Question $5(3$ points$)$

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Can there be a Bayes$-$Nash equilibrium where Bob plays N if type t$1$ and C if type t$2?$

Question $5$ options:

TrueFalse

Question $6(8$ points$)$

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Select all the strategy profiles that constitute a pooling Bayes$-$Nash equilibrium $($i$.$e$.,$ don't select it if it's not an equilibrium or if it's not pooling$).$

In the strategy profiles below the first letter is Alice's strategy, and the second and third are Bob's actions if type t$1$ and type t$2,$ respectively.$$

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$8$ points in total, $1$ point for each correct choice:

$1$ point if you select a strategy profile that is a pooling Bayes$-$Nash equilibrium,$$

$1$ point if you do not select a strategy profile that is not a pooling Bayes$-$Nash equilibrium.

Question $6$ options:

$($C$,$NN$)$

$($C$,$NC$)$

$($C$,$CN$)$

$($C$,$CC$)$

$($N$,$NN$)$

$($N$,$NC$)$

$($N$,$CN$)$

$($N$,$CC$)$

Question $7(8$ points$)$

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Select all the strategy profiles that constitute a separating Bayes$-$Nash equilibrium.$$

$8$ points in total, $1$ point for each correct choice:

$1$ point if you select a strategy profile that is a separating Bayes$-$Nash equilibrium,$$

$1$ point if you do not select a strategy profile that is not a separating Bayes$-$Nash equilibrium.

Select $1$ correct answer$($s$)$

Question $7$ options:

$($C$,$NN$)$

$($C$,$NC$)$

$($C$,$CN$)$

$($C$,$CC$)$

$($N$,$NN$)$

$($N$,$NC$)$

$($N$,$CN$)$

$($N$,$CC$)$ Incomplete information

We consider the following game. Alice and Bob must decide whether they contribute or not to a project. Alice's payoffs are common knowledge, but only Bob knows his payoffs. Bob can be either type t$1$ or type t$2.$

If Bob is type t$1,$ the game is the following $($Alice chooses the row and Bob the column$)$:

If Bob is type t$2,$ the game is the following $($again$,$ Alice chooses the row and Bob the column$)$:

The probability that Bob is type $\backslash (\backslash$mathrm$\{$t$1\}\backslash )$ is $\backslash (\backslash$mathrm$\{$p$\}=0\mathrm{.}75\backslash )$

Question $5(3$ points$)$