Dynamic games of incomplete info

Question 1

What is the difference between imperfect and incomplete information.

How do we show when their is imperfect and incomplete information.

Answer 1

In short, imperfect information concerns uncertainty about actions, whereas incom-
plete information concerns uncertainty about payoffs.

To show when there is imperfect

Question 2

To analyze games with incomplete information, game theorists make use of an idea due
to John Harsanyi. Explain this idea in words.

Answer 2

There are two parts of the idea.

The first part is to introduce a fictitious player called
“Nature”. This player does not act strategically. Instead, Nature chooses the state of the
world (which can determine what the players’ payoffs are) according to an exogenous
probability distribution.

The second part of the idea is to assume that one or more of the players cannot observe
Nature’s choice. This translates the game into one of imperfect information, for which we
have tools available to analyze it.

Question 3

What is the difference of dynamic games of complete and incomplete info

Answer 3

When we studied dynamic games of complete info, we solved for subgame-perfect NE. By doing that we ruled out empty threats.

We will also rule out empty threats in games of incomplete info, but we can not use subgame-perfect NE to solve it, due to the incomplete information reduces the amout of subgames.

Question 4

How do we solve dynamic games when they are incomplete and sequential?

Answer 4

We use Perfect Bayesian NE.

Question 5

Perfect Bayesian NE (PBNE)

Definition: A perfect Bayesian Equilibrium (PBE) is a set of straegies and beliefs such that the strategies are sequentially rational given the players´ belief and players update beliefs via Bayes rule whereever possible

What does this mean

Answer 5

Take out from the definition
1) Strategies and beliefs - Before in NE, SPNE and BNE we only looked at the strategies. Now we also include the beliefs.

2) Strategies are sequentially rational - In SPNE we were thinking, when a choice is made, and the other plaeyer needs to act, then he would go through with his choices listed in the equilibriums. In other words, his choices are credible threats. Sequentail rationality will insure the same thing happening in PBNE. 

Another way of remembering this, is that the word perfect ingoes in both SPNE and PBNE. When a game is perfect, then their is no empty threats.

The opponent could be a strong or weak type. Given that you are a strong type I will not fight you, but if you are weak then I would. Due to the incomplete information I don not no this info. Then the credibility of my threat depends on the what I belief you are - the players belief. I don not have a credible threat to fight against a strong type, but I have a credible threat to fight against a weak type. Therefore we have sequentianal rationality, My rational beliefs are arranged in a sequence of what to do (sequentianal) facing different types of opponents.

All in all the credibility of my strategies depend on my beliefs of my opponent.

3) Players update beliefs via Bayes rule - In th BNE we had prior beliefs about the opponents, but in the PBNE we have the possiblity to update our beliefs during the game. 

For example we have a game with a strong type and a weak type. The strong type will try to bully in equilibrium and the weak type will cower in fear in equlibrium. Regardless of what type your opponent are then you have got new information in the equilibrium, that you did not have before, so we would be updating our action via Bayes rule.

But it is not always possible - whereever possible - Sometimes there is an action there would never be taken in equilibrium. So sometimes we can not use Bayes rule, e.g. with pooling strategies, where we can not update our beliefs.

Question 6

When do we use Bayes rule and what does it.

Answer 6

We use Bayes rule to update our beliefs about something, where we had other beliefs in the first place

E.g. A. sports team scouts an amateur. The team has some beliefs on how good the player is, then they send out their scout to look at the player. He comes back with some new information, so the sports team now have to update their beliefs about the player. How do they do that, through Bayes rule. 

se thorughly example in brain scape mappe I one note.

Question 7

What is signaling game, and which kind of PBE is there in them.

Answer 7

Games where informed actors move first. So the informed actor´s move might “signal” something to the uninformed actor.

• Pooling equlibrium
• Seperating equilibrium
• Semi-separating/partially polling equilibrium.

Question 8

Seperating strategies

Answer 8

We seperate the types in high and low. For the high type we connect it with going to college, and for the low type we connect it with going to high school. That gives player 2 a lot of information, because she only knew a something about the types. Now she knows, if Player 1 is a high type he went to college and if he is a low type he went to high shcool.

