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I am little bit confused by the lemma preceding to minimax theorem in game theory .

I use the following material for studying GAME THEORY AND THE MINIMAX THEOREM.

gain-floor definition. A gain-floor is the minimum payoff player 1 will receive given player 2's attempt to minimize their payoff. Mathematically, the gain-floor can be represented as such: $v_1' = max_{i} \{min_{j} a_{ij}\}$.

loss-ceilling definition.A loss-ceiling is the maximimum loss player 2 can experience given player 1's attempt to maximize their payoff. Mathematically, the loss-ceiling can be represented as such: $v_2' = min_{j} \{max_{i} a_{ij}\}$.

The following Lemma 2.4 is a little bit confusing.

Lemma 2.4. Let $v_{1}'$ and $v_{2}'$ be defined as gain-floors and loss-ceiling, respectively. Since $v_{1}'=max_{i}\{min_{j} a_{ij}\}$, $v_{1}'$ is less than all other payoffs in column $j$, while $v_{2}'$ is greater than all other payoff in row $i$. Let x be the payoff in row $i$ and column $j$, i.e., $x=a_{ij}$. Since $v_{1}' \leq x \leq v_{2}', v_{1}' \leq v_{2}'$.

The question is why $v_{1}'$ is less than all other payoffs in the column $j$? As I understood the column's player first takes the action and he choices a column , then the row player according to the chosen column takes the row with the maximum payoff, therefore $v_{1}'$ should be the maximum payoff in the chosen column (according to the row player), while the wording according to $v_{2}'$ is correct.

I will appreciate any help.

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The value of $v_1'$ is less then all other payoffs in any given column $j$ by definition. So let us look at the definition again. Operation $\min_ja_{ij}$ returns a list of lowest values in each column $j$. Operation $\max_i \{\min_ja_{ij}\}$ chooses the highest value from the list calculated before. So, $v_1'$ is less then all other payoffs in any given column $j$ by the first operation, the second operation is irrelevant for the statement.

Strategically, you may think of the operation in this way: whatever $i$ player 1 chooses, player 2 will choose $j$ to minimize the payoff of the player 1. Knowing this, player 1 chooses such $i$ which will give her the highest payoff given the behavior of player 1 - which is $v_1'$.

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Thank you very much for the answer! –  tam Apr 14 '13 at 5:15
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