Game Theory 1 A game has two players, A and B and a matrix . - - PowerPoint PPT Presentation

Game Theory

1 A game has two players, A and B and a matrix

• aij
• . This is called a

two-person game.

2 Player A chooses a row. Player B chooses a column. The entry in row

i and column j, aij is paid by B to A. M = [aij] is called the payoff matrix.

3 The process is repeated.

Analyze the Game. Player A wants to maximize the amount. Player B wants to minimize the amount.

• Strategy. A systematic way to make a choice. Typically A associates

probabilities p1, . . . , pn to the choice of rows, and B associates probabilities q1, . . . , qm to the columns. Value of the Game. The expected value is piqjaij. In matrix form this is

• p1

. . . pn

• ·

   a11 . . . a1m . . . . . . . . . an1 . . . anm    ·    q1 . . . qm    . A game is called fair, if the value is 0.

Dan Barbasch Math 1105 Chapter 11, Game Theory Week of November 13 1 / 3

Notions

Zero Sum game No money enters from the outside. Whatever one player loses, the other gains. Fair Game Value is 0. Pure Strategy The player chooses a row/column consistently. The probabilites are 1 for one row/column, 0 for the others. Dominated Strategy A row Dominates another row if every entry in the

• ne row is larger than the corresponding entry in the other
• row. Similar for columns. In each row, A wants the
• maximum. In each column B wants the minimum.

Strictly Determined Games A Saddle Point is an entry which is largest in its column and smallest in its row. When there is a saddle point, the game is called strictly determined.

Dan Barbasch Math 1105 Chapter 11, Game Theory Week of November 13 2 / 3

Examples I

1

2 1 −1

• 2

2 1 −3

• 3

    2 3 1 −1 4 −7 5 2 8 −4 −1    

4

  3 8 −4 −9 −1 −2 −3 −2 6 −4 5  .

5

  −3 −2 6 2 2 5 −2 −4  

Dan Barbasch Math 1105 Chapter 11, Game Theory Week of November 13 3 / 3