3725:
1749:
414:
3878:
1197:
therefore has a palpable effect. Knowing that everyone knows does make a difference. When the outsider's public announcement (a fact already known to all, unless k=1 then the one person with blue eyes would not know until the announcement) becomes common knowledge, the blue-eyed people on this island
494:
people who have blue eyes, and the rest of the people have green eyes. At the start of the puzzle, no one on the island ever knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island at dawn; anyone not making such a discovery
3812:. For several years it has been thought that the assumption of common knowledge of rationality for the players in the game was fundamental. It turns out (Aumann and Brandenburger 1995) that, in two-player games, common knowledge of rationality is not needed as an epistemic condition for
3785:
Common knowledge was used by David Lewis in his pioneering game-theoretical account of convention. In this sense, common knowledge is a concept still central for linguists and philosophers of language (see Clark 1996) maintaining a
Lewisian, conventionalist account of language.
502:: "At least one of you has blue eyes". The outsider, furthermore, is known by all to be truthful, and all know that all know this, and so on: it is common knowledge that he is truthful, and thus it becomes common knowledge that there is at least one islander who has blue eyes (
3865:
know that the Kid knows that he knows that he knows, the chain still breaks because the Kid doesn't know that. Moments later, Rattlesnake confronts the Kid. We see the Kid realizing that his carefully constructed âedgeâ has collapsed into common knowledge.
3804:
are common knowledge, then such posterior probabilities are equal. A result based on the agreement theorem and proven by
Milgrom shows that, given certain conditions on market efficiency and information, speculative trade is impossible.
3852:
has an example of a chain of logic that is collapsed by common knowledge. The Denver Kid tells his allies that
Rattlesnake is in town, but that he has âthe edgeâ: âHe's here and I know he's here, and he knows I know he's here, but he
3822:
Computer scientists use languages incorporating epistemic logics (and common knowledge) to reason about distributed systems. Such systems can be based on logics more complicated than simple propositional epistemic logic, see
Wooldridge
2197:
2004:
3496:
The equivalence with the syntactic approach sketched above can easily be seen: consider an Aumann structure as the one just defined. We can define a correspondent Kripke structure by taking the same space
2550:
2318:
3192:
178:, then it becomes common knowledge; However, if it is transmitted to each agent in private, it becomes mutual knowledge but not common knowledge. Even if the fact that "every agent in the group knows
3492:
596: = 1 (that is, there is exactly one blue-eyed person), the person will recognize that they alone have blue eyes (by seeing only green eyes in the others) and leave at the first dawn. If
2421:
266:
864: > 1, the outsider is only telling the island citizens what they already know: that there are blue-eyed people among them. However, before this fact is announced, the fact is not
839:
398:
2134:
3285:
2058:
3415:
1087:
2619:
2093:
495:
always sleeps until after dawn. On the island, each person knows every other person's eye color, there are no reflective surfaces, and there is no communication of eye color.
2973:
2784:
3342:
2896:
2707:
2474:
1732:
4014:, 2nd edition, Oxford University Press, 1988. The first edition was published by OUP in 1972. For a discussion of both Lewis's and Schiffer's notions, see Russell Dale,
2345:
1479:
1179:
334:
3598:
773:
2645:
1694:
1557:
1391:
1313:
1278:
2920:
2731:
2494:
2441:
1925:
1901:
1880:
1615:
1583:
600: = 2, no one will leave at the first dawn, and the inaction (and the implied lack of knowledge for every agent) is observed by everyone, which then becomes
3709:
210:
172:
92:
3683:
3656:
3625:
3553:
3526:
2866:
2831:
2369:
943:
681:
566:
2669:
286:
4295:
3827:, 2000 (in which he uses a first-order logic incorporating epistemic and temporal operators) or van der Hoek et al. "Alternating Time Epistemic Logic".
4306:
2142:
1944:
568:). The problem: finding the eventual outcome, assuming all persons on the island are completely logical (every participant's knowledge obeys the
3792:
introduced a set theoretical formulation of common knowledge (theoretically equivalent to the one given above) and proved the so-called
1181:). Each blue-eyed person knows that a second blue-eyed person knows that a third blue-eyed person knows that.... (repeat for a total of
1089:). Each blue-eyed person knows that a second blue-eyed person knows that a third person has blue eyes, but no one knows that there is a
105:(1969). The sociologist Morris Friedell defined common knowledge in a 1969 paper. It was first given a mathematical formulation in a
3746:
1770:
435:
2499:
2221:
3115:
2218:
definition of common knowledge can be given. Intuitively, common knowledge is thought of as the fixed point of the "equation"
4338:
4159:
4133:
4078:
174:) which functions quite similarly to Lewis's and Friedel's 1969 "common knowledge". If a trustworthy announcement is made in
3422:
5242:
691: > 1; and also that the other blue-eyed person does not think that everyone except themself are not blue-eyed
5415:
4267:
2374:
569:
215:
5059:
4589:
4387:
5435:
4878:
4697:
4196:
4182:
4114:
4100:
4059:
3772:
2561:
1796:
498:
At some point, an outsider comes to the island, calls together all the people on the island, and makes the following
461:
3754:
1778:
782:
443:
4494:
347:
4285:
4016:
4968:
3857:
know I know he knows I know he's here.â So both protagonists know the main fact (Rattlesnake is here), but it is
2098:
4838:
4504:
4281:
4125:
3750:
3229:
3220:
Similar to the modal logic formulation above, an operator for the idea that "everyone knows can be defined as
1774:
499:
439:
175:
4677:
2012:
5019:
4432:
4407:
4292:
3793:
4303:
3347:
5425:
5369:
4795:
4544:
4534:
4469:
4275:
484:
959:
121:
in general â and of common knowledge in particular â starting in the 1980s. There are numerous
5420:
4584:
4564:
3022:. This partition represents the state of knowledge of an agent in a state. Intuitively, if two states
2578:
5410:
5303:
5054:
5024:
4682:
4519:
4514:
4151:
3711:, which is the finitary characterization of common knowledge also given by Aumann in the 1976 article.
2063:
2786:. The semantics for the common knowledge operator, then, is given by taking, for each group of agents
841:), will leave on the second dawn. Inductively, it can be reasoned that no one will leave at the first
5339:
5262:
4998:
4549:
4474:
4331:
2933:
2744:
945:). Each blue-eyed person knows that there is someone with blue eyes, but each blue eyed person does
5430:
5354:
5087:
4973:
4770:
4559:
4377:
3916:
3896:
3840:
uses the notion of common knowledge to analyze the kind of indirect speech involved in innuendoes.
3735:
3296:
2871:
2682:
2449:
1759:
1699:
869:
424:
5157:
4109:, volume 41, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1995.
5359:
4958:
4928:
4579:
4367:
3801:
3739:
2323:
1763:
1396:
1107:
428:
299:
3570:
1193:
th" blue-eyed person with that knowledge, until the outsider makes his statement. The notion of
694:
5384:
5364:
5344:
5293:
4963:
4868:
4727:
4672:
4599:
4569:
4489:
4417:
3849:
3816:
2624:
1623:
1486:
1320:
1286:
1207:
3658:
defined in the previous section corresponds to the finest common coarsening of the partitions
2905:
2716:
2479:
2426:
1910:
1886:
1865:
1588:
4843:
4828:
4397:
3986:
Morris
Friedell, "On the Structure of Shared Awareness," Behavioral Science 14 (1969): 28â39.
3901:
1562:
3688:
185:
147:
67:
5405:
5177:
5162:
5049:
5044:
4948:
4933:
4898:
4863:
4457:
4402:
4324:
4288:
at The
Twofold Gaze. Detailed description of the blue-eyed islander problem, with solution.
