2602:
1619:
2115:
1252:
1858:
2597:{\displaystyle {\begin{aligned}{dN_{i} \over {N}}&=f_{i}x_{i}dt+\sigma _{i}x_{i}dW_{i}\\-x_{i}\sum _{j}{dN_{j} \over {N}}&=-x_{i}\left(\phi dt+\sum _{j}\sigma _{j}x_{j}dW_{j}\right)\\-{(dN_{i})^{2} \over {N^{2}}}&=-\sigma _{i}^{2}x_{i}^{2}dt\\x_{i}\sum _{j}{(dN_{j})^{2} \over {N^{2}}}&=x_{i}\left(\sum _{j}\sigma _{j}^{2}x_{j}^{2}\right)dt\end{aligned}}}
3489:---are consistent with the Darwinian tenet of natural selection or any analogous evolutionary phenomena. Here, prime stands for the next time step. However, the discrete nature of the equations puts bounds on the payoff-matrix elements. Interestingly, for the simple case of two-player-two-strategy games, the type I replicator map is capable of showing
1614:{\displaystyle {\begin{aligned}dx_{i}(N_{1},...,N_{m})&={\partial x_{i} \over {\partial N_{j}}}dN_{j}+{1 \over {2}}{\partial ^{2}x_{i} \over {\partial N_{j}\partial N_{k}}}dN_{j}dN_{k}\\&={\partial x_{i} \over {\partial N_{j}}}dN_{j}+{1 \over {2}}{\partial ^{2}x_{i} \over {\partial N_{j}^{2}}}(dN_{j})^{2}\end{aligned}}}
1624:
2840:
469:
In application, populations are generally finite, making the discrete version more realistic. The analysis is more difficult and computationally intensive in the discrete formulation, so the continuous form is often used, although there are significant properties that are lost due to this smoothing.
4070:
The replicator equation or the replicator-mutator equation can be extended to include the effect of delay that either corresponds to the delayed information about the population state or in realizing the effect of interaction among players. The replicator equation can also easily be generalized to
2883:
The analysis differs in the continuous and discrete cases: in the former, methods from differential equations are utilized, whereas in the latter the methods tend to be stochastic. Since the replicator equation is non-linear, an exact solution is difficult to obtain (even in simple versions of the
2895:
In general nondegenerate cases, there can be at most one interior evolutionary stable state (ESS), though there can be many equilibria on the boundary of the simplex. All the faces of the simplex are forward-invariant which corresponds to the lack of innovation in the replicator equation: once a
465:
The replicator equation assumes a uniform population distribution; that is, it does not incorporate population structure into the fitness. The fitness landscape does incorporate the population distribution of types, in contrast to other similar equations, such as the quasispecies equation.
2110:
2607:
2899:
Phase portrait solutions for the continuous linear-fitness replicator equation have been classified in the two and three dimensional cases. Classification is more difficult in higher dimensions because the number of distinct portraits increases rapidly.
1853:{\displaystyle {\begin{aligned}{\partial x_{i} \over {\partial N_{j}}}&={1 \over {N}}\delta _{ij}-{x_{i} \over {N}}\\{\partial ^{2}x_{i} \over {\partial N_{j}^{2}}}&=-{2 \over {N^{2}}}\delta _{ij}+{2x_{i} \over {N^{2}}}\end{aligned}}}
866:
1897:
3078:
4062:
3935:
3642:
223:
579:
3210:
When one considers an unstructured infinite population with non-overlapping generations, one should work with the discrete forms of the replicator equation. Mathematically, two simple phenomenological versions---
3352:
3166:
3484:
1088:
42:
to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of
2120:
1629:
1257:
331:
1243:
645:
2835:{\displaystyle dx_{i}=x_{i}\left(f_{i}-\phi -\sigma _{i}^{2}x_{i}+\sum _{j}\sigma _{j}^{2}x_{j}^{2}\right)dt+x_{i}\left(\sigma _{i}dW_{i}-\sum _{j}\sigma _{j}x_{j}dW_{j}\right)}
725:
3509:
A generalization of the replicator equation which incorporates mutation is given by the replicator-mutator equation, which takes the following form in the continuous version:
989:
1888:
1171:
2867:
720:
416:
367:
678:
3766:
3786:
3736:
3196:
1115:
921:
477:
To simplify analysis, fitness is often assumed to depend linearly upon the population distribution, which allows the replicator equation to be written in the form:
253:
2956:
5697:
3709:
3689:
3665:
2926:
1135:
941:
894:
605:
456:
436:
387:
273:
2964:
2884:
continuous form) so the equation is usually analyzed in terms of stability. The replicator equation (in its continuous and discrete forms) satisfies the
3515:
68:
2888:
of evolutionary game theory which characterizes the stability of equilibria of the equation. The solution of the equation is often given by the set of
3946:
3801:
4277:
Pandit, Varun; Mukhopadhyay, Archan; Chakraborty, Sagar (2018). "Weight of fitness deviation governs strict physical chaos in replicator dynamics".
2929:
2105:{\displaystyle dx_{i}={dN_{i} \over {N}}-x_{i}\sum _{j}{dN_{j} \over {N}}-{(dN_{i})^{2} \over {N^{2}}}+x_{i}\sum _{j}{(dN_{j})^{2} \over {N^{2}}}}
483:
998:
4610:
5514:
5331:
4861:
4659:
4412:
3795:
The discrete version of the replicator-mutator equation may have two simple types in line with the two replicator maps written above:
5687:
5150:
4969:
3084:
4766:
3217:
5702:
5682:
5240:
5692:
4776:
5110:
3358:
5291:
4704:
4679:
3788:
is the mean fitness of the population. This equation is a simultaneous generalization of the replicator equation and the
5641:
5067:
4816:
4806:
4741:
4856:
4836:
4136:
Bomze, Immanuel M. (1983-10-01). "Lotka-Volterra equation and replicator dynamics: A two-dimensional classification".
3490:
2889:
2878:
5326:
5575:
5296:
4954:
4791:
4786:
2885:
4179:
Bomze, Immanuel M. (1995-04-01). "Lotka-Volterra equation and replicator dynamics: new issues in classification".
5611:
5534:
5270:
4821:
4746:
4603:
4560:
4076:
3498:
992:
5626:
5359:
5245:
5042:
4831:
4649:
3668:
31:
5429:
278:
5631:
5230:
5200:
4851:
4639:
1180:
5565:
5656:
5636:
5616:
5235:
5140:
4999:
4949:
4944:
4871:
4841:
4761:
4689:
610:
4669:
59:
5115:
5100:
868:
In other words, the change in the ratio is driven entirely by the difference in fitness between types.
4429:
4339:
5677:
5449:
5434:
5321:
5316:
5220:
5205:
5170:
5135:
4729:
4674:
4596:
4506:
4451:
4361:
4296:
4235:
4223:
5606:
5225:
5175:
5012:
4939:
4914:
4771:
4654:
946:
43:
3198:
is the Lotka–Volterra variable. The continuous replicator dynamic is also equivalent to the
1863:
1140:
5585:
5444:
5275:
5255:
5105:
4984:
4884:
4811:
4756:
4538:
4475:
4441:
4385:
4351:
4320:
4286:
4204:
4161:
2845:
683:
24:
5570:
5539:
5494:
5389:
5260:
5215:
5190:
5120:
4994:
4919:
4909:
4801:
4751:
4699:
4530:
4522:
4467:
4408:
4377:
4312:
4259:
4251:
4196:
4153:
4118:
1246:
392:
336:
34:. The replicator equation differs from other equations used to model replication, such as the
4096:
650:
5651:
5646:
5580:
5544:
5524:
5484:
5454:
5409:
5364:
5349:
5306:
5160:
4934:
4796:
4733:
4719:
4684:
4514:
4459:
4430:"Stability of evolutionarily stable strategies in discrete replicator dynamicswithtimedelay"
4369:
4304:
4243:
4188:
4145:
4108:
4072:
3741:
607:
holds all the fitness information for the population: the expected payoff can be written as
39:
3771:
3714:
3174:
1093:
899:
861:{\displaystyle {d \over {dt}}\left({x_{i} \over {x_{j}}}\right)={x_{i} \over {x_{j}}}\left}
231:
5549:
5509:
5464:
5379:
5374:
5095:
5047:
4929:
4694:
4664:
4634:
1891:
5414:
4340:"Periodic Orbit can be Evolutionarily Stable: Case Study of Discrete Replicator Dynamics"
2935:
2869:
terms are identically zero, the deterministic replicator dynamics equation is recovered.
