51:
2452:
3050:(VCG) mechanism. The idea in VCG is that items are assigned to maximize the sum of utilities; then each bidder pays the "opportunity cost" that their presence introduces to all the other players. This opportunity cost for a bidder is defined as the total bids of all the other bidders that would have won if the first bidder had not bid, minus the total bids of all the other actual winning bidders.
1545:
and the buyer is not the current high bidder, it is more profitable to bid than to let someone else be the winner. Thus it is a dominant strategy for a buyer to drop out of the bidding when the asking price reaches his or her valuation. Thus, just as in the
Vickrey sealed second price auction, the
1452:
The two most common auctions are the sealed first price (or high-bid) auction and the open ascending price (or
English) auction. In the former each buyer submits a sealed bid. The high bidder is awarded the item and pays his or her bid. In the latter, the auctioneer announces successively higher
631:
A Vickrey auction is decision efficient (the winner is the bidder with the highest valuation) under the most general circumstances; it thus provides a baseline model against which the efficiency properties of other types of auctions can be posited. It is only ex-post efficient (sum of transfers
3046:. The uniform-price auction does not, however, result in bidders bidding their true valuations as they do in a second-price auction unless each bidder has demand for only a single unit. A generalization of the Vickrey auction that maintains the incentive to bid truthfully is known as the
898:
2671:
1181:
then only the strategy of overbidding would win the auction. The payoff would be negative for the strategy of overbidding because they paid more than their value of the item, while the payoff for a truthful bid would be
1438:
then only the strategy of truthfully bidding would win the auction. The payoff for the truthful strategy would be positive as they paid less than their value of the item, while the payoff for an underbid bid would be
1549:
Consider then the expected payment in the sealed second-price auction. Vickrey considered the case of two buyers and assumed that each buyer's value was an independent draw from a uniform distribution with support
769:
546:. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest bid. This type of auction is strategically similar to an
3290:
Benjamin
Edelman, Michael Ostrovsky, and Michael Schwarz: "Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords". American Economic Review 97(1), 2007 pp
1443:
Thus the strategy of underbidding is dominated by the strategy of truthfully bidding. As truthful bidding dominates the other possible strategies (i.e. underbidding and overbidding), it is an optimal strategy.
1186:
Thus the strategy of bidding higher than one's true valuation is dominated by the strategy of truthfully bidding. The strategy of underbidding is also dominated by bidding truthfully. Assume that bidder
2355:
1436:
1179:
2942:
2884:
1364:
1305:
1107:
1048:
1822:
2516:
concept. The basic idea of a VCG mechanism in network routing is to pay the owner of each link or node (depending on the network model) that is part of the solution, its declared cost
623:
by bidding (revealing) their valuation of the item for sale. These type of auctions are sometimes used for specified pool trading in the agency mortgage-backed securities (MBS) market.
2766:
3057:—a minimum price below which the item is not sold at all. In some cases, setting a reservation price can substantially increase the revenue of the auctioneer. This is an example of
1731:
3032:
2833:
1050:
then the bidder would win the item with a truthful bid as well as an overbid. The bid's amount does not change the payoff so the two strategies have equal payoffs in this case.
741:
2804:
1245:
988:
2143:
2436:
2014:
1971:
1876:
2210:
2172:
2978:
1626:
1543:
1517:
2576:
928:
716:
689:
2387:
1905:
510:
3042:
The most obvious generalization to multiple or divisible goods is to have all winning bidders pay the amount of the highest non-winning bid. This is known as a
2094:
2034:
1925:
1754:
1646:
1600:
1491:
1471:
1205:
948:
761:
2074:
1678:
1580:
583:
are, in this case only, equivalent and used interchangeably. In the case of multiple identical goods, the bidders submit inverse demand curves and pay the
632:
equal to zero) if the seller is included as "player zero," whose transfer equals the negative of the sum of the other players' transfers (i.e. the bids).
2584:
893:{\displaystyle {\begin{cases}v_{i}-\max _{j\neq i}b_{j}&{\text{if }}b_{i}>\max _{j\neq i}b_{j},\\0&{\text{otherwise.}}\end{cases}}}
1827:
That is, the payment of the winner in the sealed first-price auction is equal to the expected revenue in the sealed second-price auction.
3322:
1307:
then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies have equal payoffs for this case.
661:
The dominant strategy in a
Vickrey auction with a single, indivisible item is for each bidder to bid their true value of the item.
590:
Vickrey auctions are much studied in economic literature but uncommon in practice. Generalized variants of the
Vickrey auction for
2710:. Links not in the MCF are paid nothing. This routing problem is one of the cases for which VCG is strategyproof and minimum.
3047:
603:
265:
2218:
1372:
1115:
17:
2904:
3275:
645:, that is, discovery of the market price if the buyers are unsure of their own valuations, without sequential auctions.
50:
3119:
3058:
2491:
517:
255:
2838:
2469:
1313:
1254:
1056:
997:
595:
575:
Vickrey's original paper mainly considered auctions where only a single, indivisible good is being sold. The terms
180:
1453:
asking prices and continues until no one is willing to accept a higher price. Suppose that a buyer's valuation is
1762:
175:
3086:
2473:
230:
225:
2728:
125:
1686:
2989:
450:
145:
2816:
305:
140:
2714:
569:
2774:
3363:
485:
3223:
3178:"Vickrey Auctions in Practice: From Nineteenth-Century Philately to Twenty-First-Century E-Commerce"
1210:
953:
778:
2099:
400:
1736:
We now argue that in the sealed first price auction the equilibrium bid of a buyer with valuation
721:
2462:
1109:
then the bidder would lose the item either way so the strategies have equal payoffs in this case.
