# Aumann's agreement theorem

Aumann's agreement theorem Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree",[1] which introduced the set theoretic description of common knowledge. The theorem concerns agents who share a common prior and update their probabilistic beliefs by Bayes' rule. It states that if the probabilistic beliefs of such agents, regarding a fixed event, are common knowledge then these probabilities must coincide. Thus, agents cannot agree to disagree, that is have common knowledge of a disagreement over the posterior probability of a given event.

Contents 1 The Theorem 2 Extensions 3 Dynamics 4 References The Theorem The model used in Aumann[1] to prove the theorem consists of a finite set of states {displaystyle S} with a prior probability {displaystyle p} , which is common to all agents. Agent {displaystyle a} 's knowledge is given by a partition {displaystyle Pi _{a}} of {displaystyle S} . The posterior probability of agent {displaystyle a} , denoted {displaystyle p_{a}} is the conditional probability of {displaystyle p} given {displaystyle Pi _{a}} . Fix an event {displaystyle E} and let {displaystyle X} be the event that for each {displaystyle a} , {displaystyle p_{a}(E)=x_{a}} . The theorem claims that if the event {displaystyle C(X)} that {displaystyle X} is common knowledge is not empty then all the numbers {displaystyle x_{a}} are the same. The proof follows directly from the definition of common knowledge. The event {displaystyle C(X)} is a union of elements of {displaystyle Pi _{a}} for each {displaystyle a} . Thus, for each {displaystyle a} , {displaystyle p(E|C(x))=x_{a}} . The claim of the theorem follows since the left hand side is independent of {displaystyle a} . The theorem was proved for two agents but the proof for any number of agents is similar.

Extensions Monderer and Samet[2] relaxed the assumption of common knowledge and assumed instead common {displaystyle p} -belief of the posteriors of the agents. They gave an upper bound of the distance between the posteriors {displaystyle x_{a}} . This bound approaches 0 when {displaystyle p} approaches 1.

Ziv Hellman[3] relaxed the assumption of a common prior and assumed instead that the agents have priors that are {displaystyle varepsilon } -close in a well defined metric. He showed that common knowledge of the posteriors in this case implies that they are {displaystyle varepsilon } -close. When {displaystyle varepsilon } goes to zero, Aumann's original theorem is recapitulated.

Nielsen[4] extended the theorem to non-discrete models in which knowledge is described by {displaystyle sigma } -algebras rather than partitions.

Knowledge which is defined in terms of partitions has the property of negative introspection. That is, agents know that they do not know what they do not know. However, it is possible to show that it is impossible to agree to disagree even when knowledge does not have this property. [5] Halpern and Kets[6] argued that players can agree to disagree in the presence of ambiguity, even if there is a common prior. However, allowing for ambiguity is more restrictive than assuming heterogeneous priors.

The impossibility of agreeing to disagree, in Aumann's theorem, is a necessary condition for the existence of a common prior. A stronger condition can be formulated in terms of bets. A bet is a set of random variables {displaystyle f_{a}} , one for each agent {displaystyle a} , such the {displaystyle sum _{a}f_{a}=0} . The bet is favorable to agent {displaystyle a} in a state {displaystyle s} if the expected value of {displaystyle f_{a}} at {displaystyle s} is positive. The impossibility of agreeing on the profitability of a bet is a stronger condition than the impossibility of agreeing to disagree, and moreover, it is a necessary and sufficient condition for the existence of a common prior. ,[7] [8] Dynamics A dialogue between two agents is a dynamic process in which, in each stage, the agents tell each other their posteriors of a given event {displaystyle E} . Upon gaining this new information, each is updating her posterior of {displaystyle E} . Aumann[1] suggested that such a process leads the agents to commonly know their posteriors, and hence, by the agreement theorem, the posteriors at the end of the process coincide. Geanakoplos and Polemarchakis [9] proved it for dialogues in finite state spaces. Polemarchakis[10] showed that any pair of finite sequences of the same length that end with the same number can be obtained as a dialogue. In contrast, Di Tillio et al[11] showed that infinite dialogues must satisfy certain restrictions on their variation. Scott Aaronson[12] studied the complexity and rate of convergence of various types of dialogues with more than two agents.