4510:
4455:
4365:
4300:
4239:
5489:
5479:
5469:
5404:
5394:
5384:
5369:
5165:
5145:
5130:
5125:
5085:
5052:
5037:
5032:
5022:
4826:
3694:
3674:
3650:
3199:
2911:
1174:
1120:
926:
879:
590:
441:
421:
418:
is the average population fitness (given by the weighted average of the fitness of the
372:
258:
3073:{\displaystyle x_{i}={\frac {y_{i}}{1+\sum _{j=1}^{n-1}{y_{j}}}}\quad i=1,\ldots ,n-1}
5671:
5529:
5519:
5474:
5459:
5439:
5265:
5210:
5185:
5057:
5027:
5017:
5004:
4904:
4846:
4781:
4714:
4567:
Taylor, P.D.; Jonker, L. (1978). "Evolutionary Stable
Strategies and Game Dynamics".
4389:
4165:
585:
21:
4479:
4208:
5504:
5499:
5354:
4924:
4542:
4324:
3789:
3494:
35:
4113:
3637:{\displaystyle {\dot {x_{i}}}=\sum _{j=1}^{n}{x_{j}f_{j}(x)Q_{ji}}-\phi (x)x_{i},}
218:{\displaystyle {\dot {x_{i}}}=x_{i},\quad \phi (x)=\sum _{j=1}^{n}{x_{j}f_{j}(x)}}
46:. Unlike the quasispecies equation, the replicator equation does not incorporate
5621:
5424:
5419:
5399:
5195:
5180:
4989:
4959:
4889:
4879:
4709:
4644:
4620:
4057:{\displaystyle x'_{i}={\frac {\sum _{j=1}^{n}{x_{j}f_{j}(x)Q_{ji}}}{\phi (x)}},}
3930:{\displaystyle x'_{i}=x_{i}+\sum _{j=1}^{n}{x_{j}f_{j}(x)Q_{ji}}-\phi (x)x_{i},}
4463:
4373:
5250:
4899:
4494:
27:
4526:
4255:
4200:
4157:
4122:
5155:
5075:
4894:
4075:. A recent generalization that incorporates population structure is used in
574:{\displaystyle {\dot {x_{i}}}=x_{i}\left(\left(Ax\right)_{i}-x^{T}Ax\right)}
58:
The most general continuous form of the replicator equation is given by the
4534:
4471:
4381:
4316:
4263:
4247:
2604:
Then the stochastic replicator dynamics equation for each type is given by:
470:
Note that the continuous form can be obtained from the discrete form by a
5590:
5090:
471:
458:
sum to unity by definition, the equation is defined on the n-dimensional
47:
4588:
4518:
5311:
5301:
4979:
4192:
4149:
459:
4446:
4308:
438:
types in the population). Since the elements of the population vector
2958:
dimensions. The transformation is made by the change of variables:
647:
and the mean fitness of the population as a whole can be written as
4356:
4291:
5080:
680:. It can be shown that the change in the ratio of two proportions
3497:
and it also gives a hint on how to generalize the concept of the
3792:
equation, and is used in the mathematical analysis of language.
4592:
3161:{\displaystyle x_{n}={\frac {1}{1+\sum _{j=1}^{n-1}{y_{j}}}},}
872:
Derivation of deterministic and stochastic replicator dynamics
333:
is the vector of the distribution of types in the population,
4493:
Lieberman, Erez; Hauert, Christoph; Nowak, Martin A. (2005).
2112:
Each of the components in this equation may be calculated as:
50:
and so is not able to innovate new types or pure strategies.
3347:{\displaystyle x'_{i}=x_{i}+x_{i}\left\,({\rm {type~I),}}}
1083:{\displaystyle dN_{i}=f_{i}N_{i}dt+\sigma _{i}N_{i}dW_{i}}
4582:. Economic Learning and Social Evolution, The MIT Press.
2896:
strategy becomes extinct there is no way to revive it.
4405:
Evolutionary
Dynamics: Exploring the Equations of Life
3949:
3804:
3774:
3744:
3717:
3697:
3677:
3653:
3518:
3479:{\displaystyle x'_{i}=x_{i}\left\,({\rm {type~II),}}}
3361:
3220:
3177:
3087:
2967:
2938:
2914:
2848:
2610:
2118:
1900:
1866:
1627:
1255:
1183:
1143:
1123:
1096:
1001:
991:. Assume that the change in each type is governed by
949:
929:
902:
882:
728:
686:
653:
613:
593:
486:
444:
424:
395:
375:
339:
281:
261:
234:
71:
5599:
5558:
5340:
5284:
5066:
4968:
4870:
4728:
4627:
4056:
3929:
3780:
3760:
3730:
3703:
3683:
3659:
3636:
3501:to accommodate the periodic solutions of the map.
3478:
3346:
3190:
3160:
3072:
2950:
2920:
2861:
2834:
2596:
2104:
1882:
1852:
1613:
1237:
1165:
1129:
1109:
1082:
983:
935:
915:
888:
860:
714:
672:
639:
599:
573:
450:
430:
410:
381:
361:
325:
267:
247:
217:
4338:Mukhopadhyay, Archan; Chakraborty, Sagar (2020).
4222:Page, KAREN M.; Nowak, MARTIN A. (2002-11-07).
876:Suppose that the number of individuals of type
4561:Evolutionary Dynamics and Extensive Form Games
4604:
4101:Bulletin of the American Mathematical Society
8:
923:and that the total number of individuals is
389:(which is dependent on the population), and
943:. Define the proportion of each type to be
4611:
4597:
4589:
4580:Population Games and Evolutionary Dynamics
4445:
4355:
4290:
4112:
4024:
4005:
3995:
3990:
3984:
3973:
3966:
3954:
3948:
3918:
3889:
3870:
3860:
3855:
3849:
3838:
3825:
3809:
3803:
3773:
3749:
3743:
3722:
3716:
3696:
3676:
3652:
3625:
3596:
3577:
3567:
3562:
3556:
3545:
3526:
3520:
3519:
3517:
3446:
3445:
3441:
3422:
3411:
3392:
3382:
3366:
3360:
3317:
3316:
3312:
3295:
3282:
3254:
3241:
3225:
3219:
3182:
3176:
3145:
3140:
3128:
3117:
3101:
3092:
3086:
3032:
3027:
3015:
3004:
2987:
2981:
2972:
2966:
2937:
2930:Generalized Lotka–Volterra equation
2913:
2853:
2847:
2821:
2808:
2798:
2788:
2775:
2762:
2747:
2723:
2718:
2708:
2703:
2693:
2680:
2670:
2665:
2646:
2631:
2618:
2609:
2573:
2568:
2558:
2553:
2543:
2528:
2508:
2503:
2496:
2486:
2473:
2467:
2457:
2437:
2432:
2422:
2417:
2394:
2389:
2382:
2372:
2359:
2341:
2328:
2318:
2308:
2281:
2260:
2253:
2243:
2237:
2227:
2210:
2197:
2187:
2168:
2158:
2140:
2133:
2123:
2119:
2117:
2093:
2088:
2081:
2071:
2058:
2052:
2042:
2026:
2021:
2014:
2004:
1991:
1981:
1974:
1964:
1958:
1948:
1934:
1927:
1917:
1908:
1899:
1871:
1865:
1837:
1832:
1825:
1815:
1803:
1790:
1785:
1780:
1761:
1756:
1748:
1741:
1731:
1724:
1713:
1707:
1701:
1689:
1678:
1673:
1657:
1649:
1642:
1632:
1628:
1626:
1601:
1591:
1572:
1567:
1559:
1552:
1542:
1535:
1528:
1523:
1514:
1498:
1490:
1483:
1473:
1457:
1444:
1428:
1415:
1407:
1400:
1390:
1383:
1376:
1371:
1362:
1346:
1338:
1331:
1321:
1305:
1280:
1267:
1256:
1254:
1226:
1201:
1188:
1182:
1154:
1142:
1122:
1101:
1095:
1074:
1061:
1051:
1032:
1022:
1009:
1000:
973:
967:
954:
948:
928:
907:
901:
881:
838:
816:
798:
793:
787:
781:
765:
760:
754:
748:
734:
729:
727:
706:
697:
691:
685:
658:
652:
631:
612:
592:
554:
541:
513:
494:
488:
487:
485:
443:
423:
394:
374:
344:
338:
314:
295:
280:
260:
239:
233:
199:
189:
184:
178:
167:
111:
98:
79:
73:
72:
70:
30:and non-innovative game dynamic used in
4095:Hofbauer, Josef; Sigmund, Karl (2003).
4087:
326:{\displaystyle x=(x_{1},\ldots ,x_{n})}
4428:Alboszta, Jan; Miękisz, Jacek (2004).
