37:
2903:
2484:
2951:
Consider a corrupt official who is dealing with campaign donors: Each wants him to do a favor that is worth somewhere between $ 0 and $ 1000 to them (uniformly distributed). Their actual valuations are $ 250, $ 500 and $ 750. They can only observe their own valuations. They each treat the official to
669:
In IPV bidders are symmetric because valuations are from the same distribution. These make the analysis focus on symmetric and monotonic bidding strategies. This implies that two bidders with the same valuation will submit the same bid. As a result, under symmetry, the bidder with the highest value
3665:
This example implies that the official will finally get $ 375 but only the third donor, who donated $ 281.3 will win the official's favor. Note that the other two donors know their valuations are not high enough (low chance of winning), so they do not donate much, thus balancing the possible
532:
in which every bidder must pay regardless of whether they win the prize, which is awarded to the highest bidder as in a conventional auction. As shown by Riley and
Samuelson (1981), equilibrium bidding in an all pay auction with private information is revenue equivalent to bidding in a sealed high
2898:{\displaystyle {\begin{aligned}\int _{0}^{v_{i}}F(\tau )^{n-1}d\tau &=(F(v_{i})^{n-1}\cdot v_{i}-\beta (v_{i}))-(F^{n-1}(0)\cdot 0-\beta (0))\\\beta (v_{i})&=F^{n-1}(v_{i})v_{i}-\int _{0}^{v_{i}}F(\tau )^{n-1}d\tau \\\beta (v_{i})&=\int _{0}^{v_{i}}\tau dF^{n-1}(\tau )\end{aligned}}}
626:
The analysis is based on an independent private value (IPV) environment where the valuation of each bidder is drawn independently from a uniform distribution . In the IPV environment, if my value is 0.6 then the probability that some other bidder has a lower value is also 0.6. Accordingly, the
3705:
Jehiel P, Moldovanu B (2006) Allocative and informational externalities in auctions and related mechanisms. In: Blundell R, Newey WK, Persson T (eds) Advances in
Economics and Econometrics: Volume 1: Theory and Applications, Ninth World Congress, vol 1, Cambridge University Press, chap
548:
and studies have shown that over-bidding is common. That is, the seller's revenue frequently exceeds that of the value of the prize, in hopes of securing the winning bid. In repeated games even bidders that win the prize frequently will most likely take a loss in the long run.
2351:
3445:
3314:
3183:
2068:
2472:
2952:
an expensive present - if they spend X Dollars on the present then this is worth X dollars to the official. The official can only do one favor and will do the favor to the donor who is giving him the most expensive present.
1855:
1499:
600:(also known as biological auctions), in which the highest bidder wins, but all (or more typically, both) bidders pay only the lower bid. The war of attrition is used by biologists to model conventional contests, or
3049:
2213:
2489:
1716:
3661:
3590:
3519:
2955:
This is a typical model for all-pay auction. To calculate the optimal bid for each donor, we need to normalize the valuations {250, 500, 750} to {0.25, 0.5, 0.75} so that IPV may apply.
953:
1198:
3320:
3189:
3058:
1032:
2355:
Note because F is smooth the probability of a tie is zero. This means the probability of winning the auction will be equal to the CDF raised to the number of players minus 1: i.e.,
1561:
1279:
2102:
3885:
Algorithmic Game Theory. Vazirani, Vijay V; Nisan, Noam; Roughgarden, Tim; Tardos, Eva; Cambridge, UK: Cambridge
University Press, 2007. Complete preprint on-line at
2941:
1897:
1319:
716:
1910:
658:
1395:
853:
1617:
2206:
1652:
2163:
797:
745:
2358:
496:
2183:
2136:
1739:
1584:
1359:
1339:
1221:
1114:
1094:
1074:
1054:
973:
817:
768:
718:
be the private valuations independent and identically distributed on a uniform distribution from . We wish to find a monotone increasing bidding function,
552:
The all-pay auction with complete information does not have a Nash equilibrium in pure strategies, but does have a Nash equilibrium in mixed-strategies.
3876:
Auctions: Theory and
Practice: The Toulouse Lectures in Economics; Paul Klemperer; Nuffield College, Oxford University, Princeton University Press, 2004
1746:
2185:. Given free disposal, each bidder's value is bounded below by zero. Without loss of generality, then, normalize the lowest possible value to zero.
564:, in which everyone submits a bid but both the losers and the winners pay their submitted bids. This is instrumental in describing certain ideas in
2208:. Because each player's payoff is defined as their expected gain minus their bid, we can recursively define the optimal bid function as follows:
575:
is a two player
Tullock auction, or a multiplayer game in which only the two highest bidders pay their bids. Another practical examples are the
2963:
1402:
2346:{\displaystyle \beta (v_{i})\in arg\max _{b\in \mathbb {R} }\left\{\mathbb {P} (\forall j\neq i:\beta (v_{j})\leq b)v_{i}-b\right\}}
2188:
Because the game is symmetric, the optimal bidding function must be the same for all players. Call this optimal bidding function
3916:
251:
4815:
4632:
4167:
3965:
36:
4451:
4270:
1660:
503:
241:
3596:
3449:
To get the real optimal amount that each of the three donors should give, simply multiplied the IPV values by 1000:
4072:
3525:
3454:
560:
The most straightforward form of an all-pay auction is a
Tullock auction, sometimes called a Tullock lottery after
166:
3440:{\displaystyle b_{3}(v_{3})=\left({\frac {n-1}{n}}\right){v_{3}}^{n}=\left({\frac {2}{3}}\right){0.75}^{3}=0.2813}
3309:{\displaystyle b_{2}(v_{2})=\left({\frac {n-1}{n}}\right){v_{2}}^{n}=\left({\frac {2}{3}}\right){0.50}^{3}=0.0833}
3178:{\displaystyle b_{1}(v_{1})=\left({\frac {n-1}{n}}\right){v_{1}}^{n}=\left({\frac {2}{3}}\right){0.25}^{3}=0.0104}
4983:
4541:
860:
161:
1121:
544:
and expected pay-offs are zero. The seller's expected revenue is equal to the value of the prize. However, some
4411:
4082:
978:
216:
211:
4250:
4592:
4010:
3985:
111:
4942:
4368:
4122:
4112:
4047:
3783:
436:
131:
3886:
1506:
4162:
4142:
2074:
291:
126:
1226:
4876:
4627:
4597:
4255:
4097:
4092:
4912:
4835:
4571:
4127:
4052:
3909:
471:
2080:
4978:
4927:
4660:
4546:
4343:
4137:
3955:
2910:
386:
4730:
2063:{\displaystyle \mathbb {E} =\mathbb {E} =\mathbb {E} =\int \limits _{0}^{1}v^{2}dv={\frac {1}{3}}}
4932:
4531:
4501:
4157:
3945:
1862:
1741:
constitutes a
Bayesian-Nash Equilibrium. The revenue from the all-pay auction in this example is
1284:
681:
596:
371:
181:
4957:
4937:
4917:
4866:
4536:
4441:
4300:
4245:
4177:
4147:
4067:
3995:
545:
1364:
822:
630:
4416:
4401:
3975:
1589:
431:
376:
336:
231:
226:
116:
91:
3800:
2467:{\displaystyle \mathbb {P} (\forall j\neq i:\beta (v_{j})\leq \beta (v_{i}))=F(v_{i})^{n-1}}
2191:
1622:
4750:
4735:
4622:
4617:
4521:
4506:
4471:
4436:
4035:
3980:
3902:
2141:
221:
121:
773:
721:
8:
4907:
4526:
4476:
4313:
4240:
4220:
4077:
3960:
601:
576:
441:
366:
81:
64:
4566:
3744:
4886:
4745:
4576:
4556:
4406:
4285:
4190:
4117:
4062:
3851:
3826:
3760:
2168:
2121:
1724:
1569:
1344:
1324:
1206:
1099:
1079:
1059:
1039:
958:
802:
753:
396:
236:
191:
4871:
4840:
4795:
4690:
4561:
4516:
4491:
4421:
4295:
4225:
4215:
4107:
4057:
4005:
3856:
3764:
605:
446:
401:
256:
186:
96:
76:
20:
4952:
4947:
4881:
4845:
4825:
4785:
4755:
4710:
4665:
4650:
4607:
4461:
4102:
4039:
4025:
3990:
3846:
3838:
3756:
3725:
2478:
537:
489:
476:
391:
286:
206:
176:
171:
101:
4850:
4810:
4765:
4680:
4675:
4348:
4235:
4000:
3970:
3940:
3716:
Gneezy, Uri; Smorodinsky, Rann (2006). "All-pay auctions—an experimental study".
