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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
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.
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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- Fall '13
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