3832:
3661:
3634:
3603:
3531:
3504:
2844:
2809:
2354:
882:
607:
505:
2654:
125:
based upon the concept which have been extensively investigated by mathematicians such as
8:
5334:
4953:
4903:
4740:
4667:
4642:
4499:
4382:
4088:
2211:
1823:
98:
4993:
4043:
Aumann Robert and Adam
Brandenburger (1995) "Epistemic Conditions for Nash Equilibrium"
5313:
5172:
5003:
4983:
4833:
4712:
4612:
4539:
4484:
4258:
4238:
4220:
4211:
3861:âcommon knowledgeâ. Note that this is true even if the Kid is wrong: maybe Rattlesnake
3008:
2799:
271:
126:
5298:
5267:
5222:
5117:
4988:
4943:
4918:
4848:
4722:
4647:
4637:
4529:
4479:
4427:
4192:
4178:
4155:
4129:
4110:
4096:
4074:
4055:
3883:
3797:
2791:
480:
27:
4242:
853: â 1 blue-eyed people among the others and knowing there must be at least
5379:
5374:
5308:
5272:
5252:
5212:
5182:
5137:
5092:
5077:
5034:
4888:
4662:
4524:
4461:
4447:
4412:
4230:
3970:
3911:
3813:
2676:
2210:
above defines common knowledge as an infinite conjunction of formulas, hence not a
1815:
141:
133:
114:
5277:
5237:
5192:
5107:
5102:
4823:
4775:
4657:
4422:
4392:
4362:
4310:
4299:
4262:
3919:
for the impossibility of establishing common knowledge over an unreliable channel
1819:
118:
5142:
3627:
is the event of the Aumann structure corresponding to the primitive proposition
2202:
There is, however, a complication. The languages of epistemic logic are usually
2192:{\displaystyle C\varphi \Leftrightarrow \bigwedge _{i=0}^{\infty }E^{i}\varphi }
1093:
blue-eyed person with that knowledge, until the outsider makes their statement.
584:
th dawn after the announcement, all the blue-eyed people will leave the island.
16:
Statement that players know and also know that other players know (ad infinitum)
5217:
5207:
5197:
5132:
5122:
5112:
5097:
4893:
4873:
4858:
4853:
4813:
4780:
4765:
4760:
4750:
4554:
4202:
3949:
4167:
3631:. It is not difficult to see that the common knowledge accessibility function
1999:{\displaystyle E_{G}\varphi \Leftrightarrow \bigwedge _{i\in G}K_{i}\varphi ,}
5399:
5257:
5247:
5202:
5187:
5167:
4938:
4913:
4785:
4755:
4745:
4732:
4632:
4574:
4509:
4442:
4033:
3837:
3789:
2988:
212:) is transmitted to each agent in private, it is still not common knowledge:
110:
3103:. This model excludes cases in which agents know things that are not true.)
5232:
5227:
5082:
4652:
4044:
2348:
122:
60:
2983:
Alternatively (yet equivalently) common knowledge can be formalized using
5349:
5152:
5147:
5127:
4923:
4908:
4717:
4687:
4617:
4607:
4437:
4372:
4348:
4206:
3891:
3809:
2672:
4316:
4234:
4147:
Multiagent
Systems: Algorithmic, Game-Theoretic, and Logical Foundations
4978:
4627:
4209:(1990). "Knowledge and Common Knowledge in a Distributed Environment".
2984:
106:
3040:
are elements of the same part of partition of an agent, it means that
2565:. A Kripke structure is given by a set of states (or possible worlds)
4883:
4803:
4622:
4070:
3906:
23:
3724:
3528:
that define the equivalence classes corresponding to the partitions
2675:, in each state, to each primitive proposition in the language. The
1748:
683:). The two blue-eyed people, seeing only one person with blue eyes,
479:
The idea of common knowledge is often introduced by some variant of
413:
97:
The concept was first introduced in the philosophical literature by
5318:
4818:
4225:
4065:
Fagin, Ronald; Halpern, Joseph; Moses, Yoram; Vardi, Moshe (2003).
2651:
considers possible from any given state, and a valuation function
5039:
5029:
4707:
4145:
4107:
Epistemic Logic for
Computer Science and Artificial Intelligence
1393:, with a passing day where no one leaves, implies the next day
3417:. Using this we can then define a common knowledge function,
949:
know that the other blue-eyed person has this same knowledge.
4808:
2559:
characterization is given semantic content through so-called
2207:
1822:. At the propositional level, such systems are extensions of
3833:
The Stuff of
Thought: Language as a Window into Human Nature
3209:) is the set of states where the agent will know that event
2978:
2991:
in his seminal 1976 paper). Starting with a set of states
2545:{\displaystyle C_{G}E_{G}\varphi \Rightarrow C_{G}\varphi }
2423:
from which, in the limit, we can infer common knowledge of
1809:
1189:
th person has blue eyes, but no one knows that there is a "
3058:
are indistinguishable to that agent. In general, in state
2313:{\displaystyle C_{G}\varphi ==E_{G}^{\aleph _{0}}\varphi }
3187:{\displaystyle K_{i}(e)=\{s\in S\mid P_{i}(s)\subset e\}}
1281:
2679:
for the knowledge operator is given by stipulating that
2136:, common knowledge could then be defined with the axiom
1826:. The extension consists of the introduction of a group
592:
The solution can be seen with an inductive argument. If
336:). If every agent publicly announces their knowledge of
1280:
is free (i.e. known prior to the outsider's statement)
857:, will reason that they must have blue eyes and leave.
845: â 1 dawns if and only if there are at least
4064:
3487:{\displaystyle C(e)=\bigcap _{n=1}^{\infty }E^{n}(e).}
1814:
Common knowledge can be given a logical definition in
879: = 2, it is merely "first-order" knowledge (
3691:
3664:
3637:
3606:
3573:
3555:, and a valuation function such that it yields value
3534:
3507:
3425:
3350:
3299:
3232:
3118:
2999:
can then be defined as a subset of the set of states
2936:
2908:
2874:
2847:
2812:
2747:
2719:
2685:
2657:
2627:
2581:
2502:
2482:
2452:
2429:
2377:
2357:
2326:
2224:
2145:
2101:
2066:
2015:
1947:
1913:
1903:
is a formula of the logical calculus) is read "agent
1889:
1868:
1818:
systems in which the modal operators are interpreted
1702:
1626:
1591:
1565:
1489:
1399:
1323:
1289:
1210:
1110:
962:
885:
785:
697:
610:
508:
350:
302:
274:
218:
188:
150:
70:
3996:
Ian Stewart (2004). "I Know That You Know That...".
3873:
4122:
Epistemic Logic: A Survey Of the Logic Of Knowledge
2416:{\displaystyle E_{G}(\varphi \wedge C_{G}\varphi )}
4143:
3703:
3677:
3650:
3619:
3592:
3547:
3520:
3486:
3409:
3336:
3279:
3186:
2967:
2914:
2890:
2860:
2825:
2778:
2725:
2701:
2663:
2639:
2613:
2544:
2488:
2468:
2435:
2415:
2363:
2339:
2312:
2191:
2128:
2087:
2052:
1998:
1919:
1895:
1874:
1726:
1688:
1609:
1577:
1551:
1473:
1385:
1307:
1272:
1173:
1081:
937:
833:
767:
687:that no one left on the first dawn (and thus that
675:
560:
392:
328:
292:, then it becomes common knowledge that they know
280:
261:{\displaystyle E_{G}E_{G}p\not \Rightarrow C_{G}p}
260:
204:
166:
86:
54:, they all know that they all know that they know
1145:
1141:
1053:
1049:
996:
992:
909:
905:
796:
792:
724:
720:
634:
630:
532:
528:
5397:
4256:
3948:A structurally identical problem is provided by
3289:As with the modal operator, we will iterate the
2351:. In this way, it is possible to find a formula
2214:of the language. To overcome this difficulty, a
1934:with the intended meaning of "everyone in group
956: = 3, it is "second order" knowledge (
3937:by Fagin, Halpern, Moses and Vardi (1995), and
2987:(this was the path taken by the Nobel laureate
849:blue-eyed people. Those with blue eyes, seeing
3808:The concept of common knowledge is central in
834:{\displaystyle \exists x\!\in \!(G-a)(Bl_{x})}
140:, independently developed a notion he called "
4332:
2647:, intuitively representing what states agent
393:{\displaystyle C_{G}E_{G}p\Rightarrow C_{G}p}
4201:
3181:
3141:
2446:From this definition it can be seen that if
3995:
3952:(2000); he calls it "The Women of Sevitan".
3753:. Unsourced material may be challenged and
1777:. Unsourced material may be challenged and
1198:eventually deduce their status, and leave.
442:. Unsourced material may be challenged and
4339:
4325:
4144:Shoham, Yoav; Leyton-Brown, Kevin (2009).
2129:{\displaystyle E_{G}^{0}\varphi =\varphi }
1104: − 1)th order" knowledge (
4346:
4224:
3796:through which: if two agents have common
3773:Learn how and when to remove this message
3280:{\displaystyle E(e)=\bigcap _{i}K_{i}(e)}
3110:can now be defined in the following way:
2979:Set theoretic (semantic characterization)
1797:Learn how and when to remove this message
572:) and that this too is common knowledge.
462:Learn how and when to remove this message
4276:Prof. Terence Tao's blog post (Feb 2008)
4257:Vanderschraaf, Peter; Sillari, Giacomo.