2908:The continuous replicator equation on
1238:{\displaystyle x_{i}(N_{1},...,N_{m})}
7:
5698:Mathematical and theoretical biology
1177:are assumed to be uncorrelated. For
1117:is the fitness associated with type
4407:. Belknap Press. pp. 272–273.
1137:. The average fitness of the types
640:{\displaystyle \left(Ax\right)_{i}}
4660:First-player and second-player win
3465:
3462:
3456:
3453:
3450:
3447:
3333:
3327:
3324:
3321:
3318:
1749:
1728:
1650:
1635:
1560:
1539:
1491:
1476:
1421:
1408:
1387:
1339:
1324:
14:
4495:"Evolutionary dynamics on graphs"
1894:. These relationships imply that:
1621:The partial derivatives are then:
4767:Coalition-proof Nash equilibrium
4224:"Unifying Evolutionary Dynamics"
2904:Relationships to other equations
38:equation, in that it allows the
3042:
147:
4777:Evolutionarily stable strategy
4434:Journal of Theoretical Biology
4344:Journal of Theoretical Biology
4228:Journal of Theoretical Biology
4045:
4039:
4017:
4011:
3911:
3905:
3882:
3876:
3618:
3612:
3589:
3583:
3468:
3442:
3336:
3313:
2493:
2476:
2379:
2362:
2078:
2061:
2011:
1994:
1598:
1581:
1311:
1273:
1232:
1194:
850:
844:
828:
822:
405:
399:
356:
350:
320:
288:
211:
205:
157:
151:
141:
138:
132:
123:
117:
104:
1:
4705:Simultaneous action selection
4578:Sandholm, William H. (2010).
4114:10.1090/S0273-0979-03-00988-1
984:{\displaystyle x_{i}=N_{i}/N}
5642:List of games in game theory
4817:Quantal response equilibrium
4807:Perfect Bayesian equilibrium
4742:Bayes correlated equilibrium
4097:"Evolutionary game dynamics"
3206:Discrete replicator equation
2890:evolutionarily stable states
1883:{\displaystyle \delta _{ij}}
1166:{\displaystyle \phi =x^{T}f}
5111:Optional prisoner's dilemma
4837:Self-confirming equilibrium
3491:period doubling bifurcation
2928:types is equivalent to the
2879:Evolutionarily stable state
2862:{\displaystyle \sigma _{i}}
715:{\displaystyle x_{i}/x_{j}}
5719:
5576:Principal variation search
5292:Aumann's agreement theorem
4955:Strategy-stealing argument
4862:Trembling hand equilibrium
4792:Markov perfect equilibrium
4787:Mertens-stable equilibrium
4464:10.1016/j.jtbi.2004.06.012
4374:10.1016/j.jtbi.2020.110288
2876:
255:is the proportion of type
5612:Combinatorial game theory
5271:Princess and monster game
4822:Quasi-perfect equilibrium
4747:Bayesian Nash equilibrium
4403:Nowak, Martin A. (2006).
4077:evolutionary graph theory
3671:for the mutation of type
3499:evolutionary stable state
993:geometric Brownian motion
5688:Evolutionary game theory
5627:Evolutionary game theory
5360:Antoine Augustin Cournot
5246:Guess 2/3 of the average
5043:Strictly determined game
4832:Satisfaction equilibrium
4650:Escalation of commitment
4569:Mathematical Biosciences
3669:transition probabilities
1892:Kronecker delta function
722:with respect to time is:
411:{\displaystyle \phi (x)}
362:{\displaystyle f_{i}(x)}
32:evolutionary game theory
5632:Glossary of game theory
5231:Stackelberg competition
4852:Strong Nash equilibrium
673:{\displaystyle x^{T}Ax}
369:is the fitness of type
5703:Mathematical economics
5683:Differential equations
5657:Tragedy of the commons
5637:List of game theorists
5617:Confrontation analysis
5327:Sprague–Grundy theorem
4842:Sequential equilibrium
4762:Correlated equilibrium
4248:10.1006/jtbi.2002.3112
4181:Biological Cybernetics
4138:Biological Cybernetics
4058:
3989:
3931:
3854:
3782:
3762:
3761:{\displaystyle i^{th}}
3738:is the fitness of the
3732:
3705:
3685:
3661:
3638:
3561:
3480:
3348:
3192:
3162:
3139:
3074:
3026:
2952:
2922:
2863:
2836:
2598:
2106:
1884:
1854:
1615:
1239:
1167:
1131:
1111:
1084:
985:
937:
917:
890:
862:
716:
674:
641:
601:
575:
452:
432:
412:
383:
363:
327:
269:
249:
219:
183:
5693:Evolutionary dynamics
5430:Jean-François Mertens
4558:Cressman, R. (2003).
4059:
3969:
3932:
3834:
3783:
3781:{\displaystyle \phi }
3763:
3733:
3731:{\displaystyle f_{i}}
3706:
3686:
3662:
3639:
3541:
3481:
3349:
3193:
3191:{\displaystyle y_{i}}
3163:
3113:
3075:
3000:
2953:
2923:
2864:
2837:
2599:
2107:
1885:
1855:
1616:
1240:
1168:
1132:
1112:
1110:{\displaystyle f_{i}}
1085:
986:
938:
918:
916:{\displaystyle N_{i}}
891:
863:
717:
675:
642:
602:
576:
453:
433:
413:
384:
364:
328:
270:
250:
248:{\displaystyle x_{i}}
220:
163:
60:differential equation
5559:Search optimizations
5435:Jennifer Tour Chayes
5322:Revelation principle
5317:Purification theorem
5256:Nash bargaining game
5221:Bertrand competition
5206:El Farol Bar problem
5171:Electronic mail game
5136:Lewis signaling game
4675:Hierarchy of beliefs
3947:
3802:
3772:
3742:
3715:
3695:
3675:
3651:
3516:
3359:
3218:
3175:
3085:
2965:
2936:
2912:
2846:
2608:
2116:
1898:
1864:
1625:
1253:
1181:
1141:
1121:
1094:
999:
947:
927:
900:
880:
726:
684:
651:
611:
591:
484:
442:
422:
393:
373:
337:
279:
259:
232:
69:
16:In mathematics, the
5607:Bounded rationality
5226:Cournot competition
5176:Rock paper scissors
5151:Battle of the sexes
5141:Volunteer's dilemma
5013:Perfect information
4940:Dominant strategies
4772:Epsilon-equilibrium
4655:Extensive-form game
4519:10.1038/nature03204
4511:2005Natur.433..312L
4456:2004JThBi.231..175A
4366:2020JThBi.49710288M
4301:2018Chaos..28c3104P
4240:2002JThBi.219...93P
3962:
3817:
3374:
3233:
2951:{\displaystyle n-1}
2892:of the population.
2728:
2713:
2675:
2578:
2563:
2442:
2427:
1766:
1577:
275:in the population,
18:replicator equation
5586:Paranoid algorithm
5566:Alpha–beta pruning
5445:John Maynard Smith
5276:Rendezvous problem
5116:Traveler's dilemma
5106:Gift-exchange game
5101:Prisoner's dilemma
5018:Large Poisson game
4985:Bargaining problem
4885:Backward induction
4857:Subgame perfection
4812:Proper equilibrium
4193:10.1007/BF00201420
4150:10.1007/BF00318088
4054:
3950:
3927:
3805:
3778:
3758:
3728:
3701:
3681:
3657:
3634:
3476:
3362:
3344:
3221:
3188:
3158:
3070:
2948:
2918:
2859:
2842:Assuming that the
2832:
2793:
2714:
2699:
2698:
2661:
2594:
2592:
2564:
2549:
2548:
2472:
2428:
2413:
2313:
2242:
2102:
2057:
1963:
1880:
1850:
1848:
1752:
1611:
1609:
1563:
1235:
1163:
1127:
1107:
1080:
981:
933:
913:
886:
858:
712:
670:
637:
597:
571:
448:
428:
408:
379:
359:
323:
265:
245:
215:
5665:
5664:
5571:Aspiration window
5540:Suzanne Scotchmer
5495:Oskar Morgenstern
5390:Donald B. Gillies
5332:Zermelo's theorem
5261:Induction puzzles
5216:Fair cake-cutting
5191:Public goods game
5121:Coordination game
4995:Intransitive game
4920:Forward induction
4802:Pareto efficiency
4782:Gibbs equilibrium
4752:Berge equilibrium
4700:Simultaneous game
4505:(7023): 312–316.
