37:
4836:
An important implication of the theorem is that any single-item auction which unconditionally gives the item to the highest bidder is going to have the same expected revenue. This means that, if we want to increase the auctioneer's revenue, the outcome function must be changed. One way to do this is
4468:
Now, a player simply pays what the player bids, and let's assume that players with higher values still win, so that the probability of winning is simply a player's value, as in the second price auction. We will later show that this assumption was correct. Again, a player pays nothing if he loses the
3230:
Suppose that a buyer has value v and bids b. His opponent bids according to the equilibrium bidding strategy. The support of the opponent's bid distribution is . Thus any bid of at least B(1) wins with probability 1. Therefore, the best bid b lies in the interval and so we can write this bid as b =
4672:
Note that with this bidding function, the player with the higher value still wins. We can show that this is the correct equilibrium bidding function in an additional way, by thinking about how a player should maximize his bid given that all other players are bidding using this bidding function. See
4462:
3999:
2755:
Consider the case of two buyers, each with a value that is an independent draw from a distribution with support , cumulative distribution function F(v) and probability density function f(v). If buyers behave according to their dominant strategies, then a buyer with value v wins if his opponent's
1681:
In other words, if each player bids such that they bid the expected value of second highest bid, assuming that theirs was the highest, then no player has any incentive to deviate. If this were true, then it is easy to see that the expected revenue from this auction is also
3226:
Let B(v) be the equilibrium bid function in the sealed first-price auction. We establish revenue equivalence by showing that B(v)=e(v), that is, the equilibrium payment by the winner in one auction is equal to the equilibrium expected payment by the winner in the other.
3678:
3684:
That is if B(x) is the buyer's best response it must satisfy this first order condition. Finally we note that for B(v) to be the equilibrium bid function, the buyer's best response must be B(v). Thus x=v. Substituting for x in the necessary condition,
1310:
which is
Bayesian-Nash incentive compatible; second-price auction is dominant-strategy-incentive-compatible, which is even stronger than Bayesian-Nash incentive compatible. The two mechanisms fulfill the conditions of the theorem because:
2171:
3020:
4359:
3896:
2451:
2573:
4841:
on the item. This changes the
Outcome function since now the item is not always given to the highest bidder. By carefully selecting the reservation price, an auctioneer can get a substantially higher expected revenue.
3357:
1676:
4311:
3845:
We can use revenue equivalence to predict the bidding function of a player in a game. Consider the two player version of the second price auction and the first price auction, where each player's value is drawn
2021:
4193:
3787:
4605:
3221:
2752:), a buyer's dominant strategy is to remain in the auction until the asking price is equal to his value. Then, if he is the last one remaining in the arena, he wins and pays the second-highest bid.
2676:
2889:
644:
4790:
3119:
4321:
We can use revenue equivalence to generate the correct symmetric bidding function in the first price auction. Suppose that in the first price auction, each player has the bidding function
1930:
4667:
2305:
2813:
3453:
4722:
921:
4525:
2223:
786:
1723:
1555:
1512:
1133:
4827:
4861:
When the players' valuations are inter-dependent, e.g., if the valuations depend on some state of the world that is only partially known to the bidders (this is related to the
3465:
4236:
4082:
4042:
1446:
523:
that states that given certain conditions, any mechanism that results in the same outcomes (i.e. allocates items to the same bidders) also has the same expected revenue.
2738:
1372:
In fact, we can use revenue equivalence to prove that many types of auctions are revenue equivalent. For example, the first price auction, second price auction, and the
3835:
1245:
1351:
1210:
1061:
970:
1806:
1016:
866:
2029:
701:
4348:
1780:
2700:
2597:
1866:
1846:
1826:
1743:
1469:
1406:
1265:
1081:
990:
832:
809:
721:
674:
592:
572:
549:
3880:
2900:
4457:{\displaystyle E({\text{Payment}}~|~{\text{Player 1 wins}})P({\text{Player 1 wins}})+E({\text{Payment}}~|~{\text{Player 1 loses}})P({\text{Player 1 loses}})}
3994:{\displaystyle E({\text{Payment}}~|~{\text{Player 1 wins}})P({\text{Player 1 wins}})+E({\text{Payment}}~|~{\text{Player 1 loses}})P({\text{Player 1 loses}})}
4004:
Since players bid truthfully in a second price auction, we can replace all prices with players' values. If player 1 wins, he pays what player 2 bids, or
2314:
2459:
36:
4858:
rather than risk-neutral as assumed above. In this case, it is known that first-price auctions generate more revenue than second-price auctions.
251:
3237:
1572:
4244:
4530:
By the
Revenue Equivalence principle, we can equate this expression to the revenue of the second-price auction that we calculated above:
1360:
Indeed, the expected payment for each player is the same in both auctions, and the auctioneer's revenue is the same; see the page on
4921:
3847:
1935:
503:
241:
1376:
are all revenue equivalent when the bidders are symmetric (that is, their valuations are independent and identically distributed).
4090:
166:
4536:
161:
4674:
1361:
216:
211:
3691:
4869:
generates more revenue than second-price auction, as it lets the bidders learn information from the bids of other players.
3131:
1147:
in which a player's strategy is his reported type as a function of his true type. A mechanism is said to be
Bayesian-Nash
1135:, determining how much each player should receive (a negative payment means that the player should pay a positive amount).
2824:
436:
131:
2605:
600:
291:
126:
4734:
1874:
4964:
4616:
2231:
1152:
471:
3031:
2762:
386:
3369:
3025:
Multiplying both sides by F(v) and differentiating by v yields the following differential equation for e(v).
