Question

This is the assignment I have, I need the best game theory problem example for it which is difficult. Thank you:)

Think of an example of a decision problem or a game theory problem. This time you
will have to submit the source of the story. You can take it from a newspaper, from
Internet, magazine, book...
1. Submit a photocopy or a printed version of your problem.
2. Simplify the problem and represent it using the concepts of game theory or
decision theory.
3. Solve it.
4. Compare your results with the real-life results.
5. In case the results do not coincide explain why. Which is the element that the
theory does not reflect?

Your grade will depend on:
1. The problem you have choosen (easy-difficult)
2. Your ability to represent the problem
3. Your knowledge of the methods we have used in class, that you are expected to
use to solve your problem
4. If you want an excelent grade, try to find problems in which game theory or
decision theory fails. For examples not taking into account emotions or framing
effects.

Answer 1

There are 3 examples.

1 ) There is a team of 8 people who decide to undertake a trekking. And the place where they are going is prone to frequent attacks by contagious virus and it is advice to get vaccinated if one wishes to do so. The cost of getting the vaccination is 3 units. The value of satisfaction derived from the trekking for those who take the vaccination will be 5 units and for those who do not get the vaccination will be equal to the number of people in the team who decide to get vaccinated. What is the best course of action for you to undertake.

If 7 people get vaccinated ,then you will get the maximum satisfaction if you choose not  to get vaccinated, in this case you will be getting a satisfaction of 7 units.

If 6 people get vaccinated, then you will get the maximum satisfaction if you choose not to get vaccinated, in this case you will get a satisfaction of 6 units.

If 5 people decide to get vaccinated then in this case it will not make a difference if you choose to get vaccinated you will get a satisfaction of 5 units and if you choose not to get vaccinated even then you will be getting a satisfaction of 5 units.

If below 4 people decide to get vaccinated then you are better off getting vaccinated. As you will get a satisfaction of 5 units.And if you do not get vaccinated then you will get a satisfaction of below 5 units.

We see that cases where 5,4,3,2,1,0 people decide to get vaccinated you are better off if you get vaccinated in this case you will get a satisfaction of 5 units.

But if 7,6,5 people decide to get vaccinated you are better off if you do not get vaccinated in this case you will get the maximum satisfaction of 7 units and 6 units respectively.

Though there are chances of getting maximum satisfaction when 7,6 people get vaccinated and you do not get vaccinated you would be worse off if you take that  risk as if you do not get vaccinated assuming that more then 5 people would get vaccinated and if everybody wants to maximize their welfare and acts in a similar manner and prefers to not get vaccinated then in all likelihood you would end up with a satisfaction of below 5 units. Hence you are better off to get vaccinated and ensuring  that you get at least  5 units of satisfaction because if your greed drives you to not get vaccinated and capitalize on 5 or more people getting vaccinated, you should remember that everybody are driven by this greed and prefer to not get vaccinated and try getting a satisfaction of more then 5 units by capitalizing on the teammates getting vaccinated.

2) There is a highway and you are driving at night if your headlight flashes a high beam then your visibility will be 3.

If your headlight flashes a low beam your visibility will be 2.

If you flash a high beam and the car coming in the opposite direction also chooses to flash high beam then your visibility will be reduced to 1.

If you decide to flash low beam and the car coming in the opposite direction decides to flash high beam then your visibility will be 0.

Both are better off if you and the car coming in the opposite direction turn on the low beam , since the visibility for both the cars will be good. But you do not trust that the other person would lower their beam and if you lower your beam and the other person does not lower his then your visibility will be 0 and the other person's visibility will be 3. Hence you decide not to lower your beam. The thought process of the other person will also be the same and the other person will also decide not to lower the beam and now the visibility of both the cars would be 1 had both the cars lowered the beam the visibility would have been 2 for both the cars

3) Three Legislators vote whether they allow themselves a raise in salary of 2000 Dollars per year. Since voters are observing the vote, there is some loss of face for a legislator to vote for a raise. Let's assume that the legislators estimate that loss of face is worth 1000 Dollars per year. This is a simultaneous 3- player game. Its is best visualized with two matrices. Player A chooses the matrix, B chooses the row, C chooses the column. The Payoff ( in thousands of Dollars ) are

The best options are the ones which enables the legislators earn the maximum payoff and there are 2 such options the are-

When A votes for a raise, B votes for a raise,C votes against a raise = (1,1,2)= 1000+1000+2000= total payoff 4000 Dollars

and When A votes against a raise, B votes for a raise, C votes against a raise =(2,1,1) .= 2000 + 1000+1000 =4000 Dollars total payoff

But the legislators will not choose the options which maximizes the overall benefit or the total payoff. They aim at maximizing their individual payoff which results in the best options of maximizing the total payoff being abandoned.

A will choose to to vote against the rise as it is the best choice for him to make without knowing the other legislators decision. When A chooses to vote against the rise his pay off might be 2000 or nothing determined by what other legislators decide. When he chooses to vote for the rise his pay off might be 1000 or -1000. Hence he is better off choosing to vote against the rise.

B will choose to vote against the rise  because in this case he either earns 2000 Dollars or does not earn anything at all but if he chooses to vote for the rise he might earn 1000 dollars or he might loose 1000 Dollars. Hence B will choose to vote against the rise at it the best option for him to take without knowing the other legislators decisions.

C will vote against  the rise. Because it is the best choice for her to take without knowing the other legislators vote. As when C chooses to vote against the rise her payoff might either be 2000 or nothing depending on what other legislators decide. But if she chooses to vote for the rise her pay off might be 2000 or 1000 or -1000. Hence she is better off choosing to vote against the rise.

Since all the legislators choose not to vote the final pay off will be (0,0,0) = 0

Hence payoff of all the legislators will add up to 0. Here there is a loss, they lose out on the opportunity of earning a payoff of 4000 Dollars. They try to maximize individual pay off / Benefit and end up with no pay off.