4309:10.1063/1.5011955
4049:
3704:{\displaystyle i}
3684:{\displaystyle j}
3660:{\displaystyle Q}
3647:where the matrix
3535:
3461:
3435:
3332:
3153:
3040:
2921:{\displaystyle n}
2784:
2689:
2539:
2515:
2463:
2401:
2304:
2265:
2233:
2145:
2100:
2048:
2033:
1986:
1954:
1939:
1844:
1797:
1768:
1718:
1683:
1664:
1579:
1533:
1505:
1435:
1381:
1353:
1130:{\displaystyle i}
936:{\displaystyle N}
889:{\displaystyle i}
805:
772:
742:
600:{\displaystyle A}
503:
451:{\displaystyle x}
431:{\displaystyle n}
382:{\displaystyle i}
268:{\displaystyle i}
88:
5710:
5652:Topological game
5647:No-win situation
5545:Thomas Schelling
5525:Robert B. Wilson
5485:Merrill M. Flood
5455:John von Neumann
5365:Ariel Rubinstein
5350:Albert W. Tucker
5201:War of attrition
5161:Matching pennies
4935:Pairing strategy
4797:Nash equilibrium
4720:Mechanism design
4685:Normal-form game
4640:Cooperative game
4613:
4606:
4599:
4590:
4547:
4546:
4490:
4484:
4483:
4449:
4425:
4419:
4418:
4400:
4394:
4393:
4359:
4335:
4329:
4328:
4294:
4274:
4268:
4267:
4219:
4213:
4212:
4176:
4170:
4169:
4133:
4127:
4126:
4116:
4092:
4073:asymmetric games
4063:
4061:
4060:
4055:
4050:
4048:
4034:
4033:
4032:
4031:
4010:
4009:
4000:
3999:
3988:
3983:
3967:
3958:
3936:
3934:
3933:
3928:
3923:
3922:
3898:
3897:
3896:
3875:
3874:
3865:
3864:
3853:
3848:
3830:
3829:
3813:
3787:
3785:
3784:
3779:
3767:
3765:
3764:
3759:
3757:
3756:
3737:
3735:
3734:
3729:
3727:
3726:
3710:
3708:
3707:
3702:
3690:
3688:
3687:
3682:
3666:
3664:
3663:
3658:
3643:
3641:
3640:
3635:
3630:
3629:
3605:
3604:
3603:
3582:
3581:
3572:
3571:
3560:
3555:
3537:
3536:
3531:
3530:
3521:
3485:
3483:
3482:
3477:
3475:
3474:
3459:
3440:
3436:
3434:
3427:
3426:
3416:
3415:
3410:
3406:
3393:
3387:
3386:
3370:
3353:
3351:
3350:
3345:
3343:
3342:
3330:
3311:
3307:
3300:
3299:
3287:
3286:
3281:
3277:
3259:
3258:
3246:
3245:
3229:
3197:
3195:
3194:
3189:
3187:
3186:
3167:
3165:
3164:
3159:
3154:
3152:
3151:
3150:
3149:
3138:
3127:
3102:
3097:
3096:
3079:
3077:
3076:
3071:
3041:
3039:
3038:
3037:
3036:
3025:
3014:
2992:
2991:
2982:
2977:
2976:
2957:
2955:
2954:
2949:
2927:
2925:
2924:
2919:
2868:
2866:
2865:
2860:
2858:
2857:
2841:
2839:
2838:
2833:
2831:
2827:
2826:
2825:
2813:
2812:
2803:
2802:
2792:
2780:
2779:
2767:
2766:
2752:
2751:
2733:
2729:
2727:
2722:
2712:
2707:
2697:
2685:
2684:
2674:
2669:
2651:
2650:
2636:
2635:
2623:
2622:
2603:
2601:
2600:
2595:
2593:
2583:
2579:
2577:
2572:
2562:
2557:
2547:
2533:
2532:
2516:
2514:
2513:
2512:
2502:
2501:
2500:
2491:
2490:
2474:
2471:
2462:
2461:
2441:
2436:
2426:
2421:
2402:
2400:
2399:
2398:
2388:
2387:
2386:
2377:
2376:
2360:
2351:
2347:
2346:
2345:
2333:
2332:
2323:
2322:
2312:
2286:
2285:
2266:
2264:
2259:
2258:
2257:
2244:
2241:
2232:
2231:
2215:
2214:
2202:
2201:
2192:
2191:
2173:
2172:
2163:
2162:
2146:
2144:
2139:
2138:
2137:
2124:
2111:
2109:
2108:
2103:
2101:
2099:
2098:
2097:
2087:
2086:
2085:
2076:
2075:
2059:
2056:
2047:
2046:
2034:
2032:
2031:
2030:
2020:
2019:
2018:
2009:
2008:
1992:
1987:
1985:
1980:
1979:
1978:
1965:
1962:
1953:
1952:
1940:
1938:
1933:
1932:
1931:
1918:
1913:
1912:
1889:
1887:
1886:
1881:
1879:
1878:
1859:
1857:
1856:
1851:
1849:
1845:
1843:
1842:
1841:
1831:
1830:
1829:
1816:
1811:
1810:
1798:
1796:
1795:
1794:
1781:
1769:
1767:
1765:
1760:
1747:
1746:
1745:
1736:
1735:
1725:
1719:
1717:
1712:
1711:
1702:
1697:
1696:
1684:
1682:
1674:
1665:
1663:
1662:
1661:
1648:
1647:
1646:
1633:
1620:
1618:
1617:
1612:
1610:
1606:
1605:
1596:
1595:
1580:
1578:
1576:
1571:
1558:
1557:
1556:
1547:
1546:
1536:
1534:
1532:
1524:
1519:
1518:
1506:
1504:
1503:
1502:
1489:
1488:
1487:
1474:
1466:
1462:
1461:
1449:
1448:
1436:
1434:
1433:
1432:
1420:
1419:
1406:
1405:
1404:
1395:
1394:
1384:
1382:
1380:
1372:
1367:
1366:
1354:
1352:
1351:
1350:
1337:
1336:
1335:
1322:
1310:
1309:
1285:
1284:
1272:
1271:
1244:
1242:
1241:
1236:
1231:
1230:
1206:
1205:
1193:
1192:
1175:Wiener processes
1172:
1170:
1169:
1164:
1159:
1158:
1136:
1134:
1133:
1128:
1116:
1114:
1113:
1108:
1106:
1105:
1089:
1087:
1086:
1081:
1079:
1078:
1066:
1065:
1056:
1055:
1037:
1036:
1027:
1026:
1014:
1013:
990:
988:
987:
982:
977:
972:
971:
959:
958:
942:
940:
939:
934:
922:
920:
919:
914:
912:
911:
895:
893:
892:
887:
867:
865:
864:
859:
857:
853:
843:
842:
821:
820:
806:
804:
803:
802:
792:
791:
782:
777:
773:
771:
770:
769:
759:
758:
749:
743:
741:
730:
721:
719:
718:
713:
711:
710:
701:
696:
695:
679:
677:
676:
671:
663:
662:
646:
644:
643:
638:
636:
635:
630:
626:
606:
604:
603:
598:
580:
578:
577:
572:
570:
566:
559:
558:
546:
545:
540:
536:
518:
517:
505:
504:
499:
498:
489:
457:
455:
454:
449:
437:
435:
434:
429:
417:
415:
414:
409:
388:
386:
385:
380:
368:
366:
365:
360:
349:
348:
332:
330:
329:
324:
319:
318:
300:
299:
274:
272:
271:
266:
254:
252:
251:
246:
244:
243:
224:
222:
221:
216:
214:
204:
203:
194:
193:
182:
177:
116:
115:
103:
102:
90:
89:
84:
83:
74:
40:fitness function
5718:
5717:
5713:
5712:
5711:
5709:
5708:
5707:
5668:
5667:
5666:
5661:
5595:
5581:max^n algorithm
5554:
5550:William Vickrey
5510:Reinhard Selten
5465:Kenneth Binmore
5380:David K. Levine
5375:Daniel Kahneman
5342:
5336:
5312:Negamax theorem
5302:Minimax theorem
5280:
5241:Diner's dilemma
5096:All-pay auction
5062:
5048:Stochastic game
5000:Mean-field game
4971:
4964:
4930:Markov strategy
4866:
4732:
4724:
4695:Sequential game
4680:Information set
4665:Game complexity
4635:Congestion game
4623:
4617:
4586:
4555:
4553:Further reading
4550:
4492:
4491:
4487:
4427:
4426:
4422:
4415:
4402:
4401:
4397:
4337:
4336:
4332:
4276:
4275:
4271:
4221:
4220:
4216:
4178:
4177:
4173:
4135:
4134:
4130:
4094:
4093:
4089:
4085:
4035:
4020:
4001:
3991:
3968:
3945:
3944:
3914:
3885:
3866:
3856:
3821:
3800:
3799:
3770:
3769:
3745:
3740:
3739:
3718:
3713:
3712:
3693:
3692:
3673:
3672:
3649:
3648:
3621:
3592:
3573:
3563:
3522:
3514:
3513:
3507:
3505:Generalizations
3418:
3417:
3399:
3395:
3394:
3388:
3378:
3357:
3356:
3291:
3270:
3266:
3265:
3264:
3260:
3250:
3237:
3216:
3215:
3208:
3178:
3173:
3172:
3141:
3106:
3088:
3083:
3082:
3028:
2993:
2983:
2968:
2963:
2962:
2934:
2933:
2910:
2909:
2906:
2881:
2875:
2849:
2844:
2843:
2817:
2804:
2794:
2771:
2758:
2757:
2753:
2743:
2676:
2642:
2641:
2637:
2627:
2614:
2606:
2605:
2591:
2590:
2538:
2534:
2524:
2517:
2504:
2492:
2482:
2475:
2453:
2450:
2449:
2403:
2390:
2378:
2368:
2361:
2353:
2352:
2337:
2324:
2314:
2291:
2287:
2277:
2267:
2249:
2245:
2223:
2217:
2216:
2206:
2193:
2183:
2164:
2154:
2147:
2129:
2125:
2114:
2113:
2089:
2077:
2067:
2060:
2038:
2022:
2010:
2000:
1993:
1970:
1966:
1944:
1923:
1919:
1904:
1896:
1895:
1867:
1862:
1861:
1847:
1846:
1833:
1821:
1817:
1799:
1786:
1770:
1737:
1727:
1726:
1721:
1720:
1703:
1685:
1666:
1653:
1638:
1634:
1623:
1622:
1608:
1607:
1597:
1587:
1548:
1538:
1537:
1510:
1494:
1479:
1475:
1464:
1463:
1453:
1440:
1424:
1411:
1396:
1386:
1385:
1358:
1342:
1327:
1323:
1314:
1301:
1276:
1263:
1251:
1250:
1222:
1197:
1184:
1179:
1178:
1150:
1139:
1138:
1119:
1118:
1097:
1092:
1091:
1070:
1057:
1047:
1028:
1018:
1005:
997:
996:
963:
950:
945:
944:
925:
924:
903:
898:
897:
878:
877:
874:
834:
812:
811:
807:
794:
783:
761:
750:
744:
724:
723:
702:
687:
682:
681:
654:
649:
648:
619:
615:
614:
609:
608:
589:
588:
550:
529:
525:
524:
523:
519:
509:
490:
482:
481:
440:
439:
420:
419:
391:
390:
371:
370:
340:
335:
334:
310:
291:
277:
276:
257:
256:
235:
230:
229:
226:
195:
185:
107:
94:
75:
67:
66:
56:
12:
11:
5:
5716:
5714:
5706:
5705:
5700:
5695:
5690:
5685:
5680:
5670:
5669:
5663:
5662:
5660:
5659:
5654:
5649:
5644:
5639:
5634:
5629:
5624:
5619:
5614:
5609:
5603:
5601:
5597:
5596:
5594:
5593:
5588:
5583:
5578:
5573:
5568:
5562:
5560:
5556:
5555:
5553:
5552:
5547:
5542:
5537:
5532:
5527:
5522:
5517:
5515:Robert Axelrod
5512:
5507:
5502:
5497:
5492:
5490:Olga Bondareva
5487:
5482:
5480:Melvin Dresher
5477:
5472:
5470:Leonid Hurwicz
5467:
5462:
5457:
5452:
5447:
5442:
5437:
5432:
5427:
5422:
5417:
5412:
5407:
5405:Harold W. Kuhn
5402:
5397:
5395:Drew Fudenberg
5392:
5387:
5385:David M. Kreps
5382:
5377:
5372:
5370:Claude Shannon
5367:
5362:
5357:
5352:
5346:
5344:
5338:
5337:
5335:
5334:
5329:
5324:
5319:
5314:
5309:
5307:Nash's theorem
5304:
5299:
5294:
5288:
5286:
5282:
5281:
5279:
5278:
5273:
5268:
5263:
5258:
5253:
5248:
5243:
5238:
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5198:
5193:
5188:
5183:
5178:
5173:
5168:
5166:Ultimatum game
5163:
5158:
5153:
5148:
5146:Dollar auction
5143:
5138:
5133:
5131:Centipede game
5128:
5123:
5118:
5113:
5108:
5103:
5098:
5093:
5088:
5086:Infinite chess
5083:
5078:
5072:
5070:
5064:
5063:
5061:
5060:
5055:
5053:Symmetric game
5050:
5045:
5040:
5038:Signaling game
5035:
5033:Screening game
5030:
5025:
5023:Potential game
5020:
5015:
5010:
5002:
4997:
4992:
4987:
4982:
4976:
4974:
4966:
4965:
4963:
4962:
4957:
4952:
4950:Mixed strategy
4947:
4942:
4937:
4932:
4927:
4922:
4917:
4912:
4907:
4902:
4897:
4892:
4887:
4882:
4876:
4874:
4868:
4867:
4865:
4864:
4859:
4854:
4849:
4844:
4839:
4834:
4829:
4827:Risk dominance
4824:
4819:
4814:
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4759:
4754:
4749:
4744:
4738:
4736:
4726:
4725:
4723:
4722:
4717:
4712:
4707:
4702:
4697:
4692:
4687:
4682:
4677:
4672:
4670:Graphical game
4667:
4662:
4657:
4652:
4647:
4642:
4637:
4631:
4629:
4625:
4624:
4618:
4616:
4615:
4608:
4601:
4593:
4584:
4583:
4576:
4565:
4564:The MIT Press.
4554:
4551:
4549:
4548:
4485:
4440:(2): 175–179.
4420:
4414:978-0674023383
4413:
4395:
4330:
4269:
4214:
4187:(5): 447–453.
4171:
4144:(3): 201–211.
4128:
4107:(4): 479–519.
4086:
4084:
4081:
4067:respectively.