4959:
4688:
1148:
871:
371:
181:
4475:
2179:
729:
496:
4908:
1685:
1517:
1474:
1086:
1569:, where the player with the highest bid simply pays its bid, if all players bid using a bidding function
4798:
3673:{\displaystyle {U}'(x)=f(x)(v-B(x))-F(x){B}'(x)=F(x)\left({\frac {f(x)}{F(x)}}(v-B(x))-{B}'(x)\right)=0}
431:
376:
336:
231:
226:
116:
91:
3793:
Note that this differential equation is identical to that for e(v). Since e(0)=B(0)=0 it follows that
1388:, in which the player with the highest bid pays the second highest bid. It is optimal for each player
1303:
221:
121:
3231:
B(x) where x lies in . If the opponent has value y he bids B(y). Therefore, the win probability is
1566:
1307:
1299:
653:
366:
81:
64:
4201:
4047:
4007:
3890:
The expected payment of the first player in the second price auction can be computed as follows:
1411:
396:
236:
191:
2705:
3796:
1218:
4917:
4862:
4838:
2166:{\displaystyle Pr(\max _{i>1}v_{i}<z)(v-E(\max _{i>1}v_{i}~|~v_{i}<z~\forall ~i))}
446:
401:
256:
186:
96:
76:
20:
1318:
1177:
1028:
937:
1785:
995:
841:
489:
476:
391:
286:
206:
176:
171:
101:
3363:
The buyer's expected payoff is his win probability times his net gain if he wins, that is,
679:
4900:
4866:
4725:
4324:
2749:
1756:
1385:
1373:
1140:
351:
316:
311:
306:
246:
201:
151:
146:
111:
59:
54:
4685:
Similarly, we know that the expected payment of player 1 in the second price auction is
3015:{\displaystyle e(v)={\frac {C(v)}{F(v)}}={\frac {\int \limits _{0}^{v}{}xf(x)dx}{F(v)}}}
4937:
4892:
2685:
2582:
1851:
1831:
1811:
1728:
1454:
1391:
1271:
1250:
1066:
975:
817:
794:
706:
659:
577:
557:
534:
520:
456:
421:
361:
261:
156:
136:
106:
69:
3853:
4953:
4904:
4855:
1144:
1019:
356:
141:
2446:{\displaystyle E(X~|~X\leq z)\cdot Pr(X<z)=\int _{0}^{z}Pr(X<z)-Pr(X<y)dy}
411:
406:
381:
341:
296:
196:
1353:
function is the same in both mechanisms - the highest bidder wins the item; and:
574:
agents which have different valuations for each outcome. The valuation of agent
346:
301:
281:
4896:
2568:{\displaystyle Pr(X<z)\cdot v-Pr(X<z)\cdot z+\int _{0}^{z}Pr(X<y)dy}
1267:(averaged on the types of the other players) is the same in both mechanisms;
649:
which expresses the value it has for each alternative, in monetary terms.
3352:{\displaystyle w=\Pr\{b<B(y)\}=\Pr\{B(x)<B(y)\}=\Pr\{x<y\}=F(v)}
466:
1289:
The expected revenue (- sum of payments) is the same in both mechanisms.
2740:
is monotone increasing, we verify that this is indeed a maximum point.
326:
271:
86:
44:
28:
1671:{\displaystyle b(v)=E(\max _{j\neq i}v_{j}~|~v_{j}\leq v~\forall ~j),}
4795:
Thus, the bidding function for each player in the all-pay auction is
4306:{\displaystyle {\frac {v_{1}}{2}}\cdot v_{1}={\frac {v_{1}^{2}}{2}}}
1356:
A player who values the item as 0 always pays 0 in both mechanisms.
4850:
The revenue-equivalence theorem breaks in some important cases:
1170:
For any two
Bayesian-Nash incentive compatible mechanisms, if:
2016:{\displaystyle E(\max _{i>1}v_{i}~|~v_{i}<z~\forall ~i)}
4188:{\displaystyle E(v_{2}~|~v_{2}<v_{1})P(v_{2}<v_{1})+0}
2894:
The expected payment conditional upon winning is therefore
4600:{\displaystyle b(v_{1})\cdot v_{1}={\frac {v_{1}^{2}}{2}}}
4238:
come from a uniform distribution, we can simplify this to
4084:. Since payment is zero when player 1 loses, the above is
3459:
Differentiating, the necessary condition for a maximum is
1139:
The agents' types are independent identically-distributed
1368:
Equivalence of auction mechanisms in single item auctions
4724:, and this must be equal to the expected payment in the
703:(positive or negative), then the total utility of agent
594:(also called its "type") is represented as a function:
4353:
The expected payment of player 1 in this game is then
3841:
Using revenue equivalence to predict bidding functions
3782:{\displaystyle {\frac {f(v)}{F(v)}}(v-B(v))-{B}'(v)=0}
2225:, a random variable. Then we can rewrite the above as
1471:
wins the auction, and pays the second highest bid, or
4801:
4737:
4691:
4619:
4539:
4478:
4362:
4327:
4247:
4204:
4093:
4050:
4010:
3899:
3856:
3799:
3694:
3468:
3372:
3240:
3134:
3034:
2903:
2827:
2765:
2708:
2688:
2608:
2585:
2462:
2317:
2234:
2182:
2032:
1938:
1877:
1868:
such that the player's expected payoff is maximized.
1854:
1834:
1814:
1788:
1759:
1731:
1688:
1575:
1520:
1477:
1457:
1414:
1394:
1321:
1298:
A classic example is the pair of auction mechanisms:
1253:
1221:
1180:
1089:
1069:
1031:
998:
978:
940:
874:
844:
820:
797:
732:
709:
682:
662:
603:
580:
560:
537:
3216:{\displaystyle {e}'(v)={\frac {f(v)}{F(v)}}(v-e(v))}
2884:{\displaystyle C(v)=\int \limits _{0}^{v}{}xf(x)dx}
4821:
4784:
4716:
4661:
4599:
4519:
4456:
4342:
4305:
4230:
4187:
4076:
4036:
3993:
3874:
3829:
3781:
3672:
3447:
3351:
3215:
3113:
3014:
2883:
2807:
2732:
2694:
2670:
2591:
2567:
2445:
2299:
2217:
2165:
2015:
1924:
1860:
1840:
1820:
1800:
1774:
1737:
1717:
1670:
1549:
1506:
1463:
1440:
1400:
1345:
1259:
1239:
1204:
1127:
1075:
1055:
1010:
984:
964:
915:
860:
826:
803:
780:
715:
695:
668:
638:
586:
566:
543:
2671:{\displaystyle Pr(X<z)'(v-z)=0\Rightarrow v=z}
1286:types are the same in both mechanisms, and hence:
639:{\displaystyle v_{i}:X\longrightarrow R_{\geq 0}}
4350:, where this function is unknown at this point.
3316:
3277:
3247:
2772:
2190:
2093:
2043:
1946:
1888:
1690:
1598:
1522:
1479:
1063:function, that takes as input the value-vector
791:The vector of all value-functions is denoted by
4785:{\displaystyle {\frac {v_{1}^{2}}{2}}=b(v_{1})}
2756:value x is lower. Thus his win probability is
972:function, that takes as input the value-vector
4610:From this, we can infer the bidding function:
2702:maximizes the player's expected payoff. Since
656:functions; this means that, if the outcome is
4916:. Cambridge, UK: Cambridge University Press.
