the signals about other players' estimates. Those estimates are partially revealed during the bidding process through bids in open cry auctions, in a sealed-bid auction this doesn't occur.
+Auctions with imperfect information and common value component are more realistic but more complicated to analyse as well. The main difference is that valuation of the auctioned item for player ''i'' is based not only on his estimate but also on the signals about other players' estimates. Those estimates are partially revealed during the bidding process through bids in open cry auctions, in a sealed-bid auction this doesn't occur.
-Consider an auction of mining permits mentioned at the beginning of this chapter. Each player surveys the area and formulates an estimate of the market value on which she will base her bid. However, knowledge of other players' estimates might change her bid - she might see her estimate is high or low in comparison to others and adjust it. If all the players put the same weight to their estimate as to the others' estimates, the valuation is purely common and all players will end up with the same valuation (in most auctions this is not the case).
+Consider an auction of mining permits mentioned at the beginning of this chapter. Each player surveys the area and formulates an estimate of the market value on which she will base her bid. However, knowledge of other players' estimates might change her bid - she might see her estimate is too high or too low in comparison to others and adjust it. If all the players put the same weight to their estimate as to the others' estimates, the valuation is purely common and all players will end up with the same valuation (in most auctions this is not the case).
 [[File:winners-curse.png|thumb|x115px|right|Figure 3: An illustration of the winner's curse in a common-value auction with imperfect information. The blue points represent initial estimates, the red point represents the average estimate and the green point represents the real value. The winner's curse is indicated for a situation when all the estimates were placed as bids.]]
-Important thing to note here is that when we average initial estimates (uninfluenced by other players' estimates) of all the players we should get close to the real value - some players underestimate and some overestimate it, but the average is close. A naïve bidder might bid her valuation. If she happens to be the most overestimating one, she will win the auction but with negative payoff (figure 3)! If it was a first-price sealed-bid auction with '''private values''' she would get 0, but since it is a common value auction and she's likely (but not necessarily) bidding higher than is the real value of the item she's losing money. This phenomenon is known as the '''winner's curse'''.
+Figure 3 shows number of initial estimates uninfluenced by other players' estimates. Some players are quite close to the real value, some underestimate and some overestimate it. If they had knowledge of other players' estimates (perfect information) all the estimates would converge to an average value (which in this case would be quite accurate). If they didn't have the knowledge and naïvely bid their estimates, then the player who overestimates the real value the most would win and suffer so called '''winner's curse''', i.e. she would win the auction bud with a negative payoff due to overestimating the real value.
-In the previous section with private value we observed that in first-price auctions the players bid less than was their valuation. In an auction with a common value component this applies as well, moreover, even players in a second-price auction have an incentive to shade their bids. The winning player is at risk of negative payoff if she wins the auction because she might be overestimating the real value (which is obviously a preferable outcome for the seller). More sophisticated bidders try to accommodate for the winner's curse in a way similar to that bidding strategy in first-price sealed-bid auction.
+In the previous section with private value we observed that in first-price auctions the players bid less than was their valuation. In an auction with a common value component this applies as well, moreover, even players in a second-price auction have an incentive to shade their bids. The winning player is at risk of a negative payoff (which is obviously a preferable outcome for the seller) if she wins the auction because she might be overestimating the real value. More sophisticated bidders try to avoid the winner's curse by strategy similar to shading shown above in the first-price sealed-bid auction.
 == Things to remember ==

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 * Under certain assumptions all four auction designs can be modelled by the first- and second-price sealed bid auctions.
 * Second-price sealed-bid auction reveals true private valuations and gives some information about common valuation.
-* Bids are lower in first-price sealed-bid auctions than in second-price sealed-bid auctions but the revenue is the same under certain assumptions - revenue equivalency theorem.
+* Bids are lower in first-price sealed-bid auctions than in second-price sealed-bid auctions but the revenue is the same under certain assumptions - the revenue equivalency theorem.
 * There is a risk of the winner's curse in auctions with common value component regardless of the auction mechanism.