4065:
4064:
4053:
4047:
4044:
4041:
4038:
4030:
4027:
4023:
4019:
4016:
4013:
4008:
4004:
3998:
3994:
3987:
3982:
3979:
3976:
3972:
3965:
3961:
3957:
3953:
3938:
3937:
3926:
3921:
3917:
3913:
3910:
3907:
3904:
3901:
3895:
3892:
3888:
3884:
3881:
3878:
3873:
3869:
3863:
3859:
3852:
3847:
3844:
3841:
3837:
3833:
3828:
3824:
3820:
3816:
3812:
3808:
3777:
3755:
3752:
3748:
3725:
3721:
3700:
3680:
3656:
3645:
3644:
3633:
3628:
3624:
3620:
3617:
3614:
3611:
3608:
3602:
3599:
3595:
3591:
3588:
3585:
3580:
3576:
3570:
3566:
3559:
3554:
3551:
3548:
3544:
3540:
3534:
3529:
3525:
3506:
3503:
3487:
3486:
3473:
3470:
3467:
3464:
3458:
3455:
3452:
3449:
3444:
3439:
3433:
3430:
3425:
3421:
3414:
3409:
3405:
3402:
3398:
3391:
3385:
3381:
3377:
3373:
3369:
3365:
3354:
3341:
3338:
3335:
3329:
3326:
3323:
3320:
3315:
3310:
3306:
3303:
3298:
3294:
3290:
3285:
3280:
3276:
3273:
3269:
3263:
3257:
3253:
3249:
3244:
3240:
3236:
3232:
3228:
3224:
3207:
3204:
3200:Price equation
3185:
3181:
3169:
3168:
3157:
3148:
3144:
3137:
3134:
3131:
3126:
3123:
3120:
3116:
3112:
3109:
3105:
3100:
3095:
3091:
3080:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3035:
3031:
3024:
3021:
3018:
3013:
3010:
3007:
3003:
2999:
2996:
2990:
2986:
2980:
2975:
2971:
2947:
2944:
2941:
2917:
2905:
2902:
2877:Main article:
2874:
2871:
2856:
2852:
2830:
2824:
2820:
2816:
2811:
2807:
2801:
2797:
2791:
2787:
2783:
2778:
2774:
2770:
2765:
2761:
2756:
2750:
2746:
2742:
2739:
2736:
2732:
2726:
2721:
2717:
2711:
2706:
2702:
2696:
2692:
2688:
2683:
2679:
2673:
2668:
2664:
2660:
2657:
2654:
2649:
2645:
2640:
2634:
2630:
2626:
2621:
2617:
2613:
2589:
2586:
2582:
2576:
2571:
2567:
2561:
2556:
2552:
2546:
2542:
2537:
2531:
2527:
2523:
2520:
2518:
2511:
2507:
2499:
2495:
2489:
2485:
2481:
2478:
2470:
2466:
2460:
2456:
2452:
2451:
2448:
2445:
2440:
2435:
2431:
2425:
2420:
2416:
2412:
2409:
2406:
2404:
2397:
2393:
2385:
2381:
2375:
2371:
2367:
2364:
2358:
2355:
2354:
2350:
2344:
2340:
2336:
2331:
2327:
2321:
2317:
2311:
2307:
2303:
2300:
2297:
2294:
2290:
2284:
2280:
2276:
2273:
2270:
2268:
2263:
2256:
2252:
2248:
2240:
2236:
2230:
2226:
2222:
2219:
2218:
2213:
2209:
2205:
2200:
2196:
2190:
2186:
2182:
2179:
2176:
2171:
2167:
2161:
2157:
2153:
2150:
2148:
2143:
2136:
2132:
2128:
2122:
2121:
2096:
2092:
2084:
2080:
2074:
2070:
2066:
2063:
2055:
2051:
2045:
2041:
2037:
2029:
2025:
2017:
2013:
2007:
2003:
1999:
1996:
1990:
1984:
1977:
1973:
1969:
1961:
1957:
1951:
1947:
1943:
1937:
1930:
1926:
1922:
1916:
1911:
1907:
1903:
1877:
1874:
1870:
1840:
1836:
1828:
1824:
1820:
1814:
1809:
1806:
1802:
1793:
1789:
1784:
1779:
1776:
1773:
1771:
1764:
1759:
1755:
1751:
1744:
1740:
1734:
1730:
1723:
1722:
1716:
1710:
1706:
1700:
1695:
1692:
1688:
1681:
1677:
1672:
1669:
1667:
1660:
1656:
1652:
1645:
1641:
1637:
1631:
1630:
1604:
1600:
1594:
1590:
1586:
1583:
1575:
1570:
1566:
1562:
1555:
1551:
1545:
1541:
1531:
1527:
1522:
1517:
1513:
1509:
1501:
1497:
1493:
1486:
1482:
1478:
1472:
1469:
1467:
1465:
1460:
1456:
1452:
1447:
1443:
1439:
1431:
1427:
1423:
1418:
1414:
1410:
1403:
1399:
1393:
1389:
1379:
1375:
1370:
1365:
1361:
1357:
1349:
1345:
1341:
1334:
1330:
1326:
1320:
1317:
1315:
1313:
1308:
1304:
1300:
1297:
1294:
1291:
1288:
1283:
1279:
1275:
1270:
1266:
1262:
1259:
1258:
1249:then gives us:
1234:
1229:
1225:
1221:
1218:
1215:
1212:
1209:
1204:
1200:
1196:
1191:
1187:
1162:
1157:
1153:
1149:
1146:
1126:
1104:
1100:
1077:
1073:
1069:
1064:
1060:
1054:
1050:
1046:
1043:
1040:
1035:
1031:
1025:
1021:
1017:
1012:
1008:
1004:
980:
976:
970:
966:
962:
957:
953:
932:
910:
906:
885:
873:
870:
856:
852:
849:
846:
841:
837:
833:
830:
827:
824:
819:
815:
810:
801:
797:
790:
786:
780:
776:
768:
764:
757:
753:
747:
740:
737:
733:
709:
705:
700:
694:
690:
669:
666:
661:
657:
634:
629:
625:
622:
618:
596:
582:
581:
569:
565:
562:
557:
553:
549:
544:
539:
535:
532:
528:
522:
516:
512:
508:
502:
497:
493:
447:
427:
407:
404:
401:
398:
378:
358:
355:
352:
347:
343:
322:
317:
313:
309:
306:
303:
298:
294:
290:
287:
284:
264:
242:
238:
213:
210:
207:
202:
198:
192:
188:
181:
176:
173:
170:
166:
162:
159:
156:
153:
150:
146:
143:
140:
137:
134:
131:
128:
125:
122:
119:
114:
110:
106:
101:
97:
93:
87:
82:
78:
64:
55:
52:
13:
10:
9:
6:
4:
3:
2:
5715:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5684:
5681:
5679:
5676:
5675:
5673:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5638:
5635:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5604:
5602:
5600:Miscellaneous
5598:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5572:
5569:
5567:
5564:
5563:
5561:
5557:
5551:
5548:
5546:
5543:
5541:
5538:
5536:
5535:Samuel Bowles
5533:
5531:
5530:Roger Myerson
5528:
5526:
5523:
5521:
5520:Robert Aumann
5518:
5516:
5513:
5511:
5508:
5506:
5503:
5501:
5498:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5475:Lloyd Shapley
5473:
5471:
5468:
5466:
5463:
5461:
5460:Kenneth Arrow
5458:
5456:
5453:
5451:
5448:
5446:
5443:
5441:
5440:John Harsanyi
5438:
5436:
5433:
5431:
5428:
5426:
5423:
5421:
5418:
5416:
5413:
5411:
5410:Herbert Simon
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5391:
5388:
5386:
5383:
5381:
5378:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5347:
5345:
5339:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5289:
5287:
5283:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5211:Fair division
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5189:
5187:
5186:Dictator game
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5087:
5084:
5082:
5079:
5077:
5074:
5073:
5071:
5069:
5065:
5059:
5058:Zero-sum game
5056:
5054:
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5031:
5029:
5028:Repeated game
5026:
5024:
5021:
5019:
5016:
5014:
5011:
5009:
5007:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4975:
4973:
4967:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4945:Pure strategy
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4921:
4918:
4916:
4913:
4911:
4908:
4906:
4905:De-escalation
4903:
4901:
4898:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4877:
4875:
4873:
4869:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4847:Shapley value
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4739:
4737:
4735:
4731:
4727:
4721:
4718:
4716:
4715:Succinct game
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4632:
4630:
4626:
4622:
4614:
4609:
4607:
4602:
4600:
4595:
4594:
4591:
4587:
4581:
4577:
4574:
4570:
4566:
4563:
4562:
4557:
4556:
4552:
4544:
4540:
4536:
4532:
4528:
4524:
4520:
4516:
4512:
4508:
4504:
4500:
4496:
4489:
4486:
4481:
4477:
4473:
4469:
4465:
4461:
4457:
4453:
4448:
4447:q-bio/0409024
4443:
4439:
4435:
4431:
4424:
4421:
4416:
4410:
4406:
4399:
4396:
4391:
4387:
4383:
4379:
4375:
4371:
4367:
4363:
4358:
4353:
4349:
4345:
4341:
4334:
4331:
4326:
4322:
4318:
4314:
4310:
4306:
4302:
4298:
4293:
4288:
4285:(3): 033104.