2053:{\displaystyle E_{G}E_{G}^{n-1}\varphi }
1853:) with the intended meaning that "agent
1810:Modal logic (syntactic characterization)
4119:
5398:
3963:
3410:{\displaystyle E^{n+1}(e)=E(E^{n}(e))}
1585:is thus reached iff it is reached for
288:publicly announces their knowledge of
4320:
3843:
1938:knows" by defining it with the axiom
4286:"In the Long Run We Are All Dead II"
3944:
3939:Epistemic Logic for computer science
3929:
3751:adding citations to reliable sources
3718:
3079:) obtains, but not which one. (Here
1775:adding citations to reliable sources
1742:
1082:{\displaystyle E_{G}E_{G}=E_{G}^{2}}
440:adding citations to reliable sources
407:
4268:Stanford Encyclopedia of Philosophy
117:grew an interest in the subject of
13:
4388:First-player and second-player win
4027:
3977:. Cambridge, MA: MIT, 1994. Print.
3457:
2614:{\displaystyle R_{1},\dots ,R_{n}}
2328:
2296:
2171:
1709:
1135:
1043:
986:
899:
786:
743:
714:
698:
641:
624:
570:axiom schemata for epistemic logic
522:
14:
5447:
4282:"In the Long Run We Are All Dead"
4250:
4105:J-J Ch. Meyer and W van der Hoek
4093:Convention: A Philosophical Study
3941:by Meyer and van der Hoek (1995).
3825:Reasoning about Artificial Agents
2088:{\displaystyle E_{G}^{n}\varphi }
4495:Coalition-proof Nash equilibrium
3876:
3723:
3092:) denotes the unique element of
3066:knows that one of the states in
1747:
1738:
412:
4036:(1976) "Agreeing to Disagree"
3714:
2009:By abbreviating the expression
1185: − 1 levels) a
50:, they all know that they know
4505:Evolutionarily stable strategy
4126:University of Pittsburgh Press
4004:
3989:
3980:
3800:over a certain event, and the
3478:
3472:
3435:
3429:
3404:
3401:
3395:
3382:
3373:
3367:
3331:
3325:
3316:
3310:
3274:
3268:
3242:
3236:
3172:
3166:
3135:
3129:
2968:{\displaystyle (s,t)\in R_{G}}
2949:
2937:
2779:{\displaystyle (s,t)\in R_{i}}
2760:
2748:
2526:
2410:
2388:
2279:
2276:
2260:
2241:
2152:
1961:
1683:
1673:
1669:
1653:
1646:
1642:
1546:
1536:
1532:
1516:
1509:
1505:
1468:
1452:
1448:
1432:
1425:
1421:
1380:
1370:
1366:
1350:
1343:
1339:
1267:
1257:
1253:
1237:
1230:
1226:
1168:
1165:
1149:
1132:
1076:
1073:
1057:
1040:
1019:
1016:
1000:
983:
932:
929:
913:
896:
828:
812:
809:
797:
762:
759:
740:
737:
725:
711:
670:
667:
638:
621:
555:
552:
536:
519:
374:
1:
4433:Simultaneous action selection
4304:"Green-eyed Dragons Solution"
4054:, Cambridge University Press
3956:
3559:to the primitive proposition
3337:{\displaystyle E^{1}(e)=E(e)}
2891:{\displaystyle C_{G}\varphi }
2702:{\displaystyle K_{i}\varphi }
2469:{\displaystyle E_{G}\varphi }
1927:." We can define an operator
1727:{\displaystyle i=\infty ,j=1}
5370:List of games in game theory
4545:Quantal response equilibrium
4535:Perfect Bayesian equilibrium
4470:Bayes correlated equilibrium
4293:"Green-eyed Dragons Problem"
4191:Princeton University Press.
4177:Princeton University Press.
7:
4839:Optional prisoner's dilemma
4565:Self-confirming equilibrium
3869:
3563:in all and only the states
3213:obtains. It is a subset of
2340:{\displaystyle \aleph _{0}}
1474:{\displaystyle E_{G}^{i-1}}
1174:{\displaystyle E_{G}^{k-1}}
1100: > 1, it is "(
580:The answer is that, on the
575:
329:{\displaystyle C_{G}K_{a}p}
10:
5452:
5416:Fixed points (mathematics)
5304:Principal variation search
5020:Aumann's agreement theorem
4683:Strategy-stealing argument
4590:Trembling hand equilibrium
4520:Markov perfect equilibrium
4515:Mertens-stable equilibrium
4152:Cambridge University Press
3593:{\displaystyle s\in E^{p}}
3501:, accessibility relations
2496:is also common knowledge (
2476:is common knowledge, then
768:{\displaystyle \neg K_{a}}
403:
5340:Combinatorial game theory
5327:
5286:
5068:
5012:
4999:Princess and monster game
4794:
4696:
4598:
4550:Quasi-perfect equilibrium
4475:Bayesian Nash equilibrium
4456:
4355:
4120:Rescher, Nicolas (2005).
4067:Reasoning about Knowledge
3935:Reasoning about knowledge
2640:{\displaystyle S\times S}
1689:{\displaystyle E_{G}^{i}}
1552:{\displaystyle E_{G}^{i}}
1386:{\displaystyle E_{G}^{i}}
1308:{\displaystyle i+j\leq k}
1273:{\displaystyle E_{G}^{i}}
474:
344:becomes common knowledge
5436:Concepts in epistemology
5355:Evolutionary game theory
5088:Antoine Augustin Cournot
4974:Guess 2/3 of the average
4771:Strictly determined game
4560:Satisfaction equilibrium
4378:Escalation of commitment
4168:downloadable free online
3969:Osborne, Martin J., and
3923:
3897:Mutual knowledge (logic)
2915:{\displaystyle \varphi }
2726:{\displaystyle \varphi }
2489:{\displaystyle \varphi }
2436:{\displaystyle \varphi }
1920:{\displaystyle \varphi }
1896:{\displaystyle \varphi }
1875:{\displaystyle \varphi }
1610:{\displaystyle i+j>k}
587:
490:On an island, there are
5360:Glossary of game theory
4959:Stackelberg competition
4580:Strong Nash equilibrium
4187:Gintis, Herbert (2009)
4173:Gintis, Herbert (2000)
3975:A Course in Game Theory
3802:posterior probabilities
2868:, and stipulating that
2841:, call such a relation
2574:accessibility relations
1578:{\displaystyle j\geq k}
64:. It can be denoted as
42:when all the agents in
5385:Tragedy of the commons
5365:List of game theorists
5345:Confrontation analysis
5055:SpragueâGrundy theorem
4570:Sequential equilibrium
4490:Correlated equilibrium
4050:Clark, Herbert (1996)
3850:Hot Lead and Cold Feet
3705:
3704:{\displaystyle i\in G}
3679:
3652:
3621:
3594:
3549:
3522:
3488:
3461:
3411:
3338:
3281:
3188:
2969:
2916:
2892:
2862:
2827:
2780:
2727:
2703:
2665:
2641:
2615:
2546:
2490:
2470:
2437:
2417:
2365:
2341:
2314:
2193:
2175:
2130:
2089:
2054:
2000:
1921:
1897:
1876:
1728:
1690:
1611:
1579:
1553:
1475:
1387:
1309:
1274:
1175:
1083:
939:
835:
769:
677:
562:
394:
330:
282:
262:
206:
205:{\displaystyle E_{G}p}
168:
167:{\displaystyle E_{G}p}
88:
87:{\displaystyle C_{G}p}
5158:Jean-François Mertens
4017:The Theory of Meaning
3917:Two Generals' Problem
3902:Pluralistic ignorance
3706:
3680:
3678:{\displaystyle P_{i}}
3653:
3651:{\displaystyle R_{G}}
3622:
3620:{\displaystyle E^{p}}
3595:
3550:
3548:{\displaystyle P_{i}}
3523:
3521:{\displaystyle R_{i}}
3489:
3441:
3412:
3339:
3282:
3189:
3106:A knowledge function
2970:
2917:
2893:
2863:
2861:{\displaystyle R_{G}}
2828:
2826:{\displaystyle R_{i}}
2781:
2728:
2704:
2666:
2642:
2616:
2547:
2491:
2471:
2438:
2418:
2366:
2364:{\displaystyle \psi }
2342:
2315:
2194:
2155:
2131:
2090:
2055:
2001:
1922:
1898:
1877:
1729:
1691:
1612:
1580:
1554:
1476:
1388:
1310:
1275:
1176:
1084:
940:
938:{\displaystyle E_{G}}
836:
770:
678:
676:{\displaystyle C_{G}}
563:
561:{\displaystyle C_{G}}
485:Muddy children puzzle
395:
331:
283:
263:
207:
169:
89:
38:in a group of agents
22:is a special kind of
5287:Search optimizations
5163:Jennifer Tour Chayes
5050:Revelation principle
5045:Purification theorem
4984:Nash bargaining game
4949:Bertrand competition
4934:El Farol Bar problem
4899:Electronic mail game
4864:Lewis signaling game
4403:Hierarchy of beliefs
4291:physics.harvard.edu
4189:The Bounds of Reason
4175:Game Theory Evolving
4166:. See Section 13.4;
4038:Annals of Statistics
3747:improve this section
3689:
3662:
3635:
3604:
3571:
3532:
3505:
3423:
3348:
3297:
3230:
3116:
2934:
2906:
2872:
2845:
2810:
2745:
2717:
2683:
2664:{\displaystyle \pi }
2655:
2625:
2579:
2500:
2480:
2450:
2427:
2375:
2355:
2324:
2222:
2143:
2099:
2064:
2013:
1945:
1911:
1887:
1866:
1849:= 1, ...,
1771:improve this section
1700:
1624:
1589:
1563:
1487:
1397:
1321:
1287:
1208:
1108:
960:
883:
783:
695:
608:
506:
436:improve this section
348:
300:
272:
268:. But, if any agent
216:
186:
148:
68:
5426:Social epistemology
5335:Bounded rationality
4954:Cournot competition
4904:Rock paper scissors
4879:Battle of the sexes
4869:Volunteer's dilemma
4741:Perfect information
4668:Dominant strategies
4500:Epsilon-equilibrium
4383:Extensive-form game
4235:10.1145/79147.79161
4095:Oxford: Blackburn.