@@ Line 128: / Line 128: @@
 == Auctions with imperfect information and common value component ==
-Auctions with imperfect information and common value component are more realistic but more complicated to analyze as well. The main difference is that valuation of the auctioned item for player ''i'' is based not only on his estimate but also on the signals about other players' estimates. Those estimates are partially revealed during the bidding proces through bids in open cry auctions, in sealed-bid auction this doesn't occur.
+Auctions with imperfect information and common value component are more realistic but more complicated to analyze as well. The main difference is that valuation of the auctioned item for player ''i'' is based not only on his estimate but also on the signals about other players' estimates. Those estimates are partially revealed during the bidding process through bids in open cry auctions, in a sealed-bid auction this doesn't occur.
-Consider an auction of mining permits mentioned at the beginning of this chapter. Each player surveys the area and formulates an estimate of the market value which will be a base for her bid. However, knowledge of other players' estimates might change her bid. If all the players put the same weight to their estimate as to the others' estimates, the valuation is purely common and all players will end up with the same valuation.
+Consider an auction of mining permits mentioned at the beginning of this chapter. Each player surveys the area and formulates an estimate of the market value on which she will base her bid. However, knowledge of other players' estimates might change her bid - she might see her estimate is high or low in comparison to others and adjust it. If all the players put the same weight to their estimate as to the others' estimates, the valuation is purely common and all players will end up with the same valuation (in most auctions this is not the case).
 [[File:winners-curse.png|thumb|x115px|right|Figure 3: An illustration of the winner's curse in a common-value auction with imperfect information. The blue points represent initial estimates, the red point represents the average estimate and the green point represents the real value. The winner's curse is indicated for a situation when all the estimates were placed as bids.]]
-Important thing to note here is that when we average initial estimates of all the players we should arrive close to the real value - some players underestimate and some overestimate, but the average is close. A naïve bidder might bid his valuation. If he happens to be the most overestimating one, he will win the auction but with negative payoff (figure 3)! If it was a first-price sealed-bid auction with private values he would get 0, but since it is a common value auction and he's likely (but not necessarily) bidding higher than is the real value of the item he's losing money. This phenomenon is known as the '''winner's curse'''.
+Important thing to note here is that when we average initial estimates (uninfluenced by other players' estimates) of all the players we should get close to the real value - some players underestimate and some overestimate it, but the average is close. A naïve bidder might bid her valuation. If she happens to be the most overestimating one, she will win the auction but with negative payoff (figure 3)! If it was a first-price sealed-bid auction with '''private values''' she would get 0, but since it is a common value auction and she's likely (but not necessarily) bidding higher than is the real value of the item she's losing money. This phenomenon is known as the '''winner's curse'''.
-In the previous section with private value we observed that in first-price auctions the players bid less than was their valuation. In an auction with a common value component this applies as well, moreover, even players in a second-price auction have an incentive to shade their bids. The winning player is at risk of negative payoff if she wins the auction because she might be overestimating the real value. That is obviously a preferable outcome for the seller. More sophisticated bidders try to accommodate for the winner's curse in a way similar to that bidding strategy in first-price sealed-bid auction.
+In the previous section with private value we observed that in first-price auctions the players bid less than was their valuation. In an auction with a common value component this applies as well, moreover, even players in a second-price auction have an incentive to shade their bids. The winning player is at risk of negative payoff if she wins the auction because she might be overestimating the real value (which is obviously a preferable outcome for the seller). More sophisticated bidders try to accommodate for the winner's curse in a way similar to that bidding strategy in first-price sealed-bid auction.
 == Things to remember ==

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 === First-price sealed-bid auction ===
-Coming back to first-price sealed-bid auction with imperfect information we might expect that b<sub>i</sub> < v <sub>i</sub> will weakly dominate higher bids (including b<sub>i</sub> = v<sub>i</sub>). Indeed, that is the case. Bidding below her own evaluation is at least as good as bidding higher in an imperfect information scenario for each player.
+Coming back to the first-price sealed-bid auction with imperfect information we might expect that b<sub>i</sub> < v <sub>i</sub> will weakly dominate higher bids (including b<sub>i</sub> = v<sub>i</sub>). Indeed, that is the case. For each player bidding below her own valuation is at least as good as bidding her valuation (which in turn is at least as good as bidding even higher) in an imperfect information scenario.
-Overbidding her own valuation is weakly dominated by bidding the valuation in a similar fashion as shown in previous section. Underbidding the valuation weakly dominates bidding the valuation. While equilibria for this case can be excatly calulated with certain assumptions we won't go into that and settle for a question a rational bidder should ask: Assuming my valuation is the highest, what is the highest of the other players' valuations? I should bid that amount (plus minimal increment), because it still wins but gets me better payoff.
+Overbidding her own valuation is weakly dominated by bidding the valuation in a similar fashion as shown in the previous section. Underbidding the valuation weakly dominates bidding the valuation. While equilibria for this case can be exactly calculated with certain assumptions we won't go into that and settle for a question a rational bidder should ask: Assuming my valuation is the highest, what is the highest of the other players' valuations? I should bid that amount (plus minimal increment), because it still wins but gets me better payoff.
 === Revenue equivalency in the world of imperfect information ===

