“Fundamental Theorem of Game Theory”的概念、定义、翻译、参考文献-科学参考

Fundamental Theorem of Game Theory

Mathematics

Due to Von Neumann, the theorem states:

in a matrix game, with E(x,y) denoting the expectation, where x and y are mixed strategies for the two players, then

${max}_{x} {min}_{y} E (x, y) = {min}_{y} {max}_{x} E (x, y) .$

By using a maximin strategy (see conservative strategy), one player, R, ensures that the expectation is at least as large as the left‐hand side of the equation. Similarly, by using a minimax strategy, the other player, C, ensures that the expectation is less than or equal to the right‐hand side. Such strategies may be called optimal strategies for R and C. Since, by the theorem, the two sides of the equation are equal, then if R and C use optimal strategies the expectation is equal to the common value, which is called the value of the game.

For example, consider the game given by the matrix

$[\begin{matrix} 4 & 2 \\ 3 & 4 \end{matrix}] .$

if $x^{*} = (\frac{1}{3}, \frac{2}{3})$ , it can be shown that E(x*,y)≥10/3 for all y. Also, if $y^{*} = (\frac{2}{3}, \frac{1}{3})$ , then E(x, y*)≤10/3 for all x. It follows that the value of the game is 10/3, and x* and y* are optimal strategies for the two players.

单词	Fundamental Theorem of Game Theory
释义	Fundamental Theorem of Game Theory Mathematics Due to Von Neumann, the theorem states: in a matrix game, with E(x,y) denoting the expectation, where x and y are mixed strategies for the two players, then ${max}_{x} {min}_{y} E (x, y) = {min}_{y} {max}_{x} E (x, y) .$ By using a maximin strategy (see conservative strategy), one player, R, ensures that the expectation is at least as large as the left‐hand side of the equation. Similarly, by using a minimax strategy, the other player, C, ensures that the expectation is less than or equal to the right‐hand side. Such strategies may be called optimal strategies for R and C. Since, by the theorem, the two sides of the equation are equal, then if R and C use optimal strategies the expectation is equal to the common value, which is called the value of the game. For example, consider the game given by the matrix $[\begin{matrix} 4 & 2 \\ 3 & 4 \end{matrix}] .$ if $x^{} = (\frac{1}{3}, \frac{2}{3})$ , it can be shown that E(x,y)≥10/3 for all y. Also, if $y^{} = (\frac{2}{3}, \frac{1}{3})$ , then E(x, y)≤10/3 for all x. It follows that the value of the game is 10/3, and x* and y* are optimal strategies for the two players.
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