3830:In his 2007 book,
2306:
2212:well-formed formula
2116:
2081:
2046:
1824:propositional logic
1641:
1620:The outsider gives
1504:
1420:
1338:
1225:
1131:
1039:
500:public announcement
136:, in his 1972 book
115:Computer scientists
99:David Kellogg Lewis
5421:Knowledge transfer
5314:Paranoid algorithm
5294:Alphaâbeta pruning
5173:John Maynard Smith
5004:Rendezvous problem
4844:Traveler's dilemma
4834:Gift-exchange game
4829:Prisoner's dilemma
4746:Large Poisson game
4713:Bargaining problem
4613:Backward induction
4585:Subgame perfection
4540:Proper equilibrium
4309:2014-12-01 at the
4298:2014-12-01 at the
4259:"Common Knowledge"
4212:Journal of the ACM
4010:Stephen Schiffer,
3933:See the textbooks
3844:In popular culture
3701:
3675:
3648:
3617:
3590:
3545:
3518:
3484:
3407:
3334:
3277:
3257:
3184:
3003:. For each agent
2965:
2912:
2888:
2858:
2823:
2800:transitive closure
2776:
2723:
2699:
2661:
2637:
2611:
2542:
2486:
2466:
2433:
2413:
2361:
2337:
2310:
2285:
2189:
2126:
2102:
2085:
2067:
2050:
2026:
1996:
1979:
1917:
1893:
1872:
1724:
1686:
1627:
1607:
1575:
1549:
1490:
1471:
1400:
1383:
1324:
1305:
1270:
1211:
1171:
1111:
1079:
1025:
935:
831:
765:
673:
558:
390:
326:
278:
258:
202:
164:
84:
5411:Concepts in logic
5393:
5392:
5299:Aspiration window
5268:Suzanne Scotchmer
5223:Oskar Morgenstern
5118:Donald B. Gillies
5060:Zermelo's theorem
4989:Induction puzzles
4944:Fair cake-cutting
4919:Public goods game
4849:Coordination game
4723:Intransitive game
4648:Forward induction
4530:Pareto efficiency
4510:Gibbs equilibrium
4480:Berge equilibrium
4428:Simultaneous game
4161:978-0-521-89943-7
4135:978-0-8229-4246-7
4080:978-0-262-56200-3
4047:63(5): 1161â1180.
3884:Philosophy portal
3848:The comedy movie
3798:prior probability
3794:agreement theorem
3783:
3782:
3775:
3248:
2898:is true at state
2833:, for all agents
2709:is true at state
2562:Kripke structures
1964:
1816:multi-modal logic
1807:
1806:
1799:
779:blue-eyed person
481:induction puzzles
472:
471:
464:
281:{\displaystyle a}
5443:
5380:Topological game
5375:No-win situation
5273:Thomas Schelling
5253:Robert B. Wilson
5213:Merrill M. Flood
5183:John von Neumann
5093:Ariel Rubinstein
5078:Albert W. Tucker
4929:War of attrition
4889:Matching pennies
4663:Pairing strategy
4525:Nash equilibrium
4448:Mechanism design
4413:Normal-form game
4368:Cooperative game
4341:
4334:
4327:
4318:
4317:
4272:
4263:Zalta, Edward N.
4246:
4228:
4165:
4140:. See Chapter 3.
4139:
4084:
4040:4(6): 1236â1239.
4021:
4008:
4002:
4001:
3993:
3987:
3984:
3978:
3971:Ariel Rubinstein
3967:
3947:
3932:
3912:Stephen Schiffer
3886:
3881:
3880:
3879:
3814:Nash equilibrium
3778:
3771:
3767:
3764:
3758:
3727:
3719:
3710:
3708:
3707:
3702:
3684:
3682:
3681:
3676:
3674:
3673:
3657:
3655:
3654:
3649:
3647:
3646:
3626:
3624:
3623:
3618:
3616:
3615:
3599:
3597:
3596:
3591:
3589:
3588:
3554:
3552:
3551:
3546:
3544:
3543:
3527:
3525:
3524:
3519:
3517:
3516:
3493:
3491:
3490:
3485:
3471:
3470:
3460:
3455:
3416:
3414:
3413:
3408:
3394:
3393:
3366:
3365:
3343:
3341:
3340:
3335:
3309:
3308:
3286:
3284:
3283:
3278:
3267:
3266:
3256:
3193:
3191:
3190:
3185:
3165:
3164:
3128:
3127:
2974:
2972:
2971:
2966:
2964:
2963:
2921:
2919:
2918:
2913:
2897:
2895:
2894:
2889:
2884:
2883:
2867:
2865:
2864:
2859:
2857:
2856:
2832:
2830:
2829:
2824:
2822:
2821:
2785:
2783:
2782:
2777:
2775:
2774:
2732:
2730:
2729:
2724:
2708:
2706:
2705:
2700:
2695:
2694:
2677:Kripke semantics
2670:
2668:
2667:
2662:
2646:
2644:
2643:
2638:
2620:
2618:
2617:
2612:
2610:
2609:
2591:
2590:
2551:
2549:
2548:
2543:
2538:
2537:
2522:
2521:
2512:
2511:
2495:
2493:
2492:
2487:
2475:
2473:
2472:
2467:
2462:
2461:
2442:
2440:
2439:
2434:
2422:
2420:
2419:
2414:
2406:
2405:
2387:
2386:
2370:
2368:
2367:
2362:
2346:
2344:
2343:
2338:
2336:
2335:
2319:
2317:
2316:
2311:
2305:
2304:
2303:
2293:
2272:
2271:
2259:
2258:
2234:
2233:
2198:
2196:
2195:
2190:
2185:
2184:
2174:
2169:
2135:
2133:
2132:
2127:
2115:
2110:
2094:
2092:
2091:
2086:
2080:
2075:
2059:
2057:
2056:
2051:
2045:
2034:
2025:
2024:
2005:
2003:
2002:
1997:
1989:
1988:
1978:
1957:
1956:
1926:
1924:
1923:
1918:
1902:
1900:
1899:
1894:
1881:
1879:
1878:
1873:
1838:modal operators
1802:
1795:
1791:
1788:
1782:
1751:
1743:
1733:
1731:
1730:
1725:
1695:
1693:
1692:
1687:
1676:
1668:
1667:
1649:
1640:
1635:
1616:
1614:
1613:
1608:
1584:
1582:
1581:
1576:
1558:
1556:
1555:
1550:
1539:
1531:
1530:
1512:
1503:
1498:
1480:
1478:
1477:
1472:
1455:
1447:
1446:
1428:
1419:
1408:
1392:
1390:
1389:
1384:
1373:
1365:
1364:
1346:
1337:
1332:
1314:
1312:
1311:
1306:
1279:
1277:
1276:
1271:
1260:
1252:
1251:
1233:
1224:
1219:
1195:common knowledge
1180:
1178:
1177:
1172:
1164:
1163:
1130:
1119:
1096:In general: For
1088:
1086:
1085:
1080:
1072:
1071:
1038:
1033:
1015:
1014:
982:
981:
972:
971:
944:
942:
941:
936:
928:
927:
895:
894:
870:mutual knowledge
866:common knowledge
840:
838:
837:
832:
827:
826:
774:
772:
771:
766:
758:
757:
710:
709:
682:
680:
679:
674:
666:
665:
653:
652:
620:
619:
602:common knowledge
567:
565:
564:
559:
551:
550:
518:
517:
467:
460:
456:
453:
447:
416:
408:
399:
397:
396:
391:
386:
385:
370:
369:
360:
359:
335:
333:
332:
327:
322:
321:
312:
311:
287:
285:
284:
279:
267:
265:
264:
259:
254:
253:
238:
237:
228:
227:
211:
209:
208:
203:
198:
197:
173:
171:
170:
165:
160:
159:
142:mutual knowledge
134:Stephen Schiffer
132:The philosopher
93:
91:
90:
85:
80:
79:
32:common knowledge
20:Common knowledge
5451:
5450:
5446:
5445:
5444:
5442:
5441:
5440:
5431:Epistemic logic
5396:
5395:
5394:
5389:
5323:
5309:max^n algorithm
5282:
5278:William Vickrey
5238:Reinhard Selten
5193:Kenneth Binmore
5108:David K. Levine
5103:Daniel Kahneman
5070:
5064:
5040:Negamax theorem
5030:Minimax theorem
5008:
4969:Diner's dilemma
4824:All-pay auction
4790:
4776:Stochastic game
4728:Mean-field game
4699:
4692:
4658:Markov strategy
4594:
4460:
4452:
4423:Sequential game
4408:Information set
4393:Game complexity
4363:Congestion game
4351:
4345:
4311:Wayback Machine
4300:Wayback Machine
4253:
4162:
4136:
4081:
4030:
4028:Further reading
4025:
4024:
4009:
4005:
3994:
3990:
3985:
3981:
3968:
3964:
3959:
3926:
3882:
3877:
3875:
3872:
3846:
3779:
3768:
3762:
3759:
3744:
3728:
3717:
3690:
3687:
3686:
3669:
3665:
3663:
3660:
3659:
3642:
3638:
3636:
3633:
3632:
3611:
3607:
3605:
3602:
3601:
3584:
3580:
3572:
3569:
3568:
3539:
3535:
3533:
3530:
3529:
3512:
3508:
3506:
3503:
3502:
3466:
3462:
3456:
3445:
3424:
3421:
3420:
3389:
3385:
3355:
3351:
3349:
3346:
3345:
3304:
3300:
3298:
3295:
3294:
3262:
3258:
3252:
3231:
3228:
3227:
3204:
3160:
3156:
3123:
3119:
3117:
3114:
3113:
3097:
3087:
3074:
3057:
3048:
3039:
3030:
3020:
2981:
2959:
2955:
2935:
2932:
2931:
2907:
2904:
2903:
2879:
2875:
2873:
2870:
2869:
2852:
2848:
2846:
2843:
2842:
2817:
2813:
2811:
2808:
2807:
2770:
2766:
2746:
2743:
2742:
2718:
2715:
2714:
2690:
2686:
2684:
2681:
2680:
2656:
2653:
2652:
2626:
2623:
2622:
2605:
2601:
2586:
2582:
2580:
2577:
2576:
2533:
2529:
2517:
2513:
2507:
2503:
2501:
2498:
2497:
2481:
2478:
2477:
2457:
2453:
2451:
2448:
2447:
2428:
2425:
2424:
2401:
2397:
2382:
2378:
2376:
2373:
2372:
2356:
2353:
2352:
2331:
2327:
2325:
2322:
2321:
2299:
2295:
2294:
2289:
2267:
2263:
2254:
2250:
2229:
2225:
2223:
2220:
2219:
2180:
2176:
2170:
2159:
2144:
2141:
2140:
2111:
2106:
2100:
2097:
2096:
2076:
2071:
2065:
2062:
2061:
2035:
2030:
2020:
2016:
2014:
2011:
2010:
1984:
1980:
1968:
1952:
1948:
1946:
1943:
1942:
1932:
1912:
1909:
1908:
1888:
1885:
1884:
1867:
1864:
1863:
1862:
1843:
1812:
1803:
1792:
1786:
1783:
1768:
1752:
1741:
1701:
1698:
1697:
1672:
1663:
1659:
1645:
1636:
1631:
1625:
1622:
1621:
1590:
1587:
1586:
1564:
1561:
1560:
1535:
1526:
1522:
1508:
1499:
1494:
1488:
1485:
1484:
1451:
1442:
1438:
1424:
1409:
1404:
1398:
1395:
1394:
1369:
1360:
1356:
1342:
1333:
1328:
1322:
1319:
1318:
1288:
1285:
1284:
1256:
1247:
1243:
1229:
1220:
1215:
1209:
1206:
1205:
1201:In particular:
1159:
1155:
1120:
1115:
1109:
1106:
1105:
1067:
1063:
1034:
1029:
1010:
1006:
977:
973:
967:
963:
961:
958:
957:
923:
919:
890:
886:
884:
881:
880:
822:
818:
784:
781:
780:
753:
749:
705:
701:
696:
693:
692:
661:
657:
648:
644:
615:
611:
609:
606:
605:
590:
578:
546:
542:
513:
509:
507:
504:
503:
477:
468:
457:
451:
448:
433:
417:
406:
381:
377:
365:
361:
355:
351:
349:
346:
345:
317:
313:
307:
303:
301:
298:
297:
273:
270:
269:
249:
245:
233:
229:
223:
219:
217:
214:
213:
193:
189:
187:
184:
183:
155:
151:
149:
146:
145:
119:epistemic logic
107:set-theoretical
75:
71:
69:
66:
65:
26:for a group of
17:
12:
11:
5:
5449:
5439:
5438:
5433:
5428:
5423:
5418:
5413:
5408:
5391:
5390:
5388:
5387:
5382:
5377:
5372:
5367:
5362:
5357:
5352:
5347:
5342:
5337:
5331:
5329:
5325:
5324:
5322:
5321:
5316:
5311:
5306:
5301:
5296:
5290:
5288:
5284:
5283:
5281:
5280:
5275:
5270:
5265:
5260:
5255:
5250:
5245:
5243:Robert Axelrod
5240:
5235:
5230:
5225:
5220:
5218:Olga Bondareva
5215:
5210:
5208:Melvin Dresher
5205:
5200:
5198:Leonid Hurwicz
5195:
5190:
5185:
5180:
5175:
5170:
5165:
5160:
5155:
5150:
5145:
5140:
5135:
5133:Harold W. Kuhn
5130:
5125:
5123:Drew Fudenberg
5120:
5115:
5113:David M. Kreps
5110:
5105:
5100:
5098:Claude Shannon
5095:
5090:
5085:
5080:
5074:
5072:
5066:
5065:
5063:
5062:
5057:
5052:
5047:
5042:
5037:
5035:Nash's theorem
5032:
5027:
5022:
5016:
5014:
5010:
5009:
5007:
5006:
5001:
4996:
4991:
4986:
4981:
4976:
4971:
4966:
4961:
4956:
4951:
4946:
4941:
4936:
4931:
4926:
4921:
4916:
4911:
4906:
4901:
4896:
4894:Ultimatum game
4891:
4886:
4881:
4876:
4874:Dollar auction
4871:
4866:
4861:
4859:Centipede game
4856:
4851:
4846:
4841:
4836:
4831:
4826:
4821:
4816:
4814:Infinite chess
4811:
4806:
4800:
4798:
4792:
4791:
4789:
4788:
4783:
4781:Symmetric game
4778:
4773:
4768:
4766:Signaling game
4763:
4761:Screening game
4758:
4753:
4751:Potential game
4748:
4743:
4738:
4730:
4725:
4720:
4715:
4710:
4704:
4702:
4694:
4693:
4691:
4690:
4685:
4680:
4678:Mixed strategy
4675:
4670:
4665:
4660:
4655:
4650:
4645:
4640:
4635:
4630:
4625:
4620:
4615:
4610:
4604:
4602:
4596:
4595:
4593:
4592:
4587:
4582:
4577:
4572:
4567:
4562:
4557:
4555:Risk dominance
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4512:
4507:
4502:
4497:
4492:
4487:
4482:
4477:
4472:
4466:
4464:
4454:
4453:
4451:
4450:
4445:
4440:
4435:
4430:
4425:
4420:
4415:
4410:
4405:
4400:
4398:Graphical game
4395:
4390:
4385:
4380:
4375:
4370:
4365:
4359:
4357:
4353:
4352:
4344:
4343:
4336:
4329:
4321:
4315:
4314:
4289:
4280:Carr, Kareem.