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 # Her valuation is lower than other bids (v<sub>i</sub> < b') but the increased bid is higher (b<sub>i</sub> > b'). Winning an auction with such a bid makes her payoff v<sub>i</sub> - b' < 0 (figure 2c, compare to figure 1e).
-Overbidding her valuation either doesn't change player's payoff or it leaves her worse off. An argument along the same lines can be done for underbidding her valuation. Second-price sealed-bid auction with imperfect information reveals the true valuations because bidding them is weakly dominating strategy for all players.
+Overbidding her valuation either doesn't change player's payoff or it leaves her worse off. An argument along the same lines can be made for underbidding her valuation as well. Second-price sealed-bid auction with imperfect information reveals the true valuations because bidding them is weakly dominating strategy for all players.
 === First-price sealed-bid auction ===

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 == Auctions with imperfect information ==
-Auctions discussed in the previous section were not very realistic because we assumed perfect information, i.e. each bidder knew valuations of other bidders, and independent private values. By loosening the perfect information requirement we get an auction in which each bidder knows only her private value. Precise analysis of these auctions is rather involved and can be found in [Osborne, pp. 291-295]. For this chapter we will show the only the basic idea.
+Auctions discussed in the previous section were not very realistic because we assumed independent private values and perfect information, i.e. each bidder knew valuations of other bidders. By loosening the perfect information requirement we get auctions in which each bidder knows only her private value. Precise analysis of these auctions is rather involved and can be found in (Osborne, pp. 291-295). For this chapter we will show only the basic idea.
 === Second-price sealed-bid auction ===

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 === Revenue equivalency of first-price and second-price auctions ===
-We have focused on the optimal strategies of bidders and omitted the interests of the seller - which auction design should he choose in order to maximise his revenue? Surprising thing is that it doesn't matter for single-item private-value auctions due to '''revenue equivalency theorem''' which states that the auction proceeds are the same regardless the precise mechanism (with several assumptions of course). Details of the theorem are beyond the scope of this text, further reading can be found in [MAS, pp. 337-340].
+We have focused on the optimal strategies of bidders and omitted the interests of the seller - which auction design should he choose in order to maximise his revenue? Surprising thing is that it doesn't matter for single-item private-value auctions due to the '''revenue equivalency theorem''' which states that the auction proceeds are the same regardless the precise mechanism (with several assumptions of course). Details of the theorem are beyond the scope of this text, further reading can be found in (Shoham, Leyton-Brown, 2009, pp. 337-340).
 == Auctions with imperfect information ==