References ^ Jump up to: a b c Aumann, Robert J. (1976). "Agreeing to Disagree" (PDF). The Annals of Statistics. 4 (6): 1236–1239. doi:10.1214/aos/1176343654. ISSN 0090-5364. JSTOR 2958591. ^ Monderer, dov; Dov Samet (1989). "Approximating common knowledge with common beliefs". Games and Economic Behavior. 1 (2): 170–190. ^ Hellman, Ziv (2013). "Almost Common Priors". International Journal of Game Theory. 42 (2): 399–410. doi:10.1007/s00182-012-0347-5. ^ Nielsen, Lars Tyge (1984). "Common knowledge, communication, and convergence of beliefs". Mathematical Social Sciences. 8 (1): 1–14. doi:10.1016/0165-4896(84)90057-X. ^ Samet, Dov (1990). "Ignoring ignorance and agreeing to disagree". Journal of Economic Theory. 52 (1): 190–207. doi:10.1016/0022-0531(90)90074-T. ^ Halpern, Joseph; Willemien Kets (2013-10-28). "Ambiguous Language and Consensus" (PDF). Retrieved 2014-01-13. ^ Feinberg, Yossi (2000). "Characterizing Common Priors in the Form of Posteriors". Journal of Economic Theory. 91: 127–179. doi:10.1006/jeth.1999.2592. ^ Samet, Dov (1998). "Common Priors and Separation of Convex Sets". Games and Economic Behavior. 91: 172–174. doi:10.1006/game.1997.0615. ^ Geanakoplos, John D.; Herakles M. Polemarchakis (1982). "We can't disagree forever". Journal of Economic Theory. 28 (1): 1192–200. doi:10.1016/0022-0531(82)90099-0. ^ Polemarchakis, Herakles (2022). "Bayesian dialogs" (PDF). ^ Di Tillio, Alfredo; Ehud Lehrer; Dov Samet (2022). "Monologues, dialogues, and common priors". Theoretical Economics. 17: 587–615. doi:10.3982/TE4508. ^ Aaronson, Scott (2005). The complexity of agreement (PDF). Proceedings of ACM STOC. pp. 634–643. doi:10.1145/1060590.1060686. ISBN 978-1-58113-960-0. Retrieved 2010-08-09. hide vte Topics in game theory Definitions Congestion gameCooperative gameDeterminacyEscalation of commitmentExtensive-form gameFirst-player and second-player winGame complexityGame description languageGraphical gameHierarchy of beliefsInformation setNormal-form gamePreferenceSequential gameSimultaneous gameSimultaneous action selectionSolved gameSuccinct game Equilibrium concepts Bayesian Nash equilibriumBerge equilibriumCoreCorrelated equilibriumEpsilon-equilibriumEvolutionarily stable strategyGibbs equilibriumMertens-stable equilibriumMarkov perfect equilibriumNash equilibriumPareto efficiencyPerfect Bayesian equilibriumProper equilibriumQuantal response equilibriumQuasi-perfect equilibriumRisk dominanceSatisfaction equilibriumSelf-confirming equilibriumSequential equilibriumShapley valueStrong Nash equilibriumSubgame perfectionTrembling hand Strategies Backward inductionBid shadingCollusionForward inductionGrim triggerMarkov strategyDominant strategiesPure strategyMixed strategyStrategy-stealing argumentTit for tat Classes of games Bargaining problemCheap talkGlobal gameIntransitive gameMean-field gameMechanism designn-player gamePerfect informationLarge Poisson gamePotential gameRepeated gameScreening gameSignaling gameStackelberg competitionStrictly determined gameStochastic gameSymmetric gameZero-sum game Games GoChessInfinite chessCheckersTic-tac-toePrisoner's dilemmaGift-exchange gameOptional prisoner's dilemmaTraveler's dilemmaCoordination gameChickenCentipede gameLewis signaling gameVolunteer's dilemmaDollar auctionBattle of the sexesStag huntMatching penniesUltimatum gameRock paper scissorsPirate gameDictator gamePublic goods gameBlotto gameWar of attritionEl Farol Bar problemFair divisionFair cake-cuttingCournot gameDeadlockDiner's dilemmaGuess 2/3 of the averageKuhn pokerNash bargaining gameInduction puzzlesTrust gamePrincess and monster gameRendezvous problem Theorems Arrow's impossibility theoremAumann's agreement theoremFolk theoremMinimax theoremNash's theoremPurification theoremRevelation principleZermelo's theorem Key figures Albert W. TuckerAmos TverskyAntoine Augustin CournotAriel RubinsteinClaude ShannonDaniel KahnemanDavid K. LevineDavid M. KrepsDonald B. GilliesDrew FudenbergEric MaskinHarold W. KuhnHerbert SimonHervé MoulinJohn ConwayJean TiroleJean-François MertensJennifer Tour ChayesJohn HarsanyiJohn Maynard SmithJohn NashJohn von NeumannKenneth ArrowKenneth BinmoreLeonid HurwiczLloyd ShapleyMelvin DresherMerrill M. FloodOlga BondarevaOskar MorgensternPaul MilgromPeyton YoungReinhard SeltenRobert AxelrodRobert AumannRobert B. WilsonRoger MyersonSamuel BowlesSuzanne ScotchmerThomas SchellingWilliam Vickrey Miscellaneous All-pay auctionAlpha–beta pruningBertrand paradoxBounded rationalityCombinatorial game theoryConfrontation analysisCoopetitionEvolutionary game theoryFirst-move advantage in chessGame Description LanguageGame mechanicsGlossary of game theoryList of game theoristsList of games in game theoryNo-win situationSolving chessTopological gameTragedy of the commonsTyranny of small decisions Portals: Business and economics Mathematics Categories: Bayesian statisticsEconomics theoremsGame theoryProbability theoremsRational choice theoryTheorems in statistics

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