Project #2563 - Operations research

 

Questions 1 to 3 below all relate to the following two matrices.

?.

?A  =  ?         B  =  ?           

?

 

1.

Let represent the payoff matrix in a game against Nature.  What would the optimistic (maximax) and the pessimistic(maximin) criteria suggest to the player??    [5]

 

2.

(a) Let represent the payoff matrix in a matrix game.   Solve this game using pure strategies, if possible; otherwise solve it using mixed strategies.

(b) Let represent the payoff matrix in a matrix game.   Solve this game using pure strategies, if possible; otherwise solve it using mixed strategies.?  [35]

 

3.

Let A and B represent the payoff matrices in a bimatrix game. Solve this game using pure strategies. Is this game similar to “chicken” or “prisoners’ dilemma”? ? [10]

 

 

 

 

4.

In a war, the attacking side wants to bomb a target.  Theyhave only one bomb but two bomber aircrafts, so one of them will carry the bomb and the other is used as a decoy.  They will fly in a formation, one some distance behind the other.  The defending side has a single fighter aircraft.  If the fighter engages a bomber, there is a 50% chance that the bomber will be destroyed and a 25% chance that the fighter will be destroyed.  (It is possible for the fighter not to engage a bomber, i.e. to let it pass.)  The fighter cannot see which aircraft carries the bomb.  The aim of the attackers is to drop the bomb on the target; the aim of the defenders is to prevent this.  (You may assume loss of aircraft and personnel to be of negligible importance.)  Formulate this game and solve it using mixed strategies. Show that no matter what the efficiency of the aircrafts is at destroying their opponents (assuming it is more than 0% and less than 100%), this game does not have a solution in pure strategies.    ?  [30]

 

 

 

5.

Consider the Prisoners’ Dilemma of section 7 in the module notes, with the following variation.  Prisoner 2 knows that prisoner 1 committed a murder and has evidence to convict him.  He tells prisoner 1 that if he testifies against him then he will give the prosecutors this evidence.  (Assume that if prisoner 1 does not testify then prisoner 2 will not produce this evidence.)  A murder charge carries a penalty of a life sentence, but in practice this means 30 years.  The immunity offered by the prosecutor for testifying does not protect prisoner 1 from a murder charge.  Formulate this as a noncooperative game.  What should each of the prisoners do?  Does this game still have the main characteristic of the Prisoners’ Dilemma?????  [20]

 

 

 

 

1.?Consider the 3-player cooperative game given below.  Find the core and Shapley value solutions.

v(1) = 0, v(2) = 0, v(3) = 10, v(1,2) = 20, v(1,3) = 15, v(2,3) = 25, v(1,2,3) = 30?[20]

 

 

 

2.?Three doctors wish to establish a partnership to run a surgery.  The annual rent and running cost of this surgery is £120000.  The respective customs of the three doctors bring in annual revenues of £90000, £70000 and £80000, while their respective costs are £20000, £15000 and £20000.  Xena, Ysabel and Zelda wish to use Game Theory to determine how to split the profits. Determine the relevant characteristic function.  Calculate the Shapley value solution.  Show that this solution is in the core.??[30]

 

 

 

3.?Kent County Council will soon vote on two bills to build new bypass roads in Canterbury and Maidstone, respectively.  If a bill is passed, it will cost each district of Kent £1 million, but if a bypass is built in a district, the benefit to the district will be quite significant.  Canterbury’s estimate is that eachdistrict would reap a benefit of £10 million; Maidstone’s estimate is more conservative at £9 million to Canterbury and £8.5 million to Maidstone.  Both bills are voted on simultaneously.  Neither district can get the bill passed unless they can ensure one more vote to support their bill.  Canterbury’s councillor does not trust Maidstone’s councillor (and vice versa) sufficiently to enable cooperation, although Maidstone’s councillor believes if they both supported a bill it would be carried.  Canterbury’s councillor is more pessimistic: she thinks that in the current economic climate even if both of them voted “yes” for a bill it would still only have an 80% chance.  Maidstone’s councillor also could offer a bribe to another councillor not directly affected – for £100000 Thanet’s councillor would support Maidstone’s bill, enabling it to go through in Maidstone’s opinion.  (This option of a bribe is of course secret and would come out of Maidstone’s budget as a “consultancy fee”.)  Formulate and solve this hypergame.   ?[20]

4.?The Operational Research Society is organising a conference on the Analytic Hierarchy Process.  Three universities are offering to host the conference: Aberdeen,Birmingham and Cranfield.  The society’s president decides to employ the Analytic Hierarchy Process to aid her decision.  She is interested in three objectives: ambiance of campus,location and cost.  Her pairwise comparison matrix for these objectives is:

 

The president’s pairwise comparison matrices for the three universities with respect to the above three objectives are as follows:

?Ambiance:     ?Location:     ?Cost:     

?

Which university should host the conference?  Is the president consistent in her comparisons?

?    ?[30]

 

 

Good luck!

 

Dr. Gábor Nagy

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Due By (Pacific Time) 02/23/2013 02:00 pm
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