1270:The valuation of each player is drawn from a
1212:function is the same in both mechanisms, and:
1155:in which all players report their true type.
676:and in addition the agent receives a payment
497:
8:
3331:
3319:
3310:
3280:
3271:
3250:
2787:
2775:
1925:{\displaystyle Pr(\max _{i>1}v_{i}<z)}
1753:To prove this, suppose that a player 1 bids
4662:{\displaystyle b(v_{1})={\frac {v_{1}}{2}}}
2300:{\displaystyle Pr(X<z)(v-E(X~|X\leq z))}
834:, the vector of all value-functions of the
3114:{\displaystyle {e}'(v)F(v)+e(v)f(v)=vf(v)}
2808:{\displaystyle w=\Pr\{x<v\}\equiv F(v)}
1514:. The revenue from this auction is simply
504:
490:
15:
4808:
4802:
4800:
4773:
4749:
4744:
4738:
4736:
4703:
4698:
4692:
4690:
4648:
4642:
4630:
4618:
4586:
4581:
4575:
4566:
4550:
4538:
4505:
4489:
4477:
4446:
4432:
4424:
4416:
4399:
4385:
4377:
4369:
4361:
4326:
4292:
4287:
4281:
4272:
4254:
4248:
4246:
4222:
4209:
4203:
4170:
4157:
4138:
4125:
4113:
4104:
4092:
4068:
4055:
4049:
4028:
4015:
4009:
3983:
3969:
3961:
3953:
3936:
3922:
3914:
3906:
3898:
3855:
3798:
3755:
3695:
3693:
3641:
3581:
3543:
3470:
3467:
3371:
3239:
3157:
3136:
3133:
3036:
3033:
2972:
2966:
2961:
2954:
2919:
2902:
2858:
2852:
2847:
2826:
2764:
2748:In the open ascending price auction (aka
2707:
2687:
2607:
2584:
2532:
2527:
2461:
2386:
2381:
2330:
2316:
2277:
2233:
2209:
2193:
2181:
2133:
2121:
2112:
2096:
2062:
2046:
2031:
1986:
1974:
1965:
1949:
1937:
1907:
1891:
1876:
1853:
1833:
1813:
1808:, effectively bluffing that its value is
1787:
1758:
1730:
1709:
1693:
1687:
1638:
1626:
1617:
1601:
1574:
1541:
1525:
1519:
1498:
1482:
1476:
1456:
1432:
1419:
1413:
1393:
1320:
1252:
1231:
1226:
1220:
1179:
1116:
1097:
1088:
1068:
1030:
997:
977:
939:
901:
888:
873:
849:
843:
819:
796:
772:
750:
737:
731:
708:
687:
681:
661:
627:
608:
602:
579:
559:
536:
3448:{\displaystyle U=w(v-B(x))=F(x)(v-B(x))}
4887:
4885:
4883:
4879:
455:
420:
325:
270:
43:
27:
4717:{\displaystyle {\frac {v_{1}^{2}}{2}}}
916:{\displaystyle v\equiv (v_{i},v_{-i})}
4520:{\displaystyle b(v_{1})\cdot v_{1}+0}
2218:{\displaystyle X=\max _{i>1}v_{i}}
2023:. Then a player's expected payoff is
781:{\displaystyle u_{i}:=v_{i}(x)+p_{i}}
7:
1718:{\displaystyle \max _{j\neq i}b_{j}}
1550:{\displaystyle \max _{j\neq i}b_{j}}
1507:{\displaystyle \max _{j\neq i}b_{j}}
1128:{\displaystyle (p_{1},\dots ,p_{n})}
2579:Taking derivatives with respect to
1932:. The expected cost of this bid is
1871:The probability of winning is then
4822:{\displaystyle {\frac {v^{2}}{2}}}
2148:
2001:
1653:
1083:and returns a vector of payments,
14:
1247:, the expected payment of player
4939:Approximation in Economic Design
1386:second price single item auction
35:
4779:
4766:
4675:first-price sealed-bid auction
4636:
4623:
4556:
4543:
4495:
4482:
4451:
4443:
4437:
4425:
4413:
4404:
4396:
4390:
4378:
4366:
4337:
4331:
4176:
4150:
4144:
4114:
4097:
3988:
3980:
3974:
3962:
3950:
3941:
3933:
3927:
3915:
3903:
3869:
3857:
3824:
3818:
3809:
3803:
3770:
3764:
3748:
3745:
3739:
3727:
3721:
3715:
3707:
3701:
3656:
3650:
3634:
3631:
3625:
3613:
3607:
3601:
3593:
3587:
3573:
3567:
3558:
3552:
3539:
3533:
3524:
3521:
3515:
3503:
3500:
3494:
3485:
3479:
3442:
3439:
3433:
3421:
3418:
3412:
3403:
3400:
3394:
3382:
3346:
3340:
3307:
3301:
3292:
3286:
3268:
3262:
3210:
3207:
3201:
3189:
3183:
3177:
3169:
3163:
3151:
3145:
3108:
3102:
3090:
3084:
3078:
3072:
3063:
3057:
3051:
3045:
3006:
3000:
2986:
2980:
2945:
2939:
2931:
2925:
2913:
2907:
2872:
2866:
2837:
2831:
2802:
2796:
2727:
2715:
2656:
2647:
2635:
2628:
2615:
2556:
2544:
2511:
2499:
2481:
2469:
2453:, we can rewrite the above as
2434:
2422:
2410:
2398:
2371:
2359:
2347:
2331:
2321:
2294:
2291:
2278:
2268:
2256:
2253:
2241:
2160:
2157:
2122:
2089:
2077:
2074:
2039:
2010:
1975:
1942:
1919:
1884:
1769:
1763:
1662:
1627:
1594:
1585:
1579:
1362:first-price sealed-bid auction
1143:. Thus, a mechanism induces a
1122:
1090:
910:
881:
762:
756:
620:
1:
2682:Thus bidding with your value
1848:. We want to find a value of
1163:Under these assumptions, the
2818:and his expected payment is
2311:Using the general fact that
1678:this is a Nash equilibrium.