599:
551:
385:
195:
2392:
1366:
then the bidder would win the item either way so the strategies have equal payoffs in this case.
3333:
3218:
1976:
1933:
1838:
3043:
2177:
2148:
445:
390:
350:
245:
240:
130:
105:
3300:
Hal R. Varian: "Position
Auctions". International Journal of Industrial Organization, 2006,
2953:
1605:
1522:
1496:
1447:
3069:
2554:
906:
694:
667:
235:
135:
2363:
1881:
8:
3149:
Vickrey, William (1961). "Counterspeculation, Auctions, and
Competitive Sealed Tenders".
1546:
price paid by the buyer with the highest valuation is equal to the second highest value.
555:
455:
380:
95:
78:
2016:
and so buyer 1 wins with probability 1 with any bid of 1/2 or more. Consider then a bid
3264:
3244:
3236:
3162:
2079:
2019:
1910:
1739:
1631:
1585:
1476:
1456:
1190:
933:
746:
410:
250:
205:
2039:
1651:
1553:
3271:
3115:
3054:
2174:. Under Vickrey's assumption of uniformly distributed values, the win probability is
1648:
is the high value. Then the winning payment is uniformly distributed on the interval
1582:. With buyers bidding according to their dominant strategies, a buyer with valuation
591:
460:
415:
270:
200:
110:
90:
34:
3248:
3305:
3301:
3228:
3189:
3158:
3065:
620:
584:
565:
503:
490:
405:
300:
220:
190:
185:
115:
2666:{\displaystyle p_{k}=d_{k}+\operatorname {MCF} (G-e_{k})-\operatorname {MCF} (G),}
2505:
642:
561:
547:
365:
330:
325:
320:
215:
165:
160:
73:
68:
3081:
470:
435:
375:
275:
170:
150:
120:
83:
100:
3357:
3209:
Benny
Moldovanu and Manfred Tietzel (1998). "Goethe's Second-Price Auction".
2521:
903:
The strategy of overbidding is dominated by bidding truthfully (i.e. bidding
370:
155:
1927:. We need to show that buyer 1's best response is to use the same strategy.
3318:
425:
420:
395:
355:
310:
210:
340:
3194:
3177:
3091:
2513:
1448:
Revenue equivalence of the
Vickrey auction and sealed first price auction
360:
315:
295:
3240:
2520:
its added value. In many routing problems, this mechanism is not only
2476: in this section. Unsourced material may be challenged and removed.
58:
3132:
Teck Ho, "Consumption and
Production" UC Berkeley, Haas Class of 2010.
2532:
619:
In a Vickrey auction with private values each bidder maximizes their
558:
2451:
3232:
3208:
598:
used in Google's and Yahoo!'s online advertisement programmes (not
480:
2528:
2509:
543:
285:
42:
2886:. Prior to this result, it was known that VCG overpayment in
2713:
In 2004, it was shown that the expected VCG overpayment of an
614:
649:
572:
sold a manuscript using a sealed-bid, second-price auction.
3262:
Jones, Derek (2003). "Auction Theory for the New Economy".
3072:
can be viewed as a generalization of the Vickrey auction.
2548:
of each of the links and payment is calculated as follows:
2524:, but also the minimum among all strategyproof mechanisms.
886:
656:
2680:) indicates the cost of the minimum-cost flow in graph
3110:
Peter Cramton, Yoav Shoham, Richard Steinberg (Eds),
2820:
2732:
2350:{\displaystyle U(b)=w(b)(v-b)=2b(v-b)={\tfrac {1}{2}}}
2292:
1797:
1706:
1431:{\displaystyle b_{i}<\max _{j\neq i}b_{j}<v_{i}}
1174:{\displaystyle v_{i}<\max _{j\neq i}b_{j}<b_{i}}
3175:
2992:
2956:
2907:
2841:
2819:
2777:
2731:
2587:
2557:
2395:
2366:
2221:
2180:
2151:
2102:
2082:
2042:
2022:
1979:
1936:
1913:
1884:
1841:
1765:
1742:
1689:
1654:
1634:
1608:
1588:
1556:
1525:
1499:
1479:
1459:
1375:
1316:
1257:
1213:
1193:
1118:
1059:
1000:
956:
936:
909:
772:
749:
724:
697:
670:
1835:
Suppose that buyer 2 bids according to the strategy
2937:{\displaystyle \Omega \left({\frac {1}{np}}\right)}
3263:
3026:
2972:
2936:
2878:
2827:
2798:
2760:
2665:
2570:
2430:
2381:
2349:
2204:
2166:
2137:
2088:
2068:
2028:
2008:
1965:
1919:
1899:
1870:
1816:
1748:
1725:
1672:
1640:
1620:
1594:
1574:
1537:
1511:
1485:
1465:
1430:
1358:
1299:
1239:
1199:
1173:
1101:
1042:
982:
942:
922:
892:
755:
735:
710:
683:
554:. The auction was first described academically by
3355:
3148:
1390:
1318:
1259:
1133:
1061:
1002:
841:
795:
3053:A different kind of generalization is to set a
1519:, then the buyer loses by raising his hand. If
2879:{\displaystyle np=\omega ({\sqrt {n\log n}})}
1930:Note first that if buyer 2 uses the strategy
1680:and so the expected payment of the winner is
1359:{\displaystyle \max _{j\neq i}b_{j}<b_{i}}
1300:{\displaystyle \max _{j\neq i}b_{j}>v_{i}}
1102:{\displaystyle \max _{j\neq i}b_{j}>b_{i}}
1043:{\displaystyle \max _{j\neq i}b_{j}<v_{i}}
511:
3332:. Carnegie Mellon University. Archived from
1817:{\displaystyle B(v)=e(v)={\tfrac {1}{2}}v.}
615:Self-revelation and incentive compatibility
27:Auction priced by second-highest sealed bid
2539:is calculated based on the declared costs
518:
504:
3222:
3193:
3023:
2969:
2492:Learn how and when to remove this message
2441:
2212:. Buyer 1's expected payoff is therefore
2761:{\displaystyle \scriptstyle G\in G(n,p)}
743:bid for the item. The payoff for bidder
3330:15–396: Science of the Web Course Notes
3317:
14:
3356:
1726:{\displaystyle e(v)={\tfrac {1}{2}}v.}
691:be bidder i's value for the item. Let
657:Proof of dominance of truthful bidding
3261:
3027:{\displaystyle np=\omega (\log n).\,}
2535:, a minimum-cost flow (MCF) in graph
626:
2828:{\displaystyle \scriptstyle \infty }
2474:adding citations to reliable sources
2445:
3129:, Cambridge University Press, 2004.