1850:{\displaystyle R=b(v_{1})+b(v_{2})={\frac {v_{1}^{2}}{2}}+{\frac {v_{2}^{2}}{2}}}
586:
351:
316:
311:
306:
246:
201:
151:
146:
59:
4715:
4790:
4780:
4770:
4705:
4695:
4685:
4670:
4466:
4446:
4431:
4426:
4386:
4353:
4338:
4333:
4323:
4132:
3729:
572:
561:
541:
456:
421:
361:
261:
156:
136:
106:
69:
3842:
86:
4972:
4830:
4820:
4775:
4760:
4740:
4511:
4486:
4358:
4328:
4318:
4305:
4210:
4152:
4087:
4020:
3768:
565:
356:
141:
4805:
4800:
4655:
4230:
3860:
582:
411:
406:
381:
341:
296:
196:
326:
4922:
4725:
4720:
4700:
4496:
4481:
4290:
4260:
4195:
4185:
4015:
3950:
3926:
1721:
Since this function is indeed monotone increasing, this bidding strategy
521:
346:
301:
281:
3894:
673:
4551:
4205:
3887:
http://www.cs.cmu.edu/~sandholm/cs15-892F13/algorithmic-game-theory.pdf
3684:
44:
3825:
Chatterjee, Krishnendu; Reiter, Johannes G.; Nowak, Martin A. (2012).
1654:. We can use this to show that the constant of integration is also 0.
4456:
4376:
4200:
3044:{\displaystyle b_{i}(v_{i})=\left({\frac {n-1}{n}}\right){v_{i}}^{n}}
517:
4891:
4391:
2907:
This yields the unique symmetric Nash
Equilibrium bidding function
466:
4612:
4602:
4280:
2113:
1494:{\displaystyle \implies u_{i}'(v_{i})=0\implies 2v_{i}=b'(v_{i})}
620:
Each bidder submits a bid, which only depends on their valuation.
529:
271:
28:
2077:, all auctions with 2 players will have an expected revenue of
3801:"What's a Chinese Auction? Overview & Modern Alternatives"
4381:
1096:'s expected utility when he bids as if his private value is
799:, he wins the auction only if his bid is larger than player
536:
In the simplest version, there is complete information. The
678:
Consider the two-player version of the all-pay auction and
2106:
1901:
627:
probability that two other bidders have lower value is
2138:
risk-neutral bidders. Each bidder has a private value
633:
3824:
3599:
3528:
3457:
3323:
3192:
3061:
2966:
2913:
2487:
2477:
The objective now satisfies the requirements for the
2361:
2216:
2194:
2171:
2144:
2124:
2083:
1913:
1865:
1749:
1727:
1663:
1625:
1592:
1572:
1509:
1405:
1367:
1347:
1327:
1287:
1229:
1209:
1124:
1102:
1082:
1062:
1042:
981:
961:
863:
825:
805:
776:
756:
724:
684:
674:
Using revenue equivalence to predict bidding function
3785:"Mirror Revelation" in Second-PRice Tullock Auctions
623:
Bidders do not know the valuations of other bidders.
3666:huge winning profit and the low chance of winning.
3655:
3584:
3513:
3439:
3308:
3177:
3043:
2935:
2897:
2466:
2345:
2200:
2177:
2157:
2130:
2096:
2062:
1891:
1849:
1733:
1710:
1646:
1611:
1578:
1555:
1493:
1389:
1353:
1333:
1313:
1273:
1215:
1192:
1108:
1088:
1068:
1048:
1026:
967:
947:
847:
811:
791:
762:
739:
710:
652:
616:The following analysis follows a few basic rules.
3715:
3053:The optimal bids for three donors under IPV are:
594:Other forms of all-pay auctions exist, such as a
4970:
2249:
3792:
3788:. SIDE - ISLE 2011 - Seventh Annual Conference.
3743:Hillman, Arye L.; Riley, John G. (March 1989).
3718:Journal of Economic Behavior & Organization
2165:drawn i.i.d. from a common smooth distribution
1711:{\displaystyle b(v_{i})={\frac {v_{i}^{2}}{2}}}
1036:Thus, the probability of allocation of good to
3827:"Evolutionary dynamics of biological auctions"
3682:
3656:{\displaystyle b_{3}real(v_{3}=0.75)=\$ 281.3}
2114:Bidding Function in the Generic Symmetric Case
3910:
3745:"Politically Contestable Rents and Transfers"
3585:{\displaystyle b_{2}real(v_{2}=0.50)=\$ 83.3}
3514:{\displaystyle b_{1}real(v_{1}=0.25)=\$ 10.4}
497:
3742:
555:
948:{\displaystyle \mathbb {P} =\mathbb {P} =x}
747:, that forms a symmetric Nash Equilibrium.
3917:
3903:
2958:According to the formula for optimal bid:
1450:
1446:
1410:
1406:
1193:{\displaystyle u_{i}(x|v_{i})=v_{i}x-b(x)}
504:
490:
3924:
3850:
3818:
2363:
2272:
2260:
1993:
1932:
1915:
1905:from Unif, the expected revenue is
1027:{\displaystyle v_{j}\sim \mathrm {Unif} }
913:
865:
3699:
3683:Riley, John; Samuelson, William (1981).
855:. The probability for this to happen is
3781:
4971:
3798:
664:
3898:
3709:
533:bid or open ascending price auction.
3872:
3870:
3782:Dimitri, Nicola (29 November 2011).
1556:{\displaystyle b(v_{i})=v_{i}^{2}+c}
1223:to be a Bayesian-Nash Equilibrium,
13:
3966:First-player and second-player win
3761:10.1111/j.1468-0343.1989.tb00003.x
3647:
3576:
3505:
2370:
2279:
1341:has no incentive to deviate given
1274:{\displaystyle u_{i}(x_{i}|v_{i})}
1005:
1002:
999:
996:
14:
4995:
3867:
540:is such that each bidder plays a
4073:Coalition-proof Nash equilibrium
2104:when the private valuations are
35:
3799:Carlin, Blair (5 August 2020).