1306:. First-price auction has a
4231:{\displaystyle v_{1},v_{2}}
4077:{\displaystyle p_{1}=v_{1}}
4037:{\displaystyle p_{2}=v_{2}}
3125:Rearranging this equation,
1441:{\displaystyle b_{i}=v_{i}}
1165:revenue equivalence theorem
4981:
2733:{\displaystyle Pr(X<z)}
3830:{\displaystyle B(v)=e(v)}
1282:The expected payments of
1240:{\displaystyle v_{i}^{0}}
1167:then says the following.
1153:Bayesian Nash equilibrium
472:Private electronic market
4469:auction. We then obtain
4044:. Player 1 himself bids
930:is a pair of functions:
167:Generalized second-price
4910:Algorithmic Game Theory
1346:{\displaystyle Outcome}
1205:{\displaystyle Outcome}
1056:{\displaystyle Payment}
992:and returns an outcome
965:{\displaystyle Outcome}
162:Generalized first-price
4823:
4786:
4718:
4663:
4601:
4521:
4458:
4344:
4307:
4232:
4189:
4078:
4038:
3995:
3876:
3831:
3783:
3674:
3449:
3353:
3217:
3115:
3016:
2971:
2885:
2857:
2809:
2734:
2696:
2672:
2593:
2569:
2447:
2301:
2219:
2167:
2017:
1926:
1862:
1842:
1822:
1802:
1801:{\displaystyle z<v}
1776:
1739:
1719:
1672:
1551:
1508:
1465:
1442:
1402:
1347:
1261:
1241:
1206:
1129:
1077:
1057:
1012:
1011:{\displaystyle x\in X}
986:
966:
917:
862:
861:{\displaystyle v_{-i}}
828:
805:
782:
717:
697:
670:
640:
588:
568:
551:of possible outcomes.
545:
217:Simultaneous ascending
4865:). In this scenario,
4854:When the players are
4824:
4787:
4719:
4664:
4602:
4522:
4459:
4345:
4308:
4233:
4190:
4079:
4039:
3996:
3877:
3832:
3784:
3675:
3450:
3354:
3218:
3116:
3017:
2957:
2886:
2843:
2810:
2735:
2697:
2673:
2594:
2570:
2448:
2302:
2220:
2168:
2018:
1927:
1863:
1843:
1823:
1803:
1777:
1740:
1720:
1673:
1552:
1509:
1466:
1443:
1408:to bid its own value
1403:
1348:
1262:
1242:
1207:
1130:
1078:
1058:
1018:(it is also called a
1013:
987:
967:
918:
863:
838:agents is denoted by
829:
806:
783:
718:
698:
696:{\displaystyle p_{i}}
671:
641:
589:
569:
546:
252:Vickrey–Clarke–Groves
4799:
4735:
4689:
4617:
4537:
4476:
4360:
4343:{\displaystyle b(v)}
4325:
4245:
4202:
4091:
4048:
4008:
3897:
3886:Second price auction
3854:
3797:
3692:
3466:
3370:
3238:
3132:
3032:
2901:
2825:
2763:
2706:
2686:
2606:
2583:
2460:
2315:
2232:
2180:
2030:
1936:
1875:
1852:
1832:
1812:
1786:
1775:{\displaystyle b(z)}
1757:
1729:
1686:
1573:
1518:
1475:
1455:
1412:
1392:
1380:Second price auction
1319:
1304:second price auction
1251:
1219:
1178:
1149:incentive compatible
1087:
1067:
1029:
996:
976:
938:
872:
842:
818:
795:
730:
707:
680:
660:
601:
578:
558:
535:
132:Discriminatory price
4754:
4708:
4591:
4317:First price auction
4297:
2537:
2391:
1567:first price auction
1561:First price auction
1300:first price auction
1236:
654:quasilinear utility
517:Revenue equivalence
442:Revenue equivalence
127:Deferred-acceptance
4893:Vazirani, Vijay V.
4819:
4782:
4740:
4714:
4694:
4659:
4597:
4577:
4517:
4454:
4340:
4303:
4283:
4228:
4185:
4074:
4034:
3991:
3872:
3827:
3779:
3670:
3445:
3349:
3213:
3111:
3012:
2881:
2805:
2730:
2692:
2668:
2589:
2565:
2523:
2443:
2377:
2297:
2215:
2204:
2163:
2107:
2057:
2013:
1960:
1922:
1902:
1858:
1838:
1818:
1798:
1772:
1745:wins the auction.
1735:
1715:
1704:
1668:
1612:
1547:
1536:
1504:
1493:
1461:
1438:
1398:
1343:
1257:
1237:
1222:
1202:
1125:
1073:
1053:
1008:
982:
962:
913:
858:
824:
801:
778:
713:
693:
666:
636:
584:
564:
541:
212:Sealed first-price
4936:Hartline, Jason,
4839:Reservation price
4817:
4758:
4712:
4657:
4595:
4449:
4435:
4431:
4423:
4419:
4402:
4388:
4384:
4376:
4372:
4301:
4263:
4120:
4112:
3986:
3972:
3968:
3960:
3956:
3939:
3925:
3921:
3913:
3909:
3725:
3611:
3187:
3010:
2949:
2695:{\displaystyle v}
2592:{\displaystyle z}
2337:
2329:
2276:
2189:
2153:
2147:
2128:
2120:
2092:
2042:
2006:
2000:
1981:
1973:
1945:
1887:
1861:{\displaystyle z}
1841:{\displaystyle v}
1821:{\displaystyle z}
1738:{\displaystyle i}
1689:
1658:
1652:
1633:
1625:
1597:
1521:
1478:
1464:{\displaystyle i}
1401:{\displaystyle i}
1260:{\displaystyle i}
1076:{\displaystyle v}
985:{\displaystyle v}
827:{\displaystyle i}
804:{\displaystyle v}
716:{\displaystyle i}
669:{\displaystyle x}
587:{\displaystyle i}
567:{\displaystyle n}
544:{\displaystyle X}
514:
513:
292:Cancellation hunt
242:Value of revenues
112:Click-box bidding
4972:
4965:Mechanism design
4945:
4944:
4928:
4927:
4915:
4901:Roughgarden, Tim
4889:
4828:
4826:
4825:
4820:
4818:
4813:
4812:
4803:
4791:
4789:
4788:
4783:
4778:
4777:
4759:
4753:
4748:
4739:
4723:
4721:
4720:
