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FROM BAYES RULE TO BAYESIAN DATA SHARING

Yotam Gafni

Moshe Tennenholtz Mechanism Design for Data Science Class, 12/22

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Reminder: Bayes Rule

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Solving Monty Hall with Bayes Rule

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Solving Monty Hall with Bayes Rule

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Designing Signals? [Kamenica & Gentzkow 2011]*

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* The example we revisit here is based on [Dughmi & Xu 2016]

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Full information

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No information

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A third better thing?

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What generalizes here?

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Army’s Posterior probability that student is excellent

Advisor Utility

1

1

½

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What generalizes here? II

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  • When sender’s utility is convex in receiver’s beliefs, full information is the optimal signal.
  • Otherwise:
    • When the receiver takes the sender’s least preferred action, the signal is not ‘mixed’, i.e., the underlying state is such that this action is optimal with certainty.
    • When the receiver takes the sender’s most preferred action, the signal is ‘mixed’

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‘Bayesian Persuasion’

  • This type of signal design problems are called ‘Bayesian Persuasion’
  • “Persuasion is 25% of GDP” [McCloskey and Klamer 1995], “Persuasion is 30% of GDP” [Antioch 2013]
  • What are the special characteristics of this problem formally?
    • Other than the prior, receiver has no information of his own.
    • Sender has full information (knows true realization with certainty)
    • Sender commits to the signaling scheme, and can not ‘cheat’ once it’s fixed.
  • What if information flow is not so one-sided?

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Bayesian Data-Sharing

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0

1

0

0,0 w.p. 1/3

0,1 w.p. 1/3

1

1,0 w.p. 1/6

1,1 w.p. 1/6

Firm A

Firm B

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No Information

Full Information

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Optimal Data Sharing? [Gradwohl & Tennenholtz 2020]

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