@@ Line 86: / Line 86: @@
 : The set of possible bids b<sub>i</sub> ≥ 0 is the set of actions of player .''i''
 ; Payoff
-: If b<sub>i</sub> is the highest bid (on one of the highest and determined to be the winning one by a deterministic rule) then player ''i'''s payoff is v<sub>i</sub> - b<sub>i</sub>, if b<sub>i</sub> is not the highest bid, then the payoff is 0.
+: For each player ''i'': if b<sub>i</sub> is the highest bid (on one of the highest and determined to be the winning one by a deterministic rule) then player ''i'''s payoff is v<sub>i</sub> - b<sub>i</sub>, if b<sub>i</sub> is not the highest bid, then the payoff is 0.
 First-price sealed-bid auction has many Nash equilibria as well but in all of them the winner is player 1 who values the item most highly.
@@ Line 92: / Line 92: @@
 '''Exercise 2''': Show that the profile (b<sub>1</sub>, ... , b<sub>n</sub>) = (v<sub>2</sub> + ɛ, v<sub>2</sub>, ... , v<sub>n</sub>), where ɛ is the smallest possible increment, is a Nash equilibrium. ([[#Exercise_2|Solution]])
-Broadening the argument from exercise 2 leads to the idea that if any player ''i'' other than player 1 wins, b<sub>i</sub> > b<sub>1</sub> and if b<sub>i</sub> > v<sub>2</sub> then payoff of the winning player is negative and losing the auction by bidding 0 is a better decision. If b<sub>i</sub> < v<sub>2</sub> then player 1 can bid higher and win the auction with positive payoff instead of 0.
+Broadening the argument from exercise 2 leads to the idea that if any player ''i'' other than player 1 wins, b<sub>i</sub> > b<sub>1</sub> and if b<sub>i</sub> > v<sub>2</sub> then the payoff of the winning player is negative and losing the auction by bidding 0 is a better decision. If b<sub>i</sub> < v<sub>2</sub> then player 1 can bid higher and win the auction with positive payoff instead of 0.
-We can easily see, that a bid b<sub>i</sub> > v<sub>i</sub> can be part of an equilibrium but is risky in a similar way as it is in the second-price sealed-bid auction. However a bid b<sub>i</sub> < v<sub>i</sub> weakly dominates b<sub>i</sub> = v<sub>i</sub>. If b<sub>i</sub> = v<sub>i</sub> didn't win the auction lower bid won't win it either and payoff stays 0, but if it won an auction, lower bid can win it too and with greater payoff. In a first-price sealed-bid auction with perfect information bid b<sub>i</sub> = v<sub>i</sub> weakly dominates higher bids, but it is weakly dominated by lower bids.
+We can easily see, that a bid b<sub>i</sub> > v<sub>i</sub> can be part of an equilibrium but is risky in a similar way as it is in the second-price sealed-bid auction. However a bid b<sub>i</sub> < v<sub>i</sub> weakly dominates b<sub>i</sub> = v<sub>i</sub>. If b<sub>i</sub> = v<sub>i</sub> didn't win the auction lower bid won't win it either and payoff stays 0, but if it won an auction, lower bid can win it too and with greater payoff. '''In a first-price sealed-bid auction with perfect information bid b<sub>i</sub> = v<sub>i</sub> weakly dominates higher bids, but it is weakly dominated by lower bids.'''
 === Revenue equivalency of first-price and second-price auctions ===

@@ Line 152: / Line 152: @@
 === Exercise 1 ===
-Player 2 bidding any value greater than v<sub>2</sub> the outcome stays the same. If  player 2 bids lower than vv<sub>2</sub>, she loses to player 1 a nd her payoff is 0. Player 1 would have to increase her bid above v<sub>1</sub> to win but her payoff remains 0. Any other player raising her bid below v<sub>1</sub> wouldn't change the result and raising it higher would get her negative payoff.
+If player 2 bid any value greater than v<sub>2</sub>, the outcome stays the same. If  player 2 bids lower than v<sub>2</sub>, she loses to player 1 and her payoff is 0. Player 1 would have to increase her bid above v<sub>1</sub> to win but her payoff remains 0. Any other player raising her bid below v<sub>1</sub> wouldn't change the result and raising it higher would get her negative payoff.
 === Exercise 2 ===

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 : The set of possible bids b<sub>i</sub> ≥ 0 is the set of actions of player ''i''.
 ; Payoff
-: If b<sub>i</sub> is the highest bid and b<sub>j</sub> is the second highest (or the same as the highest but by some deterministic rule decided as the second, e.g. earlier submitted bid is preferred), then player ''i'''s payoff is v<sub>i</sub> - b<sub>j</sub>, if b<sub>i</sub> is not the highest bid, then the payoff is 0.
+: For each player ''i'': if b<sub>i</sub> is the highest bid and b<sub>j</sub> is the second highest (or the same as the highest but by some deterministic rule decided as the second, e.g. earlier submitted bid is preferred), then player ''i'''s payoff is v<sub>i</sub> - b<sub>j</sub>, if b<sub>i</sub> is not the highest bid, then the payoff is 0.
 For simplicity let's order the players by their valuation v<sub>i</sub> so that v<sub>1</sub> > v<sub>2</sub> > ... > v<sub>n</sub> > 0.