4284:
4280:
4273:
4270:
4265:
4261:
4257:
4253:
4249:
4245:
4241:
4237:
4233:
4229:
4225:
4218:
4215:
4210:
4206:
4202:
4198:
4194:
4190:
4186:
4182:
4175:
4172:
4167:
4163:
4159:
4155:
4151:
4147:
4143:
4139:
4132:
4129:
4124:
4120:
4115:
4110:
4106:
4102:
4098:
4091:
4088:
4082:
4080:
4078:
4074:
4068:
4051:
4042:
4036:
4028:
4025:
4021:
4014:
4006:
4002:
3996:
3992:
3985:
3980:
3977:
3974:
3970:
3963:
3959:
3955:
3951:
3943:
3942:
3941:
3924:
3919:
3915:
3908:
3902:
3899:
3893:
3890:
3886:
3879:
3871:
3867:
3861:
3857:
3850:
3845:
3842:
3839:
3835:
3831:
3826:
3822:
3818:
3814:
3810:
3806:
3798:
3797:
3796:
3793:
3791:
3775:
3753:
3750:
3746:
3723:
3719:
3698:
3678:
3670:
3654:
3631:
3626:
3622:
3615:
3609:
3606:
3600:
3597:
3593:
3586:
3578:
3574:
3568:
3564:
3557:
3552:
3549:
3546:
3542:
3538:
3532:
3527:
3523:
3512:
3511:
3510:
3504:
3502:
3500:
3496:
3492:
3471:
3437:
3431:
3428:
3423:
3419:
3412:
3407:
3403:
3400:
3396:
3389:
3383:
3379:
3375:
3371:
3367:
3363:
3355:
3339:
3308:
3304:
3301:
3296:
3292:
3288:
3283:
3278:
3274:
3271:
3267:
3261:
3255:
3251:
3247:
3242:
3238:
3234:
3230:
3226:
3222:
3214:
3213:
3212:
3205:
3203:
3201:
3183:
3179:
3155:
3146:
3142:
3135:
3132:
3129:
3124:
3121:
3118:
3114:
3110:
3107:
3103:
3098:
3093:
3089:
3081:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3033:
3029:
3022:
3019:
3016:
3011:
3008:
3005:
3001:
2997:
2994:
2988:
2984:
2978:
2973:
2969:
2961:
2960:
2959:
2945:
2942:
2939:
2931:
2915:
2903:
2901:
2897:
2893:
2891:
2887:
2880:
2872:
2870:
2854:
2850:
2828:
2822:
2818:
2814:
2809:
2805:
2799:
2795:
2789:
2785:
2781:
2776:
2772:
2768:
2763:
2759:
2754:
2748:
2744:
2740:
2737:
2734:
2730:
2724:
2719:
2715:
2709:
2704:
2700:
2694:
2690:
2686:
2681:
2677:
2671:
2666:
2662:
2658:
2655:
2652:
2647:
2643:
2638:
2632:
2628:
2624:
2619:
2615:
2611:
2587:
2584:
2580:
2574:
2569:
2565:
2559:
2554:
2550:
2544:
2540:
2535:
2529:
2525:
2521:
2519:
2509:
2505:
2497:
2487:
2483:
2479:
2468:
2464:
2458:
2454:
2446:
2443:
2438:
2433:
2429:
2423:
2418:
2414:
2410:
2407:
2405:
2395:
2391:
2383:
2373:
2369:
2365:
2356:
2348:
2342:
2338:
2334:
2329:
2325:
2319:
2315:
2309:
2305:
2301:
2298:
2295:
2292:
2288:
2282:
2278:
2274:
2271:
2269:
2261:
2254:
2250:
2246:
2238:
2234:
2228:
2224:
2220:
2211:
2207:
2203:
2198:
2194:
2188:
2184:
2180:
2177:
2174:
2169:
2165:
2159:
2155:
2151:
2149:
2141:
2134:
2130:
2126:
2094:
2090:
2082:
2072:
2068:
2064:
2053:
2049:
2043:
2039:
2035:
2027:
2023:
2015:
2005:
2001:
1997:
1988:
1982:
1975:
1971:
1967:
1959:
1955:
1949:
1945:
1941:
1935:
1928:
1924:
1920:
1914:
1909:
1905:
1901:
1893:
1875:
1872:
1868:
1838:
1834:
1826:
1822:
1818:
1812:
1807:
1804:
1800:
1791:
1787:
1782:
1777:
1774:
1772:
1762:
1757:
1753:
1742:
1738:
1732:
1714:
1708:
1704:
1698:
1693:
1690:
1686:
1679:
1675:
1670:
1668:
1658:
1654:
1643:
1639:
1602:
1592:
1588:
1584:
1573:
1568:
1564:
1553:
1549:
1543:
1529:
1525:
1520:
1515:
1511:
1507:
1499:
1495:
1484:
1480:
1470:
1468:
1458:
1454:
1450:
1445:
1441:
1437:
1429:
1425:
1416:
1412:
1401:
1397:
1391:
1377:
1373:
1368:
1363:
1359:
1355:
1347:
1343:
1332:
1328:
1318:
1316:
1306:
1302:
1298:
1295:
1292:
1289:
1286:
1281:
1277:
1268:
1264:
1260:
1248:
1227:
1223:
1219:
1216:
1213:
1210:
1207:
1202:
1198:
1189:
1185:
1176:
1160:
1155:
1151:
1147:
1144:
1124:
1102:
1098:
1075:
1071:
1067:
1062:
1058:
1052:
1048:
1044:
1041:
1038:
1033:
1029:
1023:
1019:
1015:
1010:
1006:
1002:
994:
978:
974:
968:
964:
960:
955:
951:
930:
908:
904:
883:
871:
869:
854:
847:
839:
835:
831:
825:
817:
813:
808:
799:
795:
788:
784:
778:
774:
766:
762:
755:
751:
745:
738:
735:
731:
707:
703:
698:
692:
688:
667:
664:
659:
655:
632:
627:
623:
620:
616:
594:
587:
586:payoff matrix
567:
563:
560:
555:
551:
547:
542:
537:
533:
530:
526:
520:
514:
510:
506:
500:
495:
491:
480:
479:
478:
475:
473:
467:
463:
461:
445:
425:
402:
396:
376:
353:
345:
341:
315:
311:
307:
304:
301:
296:
292:
285:
282:
262:
240:
236:
225:
208:
200:
196:
190:
186:
179:
174:
171:
168:
164:
160:
154:
148:
144:
135:
129:
126:
120:
112:
108:
99:
95:
91:
85:
80:
76:
63:
61:
53:
51:
49:
45:
41:
37:
33:
29:
26:
23:
22:deterministic
19:
5505:Peyton Young
5500:Paul Milgrom
5415:Hervé Moulin
5355:Amos Tversky
5297:Folk theorem
5008:-player game
5005:
4925:Grim trigger
4585:
4579:
4572:
4568:
4559:
4502:
4498:
4488:
4437:
4433:
4423:
4404:
4398:
4347:
4343:
4333:
4282:
4278:
4272:
4234:(1): 93–98.
4231:
4227:
4217:
4184:
4180:
4174:
4141:
4137:
4131:
4104:
4100:
4090:
4069:
4066:
3939:
3794:
3790:quasispecies
3646:
3508:
3488:
3209:
3170:
2907:
2898:
2894:
2886:folk theorem
2882:
875:
583:
476:
468:
464:
227:
65:
57:
36:quasispecies
17:
15:
5678:Game theory
5622:Coopetition
5425:Jean Tirole
5420:John Conway
5400:Eric Maskin
5196:Blotto game
5181:Pirate game
4990:Global game
4960:Tit for tat
4890:Bid shading
4880:Appeasement
4730:Equilibrium
4710:Solved game
4645:Determinacy
4628:Definitions
4621:game theory
3493:leading to
1247:ItĂ´'s lemma
5672:Categories
5266:Trust game
5251:Kuhn poker
4915:Escalation
4910:Deterrence
4900:Cheap talk
4872:Strategies
4690:Preference
4619:Topics of
4575:: 145–156.
4357:2102.11034
4350:: 110288.
4292:1703.10767
4083:References
3667:gives the
584:where the
28:non-linear
5450:John Nash
5156:Stag hunt
4895:Collusion
4527:1476-4687
4390:216073761
4256:0022-5193
4201:1432-0770
4166:206774680
4158:1432-0770
4123:0273-0979
4037:ϕ
3971:∑
3903:ϕ
3900:−
3836:∑
3776:ϕ
3610:ϕ
3607:−
3543:∑
3533:˙
3289:−
3133:−
3115:∑
3065:−
3056:…
3020:−
3002:∑
2943:−
2851:σ
2796:σ
2786:∑
2782:−
2760:σ
2701:σ
2691:∑
2663:σ
2659:−
2656:ϕ
2653:−
2551:σ
2541:∑
2465:∑
2415:σ
2411:−
2357:−
2316:σ
2306:∑
2293:ϕ
2275:−
2235:∑
2221:−
2185:σ
2050:∑
1989:−
1956:∑
1942:−
1869:δ
1801:δ
1778:−
1750:∂
1729:∂
1699:−
1687:δ
1651:∂
1636:∂
1561:∂
1540:∂
1492:∂
1477:∂
1422:∂
1409:∂
1388:∂
1340:∂
1325:∂
1145:ϕ
1049:σ
832:−
548:−
501:˙
474:process.
397:ϕ
305:…
165:∑
149:ϕ
130:ϕ
127:−
86:˙
44:selection
5591:Lazy SMP
5285:Theorems
5236:Deadlock
5091:Checkers
4972:of games
4734:concepts
4535:15662424
4480:15308310
4472:15380382
4382:32315673
4317:29604653
4264:12392978
4209:18754189
3960:′
3815:′
3691:to type
3372:′
3231:′
2873:Analysis
472:limiting
54:Equation
48:mutation
25:monotone
5343:figures
5126:Chicken
4980:Auction
4970:Classes
4543:4386820
4507:Bibcode
4452:Bibcode
4362:Bibcode
4325:4559066
4297:Bibcode
4236:Bibcode
1890:is the
460:simplex
4541:
4533:
4525:
4499:Nature
4478:
4470:
4411:
4388:
4380:
4323:
4315:
4262:
4254:
4207:
4199:
4164:
4156:
4121:
3460:
3331:
3171:where
1860:where
1173:. The
1090:where
228:where
5081:Chess
5068:Games
4539:S2CID
4476:S2CID
4442:arXiv
4386:S2CID
4352:arXiv
4321:S2CID
4287:arXiv
4279:Chaos
4205:S2CID
4162:S2CID
3495:chaos
20:is a
4757:Core
4531:PMID
4523:ISSN
4468:PMID
4409:ISBN
4378:PMID
4313:PMID
4260:PMID
4252:ISSN
4197:ISSN
4154:ISSN
4119:ISSN
3940:and
3768:and
5341:Key
4515:doi
4503:433
4460:doi
4438:231
4370:doi
4348:497
4305:doi
4244:doi
4232:219
4189:doi
4146:doi
4109:doi
2932:in
896:is
5674::
5076:Go
4573:40
4571:,
4537:.