4278:
4273:
4252:
4251:External links
4249:
4248:
4247:
4219:(3): 549â587.
4203:Halpern, J. Y.
4199:
4185:
4171:
4160:
4141:
4134:
4117:
4103:
4086:
4079:
4062:
4052:Using Language
4048:
4041:
4034:Aumann, Robert
4029:
4026:
4023:
4022:
4003:
3988:
3979:
3961:
3960:
3958:
3955:
3954:
3953:
3950:Herbert Gintis
3942:
3925:
3922:
3921:
3920:
3914:
3909:
3904:
3899:
3894:
3888:
3887:
3871:
3868:
3845:
3842:
3781:
3780:
3731:
3729:
3722:
3716:
3713:
3700:
3697:
3694:
3672:
3668:
3645:
3641:
3614:
3610:
3587:
3583:
3579:
3576:
3542:
3538:
3515:
3511:
3483:
3480:
3477:
3474:
3469:
3465:
3459:
3454:
3451:
3448:
3444:
3440:
3437:
3434:
3431:
3428:
3406:
3403:
3400:
3397:
3392:
3388:
3384:
3381:
3378:
3375:
3372:
3369:
3364:
3361:
3358:
3354:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3307:
3303:
3276:
3273:
3270:
3265:
3261:
3255:
3251:
3247:
3244:
3241:
3238:
3235:
3200:
3183:
3180:
3177:
3174:
3171:
3168:
3163:
3159:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3126:
3122:
3095:
3083:
3070:
3053:
3044:
3035:
3026:
3018:
2980:
2977:
2962:
2958:
2954:
2951:
2948:
2945:
2942:
2939:
2911:
2887:
2882:
2878:
2855:
2851:
2820:
2816:
2773:
2769:
2765:
2762:
2759:
2756:
2753:
2750:
2722:
2698:
2693:
2689:
2660:
2636:
2633:
2630:
2608:
2604:
2600:
2597:
2594:
2589:
2585:
2541:
2536:
2532:
2528:
2525:
2520:
2516:
2510:
2506:
2485:
2465:
2460:
2456:
2432:
2412:
2409:
2404:
2400:
2396:
2393:
2390:
2385:
2381:
2360:
2334:
2330:
2309:
2302:
2298:
2292:
2288:
2284:
2281:
2278:
2275:
2270:
2266:
2262:
2257:
2253:
2249:
2246:
2243:
2240:
2237:
2232:
2228:
2206:, whereas the
2200:
2199:
2188:
2183:
2179:
2173:
2168:
2165:
2162:
2158:
2154:
2151:
2148:
2125:
2122:
2119:
2114:
2109:
2105:
2084:
2079:
2074:
2070:
2049:
2044:
2041:
2038:
2033:
2029:
2023:
2019:
2007:
2006:
1995:
1992:
1987:
1983:
1977:
1974:
1971:
1967:
1963:
1960:
1955:
1951:
1930:
1916:
1892:
1871:
1860:
1841:
1811:
1808:
1805:
1804:
1755:
1753:
1746:
1740:
1737:
1736:
1735:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1685:
1682:
1679:
1675:
1671:
1666:
1662:
1658:
1655:
1652:
1648:
1644:
1639:
1634:
1630:
1618:
1606:
1603:
1600:
1597:
1594:
1574:
1571:
1568:
1548:
1545:
1542:
1538:
1534:
1529:
1525:
1521:
1518:
1515:
1511:
1507:
1502:
1497:
1493:
1482:
1470:
1467:
1464:
1461:
1458:
1454:
1450:
1445:
1441:
1437:
1434:
1431:
1427:
1423:
1418:
1415:
1412:
1407:
1403:
1382:
1379:
1376:
1372:
1368:
1363:
1359:
1355:
1352:
1349:
1345:
1341:
1336:
1331:
1327:
1316:
1304:
1301:
1298:
1295:
1292:
1269:
1266:
1263:
1259:
1255:
1250:
1246:
1242:
1239:
1236:
1232:
1228:
1223:
1218:
1214:
1170:
1167:
1162:
1158:
1154:
1151:
1148:
1144:
1140:
1137:
1134:
1129:
1126:
1123:
1118:
1114:
1078:
1075:
1070:
1066:
1062:
1059:
1056:
1052:
1048:
1045:
1042:
1037:
1032:
1028:
1024:
1021:
1018:
1013:
1009:
1005:
1002:
999:
995:
991:
988:
985:
980:
976:
970:
966:
934:
931:
926:
922:
918:
915:
912:
908:
904:
901:
898:
893:
889:
868:, but instead
830:
825:
821:
817:
814:
811:
808:
805:
802:
799:
795:
791:
788:
764:
761:
756:
752:
748:
745:
742:
739:
736:
733:
730:
727:
723:
719:
716:
713:
708:
704:
700:
672:
669:
664:
660:
656:
651:
647:
643:
640:
637:
633:
629:
626:
623:
618:
614:
589:
586:
577:
574:
557:
554:
549:
545:
541:
538:
535:
531:
527:
524:
521:
516:
512:
476:
473:
470:
469:
420:
418:
411:
405:
402:
389:
384:
380:
376:
373:
368:
364:
358:
354:
325:
320:
316:
310:
306:
277:
257:
252:
248:
244:
241:
236:
232:
226:
222:
201:
196:
192:
163:
158:
154:
83:
78:
74:
15:
9:
6:
4:
3:
2:
5448:
5437:
5434:
5432:
5429:
5427:
5424:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5403:
5401:
5386:
5383:
5381:
5378:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5332:
5330:
5328:Miscellaneous
5326:
5320:
5317:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5295:
5292:
5291:
5289:
5285:
5279:
5276:
5274:
5271:
5269:
5266:
5264:
5263:Samuel Bowles
5261:
5259:
5258:Roger Myerson
5256:
5254:
5251:
5249:
5248:Robert Aumann
5246:
5244:
5241:
5239:
5236:
5234:
5231:
5229:
5226:
5224:
5221:
5219:
5216:
5214:
5211:
5209:
5206:
5204:
5203:Lloyd Shapley
5201:
5199:
5196:
5194:
5191:
5189:
5188:Kenneth Arrow
5186:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5168:John Harsanyi
5166:
5164:
5161:
5159:
5156:
5154:
5151:
5149:
5146:
5144:
5141:
5139:
5138:Herbert Simon
5136:
5134:
5131:
5129:
5126:
5124:
5121:
5119:
5116:
5114:
5111:
5109:
5106:
5104:
5101:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5076:
5075:
5073:
5067:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5017:
5015:
5011:
5005:
5002:
5000:
4997:
4995:
4992:
4990:
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4940:
4939:Fair division
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4914:Dictator game
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4801:
4799:
4797:
4793:
4787:
4786:Zero-sum game
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4756:Repeated game
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4735:
4731:
4729:
4726:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4706:
4705:
4703:
4701:
4695:
4689:
4686:
4684:
4681:
4679:
4676:
4674:
4673:Pure strategy
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4649:
4646:
4644:
4641:
4639:
4636:
4634:
4633:De-escalation
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4605:
4603:
4601:
4597:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4575:Shapley value
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4471:
4468:
4467:
4465:
4463:
4459:
4455:
4449:
4446:
4444:
4443:Succinct game
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4360:
4358:
4354:
4350:
4342:
4337:
4335:
4330:
4328:
4323:
4322:
4319:
4312:
4308:
4305:
4301:
4297:
4294:
4290:
4287:
4283:
4279:
4277:
4274:
4270:
4269:
4264:
4260:
4255:
4254:
4244:
4240:
4236:
4232:
4227:
4222:
4218:
4214:
4213:
4208:
4204:
4200:
4198:
4197:0-691-14052-9
4194:
4190:
4186:
4184:
4183:0-691-14051-0
4180:
4176:
4172:
4169:
4163:
4157:
4153:
4149:
4148:
4142:
4137:
4131:
4127:
4123:
4118:
4116:
4115:0-521-46014-X
4112:
4108:
4104:
4102:
4101:0-631-23257-5
4098:
4094:
4090:
4087:
4082:
4076:
4072:
4069:. Cambridge:
4068:
4063:
4061:
4060:0-521-56745-9
