introduction-lp-duality1

More formally 94 exercices 153 theorem 914 minimax

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Unformatted text preview: rom the Duality Theorem (9.10). Exercise 9.20 (Application of duality to game theory- Minimax principle (*)). In this problem, based on a lecture of Shuchi Chawla, we present an application of linear programming duality in the theory of games. In particular, we will prove the Minimax Theorem using duality. Let us first give some definition. A two-players zero-sum game is a protocol defined as follows: two players choose strategies in turn; given two strategies x and y, we have a valuation function f (x, y) which tells us what the payoff for Player one is. Since it is a zero sum game, the payoff for the Player two is exactly − f (x, y). We can view such a game as a matrix of payoffs for one of the players. As an example take the game of Rock-Paper-Scissors, where the payoff is one for the winning party or 0 if there is a tie. The matrix of winnings for player one will then be the following: 0 −1 1 0 −1 A= 1 −1 1 0 Where Ai j corresponds to the payoff for player one if player one picks the i-th element and player two the j-th element of the sequence (Rock, Paper, Scissors). We will henceforth refer to player number two as the column player and player number one as the row player. If the row player goes first, he obviously wants to minimize the possible gain of the column player. What is the payoff of the row player? If the row player plays first, he knows that the column player will choose the minimum of the line he will choose. So he has to choose the line with the maximal minimum value. That is its payoff is max min Ai j . i j Similarly, what is the payoff of the column player if he plays first? If the column player plays first, the column player knows that the row player will choose the maximum of the column that will be chosen. So the column player has to choose the column with minimal maximum value. Hence, the payoff of the row player in this case is min max Ai j . j i Compare the payoffs. It is clear that max min Ai j ≤ min max Ai j . i j j i The minimax theorem states that if we allow the players to choose probability distributions instead of a given column or row, then the payoff is the same no matter which player starts. More formally: 9.4. EXERCICES 153 Theorem 9.14 (Minimax theorem). If x and y are probability vectors, then max(min yT Ax) = min(max(yT Ax)). y z x y Let us prove the theorem. 1. Formulate the problem of maximizing its payoff as a linear program. 2. Formulate the second problem of minimzing its loss as a linear program. 3. Prove that the second problem is a dual of the first problem. 4. Conclude. Exercise 9.21. Prove the following proposition. Proposition 9.15. The dual problem of the problem Maximize cT x subject to Ax ≤ a and Bx = b and x ≥ 0 is the problem Minimize aT y + bT z subject to AT y + BT z ≥ c and y ≥ 0, z ≥ 0. 9.4.4 Modelling Combinatorial Problems via (integer) linear programming Lots of combinatorial problems may be formulated as linear programmes. Exercise 9.22 (Minimum vertex cover (Polynomial < duality)...
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