1907:is the buyer's bid for a valuation
564:in 1961 though it had been used by
24:
3163:10.1111/j.1540-6261.1961.tb02789.x
3037:
2908:
2821:
25:
3375:
3059:Bayesian-optimal mechanism design
2508:, VCG mechanisms are a family of
552:incentive to bid their true value
3211:The Journal of Political Economy
3182:Journal of Economic Perspectives
2799:{\displaystyle {\frac {p}{2-p}}}
2450:
1973:, then buyer 2's maximum bid is
1473:and the current asking price is
596:generalized second-price auction
49:
3114:, MIT Press, 2006, Chapter 1.
2461:needs additional citations for
581:second-price sealed-bid auction
536:sealed-bid second-price auction
3311:
3306:10.1016/j.ijindorg.2006.10.002
3294:
3284:
3255:
3202:
3176:Lucking-Reiley, David (2000).
3169:
3142:
3127:Putting Auction Theory to Work
3087:First-price sealed-bid auction
3017:
3005:
2966:
2960:
2873:
2854:
2754:
2742:
2657:
2651:
2639:
2620:
2527:In the case of network flows,
2425:
2419:
2376:
2370:
2344:
2335:
2319:
2303:
2285:
2273:
2261:
2249:
2246:
2240:
2231:
2225:
2190:
2184:
2112:
2106:
2063:
2043:
1989:
1983:
1946:
1940:
1894:
1888:
1851:
1845:
1790:
1784:
1775:
1769:
1699:
1693:
1667:
1655:
1569:
1557:
1240:{\displaystyle b_{i}<v_{i}}
983:{\displaystyle b_{i}>v_{i}}
13:
1:
3097:
2138:{\displaystyle B(x)=x/2<b}
1602:wins if his opponent's value
635:
609:
604:Vickrey–Clarke–Groves auction
2983:with high probability given
1831:Proof of revenue equivalence
736:{\displaystyle i{\text{'s}}}
7:
3075:
2721:nodes and edge probability
10:
3380:
3270:. Emerald Publishing Ltd.
2431:{\displaystyle b=v/2=B(v)}
570:Johann Wolfgang von Goethe
542:) is a type of sealed-bid
2076:. Let buyer 2's value be
486:Private electronic market
3136:
2715:Erdős–Rényi random graph
2389:takes on its maximum at
2096:. Then buyer 1 wins if
2009:{\displaystyle B(1)=1/2}
1966:{\displaystyle B(v)=v/2}
1871:{\displaystyle B(v)=v/2}
652:bids to increase profit.
606:(incentive compatible).
181:Generalized second-price
3107:, Academic Press, 2002.
2205:{\displaystyle w(b)=2b}
2167:{\displaystyle x<2b}
176:Generalized first-price
3151:The Journal of Finance
3112:Combinatorial Auctions
3028:
2974:
2973:{\displaystyle O(1)\,}
2938:
2880:
2829:
2800:
2762:
2667:
2572:
2442:Use in network routing
2432:
2383:
2351:
2206:
2168:
2139:
2090:
2070:
2030:
2010:
1967:
1921:
1901:
1872:
1818:
1750:
1727:
1674:
1642:
1622:
1621:{\displaystyle x<v}
1596:
1576:
1539:
1538:{\displaystyle v>b}
1513:
1512:{\displaystyle v<b}
1487:
1467:
1432:
1360:
1301:
1241:
1201:
1175:
1103:
1044:
984:
944:
930:). Assume that bidder
924:
894:
757:
737:
712:
685:
641:It does not allow for
231:Simultaneous ascending
3321:(30 September 2008).