4057:
4053:
4049:
4046:
4042:
4039:
4035:
4032:
4031:
4019:
4018:
4013:
4007:
3999:
3998:Math Hysteria
3992:
3983:
3976:
3972:
3966:
3962:
3951:
3946:
3943:
3940:
3936:
3931:
3928:
3927:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3898:
3895:
3893:
3890:
3889:
3885:
3874:
3867:
3864:
3860:
3856:
3851:
3841:
3839:
3838:Steven Pinker
3836:
3834:
3828:
3826:
3820:
3818:
3815:
3811:
3806:
3803:
3799:
3795:
3791:
3790:Robert Aumann
3787:
3777:
3774:
3766:
3756:
3752:
3748:
3742:
3741:
3737:
3732:This section
3730:
3726:
3721:
3720:
3712:
3698:
3695:
3692:
3670:
3666:
3643:
3639:
3630:
3612:
3608:
3585:
3581:
3577:
3574:
3566:
3562:
3558:
3540:
3536:
3513:
3509:
3500:
3494:
3481:
3475:
3467:
3463:
3452:
3449:
3446:
3442:
3438:
3432:
3426:
3418:
3398:
3390:
3386:
3379:
3376:
3370:
3362:
3359:
3356:
3352:
3328:
3322:
3319:
3313:
3305:
3301:
3292:
3287:
3271:
3263:
3259:
3253:
3249:
3245:
3239:
3233:
3225:
3223:
3218:
3216:
3212:
3208:
3203:
3199:
3194:
3178:
3175:
3169:
3161:
3157:
3153:
3150:
3147:
3144:
3138:
3132:
3124:
3120:
3111:
3109:
3104:
3102:
3098:
3091:
3086:
3082:
3078:
3073:
3069:
3065:
3061:
3056:
3052:
3047:
3043:
3038:
3034:
3029:
3025:
3021:
3014:
3010:
3006:
3002:
2998:
2994:
2990:
2989:Robert Aumann
2986:
2976:
2960:
2956:
2952:
2946:
2943:
2940:
2929:
2925:
2909:
2901:
2885:
2880:
2876:
2853:
2849:
2840:
2836:
2818:
2814:
2805:
2802:(modal axiom
2801:
2797:
2794:(modal axiom
2793:
2789:
2771:
2767:
2763:
2757:
2754:
2751:
2740:
2736:
2720:
2712:
2696:
2691:
2687:
2678:
2674:
2658:
2650:
2634:
2631:
2628:
2621:, defined on
2606:
2602:
2598:
2595:
2592:
2587:
2583:
2575:
2572:
2568:
2564:
2563:
2558:
2553:
2539:
2534:
2530:
2523:
2518:
2514:
2508:
2504:
2483:
2463:
2458:
2454:
2444:
2430:
2407:
2402:
2398:
2394:
2391:
2383:
2379:
2358:
2350:
2332:
2307:
2300:
2290:
2286:
2282:
2273:
2268:
2264:
2255:
2251:
2247:
2244:
2238:
2235:
2230:
2226:
2217:
2213:
2209:
2205:
2186:
2181:
2177:
2166:
2163:
2160:
2156:
2149:
2146:
2139:
2138:
2137:
2123:
2120:
2117:
2112:
2107:
2103:
2095:and defining
2082:
2077:
2072:
2068:
2047:
2042:
2039:
2036:
2031:
2027:
2021:
2017:
1993:
1990:
1985:
1981:
1975:
1972:
1969:
1965:
1958:
1953:
1949:
1941:
1940:
1939:
1937:
1933:
1914:
1906:
1890:
1882:
1869:
1857:knows." Thus
1856:
1852:
1848:
1844:
1837:
1833:
1829:
1825:
1821:
1820:epistemically
1817:
1801:
1798:
1790:
1780:
1776:
1772:
1766:
1765:
1761:
1756:This section
1754:
1750:
1745:
1744:
1739:Formalization
1721:
1718:
1715:
1712:
1706:
1703:
1680:
1677:
1664:
1660:
1656:
1650:
1637:
1632:
1628:
1619:
1604:
1601:
1598:
1595:
1592:
1572:
1569:
1566:
1543:
1540:
1527:
1523:
1519:
1513:
1500:
1495:
1491:
1483:
1465:
1462:
1459:
1456:
1443:
1439:
1435:
1429:
1416:
1413:
1410:
1405:
1401:
1377:
1374:
1361:
1357:
1353:
1347:
1334:
1329:
1325:
1317:
1302:
1299:
1296:
1293:
1290:
1283:
1264:
1261:
1248:
1244:
1240:
1234:
1221:
1216:
1212:
1204:
1203:
1202:
1199:
1196:
1192:
1188:
1184:
1160:
1156:
1152:
1146:
1142:
1138:
1127:
1124:
1121:
1116:
1112:
1103:
1099:
1094:
1092:
1068:
1064:
1060:
1054:
1050:
1046:
1035:
1030:
1026:
1022:
1011:
1007:
1003:
997:
993:
989:
978:
974:
968:
964:
955:
950:
948:
924:
920:
916:
910:
906:
902:
891:
887:
878:
873:
871:
867:
863:
858:
856:
852:
848:
844:
823:
819:
815:
806:
803:
800:
793:
789:
778:
754:
750:
746:
734:
731:
728:
721:
717:
706:
702:
690:
686:
662:
658:
654:
649:
645:
635:
631:
627:
616:
612:
603:
599:
595:
585:
583:
573:
571:
547:
543:
539:
533:
529:
525:
514:
510:
501:
496:
493:
488:
486:
482:
466:
463:
455:
445:
441:
437:
431:
430:
426:
421:This section
419:
415:
410:
409:
401:
387:
382:
378:
371:
366:
362:
356:
352:
343:
339:
323:
318:
314:
308:
304:
295:
291:
275:
255:
250:
246:
242:
239:
234:
230:
224:
220:
199:
194:
190:
181:
177:
161:
156:
152:
143:
139:
135:
130:
128:
124:
120:
116:
112:
111:Robert Aumann
109:framework by
108:
104:
101:in his study
100:
95:
81:
76:
72:
63:
62:
57:
53:
49:
45:
41:
37:
33:
29:
25:
21:
5233:Peyton Young
5228:Paul Milgrom
5143:Hervé Moulin
5083:Amos Tversky
5025:Folk theorem
4736:-player game
4733:
4653:Grim trigger
4266:
4216:
4210:
4188:
4174:
4150:. New York:
4146:
4121:
4106:
4092:
4089:Lewis, David
4066:
4051:
4045:Econometrica
4037:
4015:
4011:
4006:
3997:
3991:
3982:
3974:
3965:
3945:
3938:
3934:
3930:
3862:
3858:
3854:
3847:
3831:
3829:
3824:
3821:
3807:
3788:
3784:
3769:
3760:
3745:Please help
3733:
3715:Applications
3628:
3564:
3560:
3556:
3498:
3495:
3419:
3290:
3288:
3226:
3221:
3219:
3214:
3210:
3206:
3201:
3197:
3195:
3112:
3107:
3105:
3100:
3093:
3089:
3084:
3080:
3076:
3071:
3067:
3063:
3059:
3054:
3050:
3045:
3041:
3036:
3032:
3027:
3023:
3016:
3012:
3004:
3000:
2996:
2995:. An event
2992:
2982:
2927:
2923:
2899:
2838:
2834:
2803:
2795:
2787:
2738:
2734:
2710:
2671:assigning a
2648:
2573:
2570:
2566:
2560:
2556:
2554:
2445:
2349:Aleph-naught
2215:
2203:
2201:
2008:
1935:
1928:
1904:
1858:
1854:
1850:
1846:
1839:
1835:
1831:
1827:
1813:
1793:
1784:
1769:Please help
1757:
1200:
1194:
1190:
1186:
1182:
1101:
1097:
1095:
1090:
953:
951:
946:
876:
874:
865:
861:
859:
854:
850:
846:
842:
776:
688:
684:
601:
597:
593:
591:
581:
579:
497:
491:
489:
478:
458:
449:
434:Please help
422:
341:
337:
293:
289:
179:
137:
131:
102:
96:
61:ad infinitum
59:
58:, and so on
55:
51:
47:
43:
39:
35:
31:
19:
18:
5406:Game theory
5350:Coopetition
5153:Jean Tirole
5148:John Conway
5128:Eric Maskin
4924:Blotto game
4909:Pirate game
4718:Global game
4688:Tit for tat
4618:Bid shading
4608:Appeasement
4458:Equilibrium
4438:Solved game
4373:Determinacy
4356:Definitions
4349:game theory
4313:(Sept 2002)
3892:Global game
3810:game theory
3099:containing
3007:, define a
2922:is true at
2733:is true at
2673:truth value
2216:fixed-point
127:John Conway
30:. There is
5400:Categories
4994:Trust game
4979:Kuhn poker
4643:Escalation
4638:Deterrence
4628:Cheap talk
4600:Strategies
4418:Preference
4347:Topics of
4226:cs/0006009
3957:References
3817:strategies
3763:March 2022
3567:such that
3293:function,
2985:set theory
2930:such that
2741:such that
1787:March 2022
452:March 2022
103:Convention
5178:John Nash
4884:Stag hunt
4623:Collusion
4207:Moses, Y.