606:recourse to physical aggression
4083:Evolutionarily stable strategy
3879:
3831:Theoretical Population Biology
3775:
3736:
3676:
3641:
3622:
3570:
3551:
3499:
3480:
3347:
3334:
3216:
3203:
3085:
3072:
2990:
2977:
2930:
2917:
2888:
2882:
2828:
2815:
2787:
2780:
2739:
2726:
2700:
2687:
2677:
2674:
2668:
2653:
2647:
2628:
2622:
2619:
2606:
2572:
2558:
2552:
2524:
2517:
2449:
2435:
2426:
2423:
2410:
2401:
2388:
2367:
2319:
2310:
2297:
2276:
2233:
2220:
2097:{\displaystyle {\frac {1}{3}}}
2010:
1997:
1986:
1936:
1925:
1919:
1794:
1781:
1772:
1759:
1680:
1667:
1635:
1629:
1526:
1513:
1488:
1475:
1447:
1437:
1424:
1407:
1384:
1371:
1268:
1254:
1240:
1187:
1181:
1156:
1142:
1135:
1021:
1009:
936:
917:
906:
903:
890:
881:
875:
869:
842:
829:
786:
780:
734:
728:
1:
4011:Simultaneous action selection
3669:
2936:{\displaystyle \beta (v_{i})}
4943:List of games in game theory
4123:Quantal response equilibrium
4113:Perfect Bayesian equilibrium
4048:Bayes correlated equilibrium
7:
4412:Optional prisoner's dilemma
4143:Self-confirming equilibrium
2946:
2075:revenue equivalence theorem
1892:{\displaystyle v_{1},v_{2}}
1314:{\displaystyle x_{i}=v_{i}}
1281:should have its maximum at
711:{\displaystyle v_{i},v_{j}}
10:
5000:
4877:Principal variation search
4593:Aumann's agreement theorem
4256:Strategy-stealing argument
4168:Trembling hand equilibrium
4098:Markov perfect equilibrium
4093:Mertens-stable equilibrium
3730:10.1016/j.jebo.2004.09.013
4913:Combinatorial game theory
4900:
4859:
4641:
4585:
4572:Princess and monster game
4367:
4269:
4176:
4128:Quasi-perfect equilibrium
4053:Bayesian Nash equilibrium
4034:
3933:
3843:10.1016/j.tpb.2011.11.003
1619:, then they will bid 0;
1503:Upon integrating, we get
653:{\textstyle 0.6^{2}=0.36}
556:Forms of all-pay auctions
472:Private electronic market
4928:Evolutionary game theory
4661:Antoine Augustin Cournot
4547:Guess 2/3 of the average
4344:Strictly determined game
4138:Satisfaction equilibrium
3956:Escalation of commitment
3689:American Economic Review
2118:Suppose the auction has
1390:{\displaystyle b(v_{j})}
848:{\displaystyle b(v_{j})}
611:
167:Generalized second-price
4933:Glossary of game theory
4532:Stackelberg competition
4158:Strong Nash equilibrium
1612:{\displaystyle v_{i}=0}
1566:We know that if player
1361:sticks with his bid of
162:Generalized first-price
4984:Mathematical economics
4958:Tragedy of the commons
4938:List of game theorists
4918:Confrontation analysis
4628:Sprague–Grundy theorem
4148:Sequential equilibrium
4068:Correlated equilibrium
3749:Economics and Politics
3657:
3586:
3515:
3441:
3310:
3179:
3045:
2937:
2899:
2481:. Thus, we can write:
2468:
2347:
2202:
2201:{\displaystyle \beta }
2179:
2159:
2132:
2098:
2064:
2030:
1893:
1851:
1735:
1712:
1648:
1647:{\displaystyle b(0)=0}
1613:
1586:has private valuation
1580:
1557:
1495:
1391:
1355:
1335:
1315:
1275:
1217:
1194:
1110:
1090:
1070:
1050:
1028:
969:
949:
849:
813:
793:
764:
741:
712:
654:
602:agonistic interactions
217:Simultaneous ascending
4731:Jean-François Mertens
3658:
3587:
3516:
3442:
3311:
3180:
3046:
2938:
2900:
2469:
2348:
2203:
2180:
2160:
2158:{\displaystyle v_{i}}
2133:
2099:
2065:
2016:
1894:
1852:
1736:
1713:
1649:
1614:
1581:
1558:
1496:
1392:
1356:
1336:
1316:
1276:
1218:
1195:
1111:
1091:
1071:
1051:
1029:
970:
950:
850:
814:
794:
765:
742:
713:
655:
579:and the penny raffle
252:Vickrey–Clarke–Groves
4860:Search optimizations
4736:Jennifer Tour Chayes
4623:Revelation principle
4618:Purification theorem
4557:Nash bargaining game
4522:Bertrand competition
4507:El Farol Bar problem
4472:Electronic mail game
4437:Lewis signaling game
3981:Hierarchy of beliefs
3597:
3526:
3455:
3321:
3190:
3059:
2964:
2911:
2485:
2359:
2214:
2192:
2169:
2142:
2122:
2081:
1911:
1863:
1747:
1725:
1661:
1623:
1590:
1570:
1507:
1403:
1365:
1345:
1325:
1285:
1227:
1207:
1122:
1100:
1080:
1060:
1040:
979:
959:
861:
823:
803:
792:{\displaystyle b(x)}
774:
754:
740:{\displaystyle b(v)}
722:
682:
631:
546:economic experiments
132:Discriminatory price
4908:Bounded rationality
4527:Cournot competition
4477:Rock paper scissors
4452:Battle of the sexes
4442:Volunteer's dilemma
4314:Perfect information
4241:Dominant strategies
4078:Epsilon-equilibrium
3961:Extensive-form game
2859:
2776:
2513:
1980:
1955:
1841:
1816:
1702:
1546:
1423:
665:Symmetry Assumption
577:bidding fee auction
442:Revenue equivalence
127:Deferred-acceptance
4887:Paranoid algorithm
4867:Alpha–beta pruning
4746:John Maynard Smith
4577:Rendezvous problem
4417:Traveler's dilemma
4407:Gift-exchange game
4402:Prisoner's dilemma
4319:Large Poisson game
4286:Bargaining problem
4191:Backward induction
4163:Subgame perfection
4118:Proper equilibrium
3685:"Optimal Auctions"
3653:
3582:
3511:
3437:
3306:
3175:
3041:
2933:
2895:
2893:
2838:
2755:
2492:
2464:
2343:
2265:
2198:
2175:
2155:
2128:
2094:
2060:
1966:
1941:
1889:
1847:
1827:
1802:
1731:
1708:
1688:
1644:
1609:
1576:
1553:
1532:
1491:
1411:
1387:
1351:
1331:
1311:
1271:
1213:
1190:
1106:
1086:
1066:
1046:
1024:
965:
945:
845:
809:
789:
760:
737:
708:
650:
212:Sealed first-price
4966:
4965:
4872:Aspiration window
4841:Suzanne Scotchmer
4796:Oskar Morgenstern
4691:Donald B. Gillies
4633:Zermelo's theorem
4562:Induction puzzles
4517:Fair cake-cutting
4492:Public goods game
4422:Coordination game
4296:Intransitive game
4226:Forward induction
4108:Pareto efficiency
4088:Gibbs equilibrium
4058:Berge equilibrium
4006:Simultaneous game
3413:
3373:
3282:
3242:
3151:
3111:
3016:
2248:
2178:{\displaystyle F}
2131:{\displaystyle n}
2092:
2058:
1984:
1959:
1845:
1820:
1734:{\displaystyle b}
1706:
1579:{\displaystyle i}
1354:{\displaystyle j}
1334:{\displaystyle i}
1216:{\displaystyle b}
1109:{\displaystyle x}
1089:{\displaystyle i}
1069:{\displaystyle x}
1049:{\displaystyle i}
968:{\displaystyle b}
812:{\displaystyle j}
763:{\displaystyle i}
670:will always win.