3048:Vickrey–Clarke–Groves
3044:uniform price auction
3029:
2975:
2939:
2881:
2830:
2801:
2763:
2668:
2573:
2571:{\displaystyle e_{k}}
2512:schemes based on the
2433:
2384:
2352:
2207:
2169:
2140:
2091:
2071:
2031:
2011:
1968:
1922:
1902:
1873:
1819:
1751:
1728:
1675:
1643:
1623:
1597:
1577:
1540:
1514:
1488:
1468:
1433:
1361:
1302:
1242:
1202:
1176:
1104:
1045:
985:
945:
925:
923:{\displaystyle v_{i}}
895:
758:
738:
713:
711:{\displaystyle b_{i}}
686:
684:{\displaystyle v_{i}}
550:and gives bidders an
266:Vickrey–Clarke–Groves
3266:New Economy Handbook
3195:10.1257/jep.14.3.183
3070:revelation principle
2990:
2954:
2905:
2839:
2817:
2775:
2729:
2585:
2555:
2551:Each link (or node)
2470:improve this article
2393:
2382:{\displaystyle U(b)}
2364:
2219:
2178:
2149:
2100:
2080:
2040:
2020:
1977:
1934:
1911:
1900:{\displaystyle B(v)}
1882:
1839:
1763:
1740:
1687:
1652:
1632:
1606:
1586:
1554:
1523:
1497:
1477:
1457:
1373:
1314:
1255:
1211:
1191:
1116:
1057:
998:
954:
934:
907:
770:
747:
722:
695:
668:
600:incentive compatible
568:since 1893. In 1797
146:Discriminatory price
18:Second-price auction
2578:in the MCF is paid
594:exist, such as the
556:Columbia University
456:Revenue equivalence
141:Deferred-acceptance
3024:
2970:
2934:
2876:
2825:
2824:
2796:
2758:
2757:
2663:
2568:
2428:
2379:
2347:
2301:
2202:
2164:
2135:
2086:
2066:
2026:
2006:
1963:
1917:
1897:
1868:
1814:
1806:
1746:
1723:
1715:
1670:
1638:
1618:
1592:
1572:
1535:
1509:
1483:
1463:
1428:
1404:
1356:
1332:
1297:
1273:
1237:
1197:
1171:
1147:
1099:
1075:
1040:
1016:
980:
940:
920:
890:
885:
855:
809:
753:
733:
708:
681:
627:Ex-post efficiency
592:multiunit auctions
226:Sealed first-price
3339:on 8 October 2008
3055:reservation price
2928:
2871:
2794:
2701:without the link
2502:
2501:
2494:
2300:
2089:{\displaystyle x}
2029:{\displaystyle b}
1920:{\displaystyle v}
1805:
1749:{\displaystyle v}
1714:
1641:{\displaystyle v}
1595:{\displaystyle v}
1486:{\displaystyle b}
1466:{\displaystyle v}
1389:
1317:
1258:
1200:{\displaystyle i}
1132:
1060:
1001:
943:{\displaystyle i}
881:
840:
825:
794:
756:{\displaystyle i}
731:
528:
527:
306:Cancellation hunt
256:Value of revenues
126:Click-box bidding
16:(Redirected from
3371:
3364:Types of auction
3349:
3348:
3346:
3344:
3338:
3327:
3315:
3309:
3298:
3292:
3288:
3282:
3281:
3269:
3259:
3253:
3252:
3226:
3206:
3200:
3199:
3197:
3173:
3167:
3166:
3146:
3066:mechanism design
3033:
3031:
3030:
3025:
2979:
2977:
2976:
2971:
2943:
2941:
2940:
2935:
2933:
2929:
2927:
2916:
2885:
2883:
2882:
2877:
2872:
2858:
2834:
2832:
2831:
2826:
2805:
2803:
2802:
2797:
2795:
2793:
2779:
2767:
2765:
2764:
2759:
2697:indicates graph
2672:
2670:
2669:
2664:
2638:
2637:
2610:
2609:
2597:
2596:
2577:
2575:
2574:
2569:
2567:
2566:
2497:
2490:
2486:
2483:
2477:
2454:
2446:
2437:
2435:
2434:
2429:
2409:
2388:
2386:
2385:
2380:
2356:
2354:
2353:
2348:
2343:
2342:
2315:
2314:
2302:
2293:
2211:
2209:
2208:
2203:
2173:
2171:
2170:
2165:
2144:
2142:
2141:
2136:
2125:
2095:
2093:
2092:
2087:
2075:
2073:
2072:
2069:{\displaystyle }
2067:
2059:
2036:on the interval
2035:
2033:
2032:
2027:
2015:
2013:
2012:
2007:
2002:
1972:
1970:
1969:
1964:
1959:
1926:
1924:
1923:
1918:
1906:
1904:
1903:
1898:
1877:
1875:
1874:
1869:
1864:
1823:
1821:
1820:
1815:
1807:
1798:
1755:
1753:
1752:
1747:
1732:
1730:
1729:
1724:
1716:
1707:
1679:
1677:
1676:
1673:{\displaystyle }
1671:
1647:
1645:
1644:
1639:
1627:
1625:
1624:
1619:
1601:
1599:
1598:
1593:
1581:
1579:
1578:
1575:{\displaystyle }
1573:
1544:
1542:
1541:
1536:
1518:
1516:
1515:
1510:
1492:
1490:
1489:
1484:
1472:
1470:
1469:
1464:
1437:
1435:
1434:
1429:
1427:
1426:
1414:
1413:
1403:
1385:
1384:
1365:
1363:
1362:
1357:
1355:
1354:
1342:
1341:
1331:
1306:
1304:
1303:
1298:
1296:
1295:
1283:
1282:
1272:
1246:
1244:
1243:
1238:
1236:
1235:
1223:
1222:
1206:
1204:
1203:
1198:
1180:
1178:
1177:
1172:
1170:
1169:
1157:
1156:
1146:
1128:
1127:
1108:
1106:
1105:
1100:
1098:
1097:
1085:
1084:
1074:
1049:
1047:
1046:
1041:
1039:
1038:
1026:
1025:
1015:
989:
987:
986:
981:
979:
978:
966:
965:
949:
947:
946:
941:
929:
927:
926:
921:
919:
918:
899:
897:
896:
891:
889:
888:
882:
879:
865:
864:
854:
836:
835:
826:
823:
819:
818:
808:
790:
789:
762:
760:
759:
754:
742:
740:
739:
734:
732:
729:
717:
715:
714:
709:
707:
706:
690:
688:
687:
682:
680:
679:
648:Sellers may use
621:expected utility
585:opportunity cost
566:stamp collectors
520:
513:
506:
451:Price of anarchy
301:Calor licitantis
53:
30:
29:
21:
3379:
3378:
3374:
3373:
3372:
3370:
3369:
3368:
3354:
3353:
3352:
3342:
3340:
3336:
3325:
3316:
3312:
3299:
3295:
3289:
3285:
3278:
3260:
3256:
3224:10.1.1.560.8278
3207:
3203:
3174:
3170:
3147:
3143:
3139:
3103:Vijay Krishna,
3100:
3078:
3040:
3038:Generalizations
2991:
2988:
2987:
2955:
2952:
2951:
2920:
2915:
2911:
2906:
2903:
2902:
2857:
2840:
2837:
2836:
2818:
2815:
2814:
2783:
2778:
2776:
2773:
2772:
2730:
2727:
2726:
2709:
2696:
2633:
2629:
2605:
2601:
2592:
2588:
2586:
2583:
2582:
2562:
2558:
2556:
2553:
2552:
2547:
2506:network routing
2498:
2487:
2481:
2478:
2467:
2455:
2444:
2405:
2394:
2391:
2390:
2365:
2362:
2361:
2338:
2334:
2310:
2306:
2291:
2220:
2217:
2216:
2179:
2176:
2175:
2150:
2147:
2146:
2121:
2101:
2098:
2097:
2081:
2078:
2077:
2055:
2041:
2038:
2037:
2021:
2018:
2017:
1998:
1978:
1975:
1974:
1955:
1935:
1932:
1931:
1912:
1909:
1908:
1883:
1880:
1879:
1860:
1840:
1837:
1836:
1796:
1764:
1761:
1760:
1741:
1738:
1737:
1705:
1688:
1685:
1684:
1653:
1650:
1649:
1633:
1630:
1629:
1628:. Suppose that
1607:
1604:
1603:
1587:
1584:
1583:
1555:
1552:
1551:
1524:
1521:
1520:
1498:
1495:
1494:
1478:
1475:
1474:
1458:
1455:
1454:
1450:
1422:
1418:
1409:
1405:
1393:
1380:
1376:
1374:
1371:
1370:
1350:
1346:
1337:
1333:
1321:
1315:
1312:
1311:
1291:
1287:
1278:
1274:
1262:
1256:
1253:
1252:
1231:
1227:
1218:
1214:
1212:
1209:
1208:
1192:
1189:
1188:
1165:
1161:
1152:
1148:
1136:
1123:
1119:
1117:
1114:
1113:
1093:
1089:
1080:
1076:
1064:
1058:
1055:
1054:
1034:
1030:
1021:
1017:
1005:
999:
996:
995:
974:
970:
961:
957:
955:
952:
951:
935:
932:
931:
914:
910:
908:
905:
904:
884:
883:
878:
876:
870:
869:
860:
856:
844:
831:
827:
822:
820:
814:
810:
798:
785:
781:
774:
773:
771:
768:
767:
748:
745:
744:
728:
723:
720:
719:
702:
698:
696:
693:
692:
675:
671:
669:
666:
665:
659:
643:price discovery
638:
629:
617:
612:
577:Vickrey auction
562:William Vickrey
548:English auction
532:Vickrey auction
524:
495:
465:
430:
335:
331:Tacit collusion
280:
196:Multi-attribute
28:
23:
22:
15:
12:
11:
5:
3377:
3367:
3366:
3351:
3350:
3310:
3293:
3283:
3277:978-0123891723
3276:
3254:
3233:10.1086/250032
3217:(4): 854–859.
3201:
3188:(3): 183–192.