4071:MIT Press
3907:Stag hunt
3734:does not
3696:∈
3578:∈
3458:∞
3443:⋂
3250:⋂
3196:That is,
3176:⊂
3154:∣
3148:∈
3009:partition
2953:∈
2910:φ
2886:φ
2806:) of the
2792:reflexive
2764:∈
2721:φ
2697:φ
2659:π
2632:×
2596:…
2557:syntactic
2540:φ
2527:⇒
2524:φ
2484:φ
2464:φ
2431:φ
2408:φ
2395:∧
2392:φ
2371:implying
2359:ψ
2329:ℵ
2308:φ
2297:ℵ
2274:φ
2248:∧
2245:φ
2236:φ
2187:φ
2172:∞
2157:⋀
2153:⇔
2150:φ
2124:φ
2118:φ
2083:φ
2048:φ
2040:−
1991:φ
1973:∈
1966:⋀
1962:⇔
1959:φ
1915:φ
1891:φ
1870:φ
1834:, and of
1758:does not
1710:∞
1678:≥
1570:≥
1541:≥
1457:≥
1414:−
1375:≥
1300:≤
1262:≥
1143:∈
1136:∃
1125:−
1051:∈
1044:∃
994:∈
987:∃
907:∈
900:∃
804:−
794:∈
787:∃
744:¬
732:−
722:∈
715:∀
699:¬
642:¬
632:∈
625:∀
604:as well (
530:∈
523:∃
423:does not
375:⇒
24:knowledge
5319:Lazy SMP
5013:Theorems
4964:Deadlock
4819:Checkers
4700:of games
4462:concepts
4307:Archived
4296:Archived
4243:52151232
3870:See also
3685:for all
3600:, where
3062:, agent
2320:. Here,
2204:finitary
576:Solution
243:⇏
113:(1976).
5071:figures
4854:Chicken
4708:Auction
4698:Classes
4265:(ed.).
4091:(1969)
4020:(1996).
4012:Meaning
3855:doesn't
3755:removed
3740:sources
2926:states
2737:states
2347:is the
1883:(where
1779:removed
1764:sources
777:another
444:removed
429:sources
404:Example
138:Meaning
123:puzzles
4241:
4195:
4181:
4158:
4132:
4113:
4099:
4077:
4058:
4000:. OUP.
2798:) and
2790:, the
1907:knows
1845:(with
1832:agents
483:(e.g.
475:Puzzle
296:(viz.
176:public
28:agents
4809:Chess
4796:Games
4261:. In
4239:S2CID
4221:arXiv
3924:Notes
2555:This
2208:axiom
2060:with
1091:third
775:, so
588:Proof
46:know
4485:Core
4193:ISBN
4179:ISBN
4156:ISBN
4130:ISBN
4111:ISBN
4097:ISBN
4075:ISBN
4056:ISBN
3863:does
3738:any
3736:cite
3557:true
3344:and
3049:and
3031:and
2902:iff
2713:iff
2552:).
1762:any
1760:cite
1696:for
1602:>
1559:for
952:For
875:For
860:For
427:any
425:cite
5069:Key
4231:doi
3859:not
3749:by
3224:".
3011:on
2924:all
2837:in
2735:all
1830:of
1773:by
1282:iff
947:not
685:and
487:):
438:by
182:" (
144:" (
34:of
5402::
4804:Go
4302:,
4284:,
4237:.
4229:.
4217:37
4215:.
4205:;
4154:.
4128:.
4124:.
4073:.
3973:.
3819:.
3217:.
3015:,
2975:.
2569:,
2443:.
872:.
400:.
340:,
129:.
94:.
4734:n
4340:e
4333:t
4326:v
4271:.
4245:.
4233::
4223::
4170:.
4164:.
4138:.
4085:.
4083:.
3835:,
3776:)
3770:(
3765:)
3761:(
3757:.
3743:.
3699:G
3693:i
3671:i
3667:P
3644:G
3640:R
3629:p
3613:p
3609:E
3586:p
3582:E
3575:s
3565:s
3561:p
3541:i
3537:P
3514:i
3510:R
3499:S
3482:.
3479:)
3476:e
3473:(
3468:n
3464:E
3453:1
3450:=
3447:n
3439:=
3436:)
3433:e
3430:(
3427:C
3405:)
3402:)
3399:e
3396:(
3391:n
3387:E
3383:(
3380:E
3377:=
3374:)
3371:e
3368:(
3363:1
3360:+
3357:n
3353:E
3332:)
3329:e
3326:(
3323:E
3320:=
3317:)
3314:e
3311:(
3306:1
3302:E
3291:E
3275:)
3272:e
3269:(
3264:i
3260:K
3254:i
3246:=
3243:)
3240:e
3237:(
3234:E
3222:e
3215:e
3211:e
3207:e
3205:(
3202:i
3198:K
3182:}
3179:e
3173:)
3170:s
3167:(
3162:i
3158:P
3151:S
3145:s
3142:{
3139:=
3136:)
3133:e
3130:(
3125:i
3121:K
3108:K
3101:s
3096:i
3094:P
3090:s
3088:(
3085:i
3081:P
3077:s
3075:(
3072:i
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3064:i
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3055:2
3051:s
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3042:s
3037:2
3033:s
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3024:s
3019:i
3017:P
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3005:i
3001:S
2997:E
2993:S
2961:G
2957:R
2950:)
2947:t
2944:,
2941:s
2938:(
2928:t
2900:s
2881:G
2877:C
2854:G
2850:R
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2835:i
2819:i
2815:R
2804:4
2796:T
2788:G
2772:i
2768:R
2761:)
2758:t
2755:,
2752:s
2749:(
2739:t
2711:s
2692:i
2688:K
2649:i
2635:S
2629:S
2607:n
2603:R
2599:,
2593:,
2588:1
2584:R
2571:n
2567:S
2535:G
2531:C
2519:G
2515:E
2509:G
2505:C
2459:G
2455:E
2411:)
2403:G
2399:C
2389:(
2384:G
2380:E
2333:0
2301:0
2291:G
2287:E
2283:=
2280:]
2277:)
2269:G
2265:C
2261:(
2256:G
2252:E
2242:[
2239:=
2231:G
2227:C
2182:i
2178:E
2167:0
2164:=
2161:i
2147:C
2121:=
2113:0
2108:G
2104:E
2078:n
2073:G
2069:E
2043:1
2037:n
2032:G
2028:E
2022:G
2018:E
1994:,
1986:i
1982:K
1976:G
1970:i
1954:G
1950:E
1936:G
1931:G
1929:E
1905:i
1861:i
1859:K
1855:i
1851:n
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1840:K
1836:n
1828:G
1800:)
1794:(
1789:)
1785:(
1781:.
1767:.
1734:.
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1719:=
1716:j
1713:,
1707:=
1704:i
1684:]
1681:j
1674:|
1670:)
1665:x
1661:l
1657:B
1654:(
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1647:|
1643:[
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1605:k
1599:j
1596:+
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1573:k
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1547:]
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1533:)
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1520:B
1517:(
1514:G
1510:|
1506:[
1501:i
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1492:E
1481:.
1469:]
1466:1
1463:+
1460:j
1453:|
1449:)
1444:x
1440:l
1436:B
1433:(
1430:G
1426:|
1422:[
1417:1
1411:i
1406:G
1402:E
1381:]
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1371:|
1367:)
1362:x
1358:l
1354:B
1351:(
1348:G
1344:|
1340:[
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1303:k
1297:j
1294:+
1291:i
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1249:x
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1238:(
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1227:[
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1169:]
1166:)
1161:x
1157:l
1153:B
1150:(
1147:G
1139:x
1133:[
1128:1
1122:k
1117:G
1113:E
1102:k
1098:k
1077:]
1074:)
1069:x
1065:l
1061:B
1058:(
1055:G
1047:x
1041:[
1036:2
1031:G
1027:E
1023:=
1020:]
1017:)
1012:x
1008:l
1004:B
1001:(
998:G
990:x
984:[
979:G
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969:G
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933:]
930:)
925:x
921:l
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914:(
911:G
903:x
897:[
892:G
888:E
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847:k
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816:B
813:(
810:)
807:a
801:G
798:(
790:x
763:]
760:)
755:x
751:l
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741:(
738:)
735:a
729:G
726:(
718:x
712:[
707:a
703:K
689:k
671:]
668:)
663:x
659:l
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