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4. Solve the game with matrix (15𝑡5)for an arbitrary real number 𝑡Draw the value of the game as a function of −∞ < 𝑡 < ∞(10 points) . .
Midterm Solution 155: Game Theory Fall 2015 Graphing this gives us 5 𝑉(𝑡)1 𝑡1 5
Midterm Solution 155: Game Theory Fall 2015 5. Consider a game with 3 piles, of sizes 105, 512, 71, respectively. On their turn, a player is allowed to pick a pile and do the following actions: • remove between 1 and 9 chips from the first pile; or• remove 1, 2, 4, 8, … , 256 chips (i.e. 2?chips with 0 ≤ ? ≤ 8) from the second pile; or • remove any number of chips from the third pile, or split it into two non-empty piles; or do the same thing to piles split from this third pile. Who wins the game, the first player or the second player? What is the winning move? You do not need to prove anything rigorously, but you must show your reasoning (for example, show you have certain patterns etc). (20 points)