4715:
4713:
4707:
4702:
4693:
4681:All-pay auctions
4668:
4666:
4665:
4660:
4658:
4653:
4652:
4643:
4635:
4634:
4606:
4604:
4603:
4598:
4596:
4590:
4585:
4576:
4571:
4570:
4555:
4554:
4526:
4524:
4523:
4518:
4510:
4509:
4494:
4493:
4463:
4461:
4460:
4455:
4450:
4447:
4436:
4433:
4429:
4428:
4421:
4420:
4417:
4403:
4400:
4389:
4386:
4382:
4381:
4374:
4373:
4370:
4349:
4347:
4346:
4341:
4312:
4310:
4309:
4304:
4302:
4296:
4291:
4282:
4277:
4276:
4264:
4259:
4258:
4249:
4237:
4235:
4234:
4229:
4227:
4226:
4214:
4213:
4194:
4192:
4191:
4186:
4175:
4174:
4162:
4161:
4143:
4142:
4130:
4129:
4118:
4117:
4110:
4109:
4108:
4083:
4081:
4080:
4075:
4073:
4072:
4060:
4059:
4043:
4041:
4040:
4035:
4033:
4032:
4020:
4019:
4000:
3998:
3997:
3992:
3987:
3984:
3973:
3970:
3966:
3965:
3958:
3957:
3954:
3940:
3937:
3926:
3923:
3919:
3918:
3911:
3910:
3907:
3881:
3879:
3878:
3875:{\displaystyle }
3873:
3836:
3834:
3833:
3828:
3788:
3786:
3785:
3780:
3763:
3759:
3726:
3724:
3710:
3696:
3679:
3677:
3676:
3671:
3663:
3659:
3649:
3645:
3612:
3610:
3596:
3582:
3551:
3547:
3478:
3474:
3454:
3452:
3451:
3446:
3358:
3356:
3355:
3350:
3222:
3220:
3219:
3214:
3188:
3186:
3172:
3158:
3144:
3140:
3120:
3118:
3117:
3112:
3044:
3040:
3021:
3019:
3018:
3013:
3011:
3009:
2995:
2973:
2970:
2965:
2955:
2950:
2948:
2934:
2920:
2890:
2888:
2887:
2882:
2859:
2856:
2851:
2814:
2812:
2811:
2806:
2739:
2737:
2736:
2731:
2701:
2699:
2698:
2693:
2677:
2675:
2674:
2669:
2634:
2598:
2596:
2595:
2590:
2574:
2572:
2571:
2566:
2536:
2531:
2452:
2450:
2449:
2444:
2390:
2385:
2335:
2334:
2327:
2306:
2304:
2303:
2298:
2281:
2274:
2224:
2222:
2221:
2216:
2214:
2213:
2203:
2172:
2170:
2169:
2164:
2151:
2145:
2138:
2137:
2126:
2125:
2118:
2117:
2116:
2106:
2067:
2066:
2056:
2022:
2020:
2019:
2014:
2004:
1998:
1991:
1990:
1979:
1978:
1971:
1970:
1969:
1959:
1931:
1929:
1928:
1923:
1912:
1911:
1901:
1867:
1865:
1864:
1859:
1847:
1845:
1844:
1839:
1827:
1825:
1824:
1819:
1807:
1805:
1804:
1799:
1781:
1779:
1778:
1773:
1744:
1742:
1741:
1736:
1724:
1722:
1721:
1716:
1714:
1713:
1703:
1677:
1675:
1674:
1669:
1656:
1650:
1643:
1642:
1631:
1630:
1623:
1622:
1621:
1611:
1556:
1554:
1553:
1548:
1546:
1545:
1535:
1513:
1511:
1510:
1505:
1503:
1502:
1492:
1470:
1468:
1467:
1462:
1447:
1445:
1444:
1439:
1437:
1436:
1424:
1423:
1407:
1405:
1404:
1399:
1352:
1350:
1349:
1344:
1266:
1264:
1263:
1258:
1246:
1244:
1243:
1238:
1235:
1230:
1211:
1209:
1208:
1203:
1141:random variables
1134:
1132:
1131:
1126:
1121:
1120:
1102:
1101:
1082:
1080:
1079:
1074:
1062:
1060:
1059:
1054:
1017:
1015:
1014:
1009:
991:
989:
988:
983:
971:
969:
968:
963:
922:
920:
919:
914:
909:
908:
893:
892:
867:
865:
864:
859:
857:
856:
833:
831:
830:
825:
814:For every agent
810:
808:
807:
802:
787:
785:
784:
779:
777:
776:
755:
754:
742:
741:
722:
720:
719:
714:
702:
700:
699:
694:
692:
691:
675:
673:
672:
667:
652:The agents have
645:
643:
642:
637:
635:
634:
613:
612:
593:
591:
590:
585:
573:
571:
570:
565:
550:
548:
547:
542:
519:is a concept in
506:
499:
492:
437:Price of anarchy
287:Calor licitantis
39:
16:
4980:
4979:
4975:
4974:
4973:
4971:
4970:
4969:
4950:
4949:
4942:
4935:
4932:
4931:
4924:
4913:
4891:
4890:
4881:
4876:
4867:English auction
4848:
4834:
4804:
4797:
4796:
4769:
4733:
4732:
4726:all-pay auction
4687:
4686:
4683:
4644:
4626:
4615:
4614:
4562:
4546:
4535:
4534:
4501:
4485:
4474:
4473:
4358:
4357:
4323:
4322:
4319:
4268:
4250:
4243:
4242:
4218:
4205:
4200:
4199:
4166:
4153:
4134:
4121:
4100:
4089:
4088:
4064:
4051:
4046:
4045:
4024:
4011:
4006:
4005:
3895:
3894:
3888:
3852:
3851:
3843:
3795:
3794:
3754:
3711:
3697:
3690:
3689:
3640:
3597:
3583:
3580:
3576:
3542:
3469:
3464:
3463:
3368:
3367:
3236:
3235:
3173:
3159:
3135:
3130:
3129:
3035:
3030:
3029:
2996:
2956:
2935:
2921:
2899:
2898:
2823:
2822:
2761:
2760:
2750:English auction
2746:
2744:English auction
2704:
2703:
2684:
2683:
2627:
2604:
2603:
2581:
2580:
2458:
2457:
2313:
2312:
2230:
2229:
2205:
2178:
2177:
2129:
2108:
2058:
2028:
2027:
1982:
1961:
1934:
1933:
1903:
1873:
1872:
1850:
1849:
1830:
1829:
1810:
1809:
1784:
1783:
1755:
1754:
1751:
1727:
1726:
1705:
1684:
1683:
1634:
1613:
1571:
1570:
1563:
1537:
1516:
1515:
1494:
1473:
1472:
1453:
1452:
1428:
1415:
1410:
1409:
1390:
1389:
1382:
1374:all-pay auction
1370:
1317:
1316:
1296:
1249:
1248:
1217:
1216:
1176:
1175:
1161:
1112:
1093:
1085:
1084:
1065:
1064:
1027:
1026:
994:
993:
974:
973:
936:
935:
897:
884:
870:
869:
845:
840:
839:
816:
815:
793:
792:
768:
746:
733:
728:
727:
705:
704:
683:
678:
677:
658:
657:
623:
604:
599:
598:
576:
575:
556:
555:
533:
532:
531:There is a set
529:
510:
481:
451:
416:
321:
317:Tacit collusion
266:
182:Multi-attribute
12:
11:
5:
4978:
4976:
4968:
4967:
4962:
4960:Auction theory
4952:
4951:
4948:
4947:
4930:
4929:
4922:
4878:
4877:
4875:
4872:
4871:
4870:
4863:Winner's curse
4859:
4847:
4844:
4833:
4830:
4816:
4811:
4807:
4793:
4792:
4781:
4776:
4772:
4768:
4765:
4762:
4757:
4752:
4747:
4743:
4711:
4706:
4701:
4697:
4682:
4679:
4670:
4669:
4656:
4651:
4647:
4641:
4638:
4633:
4629:
4625:
4622:
4608:
4607:
4594:
4589:
4584:
4580:
4574:
4569:
4565:
4561:
4558:
4553:
4549:
4545:
4542:
4528:
4527:
4516:
4513:
4508:
4504:
4500:
4497:
4492:
4488:
4484:
4481:
4466:
4465:
4453:
4448:Player 1 loses
4445:
4442:
4439:
4434:Player 1 loses
4427:
4415:
4412:
4409:
4406:
4398:
4395:
4392:
4380:
4368:
4365:
4339:
4336:
4333:
4330:
4318:
4315:
4314:
4313:
4300:
4295:
4290:
4286:
4280:
4275:
4271:
4267:
4262:
4257:
4253:
4225:
4221:
4217:
4212:
4208:
4196:
4195:
4184:
4181:
4178:
4173:
4169:
4165:
4160:
4156:
4152:
4149:
4146:
4141:
4137:
4133:
4128:
4124:
4116:
4107:
4103:
4099:
4096:
4071:
4067:
4063:
4058:
4054:
4031:
4027:
4023:
4018:
4014:
4002:
4001:
3990:
3985:Player 1 loses
3982:
3979:
3976:
3971:Player 1 loses
3964:
3952:
3949:
3946:
3943:
3935:
3932:
3929:
3917:
3905:
3902:
3887:
3884:
3871:
3868:
3865:
3862:
3859:
3842:
3839:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3791:
3790:
3778:
3775:
3772:
3769:
3766:
3762:
3758:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3723:
3720:
3717:
3714:
3709:
3706:
3703:
3700:
3682:
3681:
3669:
3666:
3662:
3658:
3655:
3652:
3648:
3644:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3609:
3606:
3603:
3600:
3595:
3592:
3589:
3586:
3579:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3550:
3546:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3477:
3473:
3457:
3456:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3361:
3360:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3224:
3223:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3185:
3182:
3179:
3176:
3171:
3168:
3165:
3162:
3156:
3153:
3150:
3147:
3143:
3139:
3123:
3122:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3043:
3039:
3023:
3022:
3008:
3005:
3002:
2999:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2969:
2964:
2960:
2953:
2947:
2944:
2941:
2938:
2933:
2930:
2927:
2924:
2918:
2915:
2912:
2909:
2906:
2892:
2891:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2855:
2850:
2846:
2842:
2839:
2836:
2833:
2830:
2816:
2815:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2745:
2742:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2691:
2680:
2679:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2633:
2630:
2626:
2623:
2620:
2617:
2614:
2611:
2588:
2577:
2576:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2535:
2530:
2526:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2389:
2384:
2380:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2333:
2326:
2323:
2320:
2309:
2308:
2296:
2293:
2290:
2287:
2284:
2280:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2212:
2208:
2202:
2199:
2196:
2192:
2188:
2185:
2174:
2173:
2162:
2159:
2156:
2150:
2144:
2141:
2136:
2132:
2124:
2115:
2111:
2105:
2102:
2099:
2095:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2065:
2061:
2055:
2052:
2049:
2045:
2041:
2038:
2035:
2012:
2009:
2003:
1997:
1994:
1989:
1985:
1977:
1968:
1964:
1958:
1955:
1952:
1948:
1944:
1941:
1921:
1918:
1915:
1910:
1906:
1900:
1897:
1894:
1890:
1886:
1883:
1880:
1857:
1837:
1817:
1797:
1794:
1791:
1771:
1768:
1765:
1762:
1750:
1747:
1734:
1712:
1708:
1702:
1699:
1696:
1692:
1667:
1664:
1661:
1655:
1649:
1646:
1641:
1637:
1629:
1620:
1616:
1610:
1607:
1604:
1600:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1562:
1559:
1544:
1540:
1534:
1531:
1528:
1524:
1501:
1497:
1491:
1488:
1485:
1481:
1460:
1435:
1431:
1427:
1422:
1418:
1397:
1381:
1378:
1369:
1366:
1358:
1357:
1354:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1295:
1292:
1291:
1290:
1287:
1276:
1275:
1272:path-connected
1268:
1256:
1234:
1229:
1225:
1215:For some type
1213:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1160:
1157:
1151:if there is a
1137:
1136:
1124:
1119:
1115:
1111:
1108:
1105:
1100:
1096:
1092:
1072:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1023:
1007:
1004:
1001:
981:
961:
958:
955:
952:
949:
946:
943:
912:
907:
904:
900:
896:
891:
887:
883:
880:
877:
855:
852:
848:
823:
800:
789:
788:
775:
771:
767:
764:
761:
758:
753:
749:
745:
740:
736:
712:
690:
686:
665:
647:
646:
633:
630:
626:
622:
619:
616:
611:
607:
583:
563:
540:
528:
525:
521:auction theory
512:
511:
509:
508:
501:
494:
486:
483:
482:
480:
479:
474:
469:
463:
460:
459:
453:
452:
450:
449:
447:Winner's curse
444:
439:
434:
428:
425:
424:
418:
417:
415:
414:
409:
404:
399:
394:
389:
384:
379:
374:
369:
364:
359:
354:
349:
344:
339:
333:
330:
329:
323:
322:
320:
319:
314:
309:
304:
299:
294:
289:
284:
278:
275:
274:
268:
267:
265:
264:
259:
254:
249:
244:
239:
234:
229:
224:
219:
214:
209:
204:
199:
194:
189:
184:
179:
174:
169:
164:
159:
154:
149:
144:
139:
134:
129:
124:
119:
114:
109:
104:
99:
94:
89:
84:
79:
74:
73:
72:
67:
62:
51:
48:
47:
41:
40:
32:
31:
25:
24:
13:
10:
9:
6:
4:
3:
2:
4977:
4966:
4963:
4961:
4958:
4957:
4955:
4941:
4940:
4934:
4933:
4925:
4923:0-521-87282-0
4919:
4912:
4911:
4906:
4902:
4898:
4894:
4888:
4886:
4884:
4880:
4873:
4868:
4864:
4860:
4857:
4853:
4852:
4851:
4845:
4843:
4840:
4831:
4829:
4814:
4809:
4805:
4774:
4770:
4763:
4760:
4755:
4750:
4745:
4741:
4731:
4730:
4729:
4727:
4709:
4704:
4699:
4695:
4680:
4678:
4676:
4654:
4649:
4645:
4639:
4631:
4627:
4620:
4613:
4612:
4611:
4592:
4587:
4582:
4578:
4572:
4567:
4563:
4559:
4551:
4547:
4540:
4533:
4532:
4531:
4514:
4511:
4506:
4502:
4498:
4490:
4486:
4479:
4472:
4471:
4470:
4440:
4410:
4407:
4401:Player 1 wins
4393:
4387:Player 1 wins
4363:
4356:
4355:
4354:
4351:
4334:
4328:
4316:
4298:
4293:
4288:
4284:
4278:
4273:
4269:
4265:
4260:
4255:
4251:
4241:
4240:
4239:
4223:
4219:
4215:
4210:
4206:
4182:
4179:
4171:
4167:
4163:
4158:
4154:
4147:
4139:
4135:
4131:
4126:
4122:
4105:
4101:
4094:
4087:
4086:
4085:
4069:
4065:
4061:
4056:
4052:
4029:
4025:
4021:
4016:
4012:
3977:
3947:
3944:
3938:Player 1 wins
3930:
3924:Player 1 wins
3900:
3893:
3892:
3891:
3885:
3883:
3866:
3863:
3860:
3849:
3840:
3838:
3821:
3815:
3812:
3806:
3800:
3776:
3773:
3767:
3760:
3756:
3751:
3742:
3736:
3733:
3730:
3718:
3712:
3704:
3698:
3688:
3687:
3686:
3667:
3664:
3660:
3653:
3646:
3642:
3637:
3628:
3622:
3619:
3616:
3604:
3598:
3590:
3584:
3577:
3570:
3564:
3561:
3555:
3548:
3544:
3536:
3530:
3527:
3518:
3512:
3509:
3506:
3497:
3491:
3488:
3482:
3475:
3471:
3462:
3461:
3460:
3436:
3430:
3427:
3424:
3415:
3409:
3406:
3397:
3391:
3388:
3385:
3379:
3376:
3373:
3366:
3365:
3364:
3343:
3337:
3334:
3328:
3325:
3322:
3313:
3304:
3298:
3295:
3289:
3283:
3274:
3265:
3259:
3256:
3253:
3244:
3241:
3234:
3233:
3232:
3228:
3204:
3198:
3195:
3192:
3180:
3174:
3166:
3160:
3154:
3148:
3141:
3137:
3128:
3127:
3126:
3105:
3099:
3096:
3093:
3087:
3081:
3075:
3069:
3066:
3060:
3054:
3048:
3041:
3037:
3028:
3027:
3026:
3003:
2997:
2992:
2989:
2983:
2977:
2974:
2967:
2962:
2958:
2951:
2942:
2936:
2928:
2922:
2916:
2910:
2904:
2897:
2896:
2895:
2878:
2875:
2869:
2863:
2860:
2853:
2848:
2844:
2840:
2834:
2828:
2821:
2820:
2819:
2799:
2793:
2790:
2784:
2781:
2778:
2769:
2766:
2759:
2758:
2757:
2753:
2751:
2743:
2741:
2724:
2721:
2718:
2712:
2709:
2689:
2665:
2662:
2659:
2653:
2650:
2644:
2641:
2638:
2631:
2624:
2621:
2618:
2612:
2609:
2602:
2601:
2600:
2586:
2562:
2559:
2553:
2550:
2547:
2541:
2538:
2533:
2528:
2524:
2520:
2517:
2514:
2508:
2505:
2502:
2496:
2493:
2490:
2487:
2484:
2478:
2475:
2472:
2466:
2463:
2456:
2455:
2454:
2440:
2437:
2431:
2428:
2425:
2419:
2416:
2413:
2407:
2404:
2401:
2395:
2392:
2387:
2382:
2378:
2374:
2368:
2365:
2362:
2356:
2353:
2350:
2344:
2341:
2338:
2324:
2318:
2288:
2285:
2282:
2271:
2265:
2262:
2259:
2250:
2247:
2244:
2238:
2235:
2228:
2227:
2226:
2210:
2206:
2200:
2197:
2194:
2186:
2183:
2154:
2142:
2139:
2134:
2130:
2113:
2109:
2103:
2100:
2097:
2086:
2083:
2080:
2071:
2068:
2063:
2059:
2053:
2050:
2047:
2036:
2033:
2026:
2025:
2024:
2007:
1995:
1992:
1987:
1983:
1966:
1962:
1956:
1953:
1950:
1939:
1916:
1913:
1908:
1904:
1898:
1895:
1892:
1881:
1878:
1869:
1855:
1835:
1815:
1795:
1792:
1789:
1766:
1760:
1748:
1746:
1732:
1710:
1706:
1700:
1697:
1694:
1679:
1665:
1659:
1647:
1644:
1639:
1635:
1618:
1614:
1608:
1605:
1602:
1591:
1588:
1582:
1576:
1568:
1560:
1558:
1542:
1538:
1532:
1529:
1526:
1499:
1495:
1489:
1486:
1483:
1458:
1449:
1433:
1429:
1425:
1420:
1416:
1395:
1387:
1384:Consider the
1379:
1377:
1375:
1367:
1365:
1364:for details.