604:resolved without
514:
513:
292:Cancellation hunt
242:Value of revenues
112:Click-box bidding
4991:
4953:Topological game
4948:No-win situation
4846:Thomas Schelling
4826:Robert B. Wilson
4786:Merrill M. Flood
4756:John von Neumann
4666:Ariel Rubinstein
4651:Albert W. Tucker
4502:War of attrition
4462:Matching pennies
4103:Nash equilibrium
4026:Mechanism design
3991:Normal-form game
3946:Cooperative game
3919:
3912:
3905:
3896:
3895:
3889:
3883:
3877:
3874:
3865:
3864:
3854:
3822:
3816:
3815:
3813:
3811:
3796:
3790:
3789:
3779:
3773:
3772:
3740:
3734:
3733:
3713:
3707:
3703:
3697:
3696:
3680:
3662:
3660:
3659:
3654:
3634:
3633:
3609:
3608:
3591:
3589:
3588:
3583:
3563:
3562:
3538:
3537:
3520:
3518:
3517:
3512:
3492:
3491:
3467:
3466:
3446:
3444:
3443:
3438:
3430:
3429:
3424:
3418:
3414:
3406:
3397:
3396:
3391:
3390:
3389:
3378:
3374:
3369:
3358:
3346:
3345:
3333:
3332:
3315:
3313:
3312:
3307:
3299:
3298:
3293:
3287:
3283:
3275:
3266:
3265:
3260:
3259:
3258:
3247:
3243:
3238:
3227:
3215:
3214:
3202:
3201:
3184:
3182:
3181:
3176:
3168:
3167:
3162:
3156:
3152:
3144:
3135:
3134:
3129:
3128:
3127:
3116:
3112:
3107:
3096:
3084:
3083:
3071:
3070:
3050:
3048:
3047:
3042:
3040:
3039:
3034:
3033:
3032:
3021:
3017:
3012:
3001:
2989:
2988:
2976:
2975:
2942:
2940:
2939:
2934:
2929:
2928:
2904:
2902:
2901:
2896:
2894:
2881:
2880:
2858:
2857:
2856:
2846:
2827:
2826:
2801:
2800:
2775:
2774:
2773:
2763:
2751:
2750:
2738:
2737:
2725:
2724:
2699:
2698:
2646:
2645:
2618:
2617:
2599:
2598:
2586:
2585:
2570:
2569:
2538:
2537:
2512:
2511:
2510:
2500:
2479:envelope theorem
2473:
2471:
2470:
2465:
2463:
2462:
2447:
2446:
2422:
2421:
2400:
2399:
2366:
2352:
2350:
2349:
2344:
2342:
2338:
2331:
2330:
2309:
2308:
2275:
2264:
2263:
2232:
2231:
2207:
2205:
2204:
2199:
2184:
2182:
2181:
2176:
2164:
2162:
2161:
2156:
2154:
2153:
2137:
2135:
2134:
2129:
2103:
2101:
2100:
2095:
2093:
2085:
2069:
2067:
2066:
2061:
2059:
2051:
2040:
2039:
2029:
2024:
2009:
2008:
1996:
1985:
1979:
1974:
1965:
1960:
1954:
1949:
1940:
1935:
1918:
1898:
1896:
1895:
1890:
1888:
1887:
1875:
1874:
1856:
1854:
1853:
1848:
1846:
1840:
1835:
1826:
1821:
1815:
1810:
1801:
1793:
1792:
1771:
1770:
1740:
1738:
1737:
1732:
1717:
1715:
1714:
1709:
1707:
1701:
1696:
1687:
1679:
1678:
1653:
1651:
1650:
1645:
1618:
1616:
1615:
1610:
1602:
1601:
1585:
1583:
1582:
1577:
1562:
1560:
1559:
1554:
1545:
1540:
1525:
1524:
1500:
1498:
1497:
1492:
1487:
1486:
1474:
1463:
1462:
1436:
1435:
1419:
1396:
1394:
1393:
1388:
1383:
1382:
1360:
1358:
1357:
1352:
1340:
1338:
1337:
1332:
1320:
1318:
1317:
1312:
1310:
1309:
1297:
1296:
1280:
1278:
1277:
1272:
1267:
1266:
1257:
1252:
1251:
1239:
1238:
1222:
1220:
1219:
1214:
1199:
1197:
1196:
1191:
1171:
1170:
1155:
1154:
1145:
1134:
1133:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1055:
1053:
1052:
1047:
1033:
1031:
1030:
1025:
1008:
991:
990:
975:is monotone and
974:
972:
971:
966:
954:
952:
951:
946:
935:
934:
916:
902:
901:
868:
854:
852:
851:
846:
841:
840:
818:
816:
815:
810:
798:
796:
795:
790:
769:
767:
766:
761:
746:
744:
743:
738:
717:
715:
714:
709:
707:
706:
694:
693:
659:
657:
656:
651:
643:
642:
597:war of attrition
538:Nash equilibrium
506:
499:
492:
437:Price of anarchy
287:Calor licitantis
39:
16:
15:
4999:
4998:
4994:
4993:
4992:
4990:
4989:
4988:
4979:All-pay auction
4969:
4968:
4967:
4962:
4896:
4882:max^n algorithm
4855:
4851:William Vickrey
4811:Reinhard Selten
4766:Kenneth Binmore
4681:David K. Levine
4676:Daniel Kahneman
4643:
4637:
4613:Negamax theorem
4603:Minimax theorem
4581:
4542:Diner's dilemma
4397:All-pay auction
4363:
4349:Stochastic game
4301:Mean-field game
4272:
4265:
4236:Markov strategy
4172:
4038:
4030:
4001:Sequential game
3986:Information set
3971:Game complexity
3941:Congestion game
3929:
3923:
3893:
3892:
3884:
3880:
3875:
3868:
3823:
3819:
3809:
3807:
3797:
3793:
3780:
3776:
3741:
3737:
3714:
3710:
3704:
3700:
3681:
3677:
3672:
3629:
3625:
3604:
3600:
3598:
3595:
3594:
3558:
3554:
3533:
3529:
3527:
3524:
3523:
3487:
3483:
3462:
3458:
3456:
3453:
3452:
3425:
3420:
3419:
3405:
3401:
3392:
3385:
3381:
3380:
3379:
3359:
3357:
3353:
3341:
3337:
3328:
3324:
3322:
3319:
3318:
3294:
3289:
3288:
3274:
3270:
3261:
3254:
3250:
3249:
3248:
3228:
3226:
3222:
3210:
3206:
3197:
3193:
3191:
3188:
3187:
3163:
3158:
3157:
3143:
3139:
3130:
3123:
3119:
3118:
3117:
3097:
3095:
3091:
3079:
3075:
3066:
3062:
3060:
3057:
3056:
3035:
3028:
3024:
3023:
3022:
3002:
3000:
2996:
2984:
2980:
2971:
2967:
2965:
2962:
2961:
2949:
2924:
2920:
2912:
2909:
2908:
2892:
2891:
2870:
2866:
2852:
2848:
2847:
2842:
2831:
2822:
2818:
2809:
2808:
2790:
2786:
2769:
2765:
2764:
2759:
2746:
2742:
2733:
2729:
2714:
2710:
2703:
2694:
2690:
2681:
2680:
2635:
2631:
2613:
2609:
2594:
2590:
2575:
2571:
2565:
2561:
2545:
2527:
2523:
2506:
2502:
2501:
2496:
2488:
2486:
2483:
2482:
2452:
2448:
2442:
2438:
2417:
2413:
2395:
2391:
2362:
2360:
2357:
2356:
2326:
2322:
2304:
2300:
2271:
2270:
2266:
2259:
2252:
2227:
2223:
2215:
2212:
2211:
2193:
2190:
2189:
2170:
2167:
2166:
2149:
2145:
2143:
2140:
2139:
2123:
2120:
2119:
2116:
2084:
2082:
2079:
2078:
2050:
2035:
2031:
2025:
2020:
2004:
2000:
1992:
1975:
1970:
1964:
1950:
1945:
1939:
1931:
1914:
1912:
1909:
1908:
1883:
1879:
1870:
1866:
1864:
1861:
1860:
1836:
1831:
1825:
1811:
1806:
1800:
1788:
1784:
1766:
1762:
1748:
1745:
1744:
1726:
1723:
1722:
1697:
1692:
1686:
1674:
1670:
1662:
1659:
1658:
1624:
1621:
1620:
1597:
1593:
1591:
1588:
1587:
1571:
1568:
1567:
1541:
1536:
1520:
1516:
1508:
1505:
1504:
1482:
1478:
1467:
1458:
1454:
1431:
1427:
1415:
1404:
1401:
1400:
1378:
1374:
1366:
1363:
1362:
1346:
1343:
1342:
1326:
1323:
1322:
1305:
1301:
1292:
1288:
1286:
1283:
1282:
1262:
1258:
1253:
1247:
1243:
1234:
1230:
1228:
1225:
1224:
1208:
1205:
1204:
1166:
1162:
1150:
1146:
1141:
1129:
1125:
1123:
1120:
1119:
1101:
1098:
1097:
1081:
1078:
1077:
1061:
1058:
1057:
1041:
1038:
1037:
995:
986:
982:
980:
977:
976:
960:
957:
956:
930:
926:
912:
897:
893:
864:
862:
859:
858:
836:
832:
824:
821:
820:
804:
801:
800:
775:
772:
771:
755:
752:
751:
723:
720:
719:
702:
698:
689:
685:
683:
680:
679:
676:
667:
638:
634:
632:
629:
628:
614:
587:Chinese auction
558:
526:all-pay auction
510:
481:
451:
416:
321:
317:Tacit collusion
266:
182:Multi-attribute
12:
11:
5:
4997:
4987:
4986:
4981:
4964:
4963:
4961:
4960:
4955:
4950:
4945:
4940:
4935:
4930:
4925:
4920:
4915:
4910:
4904:
4902:
4898:
4897:
4895:
4894:
4889:
4884:
4879:
4874:
4869:
4863:
4861:
4857:
4856:
4854:
4853:
4848:
4843:
4838:
4833:
4828:
4823:
4818:
4816:Robert Axelrod
4813:
4808:
4803:
4798:
4793:
4791:Olga Bondareva
4788:
4783:
4781:Melvin Dresher
4778:
4773:
4771:Leonid Hurwicz
4768:
4763:
4758:
4753:
4748:
4743:
4738:
4733:
4728:
4723:
4718:
4713:
4708:
4706:Harold W. Kuhn
4703:
4698:
4696:Drew Fudenberg
4693:
4688:
4686:David M. Kreps
4683:
4678:
4673:
4671:Claude Shannon
4668:
4663:
4658:
4653:
4647:
4645:
4639:
4638:
4636:
4635:
4630:
4625:
4620:
4615:
4610:
4608:Nash's theorem
4605:
4600:
4595:
4589:
4587:
4583:
4582:
4580:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4539:
4534:
4529:
4524:
4519:
4514:
4509:
4504:
4499:
4494:
4489:
4484:
4479:
4474:
4469:
4467:Ultimatum game
4464:
4459:
4454:
4449:
4447:Dollar auction
4444:
4439:
4434:
4432:Centipede game
4429:
4424:
4419:
4414:
4409:
4404:
4399:
4394:
4389:
4387:Infinite chess
4384:
4379:
4373:
4371:
4365:
4364:
4362:
4361:
4356:
4354:Symmetric game
4351:
4346:
4341:
4339:Signaling game
4336:
4334:Screening game
4331:
4326:
4324:Potential game
4321:
4316:
4311:
4303:
4298:
4293:
4288:
4283:
4277:
4275:
4267:
4266:
4264:
4263:
4258:
4253:
4251:Mixed strategy
4248:
4243:
4238:
4233:
4228:
4223:
4218:
4213:
4208:
4203:
4198:
4193:
4188:
4182:
4180:
4174:
4173:
4171:
4170:
4165:
4160:
4155:
4150:
4145:
4140:
4135:
4133:Risk dominance
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4095:
4090:
4085:
4080:
4075:
4070:
4065:
4060:
4055:
4050:
4044:
4042:
4032:
4031:
4029:
4028:
4023:
4018:
4013:
4008:
4003:
3998:
3993:
3988:
3983:
3978:
3976:Graphical game
3973:
3968:
3963:
3958:
3953:
3948:
3943:
3937:
3935:
3931:
3930:
3922:
3921:
3914:
3907:
3899:
3891:
3890:
3878:
3866:
3817:
3791:
3774:
3735:
3724:(2): 255–275.