3168:
3140:
3138:
3135:
3134:
3133:
3130:
3125:Paul Milgrom,
3123:
3108:
3105:Auction Theory
3099:
3096:
3095:
3094:
3089:
3084:
3082:Auction theory
3077:
3074:
3039:
3036:
3035:
3034:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2981:
2980:
2968:
2965:
2962:
2959:
2945:
2944:
2932:
2926:
2923:
2919:
2914:
2910:
2875:
2870:
2867:
2864:
2861:
2856:
2853:
2850:
2847:
2844:
2823:
2807:
2806:
2792:
2789:
2786:
2782:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2705:
2692:
2674:
2673:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2636:
2632:
2628:
2625:
2622:
2619:
2616:
2613:
2608:
2604:
2600:
2595:
2591:
2565:
2561:
2543:
2500:
2499:
2458:
2456:
2449:
2443:
2440:
2427:
2424:
2421:
2418:
2415:
2412:
2408:
2404:
2401:
2398:
2378:
2375:
2372:
2369:
2358:
2357:
2346:
2341:
2337:
2333:
2330:
2327:
2324:
2321:
2318:
2313:
2309:
2305:
2299:
2296:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2163:
2160:
2157:
2154:
2145:, that is, if
2134:
2131:
2128:
2124:
2120:
2117:
2114:
2111:
2108:
2105:
2085:
2065:
2062:
2058:
2054:
2051:
2048:
2045:
2025:
2005:
2001:
1997:
1994:
1991:
1988:
1985:
1982:
1962:
1958:
1954:
1951:
1948:
1945:
1942:
1939:
1916:
1896:
1893:
1890:
1887:
1867:
1863:
1859:
1856:
1853:
1850:
1847:
1844:
1833:
1832:
1825:
1824:
1813:
1810:
1804:
1801:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1745:
1734:
1733:
1722:
1719:
1713:
1710:
1704:
1701:
1698:
1695:
1692:
1669:
1666:
1663:
1660:
1657:
1637:
1617:
1614:
1611:
1591:
1571:
1568:
1565:
1562:
1559:
1534:
1531:
1528:
1508:
1505:
1502:
1482:
1462:
1449:
1446:
1441:
1440:
1425:
1421:
1417:
1412:
1408:
1402:
1399:
1396:
1392:
1388:
1383:
1379:
1367:
1353:
1349:
1345:
1340:
1336:
1330:
1327:
1324:
1320:
1308:
1294:
1290:
1286:
1281:
1277:
1271:
1268:
1265:
1261:
1234:
1230:
1226:
1221:
1217:
1196:
1184:
1183:
1168:
1164:
1160:
1155:
1151:
1145:
1142:
1139:
1135:
1131:
1126:
1122:
1110:
1096:
1092:
1088:
1083:
1079:
1073:
1070:
1067:
1063:
1051:
1037:
1033:
1029:
1024:
1020:
1014:
1011:
1008:
1004:
977:
973:
969:
964:
960:
939:
917:
913:
901:
900:
887:
877:
875:
872:
871:
868:
863:
859:
853:
850:
847:
843:
839:
834:
830:
821:
817:
813:
807:
804:
801:
797:
793:
788:
784:
780:
779:
777:
752:
727:
705:
701:
678:
674:
658:
655:
654:
653:
646:
637:
634:
628:
625:
616:
613:
611:
608:
526:
525:
523:
522:
515:
508:
500:
497:
496:
494:
493:
488:
483:
477:
474:
473:
467:
466:
464:
463:
461:Winner's curse
458:
453:
448:
442:
439:
438:
432:
431:
429:
428:
423:
418:
413:
408:
403:
398:
393:
388:
383:
378:
373:
368:
363:
358:
353:
347:
344:
343:
337:
336:
334:
333:
328:
323:
318:
313:
308:
303:
298:
292:
289:
288:
282:
281:
279:
278:
273:
268:
263:
258:
253:
248:
243:
238:
233:
228:
223:
218:
213:
208:
203:
198:
193:
188:
183:
178:
173:
168:
163:
158:
153:
148:
143:
138:
133:
128:
123:
118:
113:
108:
103:
98:
93:
88:
87:
86:
81:
76:
65:
62:
61:
55:
54:
46:
45:
39:
38:
26:
9:
6:
4:
3:
2:
3376:
3365:
3362:
3361:
3359:
3335:
3331:
3324:
3320:
3319:von Ahn, Luis
3314:
3307:
3303:
3297:
3287:
3279:
3273:
3268:
3267:
3258:
3250:
3246:
3242:
3238:
3234:
3230:
3225:
3220:
3216:
3212:
3205:
3196:
3191:
3187:
3183:
3179:
3172:
3164:
3160:
3156:
3152:
3145:
3141:
3131:
3128:
3124:
3121:
3120:0-262-03342-9
3117:
3113:
3109:
3106:
3102:
3101:
3093:
3090:
3088:
3085:
3083:
3080:
3079:
3073:
3071:
3067:
3062:
3060:
3056:
3051:
3049:
3045:
3020:
3014:
3011:
3008:
3002:
2999:
2996:
2993:
2986:
2985:
2984:
2963:
2957:
2950:
2949:
2948:
2930:
2924:
2921:
2917:
2912:
2901:
2900:
2899:
2897:
2893:
2889:
2868:
2865:
2862:
2859:
2851:
2848:
2845:
2842:
2813:, approaches
2812:
2790:
2787:
2784:
2780:
2771:
2770:
2769:
2751:
2748:
2745:
2739:
2736:
2733:
2724:
2720:
2716:
2711:
2708:
2704:
2700:
2695:
2691:
2688: −
2687:
2683:
2679:
2660:
2654:
2648:
2645:
2642:
2634:
2630:
2626:
2623:
2617:
2614:
2611:
2606:
2602:
2598:
2593:
2589:
2581:
2580:
2579:
2563:
2559:
2549:
2546:
2542:
2538:
2534:
2530:
2525:
2523:
2522:strategyproof
2519:
2515:
2511:
2507:
2496:
2493:
2485:
2475:
2471:
2465:
2464:
2459:This section
2457:
2453:
2448:
2447:
2439:
2422:
2416:
2413:
2410:
2406:
2402:
2399:
2396:
2373:
2367:
2339:
2331:
2328:
2325:
2322:
2316:
2311:
2307:
2297:
2294:
2288:
2282:
2279:
2276:
2270:
2267:
2264:
2258:
2255:
2252:
2243:
2237:
2234:
2228:
2222:
2215:
2214:
2213:
2199:
2196:
2193:
2187:
2181:
2161:
2158:
2155:
2152:
2132:
2129:
2126:
2122:
2118:
2115:
2109:
2103:
2083:
2060:
2056:
2052:
2049:
2046:
2023:
2003:
1999:
1995:
1992:
1986:
1980:
1960:
1956:
1952:
1949:
1943:
1937:
1928:
1914:
1891:
1885:
1865:
1861:
1857:
1854:
1848:
1842:
1830:
1829:
1828:
1811:
1808:
1802:
1799:
1793:
1787:
1781:
1778:
1772:
1766:
1759:
1758:
1757:
1743:
1720:
1717:
1711:
1708:
1702:
1696:
1690:
1683:
1682:
1681:
1664:
1661:
1658:
1635:
1615:
1612:
1609:
1589:
1566:
1563:
1560:
1547:
1532:
1529:
1526:
1506:
1503:
1500:
1480:
1460:
1445:
1423:
1419:
1415:
1410:
1406:
1400:
1397:
1394:
1386:
1381:
1377:
1368:
1351:
1347:
1343:
1338:
1334:
1328:
1325:
1322:
1309:
1292:
1288:
1284:
1279:
1275:
1269:
1266:
1263:
1250:
1249:
1248:
1232:
1228:
1224:
1219:
1215:
1194:
1166:
1162:
1158:
1153:
1149:
1143:
1140:
1137:
1129:
1124:
1120:
1111:
1094:
1090:
1086:
1081:
1077:
1071:
1068:
1065:
1052:
1035:
1031:
1027:
1022:
1018:
1012:
1009:
1006:
993:
992:
991:
975:
971:
967:
962:
958:
937:
915:
911:
873:
866:
861:
857:
851:
848:
845:
837:
832:
828:
815:
811:
805:
802:
799:
791:
786:
782:
775:
766:
765:
764:
750:
725:
703:
699:
676:
672:
662:
651:
647:
644:
640:
639:
633:
624:
622:
607:
605:
601:
597:
593:
588:
586:
582:
578:
573:
571:
567:
563:
560:
557:
553:
549:
545:
541:
537:
533:
521:
516:
514:
509:
507:
502:
501:
499:
498:
492:
489:
487:
484:
482:
479:
478:
476:
475:
472:
469:
468:
462:
459:
457:
454:
452:
449:
447:
446:Digital goods
444:
443:
441:
440:
437:
434:
433:
427:
424:
422:
419:
417:
414:
412:
409:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
372:
369:
367:
364:
362:
359:
357:
354:
352:
349:
348:
346:
345:
342:
339:
338:
332:
329:
327:
324:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
293:
291:
290:
287:
284:
283:
277:
274:
272:
269:
267:
264:
262:
259:
257:
254:
252:
249:
247:
246:Uniform price
244:
242:
241:Traffic light
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
137:
134:
132:
131:Combinatorial
129:
127:
124:
122:
119:
117:
114:
112:
109:
107:
106:Best/not best
104:
102:
101:Barter double
99:
97:
94:
92:
89:
85:
82:
80:
77:
75:
72:
71:
70:
67:
66:
64:
63:
60:
57:
56:
52:
48:
47:
44:
41:
40:
36:
32:
31:
19:
3341:. Retrieved
3334:the original
3329:
3313:
3296:
3286:
3265:
3257:
3214:
3210:
3204:
3185:
3181:
3171:
3154:
3150:
3144:
3126:
3111:
3104:
3063:
3052:
3041:
2982:
2946:
2895:
2891:
2887:
2810:
2808:
2722:
2718:
2712:
2706:
2702:
2698:
2693:
2689:
2685:
2681:
2677:
2675:
2550:
2544:
2540:
2536:
2526:
2517:
2503:
2488:
2479:
2468:Please help
2463:verification
2460:
2359:
1929:
1834:
1826:
1735:
1548:
1451:
1442:
1185:
902:
663:
660:
630:
618:
589:
580:
576:
574:
539:
535:
531:
529:
381:Domain names
260:
236:Single-price
136:Common value
3157:(1): 8–37.
3092:VCG auction
2768:approaches
2514:added value
96:Anglo-Dutch
79:Bidding fee
3343:6 November
3323:"Auctions"
3098:References
2676:where MCF(
2360:Note that
880:otherwise.
718:be bidder
636:Weaknesses
610:Properties
602:) and the
351:Algorithms
251:Unique bid
206:No-reserve
3219:CiteSeerX
3012:
3003:ω
2909:Ω
2866:
2852:ω
2822:∞
2788:−
2737:∈
2649:
2643:−
2627:−
2618:
2533:multicast
2326:−
2317:−
2280:−
2256:−
1398:≠
1326:≠
1267:≠
1141:≠
1069:≠
1010:≠
849:≠
803:≠
792:−
559:professor
416:Virginity
271:Walrasian
201:Multiunit
111:Brazilian
91:Amsterdam
3358:Category
3291:242–259.
