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Part 1: Perfect Bayesian Equilibrium
Part 2: Separating and Pooling
Bayes' Rule and Conditional Beliefs
Sequential Rationality and Consistency of Beliefs
Guided exercise from Watson
Suppose you don't know whether it's raining out,
but you can observe whether I'm carrying an umbrella or not.
Ex ante, you believe the joint probabilities of these events
are given by this table:
Bayes' Rule:
Before you see whether I'm carrying an umbrella, with what probability do you believe it's raining?
Suppose you see me with an umbrella. Now with what probability do you think it's raining?
Suppose player 2's payoffs were as shown.
If player 3 comes to the information set, what should their beliefs be?
"Gift Giving Game"
Nature determines whether player 1 is a "friend" or "enemy" to player 2.
Player 1, knowing their type, can decide to give a gift to player 2 or not.
If player 1 gives a gift, player 2 can choose to accept it or not. Player 2 wants to accept a gift from a friend, but not from an enemy.
Whenever a player reaches an information set, they have some updated beliefs over which node they are.
Based on these beliefs, they should choose the action that maximizes their expected payoff.
In equilibrium, players' beliefs should be consistent with the strategies being played.
What is \(q\) if player 1 plays \(G^FN^E\)?
What is \(q\) if player 1 plays \(N^FG^E\)?
What is \(q\) if player 1 plays \(G^FG^E\)?
What is \(q\) if player 1 plays \(N^FN^E\)?
What if there were three types?
Calculating \(q\) in general:
Let \(r^F\) be the probability that
Nature selects F and player 1 selects G.
Let \(r^E\) be the probability that
Nature selects E and player 1 selects G.
Suppose there are 3 types, and two choose to give?
Consider a strategy profile for the players, as well as beliefs over the nodes at all information sets.
(1) each player's strategy specifies optimal actions, given his beliefs
These are called a perfect Bayesian equilibrium (PBE) if:
(2) the beliefs are consistent with Bayes' rule wherever possible
More simply: strategies must be optimal given beliefs, and
beliefs must be consistent with the strategies being played.
A perfect Bayesian equilibrium consists of
strategy profiles for the players
AND
beliefs over nodes at all information sets
SUCH THAT
(1) each player's strategy specifies optimal actions, given their beliefs
(2) beliefs are consistent with Bayes' rule wherever possible
Steps for calculating perfect Bayesian equilibria:
Guided Exercise from Watson (p. 385)
Amy wants Brenda to take her to the mall
(choose Y rather than F)
Brenda only wants to go
if her favorite shoes are on sale
(nature has chosen S rather than N)
Amy has access to a newspaper that shows whether the shows are on sale or not
(knows Nature's move);
she can choose to
take the newspaper (T) or not (D). Taking the newspaper reduces her payoff by 2.
Find the separating and pooling equilibria of the game.