5. Keat and Sunday are neighbors. They want to live in a clean environment and want their street to be clean. The utility from the clean street is same for both at 10 if only one person clean it and the utility will be same for both at 12 if both people clean it. The problem is someone has to clean the street in order to have a clean street and cleaning the street is a public good. Keat's cost of cleaning the street is c, and Sunday's cost is c,. Both ch and c; can have two values, either 5 or 12. The probability that the cost will be 5 is 3/4. The payoffs will be as the following table. Sunday Clean Don't Keat Clean 12- сh, 12- сs 10- сh, 10 Don't 10, 10- cj 0,0 Find three Bayesian equilibria of the game.
the question is attached as photo and answer typed below. please help me show the calculations used to support my answer.
the answer
Bayesian Nash equilibrium has been refined in this way (BNE). An equilibrium idea relevant to dynamic games with partial knowledge is known as the "Perfect Bayesian Equilibrium" or "PBE" (sequential Bayesian games). The strategies and beliefs that make up a perfect Bayesian equilibrium are as follows:
Depending on the information available, a player's strategy will dictate what actions he will do based on that data (on actions taken previously in the game). In some ways, it's like playing through a series of levels in a video game.
Players' beliefs in a given information set influence what node in that information set they think the game has reached. Probabilities of the nodes in the data set and the other players' types are often used to calculate the belief's probabilities. From an a priori perspective, every node in the game has a possibility of 1. Hence a belief system is simply an assignment of probabilities to each player's node.
The tactics and beliefs must meet the following requirements:
Given the beliefs, each method should yield the best possible outcome.
To maintain consistency, each belief should be revised in light of the equilibrium strategy, observable actions, and the Bayes' rule for every path that has been achieved in equilibrium with a positive probability. Known as off-equilibrium paths, paths with zero probability require beliefs to be provided but are free to be arbitrary.
A Nash equilibrium is always a perfect Bayesian equilibrium.
Only Ch and Ci can have distinct values, as we know. They cannot have the same values since the cost probability is 3/4.
Consider Scenario 1 for a moment.
Ci = 12 and Ch = 5.
Our payment table will look like this:
|
|
|
Sunday |
|
|
|
|
cleans |
Doesn’t clean |
|
KEAT |
cleans |
7,0 |
5, 10 |
|
Doesn’t clean |
10, -2 |
0,0 |
|
As we can see, if Sunday cleans, Keat's utility will be higher than if she doesn't (10>7).
If Sunday fails to clean, Keat's utility will be higher when she does (5>0).
Now consider it from Keat's point of view.
Sunday decides not to clean if Keat cleans (10>0).
Moreover, if Keat does not clean, Sunday does not clean (0>-2).
Let us now move to Scenario 2
Ch = 12 and Ci = 5
The payoff table is as follows
|
|
|
Sunday |
|
|
|
|
Cleans |
Doesn’t clean |
|
KEAT |
Cleans |
0, 7 |
-2, 10 |
|
Doesn’t clean |
10, 5 |
0, 0 |
|
As we can see, the majority of our outcomes remain consistent.
Sunday decides not to clean if Keat cleans.
Sunday decides to clean if Keat does not.
If Sunday doesn't clean, Keat decides not to clean either.
To summarize, the three Bayesian equilibria of the game are
- Keat cleans, whereas Sunday does not.
- Sunday is a cleaning day, but Keat is not cleaning.
- Neither of them clean
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