4529:.
4521:.
4513:.
4501:.
4497:.
4474:.
4466:.
4458:.
4450:.
4436:.
4432:.
4384:.
4376:.
4368:.
4360:.
4346:.
4342:.
4319:.
4311:.
4303:.
4295:.
4283:28
4281:.
4258:.
4250:.
4242:.
4230:.
4226:.
4203:.
4195:.
4185:72
4183:.
4160:.
4152:.
4142:48
4140:.
4117:.
4105:40
4103:.
4099:.
4079:.
3711:,
3202:.
1245:,
462:.
5006:n
4612:e
4605:t
4598:v
4545:.
4517::
4509::
4482:.
4462::
4454::
4444::
4417:.
4392:.
4372::
4364::
4354::
4327:.
4307::
4299::
4289::
4266:.
4246::
4238::
4211:.
4191::
4168:.
4148::
4125:.
4111::
4052:,
4046:)
4043:x
4040:(
4029:i
4026:j
4022:Q
4018:)
4015:x
4012:(
4007:j
4003:f
3997:j
3993:x
3986:n
3981:1
3978:=
3975:j
3964:=
3956:i
3952:x
3925:,
3920:i
3916:x
3912:)
3909:x
3906:(
3894:i
3891:j
3887:Q
3883:)
3880:x
3877:(
3872:j
3868:f
3862:j
3858:x
3851:n
3846:1
3843:=
3840:j
3832:+
3827:i
3823:x
3819:=
3811:i
3807:x
3754:h
3751:t
3747:i
3724:i
3720:f
3699:i
3679:j
3655:Q
3632:,
3627:i
3623:x
3619:)
3616:x
3613:(
3601:i
3598:j
3594:Q
3590:)
3587:x
3584:(
3579:j
3575:f
3569:j
3565:x
3558:n
3553:1
3550:=
3547:j
3539:=
3528:i
3524:x
3472:,
3469:)
3466:I
3463:I
3457:e
3454:p
3451:y
3448:t
3443:(
3438:]
3432:x
3429:A
3424:T
3420:x
3413:i
3408:)
3404:x
3401:A
3397:(
3390:[
3384:i
3380:x
3376:=
3368:i
3364:x
3340:,
3337:)
3334:I
3328:e
3325:p
3322:y
3319:t
3314:(
3309:]
3305:x
3302:A
3297:T
3293:x
3284:i
3279:)
3275:x
3272:A
3268:(
3262:[
3256:i
3252:x
3248:+
3243:i
3239:x
3235:=
3227:i
3223:x
3184:i
3180:y
3156:,
3147:j
3143:y
3136:1
3130:n
3125:1
3122:=
3119:j
3111:+
3108:1
3104:1
3099:=
3094:n
3090:x
3068:1
3062:n
3059:,
3053:,
3050:1
3047:=
3044:i
3034:j
3030:y
3023:1
3017:n
3012:1
3009:=
3006:j
2998:+
2995:1
2989:i
2985:y
2979:=
2974:i
2970:x
2946:1
2940:n
2916:n
2855:i
2829:)
2823:j
2819:W
2815:d
2810:j
2806:x
2800:j
2790:j
2777:i
2773:W
2769:d
2764:i
2755:(
2749:i
2745:x
2741:+
2738:t
2735:d
2731:)
2725:2
2720:j
2716:x
2710:2
2705:j
2695:j
2687:+
2682:i
2678:x
2672:2
2667:i
2648:i
2644:f
2639:(
2633:i
2629:x
2625:=
2620:i
2616:x
2612:d
2588:t
2585:d
2581:)
2575:2
2570:j
2566:x
2560:2
2555:j
2545:j
2536:(
2530:i
2526:x
2522:=
2510:2
2506:N
2498:2
2494:)
2488:j
2484:N
2480:d
2477:(
2469:j
2459:i
2455:x
2447:t
2444:d
2439:2
2434:i
2430:x
2424:2
2419:i
2408:=
2396:2
2392:N
2384:2
2380:)
2374:i
2370:N
2366:d
2363:(
2349:)
2343:j
2339:W
2335:d
2330:j
2326:x
2320:j
2310:j
2302:+
2299:t
2296:d
2289:(
2283:i
2279:x
2272:=
2262:N
2255:j
2251:N
2247:d
2239:j
2229:i
2225:x
2212:i
2208:W
2204:d
2199:i
2195:x
2189:i
2181:+
2178:t
2175:d
2170:i
2166:x
2160:i
2156:f
2152:=
2142:N
2135:i
2131:N
2127:d
2095:2
2091:N
2083:2
2079:)
2073:j
2069:N
2065:d
2062:(
2054:j
2044:i
2040:x
2036:+
2028:2
2024:N
2016:2
2012:)
2006:i
2002:N
1998:d
1995:(
1983:N
1976:j
1972:N
1968:d
1960:j
1950:i
1946:x
1936:N
1929:i
1925:N
1921:d
1915:=
1910:i
1906:x
1902:d
1876:j
1873:i
1839:2
1835:N
1827:i
1823:x
1819:2
1813:+
1808:j
1805:i
1792:2
1788:N
1783:2
1775:=
1763:2
1758:j
1754:N
1743:i
1739:x
1733:2
1715:N
1709:i
1705:x
1694:j
1691:i
1680:N
1676:1
1671:=
1659:j
1655:N
1644:i
1640:x
1603:2
1599:)
1593:j
1589:N
1585:d
1582:(
1574:2
1569:j
1565:N
1554:i
1550:x
1544:2
1530:2
1526:1
1521:+
1516:j
1512:N
1508:d
1500:j
1496:N
1485:i
1481:x
1471:=
1459:k
1455:N
1451:d
1446:j
1442:N
1438:d
1430:k
1426:N
1417:j
1413:N
1402:i
1398:x
1392:2
1378:2
1374:1
1369:+
1364:j
1360:N
1356:d
1348:j
1344:N
1333:i
1329:x
1319:=
1312:)
1307:m
1303:N
1299:,
1296:.
1293:.
1290:.
1287:,
1282:1
1278:N
1274:(
1269:i
1265:x
1261:d
1233:)
1228:m
1224:N
1220:,
1217:.
1214:.
1211:.
1208:,
1203:1
1199:N
1195:(
1190:i
1186:x
1161:f
1156:T
1152:x
1148:=
1125:i
1103:i
1099:f
1076:i
1072:W
1068:d
1063:i
1059:N
1053:i
1045:+
1042:t
1039:d
1034:i
1030:N
1024:i
1020:f
1016:=
1011:i
1007:N
1003:d
995::
979:N
975:/
969:i
965:N
961:=
956:i
952:x
931:N
909:i
905:N
884:i
855:]
851:)
848:x
845:(
840:j
836:f
829:)
826:x
823:(
818:i
814:f
809:[
800:j
796:x
789:i
785:x
779:=
775:)
767:j
763:x
756:i
752:x
746:(
739:t
736:d
732:d
708:j
704:x
699:/
693:i
689:x
668:x
665:A
660:T
656:x
633:i
628:)
624:x
621:A
617:(
595:A
568:)
564:x
561:A
556:T
552:x
543:i
538:)
534:x
531:A
527:(
521:(
515:i
511:x
507:=
496:i
492:x
446:x
426:n
406:)
403:x
400:(
377:i
357:)
354:x
351:(
346:i
342:f
321:)
316:n
312:x
308:,
302:,
297:1
293:x
289:(
286:=
283:x
263:i
241:i
237:x
212:)
209:x
206:(
201:j
197:f
191:j
187:x
180:n
175:1
172:=
169:j
161:=
158:)
155:x
152:(
145:,
142:]
139:)
136:x
133:(
124:)
121:x
118:(
113:i
109:f
105:[
100:i
96:x
92:=
81:i
77:x
62::
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