Question 9

Polling strategies

Answer 9

Pooling strategies (not pulling strategies, but pool)

With pooling strategies we separate the game into pools. In this case Player 1 is not able to disitinguish that one type went to college and the other went to high school. Here both the high and low type went to high school. Player 2 now have to worry about that if she chooses the high type, if Player 1 in fact went to college or high school.

Here we then need to worry about OF EP beliefs, because its only poosible for player 1 to go to high school in this specific pooling game, but we need to consider what would happen if he went to college, if we wanna find a Perfect Bayesian Nash Equilibrium.

Question 10

What is the difference on the sender and the receiver in a signalling game.

Answer 10

The sender see the information from nature, where the receiver can only condition their action on the message from the sender. R does not know the type.

Question 11

PBE - Generel signaling games Cook book

Answer 11

PBE - Generel signaling games Cook book

Signaling requriment 1 (SR1): Write up the S´s possible strategies

SR2R: R maxes exp payoff given beliefs (p and q)

SR2S: S maxes exp payoff given R´s BR

SR3: On EP, beliefs are determined by Bayes rule + equlibrium structure.

SR4: OFF EP, –II– , where possible.

BUUUUUUT When you make the calculation you would use this approach

1. SR1
2. SR3/4
3. SR2R
4. SR2S
5. PBE?

Lastly we exhaust all other SR1 strategies connected to either separating or pooling.

Question 12

What is a strictly dominating strategy in a signaling game.

Answer 12

If one types payoff of one of his two actions are higher than the other actions payoff, then we have a strictly dominating strategy.

Question 13

What should you do if we had a pooling equilibrium with 3 types.

Answer 13

Then we would do the same, but with p or q equaling 1/3.

And there would be p_1, p_2 and 1-p_1-p_2.

Question 14

What is a cheap talk game.

Answer 14

A game where payoffs doesn’t depend on the message. Therefore all payoffs will be the same. E.g. for type 1 the payoffs would be the same even though S send his two different actions, R will not change his actions.

Question 15

What is then the difference on a signaling game with and without cheap talk.

Answer 15

In a signaling game, the Sender´s message is more costly than the recievers, therefore the cost is not the same for all types, but in a cheap talk game the cost is the same for all.

Question 16

What are the special cases for seperating and pooling equilibrium in cheap talk games.

Answer 16

For seperating PBE to exits in a cheap-talk game (or at least be useful), the following conditions must hold:
1) R (reciever) must prefer different actions depending on S´s (Sender´s) type.
2) Different S types must have different preferences over R´s actions
3) The preferences of S and R must not be completely opposed.

A pooling equilibrium always exist in a cheap talk game.

Therefore we can easily have a equilibrium in a pooling game, but we need to enforce some conditions on separating PBE, before it becomes informative.

Question 17

In a cheap talk game with seperating strategies we had some conditions that needed to be uphold. Why is that important?

Answer 17

Because otherwise we cannot say anything informative from the game.

A common case for it to be non informative is when the types at are indifferent between which signals to send. The the PBE is also weak separated.

Question 18

We had a case in cheap talk seperating game, where there wore no PBE. Applicants can send three different signals to a firm, (High, medium, low), and they want to hire the high worker to a high job and so on. BUUUUT the low applicant worker get a higher payoff sending a medium signal, than sending a low signal. What happens to the game, and what is the intuition behind he can lie about himself.

Answer 18

In a cheap talk game, the receiver beliefs anything you say, and the cost of the sender is the same as for the receiver. Therefore it does not cost the low applicant worker anything to send a signal, so he will send the signal that maximizes his payoff.

In this case we do not have separating equilibrium.

Question 19

Semi-pooling equlibrium signaling strategy, hybrid game.

Answer 19

Normally there will be a strictly dominating strategy.

Then there would only be one possible hydbrid equilibrium, where 1 type can only choose one signal to send and the other type has to randomizes between the two signals. In that case we get a pooling game in one side, and one branch more, which make it semi pooling.

Question 20

Which of the equilibria you found in parts (b) and (c) seems most reasonable?

We have a pooling LL, where there one of the signal L is strictly dominating.

Pooling RR, which is dominated, and not a real equilibrium, because it is dominated.

Separating equilibrium LR, where there one of the signal L is strictly dominating.

Answer 20

Pooling on R might seem more reasonable than pooling on L, since the latter does not satisfy Signaling Requirement 6. One could also argue that the separating equilibrium is the most reasonable of all, since it is Pareto dominant.