An abstract game in mixed strategies Consider the following matrix game. The set

Question

An abstract game in mixed strategies Consider the following matrix game. The sets of pure strategies are {U: D} for Row and {L: M: R} for Column. Row's strategy is (p_U, P_D) and Column's is (q_L, q_M, q_R), and a strategy combination is ((p_U, p_D), (q_L, q_M,q_R)). In addition, it must be the case that p_U + p_D = 1 and + p_L + p_M + p_R = 1. And all five of these probabilities must be in the closed interval [0, 1]. In this game, both players' payoffs (utilities) could be calculated under the Expected utility hypothesis. (1) Find the pure-strategy Nash equilibrium. (2) Considering the mixed-strategy case, what are both players' expected payoffs associated with pure strategies? (3) Is there an equilibrium in which p_U > 0, p_D > 0 and q_L > 0, q_M > 0, q_R > 0? Find it if it exists, or explain why if not. (4) Is there an equilibrium in which p_U > 0, p_D > 0 and q_L > 0, q_M = 0, q_R > 0? Find it if it exists, or explain why if not.

Explanation / Answer

Let us assume n = 3 k for some integer k.

From the recursive equation in question, we get:

T(n) = 3T(n/3) + 1

T(n/3) = 3T(n/9) + 1

T(n/9) = 3T(n/27) + 1 and so on.

Using these equations, we get:

T(n) = 3T(n/3) + 1

= 3(3T(n/9) + 1) + 1

= 9T(n/9) + 3 + 1

= 9(3T(n/27) + 1) + 3 + 1

= 27T(n/27) + 9 + 3 + 1

and so on and finally we get:

T(n) = 3 kT(n/3k ) + 3 k1 + 3 k2 + . . . + 9 + 3 + 1

= 3 kT(1) + 3 k1 + 3 k2 + . . . + 9 + 3 + 1

= 3 k + 3 k1 + 3 k2 + . . . + 3 2 + 3 + 3 0

= (3 k+1 – 1) / (3 1)

= (3 × 3 k – 1) / 2

= 3(n 1)/2

Now, we need to verify the solutions obtained:

T(n) = (3n – 1) / 2

Note that verification step is essential as this finally proves that our solution is correct.

For base case, n = 1, T(1) = (31) / 2 = 1.

Assume that the statement is true for n/3, that is T(n/3) = (3(n/3)1) / 2 = (n1) / 2 .

With this, we get:

T(n) = 3T(n/3) + 1

= 3 × (n – 1) / 2 + 1

= (3n 3 + 2) / 2

= (3n 1) / 2

Hence, using induction, we can conclude the proof that T(n) = (3n1) / 2 .

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