3708:
3698:
3674:
3673:
3671:
3668:
3652:
3649:
3646:
3643:
3640:
3637:
3632:
3628:
3624:
3621:
3618:
3615:
3612:
3607:
3603:
3581:
3578:
3575:
3572:
3569:
3566:
3561:
3557:
3553:
3550:
3547:
3544:
3541:
3536:
3532:
3510:
3507:
3504:
3501:
3498:
3495:
3490:
3486:
3482:
3479:
3476:
3473:
3470:
3465:
3461:
3436:
3433:
3428:
3423:
3417:
3412:
3409:
3404:
3400:
3395:
3388:
3384:
3377:
3372:
3368:
3365:
3362:
3356:
3352:
3349:
3344:
3340:
3336:
3331:
3327:
3305:
3302:
3297:
3292:
3286:
3281:
3278:
3273:
3269:
3264:
3257:
3253:
3246:
3241:
3237:
3234:
3231:
3225:
3221:
3218:
3213:
3209:
3205:
3200:
3196:
3174:
3171:
3166:
3161:
3155:
3150:
3147:
3142:
3138:
3133:
3126:
3122:
3115:
3110:
3106:
3103:
3100:
3094:
3090:
3087:
3082:
3078:
3074:
3069:
3065:
3038:
3031:
3027:
3020:
3015:
3011:
3008:
3005:
2999:
2995:
2992:
2987:
2983:
2979:
2974:
2970:
2948:
2945:
2932:
2927:
2923:
2919:
2916:
2890:
2887:
2884:
2879:
2876:
2873:
2869:
2865:
2862:
2855:
2851:
2845:
2841:
2837:
2834:
2832:
2830:
2825:
2821:
2817:
2814:
2811:
2810:
2807:
2804:
2799:
2796:
2793:
2789:
2785:
2782:
2779:
2772:
2768:
2762:
2758:
2754:
2749:
2745:
2741:
2736:
2732:
2728:
2723:
2720:
2717:
2713:
2709:
2706:
2704:
2702:
2697:
2693:
2689:
2686:
2683:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2644:
2641:
2638:
2634:
2630:
2627:
2624:
2621:
2616:
2612:
2608:
2605:
2602:
2597:
2593:
2589:
2584:
2581:
2578:
2574:
2568:
2564:
2560:
2557:
2554:
2551:
2548:
2546:
2544:
2541:
2536:
2533:
2530:
2526:
2522:
2519:
2516:
2509:
2505:
2499:
2495:
2491:
2490:
2461:
2458:
2455:
2451:
2445:
2441:
2437:
2434:
2431:
2428:
2425:
2420:
2416:
2412:
2409:
2406:
2403:
2398:
2394:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2365:
2341:
2337:
2334:
2329:
2325:
2321:
2318:
2315:
2312:
2307:
2303:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2274:
2269:
2262:
2258:
2255:
2251:
2247:
2244:
2241:
2238:
2235:
2230:
2226:
2222:
2219:
2197:
2174:
2152:
2148:
2127:
2115:
2112:
2091:
2088:
2057:
2054:
2049:
2046:
2043:
2038:
2034:
2028:
2023:
2019:
2015:
2012:
2007:
2003:
1999:
1995:
1991:
1988:
1983:
1978:
1973:
1969:
1963:
1958:
1953:
1948:
1944:
1938:
1934:
1930:
1927:
1924:
1921:
1917:
1886:
1882:
1878:
1873:
1869:
1844:
1839:
1834:
1830:
1824:
1819:
1814:
1809:
1805:
1799:
1796:
1791:
1787:
1783:
1780:
1777:
1774:
1769:
1765:
1761:
1758:
1755:
1752:
1730:
1705:
1700:
1695:
1691:
1685:
1682:
1677:
1673:
1669:
1666:
1643:
1640:
1637:
1634:
1631:
1628:
1608:
1605:
1600:
1596:
1575:
1552:
1549:
1544:
1539:
1535:
1531:
1528:
1523:
1519:
1515:
1512:
1490:
1485:
1481:
1477:
1473:
1470:
1466:
1461:
1457:
1453:
1449:
1445:
1442:
1439:
1434:
1430:
1426:
1422:
1418:
1414:
1409:
1386:
1381:
1377:
1373:
1370:
1350:
1330:
1308:
1304:
1300:
1295:
1291:
1270:
1265:
1261:
1256:
1250:
1246:
1242:
1237:
1233:
1212:
1189:
1186:
1183:
1180:
1177:
1174:
1169:
1165:
1161:
1158:
1153:
1149:
1144:
1140:
1137:
1132:
1128:
1105:
1085:
1065:
1045:
1023:
1020:
1017:
1014:
1011:
1007:
1004:
1001:
998:
994:
989:
985:
964:
944:
941:
938:
933:
929:
925:
922:
919:
915:
911:
908:
905:
900:
896:
892:
889:
886:
883:
880:
877:
874:
871:
867:
844:
839:
835:
831:
828:
808:
788:
785:
782:
779:
759:
736:
733:
730:
727:
705:
701:
697:
692:
688:
675:
672:
666:
663:
662:
661:
649:
646:
641:
637:
624:
621:
613:
610:
573:dollar auction
562:Gordon Tullock
557:
554:
542:mixed strategy
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:
9:
6:
4:
3:
2:
4996:
4985:
4982:
4980:
4977:
4976:
4974:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4934:
4931:
4929:
4926:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4905:
4903:
4901:Miscellaneous
4899:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4864:
4862:
4858:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4836:Samuel Bowles
4834:
4832:
4831:Roger Myerson
4829:
4827:
4824:
4822:
4821:Robert Aumann
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4776:Lloyd Shapley
4774:
4772:
4769:
4767:
4764:
4762:
4761:Kenneth Arrow
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4741:John Harsanyi
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4711:Herbert Simon
4709:
4707:
4704:
4702:
4699:
4697:
4694:
4692:
4689:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4662:
4659:
4657:
4654:
4652:
4649:
4648:
4646:
4640:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4590:
4588:
4584:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4512:Fair division
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4487:Dictator game
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4374:
4372:
4370:
4366:
4360:
4359:Zero-sum game
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4329:Repeated game
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4310:
4308:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4278:
4276:
4274:
4268:
4262:
4259:
4257:
4254:
4252:
4249:
4247:
4246:Pure strategy
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4211:De-escalation
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4183:
4181:
4179:
4175:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4153:Shapley value
4151:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4045:
4043:
4041:
4037:
4033:
4027:
4024:
4022:
4021:Succinct game
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3977:
3974:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3939:
3938:
3936:
3932:
3928:
3920:
3915:
3913:
3908:
3906:
3901:
3900:
3897:
3888:
3882:
3873:
3871:
3862:
3858:
3853:
3848:
3844:
3840:
3836:
3832:
3828:
3821:
3806:
3802:
3795:
3787:
3786:
3778:
3770:
3766:
3762:
3758:
3754:
3750:
3746:
3739:
3731:
3727:
3723:
3719:
3712:
3702:
3695:(3): 381–392.
