09 StaticIncompleteInfoGames2(1).pdf

09 StaticIncompleteInfoGames2(1).pdf - Lecture 9 Static...

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Lecture 9: Static games with incomplete information II October 17, 2016 1 Recap How to nd the Bayesian Nash equilibrium if one or more players' type is uncertain? Construct a common prior belief about the players' possible types A player chooses an optimal strategy for each of his types to maximize expected utility 2 Application: Strategic voting 2.1 Set up: three voters A jury is made up of three jurors Jurors simultaneously vote to decide whether to acquit (A) or convict (C) a defendant Whether the defendant is guilty (G) or innocent (I) is uncertain. Without any addi- tional information, the jurors have a common prior belief that the defendant is guilty with probability 0.6. The jurors have identical payo functions: u ( convict the innocent ) = u ( acquit the guilty ) = 0 , u ( convict the guilty ) = u ( acquit the innocent ) = 1 . This means that each juror prefers to convict the defendant if and only if Pr( G ) 1 2 : EU ( convict ) = u ( convict the guilty ) · Pr( G ) + u ( convict the innocent ) · Pr( I ) = Pr( G ) ( acquit ) = u ( acquit the guilty ) · Pr( G ) + u ( acquit the innocent ) · Pr( I ) = Pr( I ) ( convict ) ( acquit ) Pr( G ) Pr( I ) Pr( G ) 1 2 1
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Since the common prior belief is Pr( G ) = 0 . 6 , if no additional evidence is presented, each juror's default action is to convict. Now suppose that some evidence is brought to court. Imagine that each juror has a di erent expertise and, when observing the evidence, gets a private signal s i ∈ { s I ,s G } . Think of the signal as a juror's personal judgement that suggests whether I or G is more likely. Assume that the private signals are i.i.d. (indenpendently and identically distributed) with the following conditional distribution: Pr( s I | I ) = Pr( s G | G ) = 0 . 7 Pr( s I | G ) = Pr( s G | I ) = 0 . 3 In other words, a juror's private signal correctly re ects the true state 70% of the time. 2.2 Unanimous rule vs majority rule If there were only one juror , he will vote for convict if and only if his private signal is s G . To see why: The Bayes' rule state that since Pr( A and B ) = Pr( A | B ) Pr( B ) = Pr( B | A ) Pr( A ) , Pr( A | B ) = Pr( B | A ) Pr( A ) Pr( B ) . Applying the Bayes' rule, we know that Pr( G | s G ) = Pr( s G | G ) Pr( G ) Pr( s G ) = Pr( s G | G ) Pr( G ) Pr( s G | G ) Pr( G ) + Pr( s G | I ) Pr( I ) = 0 . 7 × 0 . 6 0 . 7 × 0 . 6 + 0 . 3 × 0 . 4 = 0 . 778 > 1 2 Therefore, the unique juror strictly prefers to convict when his signal is s G . Intuitively, a private signal s G strengthens the juror's prior belief that G is more likely, making the juror more con dent that he should vote for conviction.
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