# Learning Goal: I’m working on a game theory multi-part question and need an expl

Learning Goal: I’m working on a game theory multi-part question and need an explanation and answer to help me learn.BID ONLY IF YOU CAN DO. I DONT WANT JOKERSGTMDM 2021–2022Problem sheet 4Game Theory questions:1. Consider the following three player parametric game in normal form:Strategy triple Payoff(1;1;1) (1;2;3)(1;1;2) (2;1;3)(1;2;1) (1;3;2)(1;2;2) (3;2;1)(2;1;1) (2;3;1)(2;1;2) (3;1;2)(2;2;1) (1;2;3)(2;2;2) (x; y; z)where x; y; z are arbitrary real numbers satisfying x ≥ 1, y ≥ 1, z ≥ 1 and x + y + z ≤ 5. For thisgame:(i) Write out the characteristic function of this game, giving the details of your computations. Notethat some of the coalition values may appear as functions of (some or all of) the parameters.(ii) Find out if for some values of the parameters x; y; z (where x ≥ 1, y ≥ 1, z ≥ 1 and x+y+z ≤ 5)the game in CFF found in (i) is inessential, constant-sum or has a non-empty core. Give thedetails of your arguments. (iii) Using the strategic equivalence find the (0;1)-reduced form of the characteristic function writtenout in (i). Note that some of the coalition values may appear as functions of (some or all of)the parameters. (iv) Assuming x = y = z = 1 compute the Shapley value of the game in characteristic functionform, which you wrote out in (i). Give the details of your computation. 2. Suppose that we have a set G of gin merchants and a set W of tonic water merchants. For i 2 G,the ith gin merchant has αi litres of gin, and for j 2 W , the jth tonic water merchant has βj litresof tonic water. Assume that αi and βj are real positive numbers for each i and j (also mind that, ingeneral, αi’s are different for different i and βj’s are different for different j).The merchants can form any coalition (i.e., any subset of G [ W ) and any coalition has to use 2volume units of tonic water and 1 volume unit of gin to produce 3 volume units of gin and toniccocktail (in other words, the ratio of tonic water to gin has to be 2 : 1).Suppose S ⊆ G [ W is a coalition of merchants. Let the value of S be defined as the (maximal)volume of gin and tonic cocktail which this coalition S can produce. Write out the formula for thevalue of S. Does this formula define a game in characteristic function form? If so then check if thisgame is essential and if it is constant-sum. Prove all your answers. Hint: You may use the following algebraic property: a + min(b; c) = min(a + b; a + c) for any realnumbers a; b; c.Multicriteria Decision Making questions:For all of the below questions you need to justify your answers and show your working.13. Consider the conesK = (x1; x2; x3)> 2 R3 : x3 ≥ q2x2 1 + 3×2 2 ;M = ((x1; x2; x3)> 2 R3 : x3 ≥ rx 22 1 + x 32 2) :Prove that M = K∗.Hint: Adapt the proof from the lecture notes for finding the dual of the Lorentz cone. Alternatively, prove the formula (AL)∗ = (A>)-1L∗, for any cone L ⊆ R3 and any A 2 R3×3 nonsingularmatrix, where AL = fAx 2 R3 : x 2 Lg, and apply it to the 3-dimensional Lorentz cone with anappropriately chosen matrix A. 4. LetM1 = f(x1; x2)> 2 R2 : 2 ≤ x1 ≤ 8; 2 ≤ x2 ≤ 8g;M2 = f(x1; x2)> 2 R2 : 4 ≤ x1 ≤ 6; 0 ≤ x2 ≤ 10g;M3 = f(x1; x2)> 2 R2 : 0 ≤ x1 ≤ 10; 4 ≤ x2 ≤ 6g;M = M1 [ M2 [ M3 and K = cone (1; 2)>; (1; 3)> ⊆ R2.(a) Determine the set E(M; K) of efficient points of M with respect to K. (b) Determine the set P(M; K) of properly efficient points of M with respect to K. 2