3694:
3690:
3686:
3679:
3675:
3667:
3663:
3650:
3644:
3638:
3635:
3630:
3626:
3619:
3616:
3613:
3610:
3605:
3601:
3592:
3579:
3573:
3567:
3564:
3559:
3555:
3548:
3545:
3542:
3539:
3534:
3530:
3521:
3508:
3502:
3496:
3493:
3488:
3484:
3477:
3474:
3471:
3468:
3463:
3459:
3450:
3447:
3434:
3431:
3426:
3421:
3415:
3410:
3407:
3402:
3398:
3393:
3386:
3382:
3375:
3370:
3366:
3363:
3360:
3354:
3350:
3342:
3338:
3329:
3325:
3316:
3303:
3300:
3295:
3290:
3284:
3279:
3276:
3271:
3267:
3262:
3255:
3251:
3244:
3239:
3235:
3232:
3229:
3223:
3219:
3211:
3207:
3198:
3194:
3185:
3172:
3169:
3164:
3159:
3153:
3148:
3145:
3140:
3136:
3131:
3124:
3120:
3113:
3108:
3104:
3101:
3098:
3092:
3088:
3080:
3076:
3067:
3063:
3054:
3051:
3036:
3029:
3025:
3018:
3013:
3009:
3006:
3003:
2997:
2993:
2985:
2981:
2972:
2968:
2959:
2956:
2953:
2944:
2925:
2921:
2914:
2905:
2885:
2877:
2874:
2871:
2867:
2863:
2860:
2853:
2849:
2843:
2839:
2835:
2833:
2823:
2819:
2812:
2805:
2802:
2797:
2794:
2791:
2783:
2777:
2770:
2766:
2760:
2756:
2752:
2747:
2743:
2734:
2730:
2721:
2718:
2715:
2711:
2707:
2705:
2695:
2691:
2684:
2671:
2665:
2662:
2659:
2656:
2650:
2642:
2639:
2636:
2632:
2625:
2614:
2610:
2603:
2600:
2595:
2591:
2587:
2582:
2579:
2576:
2566:
2562:
2555:
2549:
2547:
2542:
2539:
2534:
2531:
2528:
2520:
2514:
2507:
2503:
2497:
2493:
2480:
2475:
2459:
2456:
2453:
2443:
2439:
2432:
2429:
2418:
2414:
2407:
2404:
2396:
2392:
2385:
2382:
2379:
2376:
2373:
2353:
2339:
2335:
2332:
2327:
2323:
2316:
2313:
2305:
2301:
2294:
2291:
2288:
2285:
2282:
2267:
2256:
2253:
2245:
2242:
2239:
2236:
2228:
2224:
2217:
2209:
2195:
2186:
2172:
2150:
2146:
2125:
2111:
2109:
2108:
2089:
2086:
2076:
2071:
2055:
2052:
2047:
2044:
2041:
2036:
2032:
2026:
2021:
2017:
2013:
2005:
2001:
1989:
1981:
1976:
1971:
1967:
1961:
1956:
1951:
1946:
1942:
1928:
1922:
1906:
1904:
1903:
1884:
1880:
1876:
1871:
1867:
1857:
1842:
1837:
1832:
1828:
1822:
1817:
1812:
1807:
1803:
1797:
1789:
1785:
1778:
1775:
1767:
1763:
1756:
1753:
1750:
1742:
1728:
1719:
1703:
1698:
1693:
1689:
1683:
1675:
1671:
1664:
1657:Thus, we get
1655:
1641:
1638:
1632:
1626:
1606:
1603:
1598:
1594:
1573:
1564:
1550:
1547:
1542:
1537:
1533:
1529:
1521:
1517:
1510:
1501:
1483:
1479:
1471:
1468:
1464:
1459:
1455:
1451:
1443:
1440:
1432:
1428:
1420:
1416:
1412:
1398:
1379:
1375:
1368:
1348:
1328:
1306:
1302:
1298:
1293:
1289:
1263:
1259:
1248:
1244:
1235:
1231:
1210:
1201:
1184:
1178:
1175:
1172:
1167:
1163:
1159:
1151:
1147:
1138:
1130:
1126:
1117:
1103:
1083:
1063:
1043:
1034:
1018:
1015:
1012:
992:
987:
983:
962:
942:
939:
931:
927:
923:
920:
909:
898:
894:
887:
884:
878:
872:
856:
837:
833:
826:
806:
783:
777:
757:
748:
731:
725:
703:
699:
695:
690:
686:
671:
647:
644:
639:
635:
625:
622:
619:
618:
617:
609:
607:
603:
599:
598:
592:
590:
588:
584:
578:
574:
569:
567:
566:public choice
563:
553:
550:
547:
543:
539:
534:
531:
527:
523:
519:
507:
502:
500:
495:
493:
488:
487:
485:
484:
478:
475:
473:
470:
468:
465:
464:
462:
461:
458:
455:
454:
448:
445:
443:
440:
438:
435:
433:
432:Digital goods
430:
429:
427:
426:
423:
420:
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:
325:
324:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
283:
280:
279:
277:
276:
273:
270:
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:
43:
42:
38:
34:
33:
30:
27:
26:
22:
18:
17:
4806:Peyton Young
4801:Paul Milgrom
4716:Hervé Moulin
4656:Amos Tversky
4598:Folk theorem
4396:
4309:-player game
4306:
4231:Grim trigger
3881:
3837:(1): 69–80.
3834:
3830:
3820:
3808:. Retrieved
3804:
3794:
3784:
3777:
3755:(1): 17–39.
3752:
3748:
3738:
3721:
3717:
3711:
3701:
3692:
3688:
3678:
3664:
3593:
3522:
3451:
3448:
3317:
3186:
3055:
3052:
2960:
2957:
2954:
2950:
2906:
2476:
2354:
2210:
2187:
2117:
2105:
2072:
1907:
1900:
1858:
1743:
1720:
1656:
1565:
1502:
1399:
1202:
1118:
1116:is given by
1035:
857:
749:
677:
668:
615:
595:
593:
585:known as a "
583:pejoratively
580:
570:
559:
551:
535:
525:
515:
367:Domain names
222:Single-price
122:Common value
54:
4923:Coopetition
4726:Jean Tirole
4721:John Conway
4701:Eric Maskin
4497:Blotto game
4482:Pirate game
4291:Global game
4261:Tit for tat
4196:Bid shading
4186:Appeasement
4036:Equilibrium
4016:Solved game
3951:Determinacy
3934:Definitions
3927:game theory
2110:from Unif.
2073:Due to the
568:economics.
522:game theory
82:Anglo-Dutch
65:Bidding fee
4973:Categories
4567:Trust game
4552:Kuhn poker
4221:Escalation
4216:Deterrence
4206:Cheap talk
4178:Strategies
3996:Preference
3925:Topics of
3670:References
1899:are drawn
750:If player
337:Algorithms
237:Unique bid
192:No-reserve
4751:John Nash
4457:Stag hunt
4201:Collusion
3769:0954-1985
3648:$
3577:$
3506:$
3364:−
3233:−
3102:−
3007:−
2915:β
2886:τ
2875:−
2861:τ
2840:∫
2813:β
2806:τ
2795:−
2784:τ
2757:∫
2753:−
2719:−
2685:β
2666:β
2663:−
2657:⋅
2640:−
2626:−
2604:β
2601:−
2588:⋅
2580:−
2543:τ
2532:−
2521:τ
2494:∫
2457:−
2408:β
2405:≤
2386:β
2377:≠
2371:∀
2333:−
2314:≤
2295:β
2286:≠
2280:∀
2257:∈
2237:∈
2218:β
2196:β
2018:∫
1448:⟹
1408:⟹
1176:−
993:∼
518:economics
402:Virginity
257:Walrasian
187:Multiunit
97:Brazilian
77:Amsterdam
4892:Lazy SMP
4586:Theorems
4537:Deadlock
4392:Checkers
4273:of games
4040:concepts
3861:22120126
3805:OneCause
2947:Examples
1472:′
1421:′
1321:so that
1076:. Thus,
955:, since
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
4644:figures
4427:Chicken
4281:Auction
4271:Classes
3852:3279759
1859:Since
819:'s bid
530:auction
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
3859:
3849:
3767:
3435:0.2813
3304:0.0833
3173:0.0104
528:is an
457:Online
422:Theory
397:Stamps
387:Slaves
262:Yankee
157:French
137:Double
107:Candle
70:Dollar
4382:Chess
4369:Games
3810:2 May
3651:281.3
770:bids
612:Rules
524:, an
412:Wives
377:Loans
342:Autos
142:Dutch
45:Types
4063:Core
3857:PMID
3812:2024
3765:ISSN
3639:0.75
3580:83.3
3568:0.50
3509:10.4
3497:0.25
3422:0.75
3291:0.50
3160:0.25
1203:For
924:>
885:>
648:0.36
571:The
520:and
407:Wine
382:Scam
297:Jump
197:Rank
4642:Key
3847:PMC
3839:doi
3757:doi
3726:doi
2943:.
