Minimax solution for Zero-Sum Game

Question

I try to understand the way to finding the minimax solution to zero-sum game.

The following example is takes from Wikipedia. Minimax

Wikipedia: The following example of a zero-sum game, where A and B make simultaneous moves, illustrates minimax solutions.Suppose each player has three choices and consider the payoff matrix for A displayed at right. Assume the payoff matrix for B is the same matrix with the signs reversed (i.e. if the choices are A1 and B1 then B pays 3 to A). Then, the minimax choice for A is A2 since the worst possible result is then having to pay 1, while the simple minimax choice for B is B2 since the worst possible result is then no payment.

Question: Why the minimax choice for A is actually A2, the worst possible result is actually A3 (-4), and the worst possible result for B is B3 (+4). In addition the wording "simple minimax choice for B" is very confusing, is it different from the standard minimax choice?

Answer 1

The worst A takes when taking (across the rows):

• A1: -2
• A2: -1
• A3: -4

The minimum loss is A2.

The worst B takes when taking (along the columns):

• B1: -3
• B2: 0
• B3: -4

It is best for B to pick B2.

Does that help? :)

Answer 2

It is A2 because the following. A asks oneself "What is the worst thing that can happen to me if I play A1?" The answer to that is -2. "What is the worst thing that can happen to me if I play A2$" The answer to that is -1. And finally "What is the worst thing that can happen to me if I play A3" The answer to that is -4. Hence, I'm better off in a minimax sense if I play A2. The other thinks the same, her payoffs tho are just the negatives of those above (since one pays them the other receives)

Answer 3

Let consider a matrix consisting of m rows and n columns. The two players A and B know the content of the payout matrix $A$ and the game consists for each player $k=A,B$ in choosing a strategy $ a_ {k, i} \in A$ without knowing the choice of the other.

Let $A_A=A$ be the payment matrix of player A, so $A_B=-A$: note that $A_A + A_B = 0$ (zero sum game).

The choice of a strategy by player A corresponds to the choice of a row, while for player B it corresponds to the choice of a column. Player B will pay player A the sum indicated at the intersection of the row chosen by player A with the column chosen by player B. Therefore, player B will desire to minimize the payment due to player A, i.e. player B want to maximize $A_B=-A$. So, player B will try to minimize $A$. On the other hand, player A desires to maximize the payout received by player B, i.e. player A want to maximize $ A_A = A$ and therefore will try to maximize $A$. Basically the behavior of the two players is exactly the opposite: player A is called the maximizing player, while player B is called the minimizing player.

Symmetrically if player B were given the matrix of the payments he should receive from player A $A_B=A$, then player A would become the minimizing player and player B the maximizing player.

At this point the attention of the two players turns to the same object: the $A$ matrix. Player A tries to maximize $A$, but his control is limited only to choose a row. The same for player B who tries to minimize $A$: his control is limited only to the choice of a column. The two antagonistic players what criteria will adopt?

John von Neumann answered this question using the following criterion: player A first detects the smallest number of each row and decides to choose the row that contains the largest of the monetary values considered: in this way he is certain that his payout will not be less than the value:

$ \max_i (\min_j a_{i,j})$

whatever player B choices.

Player B considers the largest number of each column and decides to choose the column that contains the smallest of the numbers considered: so he is certain that his loss will not exceed the value

$ \min_j (\max_j a_{i,j})$

whatever the choice of player A.

What it has just been described represents the definition of rational behavior given by John von Neumann and it is commonly referred to as the principle of minimax choice.