3249:53490333
3076:See also
2482:May 2009
1878:, where
824:if
491:Software
481:Ebidding
406:Spectrum
371:Children
341:Contexts
221:Scottish
191:Knapsack
186:Japanese
116:Calcutta
43:Auctions
35:a series
33:Part of
3241:2990730
2894:,
2529:unicast
2510:payment
544:auction
386:Flowers
376:Players
366:Charity
326:Suicide
321:Sniping
316:Rigging
296:Shading
286:Bidding
261:Vickrey
216:Reverse
166:Forward
161:English
74:Chinese
69:All-pay
3274:
3247:
3239:
3221:
3118:
3068:, the
2835:, for
471:Online
436:Theory
411:Stamps
401:Slaves
276:Yankee
171:French
151:Double
121:Candle
84:Dollar
3337:(PDF)
3326:(PDF)
3245:S2CID
3237:JSTOR
3137:Notes
2898:) is
2717:with
1493:. If
1439:zero.
1207:bids
1182:zero.
950:bids
650:shill
540:SBSPA
426:Wives
391:Loans
356:Autos
156:Dutch
59:Types
3345:2008
3272:ISBN
3116:ISBN
2947:and
2684:and
2518:plus
2156:<
2130:<
1613:<
1530:>
1504:<
1416:<
1387:<
1344:<
1285:>
1225:<
1159:<
1130:<
1087:>
1028:<
968:>
838:>
664:Let
579:and
421:Wine
396:Scam
311:Jump
211:Rank
3302:doi
3229:doi
3215:106
3190:doi
3159:doi
3064:In
3009:log
2863:log
2809:as
2646:MCF
2615:MCF
2531:or
2504:In
2472:by
1756:is
1391:max
1369:If
1319:max
1310:If
1260:max
1251:If
1134:max
1112:If
1062:max
1053:If
1003:max
994:If
842:max
796:max
763:is
534:or
361:Art
3360::
3328:.
3243:.
3235:.
3227:.
3213:.
3186:14
3184:.
3180:.
3155:16
3153:.
3061:.
2725:,
2438:.
1247:.
990:.
730:'s
587:.
530:A
37:on
3347:.
3308:.
3304::
3280:.
3251:.
3231::
3198:.
3192::
3165:.
3161::
3122:.
3021:.
3018:)
3015:n
3006:(
3000:=
2997:p
2994:n
2967:)
2964:1
2961:(
2958:O
2931:)
2925:p
2922:n
2918:1
2913:(
2896:p
2892:n
2890:(
2888:G
2874:)
2869:n
2860:n
2855:(
2849:=
2846:p
2843:n
2811:n
2791:p
2785:2
2781:p
2755:)
2752:p
2749:,
2746:n
2743:(
2740:G
2734:G
2723:p
2719:n
2707:k
2703:e
2699:G
2694:k
2690:e
2686:G
2682:G
2678:G
2661:,
2658:)
2655:G
2652:(
2640:)
2635:k
2631:e
2624:G
2621:(
2612:+
2607:k
2603:d
2599:=
2594:k
2590:p
2564:k
2560:e
2545:k
2541:d
2537:G
2495:)
2489:(
2484:)
2480:(
2466:.
2426:)
2423:v
2420:(
2417:B
2414:=
2411:2
2407:/
2403:v
2400:=
2397:b
2377:)
2374:b
2371:(
2368:U
2345:]
2340:2
2336:)
2332:b
2329:2
2323:v
2320:(
2312:2
2308:v
2304:[
2298:2
2295:1
2289:=
2286:)
2283:b
2277:v
2274:(
2271:b
2268:2
2265:=
2262:)
2259:b
2253:v
2250:(
2247:)
2244:b
2241:(
2238:w
2235:=
2232:)
2229:b
2226:(
2223:U
2200:b
2197:2
2194:=
2191:)
2188:b
2185:(
2182:w
2162:b
2159:2
2153:x
2133:b
2127:2
2123:/
2119:x
2116:=
2113:)
2110:x
2107:(
2104:B
2084:x
2064:]
2061:2
2057:/
2053:1
2050:,
2047:0
2044:[
2024:b
2004:2
2000:/
1996:1
1993:=
1990:)
1987:1
1984:(
1981:B
1961:2
1957:/
1953:v
1950:=
1947:)
1944:v
1941:(
1938:B
1915:v
1895:)
1892:v
1889:(
1886:B
1866:2
1862:/
1858:v
1855:=
1852:)
1849:v
1846:(
1843:B
1812:.
1809:v
1803:2
1800:1
1794:=
1791:)
1788:v
1785:(
1782:e
1779:=
1776:)
1773:v
1770:(
1767:B
1744:v
1721:.
1718:v
1712:2
1709:1
1703:=
1700:)
1697:v
1694:(
1691:e
1668:]
1665:v
1662:,
1659:0
1656:[
1636:v
1616:v
1610:x
1590:v
1570:]
1567:1
1564:,
1561:0
1558:[
1533:b
1527:v
1507:b
1501:v
1481:b
1461:v
1424:i
1420:v
1411:j
1407:b
1401:i
1395:j
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1339:j
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1019:b
1013:i
1007:j
976:i
972:v
963:i
959:b
938:i
916:i
912:v
874:0
867:,
862:j
858:b
852:i
846:j
833:i
829:b
816:j
812:b
806:i
800:j
787:i
783:v
776:{
751:i
726:i
704:i
700:b
677:i
673:v
538:(
519:e
512:t
505:v
20:)
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