2474:.
2250:max
2107:iid
1902:iid
1056:is
636:0.6
591:).
516:In
347:Art
4975::
4377:Go
3869:^
3855:.
3845:.
3835:81
3833:.
3829:.
3803:.
3763:.
3751:.
3747:.
3722:61
3720:.
3693:71
3691:.
3687:.
2070:.
1718:.
1563:.
1397:.
1200:.
608:.
23:on
4307:n
3918:e
3911:t
3904:v
3863:.
3841::
3814:.
3771:.
3759::
3753:1
3732:.
3728::
3706:3
3645:=
3642:)
3636:=
3631:3
3627:v
3623:(
3620:l
3617:a
3614:e
3611:r
3606:3
3602:b
3574:=
3571:)
3565:=
3560:2
3556:v
3552:(
3549:l
3546:a
3543:e
3540:r
3535:2
3531:b
3503:=
3500:)
3494:=
3489:1
3485:v
3481:(
3478:l
3475:a
3472:e
3469:r
3464:1
3460:b
3432:=
3427:3
3416:)
3411:3
3408:2
3403:(
3399:=
3394:n
3387:3
3383:v
3376:)
3371:n
3367:1
3361:n
3355:(
3351:=
3348:)
3343:3
3339:v
3335:(
3330:3
3326:b
3301:=
3296:3
3285:)
3280:3
3277:2
3272:(
3268:=
3263:n
3256:2
3252:v
3245:)
3240:n
3236:1
3230:n
3224:(
3220:=
3217:)
3212:2
3208:v
3204:(
3199:2
3195:b
3170:=
3165:3
3154:)
3149:3
3146:2
3141:(
3137:=
3132:n
3125:1
3121:v
3114:)
3109:n
3105:1
3099:n
3093:(
3089:=
3086:)
3081:1
3077:v
3073:(
3068:1
3064:b
3037:n
3030:i
3026:v
3019:)
3014:n
3010:1
3004:n
2998:(
2994:=
2991:)
2986:i
2982:v
2978:(
2973:i
2969:b
2931:)
2926:i
2922:v
2918:(
2889:)
2883:(
2878:1
2872:n
2868:F
2864:d
2854:i
2850:v
2844:0
2836:=
2829:)
2824:i
2820:v
2816:(
2803:d
2798:1
2792:n
2788:)
2781:(
2778:F
2771:i
2767:v
2761:0
2748:i
2744:v
2740:)
2735:i
2731:v
2727:(
2722:1
2716:n
2712:F
2708:=
2701:)
2696:i
2692:v
2688:(
2678:)
2675:)
2672:0
2669:(
2660:0
2654:)
2651:0
2648:(
2643:1
2637:n
2633:F
2629:(
2623:)
2620:)
2615:i
2611:v
2607:(
2596:i
2592:v
2583:1
2577:n
2573:)
2567:i
2563:v
2559:(
2556:F
2553:(
2550:=
2540:d
2535:1
2529:n
2525:)
2518:(
2515:F
2508:i
2504:v
2498:0
2460:1
2454:n
2450:)
2444:i
2440:v
2436:(
2433:F
2430:=
2427:)
2424:)
2419:i
2415:v
2411:(
2402:)
2397:j
2393:v
2389:(
2383::
2380:i
2374:j
2368:(
2364:P
2340:}
2336:b
2328:i
2324:v
2320:)
2317:b
2311:)
2306:j
2302:v
2298:(
2292::
2289:i
2283:j
2277:(
2273:P
2268:{
2261:R
2254:b
2246:g
2243:r
2240:a
2234:)
2229:i
2225:v
2221:(
2173:F
2151:i
2147:v
2126:n
2090:3
2087:1
2056:3
2053:1
2048:=
2045:v
2042:d
2037:2
2033:v
2027:1
2022:0
2014:=
2011:]
2006:2
2002:v
1998:[
1994:E
1990:=
1987:]
1982:2
1977:2
1972:2
1968:v
1962:+
1957:2
1952:2
1947:1
1943:v
1937:[
1933:E
1929:=
1926:]
1923:R
1920:[
1916:E
1885:2
1881:v
1877:,
1872:1
1868:v
1843:2
1838:2
1833:2
1829:v
1823:+
1818:2
1813:2
1808:1
1804:v
1798:=
1795:)
1790:2
1786:v
1782:(
1779:b
1776:+
1773:)
1768:1
1764:v
1760:(
1757:b
1754:=
1751:R
1729:b
1704:2
1699:2
1694:i
1690:v
1684:=
1681:)
1676:i
1672:v
1668:(
1665:b
1642:0
1639:=
1636:)
1633:0
1630:(
1627:b
1607:0
1604:=
1599:i
1595:v
1574:i
1551:c
1548:+
1543:2
1538:i
1534:v
1530:=
1527:)
1522:i
1518:v
1514:(
1511:b
1489:)
1484:i
1480:v
1476:(
1469:b
1465:=
1460:i
1456:v
1452:2
1444:0
1441:=
1438:)
1433:i
1429:v
1425:(
1417:i
1413:u
1385:)
1380:j
1376:v
1372:(
1369:b
1349:j
1329:i
1307:i
1303:v
1299:=
1294:i
1290:x
1269:)
1264:i
1260:v
1255:|
1249:i
1245:x
1241:(
1236:i
1232:u
1211:b
1188:)
1185:x
1182:(
1179:b
1173:x
1168:i
1164:v
1160:=
1157:)
1152:i
1148:v
1143:|
1139:x
1136:(
1131:i
1127:u
1104:x
1084:i
1064:x
1044:i
1022:]
1019:1
1016:,
1013:0
1010:[
1006:f
1003:i
1000:n
997:U
988:j
984:v
963:b
943:x
940:=
937:]
932:j
928:v
921:x
918:[
914:P
910:=
907:]
904:)
899:j
895:v
891:(
888:b
882:)
879:x
876:(
873:b
870:[
866:P
843:)
838:j
834:v
830:(
827:b
807:j
787:)
784:x
781:(
778:b
758:i
735:)
732:v
729:(
726:b
704:j
700:v
696:,
691:i
687:v
660:.
645:=
640:2
589:"
581:(
505:e
498:t
491:v
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
.jpg)