1363:
1355:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1314:
1313:
1312:
1309:
1305:
1301:
1293:
1288:
1285:
1281:
1280:
1279:
1273:
1269:
1254:
1232:
1227:
1223:
1214:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1173:
1172:
1171:
1168:
1166:
1158:
1156:
1154:
1150:
1146:
1145:Bayesian game
1142:
1117:
1113:
1109:
1106:
1103:
1098:
1094:
1070:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1024:
1021:
1020:social choice
1005:
1002:
999:
979:
959:
956:
953:
950:
947:
944:
941:
933:
932:
931:
929:
924:
905:
902:
898:
894:
889:
885:
878:
875:
853:
850:
846:
837:
821:
812:
798:
773:
769:
765:
759:
751:
747:
743:
738:
734:
726:
725:
724:
710:
688:
684:
663:
655:
650:
631:
628:
624:
617:
614:
609:
605:
597:
596:
595:
581:
561:
552:
538:
526:
524:
522:
518:
507:
502:
500:
495:
493:
488:
487:
485:
484:
478:
475:
473:
470:
468:
465:
464:
462:
461:
458:
454:
448:
445:
443:
440:
438:
435:
433:
432:Digital goods
430:
429:
427:
426:
423:
419:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
388:
385:
383:
380:
378:
375:
373:
370:
368:
365:
363:
360:
358:
355:
353:
350:
348:
345:
343:
340:
338:
335:
334:
332:
331:
328:
324:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
283:
280:
279:
277:
276:
273:
269:
263:
260:
258:
255:
253:
250:
248:
245:
243:
240:
238:
235:
233:
232:Uniform price
230:
228:
227:Traffic light
225:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
117:Combinatorial
115:
113:
110:
108:
105:
103:
100:
98:
95:
93:
92:Best/not best
90:
88:
87:Barter double
85:
83:
80:
78:
75:
71:
68:
66:
63:
61:
58:
57:
56:
53:
52:
50:
49:
46:
42:
38:
34:
33:
30:
26:
22:
18:
17:
4938:
4909:
4849:
4835:
4832:Implications
4794:
4684:
4673:the page on
4671:
4609:
4529:
4467:
4352:
4320:
4197:
4003:
3889:
3844:
3792:
3683:
3458:
3362:
3229:
3225:
3124:
3024:
2893:
2817:
2754:
2747:
2681:
2599:, we obtain
2578:
2310:
2175:
1870:
1828:rather than
1752:
1680:
1564:
1450:
1383:
1371:
1359:
1297:
1283:
1277:
1169:
1164:
1162:
1138:
927:
925:
835:
813:
790:
651:
648:
553:
530:
516:
515:
441:
367:Domain names
222:Single-price
122:Common value
4905:Tardos, Éva
4897:Nisan, Noam
4856:risk-averse
4846:Limitations
82:Anglo-Dutch
65:Bidding fee
4954:Categories
4874:References
4464:(as above)
1022:function);
554:There are
337:Algorithms
237:Unique bid
192:No-reserve
4837:to set a
4560:⋅
4499:⋅
4266:⋅
3848:uniformly
3752:−
3734:−
3638:−
3620:−
3528:−
3510:−
3428:−
3389:−
3196:−
2959:∫
2845:∫
2791:≡
2657:⇒
2642:−
2525:∫
2515:⋅
2491:−
2485:⋅
2414:−
2379:∫
2351:⋅
2342:≤
2286:≤
2263:−
2149:∀
2084:−
2002:∀
1698:≠
1654:∀
1645:≤
1606:≠
1530:≠
1487:≠
1159:Statement
1107:…
1003:∈
928:mechanism
903:−
879:≡
851:−
629:≥
621:⟶
402:Virginity
257:Walrasian
187:Multiunit
97:Brazilian
77:Amsterdam
4907:(2007).
3761:′
3647:′
3549:′
3476:′
3142:′
3042:′
2632:′
1451:Suppose
527:Notation
477:Software
467:Ebidding
392:Spectrum
357:Children
327:Contexts
207:Scottish
177:Knapsack
172:Japanese
102:Calcutta
29:Auctions
21:a series
19:Part of
4728:, i.e.
4418:Payment
4371:Payment
3955:Payment
3908:Payment
1565:In the
1308:variant
1294:Example
372:Flowers
362:Players
352:Charity
312:Suicide
307:Sniping
302:Rigging
282:Shading
272:Bidding
247:Vickrey
202:Reverse
152:Forward
147:English
60:Chinese
55:All-pay
4920:
4430:
4422:
4383:
4375:
4198:Since
4119:
4111:
3967:
3959:
3920:
3912:
2336:
2328:
2275:
2152:
2146:
2127:
2119:
2005:
1999:
1980:
1972:
1782:where
1657:
1651:
1632:
1624:
1278:then:
457:Online
422:Theory
397:Stamps
387:Slaves
262:Yankee
157:French
137:Double
107:Candle
70:Dollar
4943:(PDF)
4914:(PDF)
3850:from
1749:Proof
868:. So
836:other
412:Wives
377:Loans
342:Autos
142:Dutch
45:Types
4918:ISBN
4164:<
4132:<
3326:<
3296:<
3257:<
2782:<
2722:<
2622:<
2551:<
2506:<
2476:<
2429:<
2405:<
2366:<
2248:<
2198:>
2176:Let
2140:<
2101:>
2069:<
2051:>
1993:<
1954:>
1914:<
1896:>
1793:<
1315:The
1302:and
1274:set,
1174:The
723:is:
407:Wine
382:Scam
297:Jump
197:Rank
2191:max
2094:max
2044:max
1947:max
1889:max
1725:if
1691:max
1599:max
1523:max
1480:max
1284:all
934:An
347:Art
4956::
4903:;
4899:;
4895:;
4882:^
4677:.
3882:.
3837:.
3317:Pr
3278:Pr
3248:Pr
2773:Pr
1557:.
1448:.
1025:A
926:A
923:.
811:.
744::=
23:on
4946:.
4926:.
4815:2
4810:2
4806:v
4780:)
4775:1
4771:v
4767:(
4764:b
4761:=
4756:2
4751:2
4746:1
4742:v
4710:2
4705:2
4700:1
4696:v
4655:2
4650:1
4646:v
4640:=
4637:)
4632:1
4628:v
4624:(
4621:b
4593:2
4588:2
4583:1
4579:v
4573:=
4568:1
4564:v
4557:)
4552:1
4548:v
4544:(
4541:b
4515:0
4512:+
4507:1
4503:v
4496:)
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3978:P
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505:e
498:t
491:v
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