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Rachel E. Kranton on Communication Equilibria - Dictionary of Arguments

Kranton I 421
Communication Equilibria/Bloch/Demange/Kranton:
Full Communication Equilibrium/FCE: (…) a full communication equilibrium (FCE) [is a state], where all biased and unbiased agents transmit messages and, therefore, spread possibly false rumors. They do so because there is a sufficiently large probability the rumor is true. The equilibrium conditions rely on the number and distribution of biased and unbiased agents in the population. In a network, for any agent, the set of possible senders of a message must contain sufficiently few biased agents.
Kranton I 428
Example: Consider five agents in a line, (…), where the links allow communication in either direction, with four unbiased agents and one biased agent in the middle. Assume that unbiased agents create true messages upon receiving a signal and transmit any message they receive. Can these strategies form an equilibrium? [Yes], strategies in which all unbiased agents transmit all messages form a [full communication] equilibrium [FCE].
Kranton I 429
Strategies: Upon receipt of the signal, every biased agent i creates a message that matches her
bias, that is, Mi(s) = 1. Every biased agent only transmits a message if the message is 1; that is,
ti(0) = ∅, ti(1) = 1. Every unbiased agent i creates a true message upon receiving a signal; that
is, Mi(s) = s, and transmits any received message, that is, ti(m) = m.
Beliefs: (1) For an agent i who has received a message m = 0 from an unbiased neighbor j, ρi(0(j )) = 0. (2) For an agent i who has received a message m = 1.
These strategies and beliefs constitute an equilibrium of the network game when the posterior
belief of an agent receiving message m(j) = 1 is willing to pass it on; that is, when ρi(1(j )) ≥ 1/2.
Kranton I 430
Conclusion: Roughly (….), an FCE exists if biased agents are few in number and dispersed through the network.
Kranton I 422
Maximal Communication Equilibrium/MCE/Bloch/Demange/Kranton: [If the condition for the] full communication [fails], there is an equilibrium, called maximal communication equilibrium (MCE), in which communication is maximized: In any equilibrium, information flows on an edge only if it flows in this MCE. A main feature of this equilibrium is that information can flow from one part of the network to another but not in the reverse direction. Unbiased agents maintain the credibility of messages by blocking those that come from a part of the network that contains too many biased agents
Kranton I 432
[thus] limiting the influence of localized biased agents.
Kranton I 422
This same agent, however, will transmit messages coming from another direction. These MCEs yield the highest expected payoffs of all perfect Bayesian equilibria of the game.
Kranton I 432
In particular, two unbiased agents always communicate to each other in an MCE (…).
Kranton I 431
Strategies: Biased agents, upon receipt of the signal, create a message that matches their bias, that is, M(s) = 1. Biased agents only transmit messages that match their bias, that is, t(0) = Ø , (Ø = message blocked) t(1) = 1. Unbiased agents, upon receipt of a signal, create true messages; that is, M(s) = s. All unbiased agents i transmit message 1 received from agent j if (j, i) ∈ G∗; (>Terminology/Kranton) otherwise agent i
Kranton I 432
does not transmit the message. All unbiased agents transmit messages m = 0 received from any agent.
Beliefs: The only event for which beliefs need to be specified is when an agent receives a message 0 from a biased agent. As previously, we suppose i’s posterior belief is equal to his prior in this case.
Conclusion: The above strategies and beliefs form an equilibrium of the network game. We call this equilibrium the “MCE” as communication is maximal among all equilibria in the following sense: In any equilibrium, if (j, i) / ∈ G∗ (equivalently (j, i) ∈ W), then j is biased and i does not transmit m = 1 received from j. (>Terminology/Kranton).
Kranton I 434
Agents/Maximal Communication Equilibrium/MCE: (…) biased agents always prefer the MCE to any equilibrium with partial communication and any equilibrium with partial communication to an equilibrium without communication. The expected utility of unbiased agents is more difficult to rank. (…) the expected utility of unbiased agents is the highest in the MCE (i.e., in the FCE when it exists) and is the lowest in an equilibrium without communication.

Francis Bloch, Gabrielle Demange & Rachel Kranton, 2018. "Rumors And Social Networks," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 59(2), pages 421-448.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments
The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

Kranton I
Rachel E. Kranton
Francis Bloch
Gabrielle Demange,
Rumors And Social Networks 2018

Kranton II
Rachel E. Kranton
George A. Akerlof
Identity Economics: How Our Identities Shape Our Work, Wages, and Well-Being Princeton 2011

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