How to dynamic games of incomplete info Flashcards
Find all NE, SPNE, PBNE
Write up the strategies
Write the extensive game in the normal form to find PSNE.
Find SPNE
- Find proper subgames, if proper subgames due backwards induction.
- If no proper subgames, then PSNE=SPNE
- Notice by backwards induction, which PSNE is ON and OFF EP. You need that for later.
Show for which probability the player who blindly moves would chose the needed strategy for the given PBE.
- We calculate the payoffs of from the those strategy nodes, which have a cut of information.
Conclude on the requirements for PBE:
Req2: Write up the belief that satisfy PBE.
Req3: If PBE is ON the EP update the belief to 1, so we are playing that strategy 100 procent.
Req4: Bayes rule, not applied
Then write up the PBE
How to find a mixed PBNE
We need to show that there is no PS (perfect strategy) PBE, by showing that there is no NE in the normal form.
If there is no NE, then there is no PS PBNE, so we have to look for mixed.
Seperating strategy
SR1: S´s strategy (if two players, you would have 2 strategies to look into)
SR3/4: When two players we set p=1 and q=0 in the first case and p=0 and q=1 in the second case.
SR2R: Find BR for R given S´s BR. e.g. x|A and y|B
SR2S: check if t_1, want to deviate given R´s BR.
–> no, if the payoff given BR is higher then what S will get from deviating, vice versa.
check if t_2.
–> no, if the payoff given BR is higher then what S will get from deviating, vice versa.
PBNE: (S´s strategy) (R´s actions), given what p and q is.
Separating strategy
SR1: S´s strategy (if two players, you would have 2 strategies to look into)
SR3/4: When two players we set p=1 and q=0 in the first case and p=0 and q=1 in the second case.
SR2R: Find BR for R given S´s BR. e.g. x|A and y|B
SR2S: check if t_1, want to deviate given R´s BR.
–> no, if the payoff given BR is higher then what S will get from deviating, vice versa.
check if t_2.
–> no, if the payoff given BR is higher then what S will get from deviating, vice versa.
PBNE: (S´s strategy) (R´s actions), given what p and q is.
Pooling strategy
SR1: S´s strategy (if two players, you would have 2 strategies to look into)
SR3/4: When two players we set p=1/2 and q\epsilon[0,1], vice versa. In a pooling game we need to have 50/50 procent of t_1 or t_2.
SR2R: Find BR for R given S´s BR. Here we can only find the one side BR, e.g. y|B. Check it by looking into the which of the R´s choices that yields the highest expected payoff, that would be R´s BR to S strategy.
SR2S: check if t_1, want to deviate given R´s BR.
–> no, if the payoff given S BR to R´s BR is higher then what he will get from deviating, vice versa. If the payoff given S BR to R´s BR yields a lower payoff the one of the alternatives, then calculate for which procent he will choose the option that supports a PBNE.
–> easy yes, if the the other strategy of S is strictly dominating the strategy we look at in the pooling problem.
check if t_2.
–> same her.
If PBNE then : (S´s strategy) (R´s actions), given what p and q is.
How to do a cheap talk separating game.
Same procedure as signaling.
But conclude whether it’s informative or not. If the types are indifferent between signaling then the separating PBNE is weak.
How to do a cheap talk pooling game
Same procedure as signaling. AND it is always possible to get a pooling game in cheap talk, because the the payoff in both sides are symmetric.
How to do a semi-pooling equilibrium
SR1: write up the strategy, but more argumented her than normal.
- Normally there would be a strictly dominating strategy, which would limit the amount of signals for one of the types.
SR3/4: here we assign e.g. q and 1-q to the same type, where we before did opposite types. no p before later.
Computing a special updating beliefs formula, where we use natures probabilities.
The in the end we would look at the case if the type, who is not strictly dominated wanted to deviate.
